TEST BANK for College Algebra, 2nd Edition by Julie Miller, Donna Gerken. ISBN13: 9780077836344. by StudyGuide

Chapter 0 Review of Prerequisites 0.1 Sets and the Real Number Line 0 Concept Connections

Provide the missing information. 1) A is a collection of items called elements. Answer: set Type: SA Var: 1 Objective: Concept Connections

2) W = {0, 1, 2, 3, ...} is called the set of

numbers.

Answer: whole Type: SA Var: 1 Objective: Concept Connections

3) N = {1, 2, 3, ...} is called the set of

numbers.

Answer: natural Type: SA Var: 1 Objective: Concept Connections

4) Z = {... , -3, -2, -1, 0, 1, 2, 3, ...} is called the set of

Answer: integers Type: SA Var: 1 Objective: Concept Connections

5) A set can be defined using

notation by using a description of the set.

Answer: set, builder Type: SA Var: 1 Objective: Concept Connections

6) Listing elements in a set within set braces is called the

method to define a set.

Answer: roster Type: SA Var: 1 Objective: Concept Connections

7) Real numbers that can be expressed as a ratio of two integers are called

numbers.

Answer: rational Type: SA Var: 1 Objective: Concept Connections

8) An

number is a real number that cannot be expressed as a ratio of two integers.

Answer: irrational Type: SA Var: 1 Objective: Concept Connections

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9) The statement x < y means that x lies to the

of y on the number line.

Answer: left Type: SA Var: 1 Objective: Concept Connections

10) The

of x is denoted by x .

Answer: absolute, value Type: SA Var: 1 Objective: Concept Connections

11) Write an absolute value expression to represent the distance between a and b on the number line: . Answer: a - b or b - a Type: SA Var: 1 Objective: Concept Connections

12) Given the expression bn, the value of b is called the

and n is called the

Answer: base, exponent or power Type: SA Var: 1 Objective: Concept Connections

13) The symbol x represents the principal

root of x.

Answer: square Type: SA Var: 1 Objective: Concept Connections

14) The expression

equals

, whereas

Answer: 0, undefined Type: SA Var: 1 Objective: Concept Connections 1 Identify Subsets of the Set of Real Numbers

Determine whether the statement is true or false. 1) 3 N A) True Answer: A

B) False

Type: BI Var: 10 Objective: Identify Subsets of the Set of Real Numbers

2) 7.5 ∉ Z A) True Answer: A Type: BI Var: 50+ Objective: Identify Subsets of the Set of Real Numbers

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B) False

3) 6 ∈ N A) False

B) True

Answer: B Type: BI Var: 20 Objective: Identify Subsets of the Set of Real Numbers

4) 0.15 ∈ Z A) False

B) True

Answer: A Type: BI Var: 36 Objective: Identify Subsets of the Set of Real Numbers 2 Use Inequality Symbols and Interval Notation

Write the statement as an inequality. 1) u is at least 10. A) u < 10 B) u ≤ 10

C) u > 10

D) u ≥ 10

C) t + 4 ≥ 31

D) t + 4 > 31

Answer: D Type: BI Var: 10 Objective: Use Inequality Symbols and Interval Notation

2) The quantity (t + 4) exceeds 31. A) t + 4 ≤ 31 B) t + 4 < 31 Answer: D Type: BI Var: 50+ Objective: Use Inequality Symbols and Interval Notation

Determine whether the statement is true or false. 3) 13 < - 13 A) True

B) False

Answer: B Type: BI Var: 18 Objective: Use Inequality Symbols and Interval Notation

4) -3.51 > -3.51 A) False

B) True

Answer: A Type: BI Var: 50+ Objective: Use Inequality Symbols and Interval Notation

Express the set in interval notation. 5) A) [-3, ∞]

-3 B) [-3, ∞)

Answer: D Type: BI Var: 40 Objective: Use Inequality Symbols and Interval Notation

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C) (-∞, -3)

D) (-3, ∞)

6) 9 B) (-∞, 9]

A) [-∞, 9]

C) (-∞, 9)

D) (9, ∞)

C) (-∞, -9)

D) [-9, -7]

C) (6, 7]

D) (-∞, 6)

C) [3, 9)

D) (-∞, 3)

Answer: B Type: BI Var: 40 Objective: Use Inequality Symbols and Interval Notation

7) -9 A) (-9, ∞)

-7 B) (-9,-7]

Answer: D Type: BI Var: 36 Objective: Use Inequality Symbols and Interval Notation

8) 6 A) [6, 7]

7 B) [6, 7)

Answer: B Type: BI Var: 36 Objective: Use Inequality Symbols and Interval Notation

9) 3 A) [3,9]

9 B) (3, 9]

Answer: B Type: BI Var: 36 Objective: Use Inequality Symbols and Interval Notation

Graph the set and express it in interval notation. 10) {x | x > 5} Answer: (5, ∞) Type: SA Var: 1 Objective: Use Inequality Symbols and Interval Notation

11) {x | x < –1} Answer: (–∞, –1] Type: SA Var: 1 Objective: Use Inequality Symbols and Interval Notation

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Write the subset of real numbers in set-builder notation. 12) (-4, 4] A) {x -4 < x ≤ 4} B) {x -4 ≤ x < 4}

C) {x -4 ≤ x ≤ 4}

D) {x -4 < x < 4}

Answer: A Type: BI Var: 50+ Objective: Use Inequality Symbols and Interval Notation

For the following exercise, interval notation is given for several sets of real numbers. Graph the set and write the corresponding set-builder notation. 13) (-∞, 1.1 A)

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

x | x < 1.1 B)

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

x | x > 1.1 C)

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

x | x ≤ 1.1 D)

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

x | x ≥ 1.1 Answer: C Type: BI Var: 50+ Objective: Use Inequality Symbols and Interval Notation

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3 Find the Union and Intersection of Sets

List the elements of . 1) A = {-29, 14, 13, -20, 11, -2} and B = {-27, -10, 13, 11} A) {13} B) { } C) {13, 11} D) {-29, 14, 13, -20, 11, -2, -27, -10} Answer: C Type: BI Var: 50+ Objective: Find the Union and Intersection of Sets

Determine the intersection X ∩ Y. Express the answer in interval notation. 2) X = {x x ≥ 18} and Y = {x x < 11} A) (-∞, 11] ∪ (18, ∞) B) [11, 18) C) (11, 18] D) { } Answer: D Type: BI Var: 50+ Objective: Find the Union and Intersection of Sets

Determine the union X ∪ Y. Express the answer in interval notation. 3) X = {x x > 14} and Y = {x x ≤ 11} A) (11, 14] B) [11, 14) C) (-∞, 11] ∪ (14,∞) D) { } Answer: C Type: BI Var: 50+ Objective: Find the Union and Intersection of Sets

4) X = {x x > -9} and Y = {x x ≤ 16} A) All real numbers C) (16, -9]

B) { } D) (-∞, 16] ∪ (-9,∞)

Answer: A Type: BI Var: 50+ Objective: Find the Union and Intersection of Sets

Determine the intersection X ∩ Y. Express the answer in interval notation. 5) X = {x x ≥ -2} and Y = {x x < -6} A) { } B) [-6, -2) C) (-∞, -6] ∪ (-2, ∞) D) (-6, -2] Answer: A Type: BI Var: 50+ Objective: Find the Union and Intersection of Sets

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4 Evaluate Absolute Value Expressions

Simplify by writing the expression without absolute value bars. 1) 1 A) 1 B) -1 Answer: A Type: BI Var: 20 Objective: Evaluate Absolute Value Expressions

2) w - 2 for w < 2 A) w - 2

B) -w + 2

C) -w - 2

D) w + 2

C) x - 7

D) x + 7

C) -1

D) -15

Answer: B Type: BI Var: 20 Objective: Evaluate Absolute Value Expressions

3) x + 7 for x ≥ -7 A) - x +7

B) - x - 7

Answer: D Type: BI Var: 17 Objective: Evaluate Absolute Value Expressions

15 - z 4)

15 - z

for z < 15

A) 15

B) 1

Answer: B Type: BI Var: 18 Objective: Evaluate Absolute Value Expressions 5 Use Absolute Value to Represent Distance

Write an absolute value expression to represent the distance between the two points on the number line and simplify. 1) -3 and 2 A) -3 - 2 ; 5 B) -3 + 2 ; 1 C) -3 - 2 ; -5 D) -3 - 2 ; -1 Answer: A Type: BI Var: 50+ Objective: Use Absolute Value to Represent Distance 6 Apply the Order of Operations

Evaluate the expression. 1) (-11)2 A) -9

B) -121

Answer: D Type: BI Var: 10 Objective: Apply the Order of Operations

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C) -22

D) 121

Evaluate the root without using a calculator or note that root is not a real number. 2)

64 A) 4 C) -4

B) 5 D) Not a real number

Answer: A Type: BI Var: 5 Objective: Apply the Order of Operations 3

-125 A) –5 C) 5 Answer: A

B) 6 D) Not a real number

Type: BI Var: 1 Objective: Apply the Order of Operations 7 Simplify Algebraic Expressions

Apply the associative property of addition. 1) (r + 3) + 7 A) r + 3 B) r + 7

C) r + 10

D) r +4

C) -55p

D) 6p

6 C) w 5

Answer: C Type: BI Var: 24 Objective: Simplify Algebraic Expressions

Apply the associative property of multiplication. 5 11 2) p 5 11 A) -5p

B) p

Answer: B Type: BI Var: 20 Objective: Simplify Algebraic Expressions

Apply the commutative property of multiplication. 5 3) w · 6 5 6 B) w · A) - w 6 5 Answer: D Type: BI Var: 18 Objective: Simplify Algebraic Expressions

Clear parentheses and combine like terms. 4) -11z3 - 8z3 - z3 Answer: -20z3 Type: SA Var: 50+ Objective: Simplify Algebraic Expressions

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5 6

5) 3(2x - 5) + 14x A) 6x - 15 + 14x

B) 20x - 5

C) 20x - 15

D) 15x

C) 4s + 9t + 7

D) 4s - 9t + 7

Answer: C Type: BI Var: 40 Objective: Simplify Algebraic Expressions

6) -2x(4 - 3x) + 16x2 - 7x Answer: 22x2 - 15x Type: SA Var: 50+ Objective: Simplify Algebraic Expressions

7) 2[3.5(2.5 - 3x) - x(5 + 0.5x)] + 7.5x2 Answer: 6.5x2 - 31x + 17.5 Type: SA Var: 50+ Objective: Simplify Algebraic Expressions

Clear parentheses by applying the distributive property. 8) -(-4s + 9t + 7) A) 4s - 9t - 7 B) -4s - 9t - 7 Answer: A Type: BI Var: 50+ Objective: Simplify Algebraic Expressions

9) 4(4s - 4)- 1(8t - 2u) A) 16s - 4 - 8t - 2u C) 16s - 16 - 8t - 2u Answer: D Type: BI Var: 50+ Objective: Simplify Algebraic Expressions

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B) 16s - 4 - 8t + 2u D) 16s - 16 - 8t + 2u

8 Write Algebraic Models

Solve the problem. 1) A new diet program guarantees you will lose 1.6 lb per week. A male with a starting weight of 220 lb is modeled by W = 220 - 1.6t where t is the number of weeks after starting the diet. Use the model to determine the male's weight after 19 weeks. 250 200 150 W (lb) 100 50

8 10 12 14 16 18

t (weeks)

A) 170.6 lb

B) 189.6 lb

C) 218.4 lb

D) 208.6 lb

Answer: B Type: BI Var: 50+ Objective: Write Algebraic Models

2) A tool rental store charges a flat fee of $6.50 to rent a chain saw, and $3.75 for each day, including the first. Write an equation that expresses the cost y of renting this saw if it is rented for x days. A) y = 6.50x + 3.75 B) y = 3.75(x + 6.50) C) y = 3.75x + 6.50 D) y = 3.75x - 6.50 Answer: C Type: BI Var: 50+ Objective: Write Algebraic Models

3) A tool rental store charges a flat fee of $8.50 to rent a chain saw, and $4.00 for each day, including the first. Use a linear equation to find the cost of renting the saw for one week. A) $32.50 B) $36.50 C) $12.50 D) $28.00 Answer: B Type: BI Var: 50+ Objective: Write Algebraic Models

4) A tool rental store charges a flat fee of $9.00 to rent a chain saw, and $4.00 for each day, including the first. If you need to rent the saw and absolutely refuse to spend more than $49.00, what's the maximum number of days you can keep the saw? A) 7 days B) 5 days C) 14 days D) 10 days Answer: D Type: BI Var: 50+ Objective: Write Algebraic Models

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5) The width of a rectangle is 2 ft less than 4 times the length. Write a model for the width W in terms of the length L. A) W = 2L - 4 B) W = 2L + 4 C) W = 4L + 2 D) W = 4L - 2 Answer: D Type: BI Var: 15 Objective: Write Algebraic Models 9 Mixed Exercises

Evaluate the expression for the given values of the variables. -q 1) for q = -3, t = 2 4t 3 3 3 B) C) A) 8 2 8

D) -

1 8

Answer: A Type: BI Var: 50+ Objective: Mixed Exercises

2) Under selected conditions, a sports car gets 14 mpg in city driving and 19 mpg for highway driving. 1 1 The model G = c + h represents the amount of gasoline used (in gal) for c miles driven in the 14 19 city and h miles driven on the highway. Determine the amount of gas required to drive 98 mi in the city and 399 mi on the highway. A) 37 gal B) 28 gal C) 33 gal D) 34 gal Answer: B Type: BI Var: 50+ Objective: Mixed Exercises 10 Expanding Your Skills

Write the set as a single interval. 1) -∞, 5 ∩ -6, 8 ∩ [1,6] A) 1, 5

B) 1, 6

C) -6, 5

D) -∞, 8

2) -∞, 7 ∪ 12, ∞ ∩ 10, 15 A) 10, 15 B) 10, ∞

C) -∞, 15

D) 12, 15

Answer: A Type: BI Var: 5 Objective: Expanding Your Skills

Answer: D Type: BI Var: 16 Objective: Expanding Your Skills

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11 Technology Connections

Solve the problem. 1) If n > 0, then 9n - 9n = A) n

B) 9n

C) 0

D) 1

C) 0

D) 1

Answer: C Type: BI Var: 8 Objective: Technology Connections

2) If z > 0, then 5z + 5z = A) 10z

B) 5z

Answer: A Type: BI Var: 8 Objective: Technology Connections

0.2 Integer Exponents and Scientific Notation 0 Concept Connections

Provide the missing information. 1) For a nonzero real number b, the value of b0 =

Answer: 1 Type: SA Var: 1 Objective: Concept Connections

2) For a nonzero real number b, the value b□ =

. bn

Answer: -n Type: SA Var: 1 Objective: Concept Connections

3) A number expressed in the form a × 10n, where 1 ≤ a < 10 and n is an integer is said to be written in notation. Answer: scientific Type: SA Var: 1 Objective: Concept Connections

4) From the properties of exponents, bmbn = b□. Answer: m + n Type: SA Var: 1 Objective: Concept Connections

5) If b ≠ 0, then

□ n =b .

b Answer: m - n

Type: SA Var: 1 Objective: Concept Connections

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6) From the properties of exponents, bm n = b□. Answer: m · n Type: SA Var: 1 Objective: Concept Connections 1 Simplify Expressions with Zero and Negative Exponents

Simplify. 1)

19 0 9 A) 1

C) 0

1 9

Answer: A Type: BI Var: 50+ Objective: Simplify Expressions with Zero and Negative Exponents

2) 4-2 A)

1 16

B) 2

C) -16

D) -8

Answer: A Type: BI Var: 14 Objective: Simplify Expressions with Zero and Negative Exponents

4 -3 3) 5 A)

1 2

B) -

12 5

125 64

D) -

256 C) x 4

4 D) x 4

64 125

Answer: C Type: BI Var: 33 Objective: Simplify Expressions with Zero and Negative Exponents

4) 4x -4 1 A) 256x 4

1 B)4x 4

Answer: D Type: BI Var: 27 Objective: Simplify Expressions with Zero and Negative Exponents

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6p -3r 6 q -9 9 6

6q r A)

6q B)

p 3r 6

3 6

6p r

3 6 9

C) 6p r q

Answer: A Type: BI Var: 50+ Objective: Simplify Expressions with Zero and Negative Exponents 2 Apply Properties of Exponents

Simplify the expression. Write your answer with positive exponents only. 1) 64 · 67 A) 6-3 B) 611 C) 3611

D) 628

Answer: B Type: BI Var: 1 Objective: Apply Properties of Exponents

1812 2)

185 A) 1817

B) 187

C) 1812 - 185

D) 1812/5

Answer: B Type: BI Var: 19 Objective: Apply Properties of Exponents

3) (x9)4 36

A) x

B) x

6,561

C) x

D) x

Answer: A Type: BI Var: 25 Objective: Apply Properties of Exponents

z2y3 4)

y-3z6

1 A)

Answer: B Type: BI Var: 44 Objective: Apply Properties of Exponents

5) (-2a2b4c4)3 Answer: -8a6b12c12 Type: SA Var: 50+ Objective: Apply Properties of Exponents

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4 3

C) z y

z4y6

2x-4y4 -3 6) 5x5y-5 125x27 Answer:

8y27

Type: SA Var: 50+ Objective: Apply Properties of Exponents

7) (t3z)2(3t-2z4)-2 z7 t A) 9

t10 B)

1 C) 81t4z20

9z6

9z10 D)

Answer: B Type: BI Var: 1 Objective: Apply Properties of Exponents

-2 20 1 -23 12 10x 2y -5 x y

-1

x 27y 22

5y 2

x 19

2 B)

5y 22

x 19

x 27 D)

2y 2

Answer: C Type: BI Var: 50+ Objective: Apply Properties of Exponents

(4vw-3x2)2 9)

3 2 -4 -3 3 2 -2 4 · (-v w x )

(2v w x ) x24 A)

v23w8

x24 B) -

v19w20

x16 C)

v19w8

4x16 D) -

v23w20

Answer: B Type: BI Var: 50+ Objective: Apply Properties of Exponents 3 Apply Scientific Notation

Write the number in scientific notation. 1) 528,000,000 Answer: 5.28 × 108 Type: SA Var: 50+ Objective: Apply Scientific Notation

2) 0.00000063 A) 63 × 10-8

B) 6.3 × 10-7

Answer: B Type: BI Var: 1 Objective: Apply Scientific Notation

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C) 0.63 × 10-6

D) 6.3 × 107

Perform the indicated operation and write the answer in scientific notation. 3) (1.2 ×106)(3.8 ×107) A) 5 × 1013 B) 4.56 × 10013 C) 4.56 ×1042

D) 4.56 × 1013

Answer: D Type: BI Var: 50+ Objective: Apply Scientific Notation

Answer the question. 4) How old (in whole years) is someone that has been alive for 1.6 × 104 days? Answer: 43 years Type: SA Var: 20 Objective: Apply Scientific Notation

5) How many miles (to the nearest hundredth of a mile) is a trip that is 1.5 × 106 inches? A) 23.67 miles B) 11.835 miles C) 2,367.42 miles D) 47.34 miles Answer: A Type: BI Var: 21 Objective: Apply Scientific Notation

Perform the indicated operation and write the answer in scientific notation. 31.5 × 10 16 6) 4.5 × 10 6 A) 7 × 10 10

B) 7 × 10 22

C) 27 × 10 22

D) 27 × 10 10

Answer: A Type: BI Var: 50+ Objective: Apply Scientific Notation

0.3 Rational Exponents and Radicals 0 Concept Connections

Provide the missing information. 1) b is an nth-root of a if b□ = a. Answer: n Type: SA Var: 1 Objective: Concept Connections n

2) Given the expression

a, the value a is called the

and n is called the

Answer: radicand, index Type: SA Var: 1 Objective: Concept Connections

3) The expression am/n can be written in radical notation as number. Answer:

n m

Type: SA Var: 1 Objective: Concept Connections

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, provided that

a is a real

4) The expression a1/n can be written in radical notation as number. Answer:

, provided that

a is a real

Type: SA Var: 1 Objective: Concept Connections

5) If x represents any real number, then x2 =

Answer: x Type: SA Var: 1 Objective: Concept Connections

6) If x represents any real number, then

3 3

x =

Answer: x Type: SA Var: 1 Objective: Concept Connections

7) The product property of radicals indicates that represent real numbers. Answer:

a·

provided that

a and

a· b

Type: SA Var: 1 Objective: Concept Connections

8) Removing a radical from the denominator of a fraction is called

the denominator.

Answer: rationalizing Type: SA Var: 1 Objective: Concept Connections 1 Evaluate nth-Roots

Evaluate the root without using a calculator or note that root is not a real number. 1)

8 A) -2 C) 2

B) 3 D) Not a real number

Answer: C Type: BI Var: 5 Objective: Evaluate nth-Roots

625 A) 5 C) -5

Answer: A Type: BI Var: 5 Objective: Evaluate nth-Roots

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B) 6 D) Not a real number

3) -256 A) -4 C) 5

B) 4 D) Not a real number

Answer: D Type: BI Var: 5 Objective: Evaluate nth-Roots 2 Simplify Expressions of the Forms a1/n and am/n

Convert the expression to radical form and simplify. 1) 91/2 A) 3 C)

B) 81

D) Not a real number

Answer: A Type: BI

Var: 6

Objective: Simplify Expressions of the Forms a1/n and am/n

Simplify the expression, if possible. 2) 1,2961/4 1 A) 5,184 C) 6

B) 5,184 D) Not a real number

Answer: C Type: BI Var: 5 Objective: Simplify Expressions of the Forms a1/n and am/n

3) 1963/2 A) 2,754 C) 294

B) 2,744 D) Not a real number

Answer: B Type: BI

Var: 7

Objective: Simplify Expressions of the Forms a1/n and am/n

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4) (-81)5/2 405 2

A) 59,049

B) -

C) -59,049

D) Not a real number

Answer: D Type: BI

Var: 5

Objective: Simplify Expressions of the Forms a1/n and am/n

Convert the expression to radical notation. 5) x1/7 A)

B) 7 x

x 7

Answer: A Type: BI

Var: 8

Objective: Simplify Expressions of the Forms a1/n and am/n

Write the expression by using rational exponents rather than radical notation. 6)

16t

Answer: (16t)1/13 Type: SA Var: 50+ Objective: Simplify Expressions of the Forms a1/n and am/n 3 Simplify Expressions with Rational Exponents

Simplify the expression by using the properties of rational exponents. Write the final answer using positive exponents only. 1) (x4y8)2/3 2 A) x4y8 B) x14/3y26/3 C) x4y16/3 D) x8/3y16/3 3 Answer: D Type: BI Var: 1 Objective: Simplify Expressions with Rational Exponents

2) A)

t2 1/2 t-6 t7

B) t-5

C) t-2

Answer: D Type: BI Var: 9 Objective: Simplify Expressions with Rational Exponents

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D) t4

81s12r-4 3/4 16s-4r4 A)

27s6

27s12 B)

3s12 C)

8r6

3s16r8 D) 2

2r6

Answer: B Type: BI Var: 1 Objective: Simplify Expressions with Rational Exponents

Simply the expression. Assume that all variable expressions represent positive real numbers. c 6 -3 c 6 6 4) c-d c-d c

11 11 6

B) c

c-d

11 6

c-d C) c 11

11 6

D) c

c-d

c-d Answer: C Type: BI Var: 50+ Objective: Simplify Expressions with Rational Exponents 4 Simplify Radicals

Simplify the radical. 1)

72 3

A) 8 36 Answer: B

B) 2 9

C) 8 9

D) 6 2

Type: BI Var: 1 Objective: Simplify Radicals

Simplify the radical. Assume that all variables represent positive real numbers. 54z15 2) 3z4 A) 3z5 2z Answer: A Type: BI Var: 42 Objective: Simplify Radicals

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B) 3z 2z

C) 9z5 2z

11 D) 3 2z

11 6

3) 9 3

c 18d 20 64

9 4

cd 3 c 18d 20

c 18d 20 cd

9 C) c 6d 6 3 d 2 4

c 6d 7

Answer: C Type: BI Var: 50+ Objective: Simplify Radicals 5 Multiply Single-Term Radical Expressions

Multiply the radical expressions and simplify your answer. 1) 9 · 8 A) 36 2 B) 17 C) 6 2

D) 2 6

Answer: C Type: BI Var: 6 Objective: Multiply Single-Term Radical Expressions 6

2) -3 6x y3 · 2 10xy6 5 7

A) -24xy x y

B) -12x3y4 15xy

C) -24x3y4 15xy

Answer: B Type: BI Var: 50+ Objective: Multiply Single-Term Radical Expressions 6 Add and Subtract Radicals

Add the radical expressions. 1) 9 10 + 6 10 A) 54 10 C) 15 10

B) 15 20 D) Cannot be simplified further

Answer: C Type: BI Var: 50+ Objective: Add and Subtract Radicals

2) z2 18z + 7 98z A) (3z2 + 49) 2z C) (18z2 + 686) z Answer: A Type: BI Var: 50+ Objective: Add and Subtract Radicals

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7 9

D) -12 15x y

B) (z2 + 7) 116z D) Cannot be simplified further

Subtract the radical expressions. Assume that all variables represent positive real numbers. 3) Subtract the radical expressions. Assume that all variables represent positive real numbers. 3 72y3 - 2y A) (18y - 1) 2y

B) 17 2y3 - 2y

C) 3 72y3 - 2y

D) Cannot be simplified further

Answer: A Type: BI Var: 1 Objective: Add and Subtract Radicals

Add or subtract the radical expressions as indicated. Assume that all variables represent positive real numbers. 4) 2 3y - 12y5 + y2 48y Answer: (2 + 2y2) 3y Type: SA Var: 10 Objective: Add and Subtract Radicals 7 Mixed Exercises

Use the Pythagorean theorem to determine the length of the missing side. Write the answer as a simplified radical. 1)

14 m

7m A) 7 2 m Answer: C Type: BI Var: 9 Objective: Mixed Exercises

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B) 14 3 m

C) 7 3 m

D) 7 5 m

Solve the problem. 2) The lateral surface area A of a right circular cone is given by A = πr r2 + h2, where r and h are the radius and height of the cone. Determine the exact value (in terms of π) of the lateral surface area of a cone with radius 3 m and height 5 m. Then give a decimal approximation to the nearest square meter.

3m A) 6π 2 m2≈ 27 m2

B) 5π 34 m2≈ 92 m2

C) 10π 2 m2≈ 44 m2

D) 3π 34 m2≈ 55 m2

Answer: D Type: BI Var: 6 Objective: Mixed Exercises

0.4 Polynomials and Multiplication of Radicals 0 Concept Connections

Provide the missing information. 1) A in the variable x is a finite sum of terms of the form axn where a is a real number and n is a whole number. Answer: polynomial Type: SA Var: 1 Objective: Concept Connections

2) The polynomial 5x3 - 2x2 + 4 is written in

order by degree.

Answer: descending Type: SA Var: 1 Objective: Concept Connections

3) The

term of a polynomial is the term of highest degree.

Answer: leading Type: SA Var: 1 Objective: Concept Connections

4) The leading

of a polynomial is the numerical factor of the leading term.

Answer: coefficient Type: SA Var: 1 Objective: Concept Connections

5) A

is a polynomial that has two terms, and a

Answer: binomial, trinomial Type: SA Var: 1 Objective: Concept Connections

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is a polynomial with three terms.

6) The expanded form of the square of a binomial is a trinomial called a

square trinomial.

Answer: perfect Type: SA Var: 1 Objective: Concept Connections

results in a difference of squares a2 - b2.

7) The product of conjugates (a + b) · Answer: (a - b) Type: SA Var: 1 Objective: Concept Connections

8) The conjugate of 3 - x is

Answer: 3 + x Type: SA Var: 1 Objective: Concept Connections 1 Identify Key Elements of a Polynomial

Write the polynomial in descending order. Then identify the leading coefficient and the degree. 1) -9x + 7 - 4x6 A) -4x6 - 9x + 7; leading coefficient: -4; degree: 3 B) 7 - 4x6 - 9x; leading coefficient: 7; degree: 6 C) -4x6 - 9x + 7; leading coefficient: -4; degree: 6 D) 7 - 4x6 - 9x; leading coefficient: 7; degree: 3 Answer: C Type: BI Var: 50+ Objective: Identify Key Elements of a Polynomial

Choose the polynomial that is described. 2) A monomial in one variable of degree 4 A) 4n C) -9t4

B) -9t3 - 3n D) -9t3 - 4t2 - 3n - 8

Answer: C Type: MC Var: 50+ Objective: Identify Key Elements of a Polynomial

3) A trinomial in one variable of degree 3 A) 4r3 B) 4r2 - 9t + 7 Answer: D Type: MC Var: 50+ Objective: Identify Key Elements of a Polynomial

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C) 3r2 + 4t - 9

D) 4r3 - 9t + 7

2 Add and Subtract Polynomials

Add the polynomials and simplify. 1) (4m4 - 8m3 - 7m) + (4m4 - 7m2 - 5m) A) 4m4 - 4m3 - 7m2 - 12m C) 8m4 - 8m3 - 7m2 - 2m

B) 8m4 - 8m3 - 7m2 - 12m D) 8m4 - 8m3 - 14m2 - 5m

Answer: B Type: BI Var: 50+ Objective: Add and Subtract Polynomials

1 3 1 1 9 2) - n + n2 + 4.6n + - n3 + n2 + 4.5n 2 4 4 2 11 3 A) - n3 + n2 + 9.1n 4 4 11 1 C) - n3 + n2 + 0.1n 4 4

11 3 1 2 n - n + 0.1n 4 4 11 D) - n3 - 3 n2 + 9.1n 4 4 B) -

Answer: A Type: BI Var: 50+ Objective: Add and Subtract Polynomials

Subtract the polynomials and simplify. 3) (-6m2 - 7m + 8) - (2m2 - 5m - 9) A) -4m2 - 12m - 1 C) -8m2 - 7m2 + 5m + 17

B) -8m2 - 12m - 1 D) -8m2 - 2m + 17

Answer: D Type: BI Var: 50+ Objective: Add and Subtract Polynomials

2 2 1 p - pq + 3 q2 + 14 - 9 p2 + 4 pq - 5 q2 + 11 2 14 14 14 7 7 5 2 15 pq + 4 q2 + 3 p 14 14 7 5 1 C) - p2 + pq - 1 q2 + 25 14 14 7

A) -

B) D)

5 2 15 pq + 4 2 + 25 p q 14 14 7

13 2 1 pq - 1 2 + 25 p + q 14 14 7

Answer: A Type: BI Var: 50+ Objective: Add and Subtract Polynomials

5) Subtract (-7m3 - 6m - 6) from (-5m3 - 4m - 8). A) -5m3 + 7m2 + 2m - 2 C) 2m3 + 2m - 2 Answer: C Type: BI Var: 50+ Objective: Add and Subtract Polynomials

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B) 2m3 - 10m - 14 D) -2m3 - 2m + 2

3 Multiply Polynomials

Multiply the polynomials by using the distributive property. 1) (8t7u3)(3t4u5) A) 11t11u8 B) 24t3u-2 C) 24t11u8

D) 24t28u15

Answer: C Type: BI Var: 50+ Objective: Multiply Polynomials

2) (n - 6p)(n2 - 5np + 3p2) A) n3 - 6p - 5np - 18p3 C) n3 - 11n2p + 33np2 - 18p3

B) n3 - 6np + 33np2 - 18p3 D) n3 + 22n2p2 - 18p3

Answer: C Type: BI Var: 50+ Objective: Multiply Polynomials

Write an expression for the area and simplify your answer. 3) Rectangle

4x - 9

3x + 6 [The figure is not necessarily drawn to scale.] B) 12x2 - 3x - 54 C) 12x2 + 51x - 54

A) 12x2 - 54 Answer: B

Type: BI Var: 50+ Objective: Multiply Polynomials

Write an expression for the volume and simplify your answer. 4)

x+4 x+1 A) 3x + 12x C) 3x3 + 15x2 + 12x 3

Answer: C Type: BI Var: 50+ Objective: Multiply Polynomials

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B) 3x3 + 15x +12 D) x3 + 5x2 + 4x

D) 12x2 + 3x - 54

4 Identify and Simplify Special Case Products

Multiply. 1) (-6m2 - 9n)2 A) 36m2 + 108mn + 81n2 C) 36m4 + 81n2

B) 36m4 + 108m2n + 81n2 D) 36m4 - 81n2

Answer: B Type: BI Var: 50+ Objective: Identify and Simplify Special Case Products

2) [(-8y - 9) + z][(-8y - 9)- z] A) 64y2 + 81 - z2 C) 64y2 + 288y + 81 - z2

B) 64y2 + 144y + 81 - z2 D) 64y2 + 144y - 16yz + 81- 19z2

Answer: B Type: BI Var: 50+ Objective: Identify and Simplify Special Case Products

3) (5m - 4n)3 A) 125m3 - 200m2n + 160mn2 - 64n3 C) 125m3 - 300m2n + 240mn2 - 64n3

B) 125m3 - 64n3 D) 125m3 - 40mn - 64n3

Answer: C Type: BI Var: 24 Objective: Identify and Simplify Special Case Products 5 Multiply Radical Expressions Involving Multiple Terms

Multiply the radical expressions and simplify your answer. 1) 2( 2 - 3) A) 2 - 6 B) 2 2 - 6 C) 2 - 3

D) 2 + 6

Answer: A Type: BI Var: 5 Objective: Multiply Radical Expressions Involving Multiple Terms

2) ( 108 - 1)( 3 + 5) A) 321 - 5

B) 319 + 5 108 - 3

C) 13

D) 13 + 29 3

Answer: D Type: BI Var: 5 Objective: Multiply Radical Expressions Involving Multiple Terms

x+5 -4 A) x - 9

x+5+4 B) x - 11

C) x + 11

Answer: B Var: 50+ Type: BI Objective: Multiply Radical Expressions Involving Multiple Terms

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D) x + 9

y+6 -7

A) y + 14 y + 6 - 55

B) y - 43

C) y - 14 y + 6 + 55

D) y + 55

Answer: C Type: BI Var: 50+ Objective: Multiply Radical Expressions Involving Multiple Terms

0.5 Factoring 0 Concept Connections

Provide the missing information. 1) The binomial a3 + b3 is called a sum of

and factors as

Answer: cubes, (a + b)(a2 - ab + b2) Type: SA Var: 1 Objective: Concept Connections

2) The binomial a3 - b3 is called a

of cubes and factors as

Answer: difference, (a - b)(a2 + ab + b2) Type: SA Var: 1 Objective: Concept Connections

3) The trinomial a2 + 2ab + b2 is a

square trinomial. Its factored form is

Answer: perfect, (a + b)2 Type: SA Var: 1 Objective: Concept Connections

4) The binomial a2 - b2 is called a difference of

and factors as

Answer: squares, (a + b)(a - b) Type: SA Var: 1 Objective: Concept Connections 1 Factor Out the Greatest Common Factor

Factor out the greatest common factor. 1) 12r2s - 24rs2 + 18rs A) 2rs(6r - 12s - 12) C) 6rs(2r - 4s + 3)

B) 3rs(4r - 8s + 6) D) 6r(2rs - 4s2 + 3s)

Answer: C Type: BI Var: 50+ Objective: Factor Out the Greatest Common Factor

Factor out the indicated quantity. 2) -40q4r + 56q3r - 24q2r: Factor out the quantity -8q2r A) -8q2r(5q2 - 7q + 3) B) -8q2r(-5q2 - 7q + 3) C) -8q2r(5q3 - 7q2 + 3) D) -8q2r(-5q2 + 7q - 3) Answer: A Type: BI Var: 50+ Objective: Factor Out the Greatest Common Factor

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Factor out the greatest common factor. 3) 14k 2 10k 2 + 5 - 2k 10k 2 + 5 A) k 10k 2 + 5 14k - 2 C) 10k 2 + 5 14k 2 - 2k

B) 2k 10k 2 + 5 7k - 1 D) 2 10k 2 + 5 7k 2 - k

Answer: B Type: BI Var: 50+ Objective: Factor Out the Greatest Common Factor 2 Factor by Grouping

Factor the polynomial by grouping (if possible). 1) 3rs - 5cr - 15cs + 25c2 A) r(3s - 5c) - 5(3cs - 5c2) C) (3s - 5c)(r - 5c)

B) c(3s - 5c)(r - 5) D) Cannot be factored

Answer: C Type: BI Var: 50+ Objective: Factor by Grouping

2) 5n2p - 10np - 5n2 + 10n A) 5np(n - 2) - 5n(n - 2) C) (5n2 - 10n)(p - 1)

B) 5n(n - 2)(p - 1) D) Cannot be factored

Answer: B Type: BI Var: 50+ Objective: Factor by Grouping 3 Factor Quadratic Trinomials

Factor the trinomial completely by using any method. Remember to look for a common factor first. 1) 4p2 + 3p - 5 A) (4p + 5)(p + 3) B) p(4p + 3) - 5 C) (4p + 3)(p + 5) D) Prime Answer: D Type: BI Var: 50+ Objective: Factor Quadratic Trinomials

2) 2w3 - 16w2x + 32wx2 A) 2w(w - 4x)(w + 4x) C) 2w(w - 4x)2 Answer: C Type: BI Var: 32 Objective: Factor Quadratic Trinomials

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B) 2w2(1 - 4x)2 D) Prime

4 Factor Binomials

Factor completely. 1) 9x2 - 25y2 A) (3x - 5y)2 C) (9x + 5y)(x - 5y)

B) (3x + 5y)2 D) (3x + 5y)(3x - 5y)

Answer: D Type: BI Var: 44 Objective: Factor Binomials

2) 12x5 - 27x3 A) 3x3(4x2 - 9) C) 3x3(2x - 3)2

B) 3x3(2x + 3)(2x - 3) D) Prime.

Answer: B Type: BI Var: 1 Objective: Factor Binomials

3) 27u6 - 8v6 A) (3r2 + 2s2)(9r4 + 6r2s2 + 4s4) C) (3r2 - 2s2)(9r4 + 6r2s2 + 4s4)

B) (3r2 + 2s2)(9r4 - 6r2s2 + 4s4) D) Prime

Answer: C Type: BI Var: 12 Objective: Factor Binomials

Factor the sum or difference of cubes. 4) 125r3 + 8 A) (5r + 2)(25r2 - 20r + 4) C) (5r - 2)(25r2 + 10r + 4)

B) (5r + 2)2 (5r - 2) D) (5r + 2)(25r2 - 10r + 4)

Answer: D Type: BI Var: 16 Objective: Factor Binomials 5 Apply a General Strategy to Factor Polynomials

Factor completely. 1) 9x4 + 21x3 - 9x2 - 21x A) 3x(3x + 7)(x + 1)2 C) 3x(3x + 7)(x +1)(x - 1)

B) 3x(3x - 7)(x + 1)(x - 1) D) -3x(3x + 7)(x + 1)(x - 1)

Answer: C Type: BI Var: 50+ Objective: Apply a General Strategy to Factor Polynomials

2) x2 + 3xy - 4y2 - 5y + 5x A) (x - y)(x + 4y - 5) C) (x - y)(x + y + 5) Answer: D Type: BI Var: 18 Objective: Apply a General Strategy to Factor Polynomials

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B) (x - y)(x - 4y + 5) D) (x - y)(x + 4y + 5)

3) x 2 - 4 2 - 4 x 2 - 4 - 96 A) x 2 + 4 x + 4 2 C) x 2 + 4 x + 4 x - 4

B) x 2 - 4 x + 4 x - 4 D) x 4 - 12x 2 - 64

Answer: C Type: BI Var: 21 Objective: Apply a General Strategy to Factor Polynomials

4) a + 7 2 - y 2 A) a - 7 + y 2 C) a - 7 - y a + 7 - y

B) a + 7 - y 2 D) a + 7 + y a + 7 - y

Answer: D Type: BI Var: 14 Objective: Apply a General Strategy to Factor Polynomials

5) c - 9 2 - 4c - 2 2 A) -15c 2 - 34c + 85 C) -15c 2 - 2c + 77

B) 5c + 11 -3c + 7 D) 5c - 11 -3c - 7

Answer: D Type: BI Var: 50+ Objective: Apply a General Strategy to Factor Polynomials

6) m 6 + 124m 3 - 125 A) m + 4 2m 2 - 4m + 26 C) m + 5 m 2 - 5m + 25 m - 1 m 2 + m + 1

B) m + 5 m 2 + 5m + 25 m - 1 m 2 - m - 1 D) m 3 - 125 m 3 - 1

Answer: C Type: BI Var: 4 Objective: Apply a General Strategy to Factor Polynomials 6 Factor Expressions Containing Negative and Rational Exponents

Factor completely. 1) 8x-7 + 4x-6 + x-5 8x2 + 4x + 1 A) x5

x2 + 4x + 8 B)

8x2 + 4x + 1 C)

x2 + 4x + 8 D) x7

Answer: D Type: MC Var: 50+ Objective: Factor Expressions Containing Negative and Rational Exponents

2) 12x-6 + 7x-5 + x-4 (x - 4)(x - 3) A) x6

(x + 4)(x + 3) B)

(x - 4)(x - 3) C)

Answer: D Type: MC Var: 50+ Objective: Factor Expressions Containing Negative and Rational Exponents

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(x + 4)(x + 3) x6

3) 3c9/5 - 3c4/5 3(c - 1) A) 4/5 c

B) 3c9/5(c - 1)

C) 3c4/5(c - 1)

D) c4/5(3c - 3)

Answer: C Type: BI Var: 50+ Objective: Factor Expressions Containing Negative and Rational Exponents

4) x(2x + 1)-2/3 + (2x + 1)1/3 3x + 1 A) (2x + 1)2/3

3x B) (2x + 1)2/3

C) 1

3x + 11/3 (2x + 1)2/3

Answer: A Type: BI Var: 48 Objective: Factor Expressions Containing Negative and Rational Exponents

5 5) 5x 7/2 - x -7/2 2 5 2x7 - 1 A) 2x 7/2

5x 7/2 2x - 1 B)

5x 7/2 2 2x - 1

5 2

Answer: A Type: BI Var: 50+ Objective: Factor Expressions Containing Negative and Rational Exponents

0.6 Rational Expressions and More Operations on Radicals 0 Concept Connections

Provide the missing information. 1) A expression is a ratio of two polynomials. Answer: rational Type: SA Var: 1 Objective: Concept Connections

2) The restricted values of the variable for a rational expression are those that make the denominator equal to . Answer: zero Type: SA Var: 1 Objective: Concept Connections

3) The expression

5(x + 2) (x + 2)(x - 1)

Answer: -2, 1 Type: SA Var: 1 Objective: Concept Connections

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equals

5 x-1

provided that x ≠

and x ≠

4) The ratio of a polynomial and its opposite equals

Answer: -1 Type: SA Var: 1 Objective: Concept Connections

5) A fraction is an expression that contains one or more fractions in the numerator or denominator. Answer: complex (or compound) Type: SA Var: 1 Objective: Concept Connections

6) The process to remove a radical from the denominator of a fraction is called denominator. Answer: rationalizing Type: SA Var: 1 Objective: Concept Connections 1 Determine Restricted Values for a Rational Expression

Write the domain of the rational expression in set-builder notation. 1 1) x - 14 A) {x x = 14} C) {x x ≠ -14}

B) {x x ≠ 14} D) {x x ≠ 14, x ≠ -1}

Answer: B Type: BI Var: 50+ Objective: Determine Restricted Values for a Rational Expression

Write the domain of the rational expression in interval notation. 5-t 2) 2 t +4 A) (–∞, –2) ∪ (–2, ∞) C) (–∞, 5) ∪ (5, ∞) Answer: D Type: BI Var: 1 Objective: Determine Restricted Values for a Rational Expression

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B) (–∞, –2) ∪ (–2, 2) ∪ (2, ∞) D) (–∞, ∞)

the

2 Simplify Rational Expressions

Simplify the rational expression. y2 + 5y - 36 1) y2 - 16 y+9

(Provided y ≠ -4)

y-4

y+9

(Provided y ≠ 4)

y+4

5y - 36 (Provided y ≠ -4) 16 y2 + 5y - 36 ; cannot be simplified

y2 - 16

Answer: C Type: BI Var: 14 Objective: Simplify Rational Expressions

x3 - 5x2 + 2x - 10 3x2 - 75 x2 + 2 A) 3x - 15 C)

x2 + 2

(Provided x ≠ -5)

x3 + 2x - 7 -12

(Provided x ≠ -5)

3(x + 5)

(Provided x ≠ 5)

x3 - 5x2 + 2x - 10 3x2 - 75

; cannot be simplified

Answer: B Type: BI Var: 9 Objective: Simplify Rational Expressions 3 Multiply and Divide Rational Expressions

Multiply the rational expressions. 2x2 - 2x - 12 x2 - 25 1) · 2 x+5 2x + 4x A)

(x - 3)(x - 5) x

(2x + 4)(x - 3) x+4

(x - 3)(x - 5)2 x(x + 5)

(x - 3)(x + 5) 2x2 + 5x

16y2 - 1 y-2

Answer: A Type: BI Var: 32 Objective: Multiply and Divide Rational Expressions

Divide the rational expressions. 4y - 1 1 - 4y 2) 2 ÷ y+2 y -4 A)

1 2-y

1 y-2

Answer: A Type: BI Var: 50+ Objective: Multiply and Divide Rational Expressions

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(4y - 1)(1 - 4y) (y - 2)(y + 2)2

x 3 - 125 25x - x A)

3 ÷

2x 2 + 10x + 50 x 2 - 2x - 35

x+7 x

B) -

x+7

x+7 2x

1 2x

t2 + 4t + 5 t-3

Answer: B Type: BI Var: 50+ Objective: Multiply and Divide Rational Expressions 4 Add and Subtract Rational Expressions

Add or subtract as indicated and simplify if possible. t2 - 8 -4t + 13 1) t-3 t-3 t2 - 4t + 5 t+7 B) A) 2 t-3

C) t + 7

Answer: C Type: BI Var: 1 Objective: Add and Subtract Rational Expressions

1 -7 2 + 6y 8y -6 A)

-6 3

14y

48y

-21y + 4 24y2

-17 24y2

-y - 3 4y + 6y - 10

1-y (2y + 5)(y - 2)

Answer: C Type: BI Var: 50+ Objective: Add and Subtract Rational Expressions

-2y y-3 + 2 2y + 5y 2y + y - 10 2

-3y2 + y + 1 y(2y + 5)(y - 2)

-2y2 + y - 3 3

2y + 5y - 10

Answer: D Type: BI Var: 1 Objective: Add and Subtract Rational Expressions

3 - x2 x 4) x + 2 x2 - 4 -2x - 3 A) 2 x -4

x2 + x - 3 -x2 + x + 6

Answer: D Type: BI Var: 8 Objective: Add and Subtract Rational Expressions

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x3 + 2x - 3 (x2 - 4)(x + 2)

2x2 - 2x - 3 x2 - 4

6 1 2 2 2 9a - b 3a - ab a A) 3a + b

1 B) a 3a + b

9a - 3a - b C)

a 3a + b 3a - b

a+b a 3a + b

Answer: B Type: BI Var: 6 Objective: Add and Subtract Rational Expressions

3w w-3

2w + 3 3-w w+3

A) -1

w-3

5w - 3 3-w

D) 1

Answer: D Type: BI Var: 50+ Objective: Add and Subtract Rational Expressions

5 4 - 2 x + 1x - 6 x + 0x - 4 2

x A) x - 2 x + 3 x + 2 C)

1 x+3x+2

1 x + 22 1 x-2 x+3x+2

Answer: C Type: BI Var: 50+ Objective: Add and Subtract Rational Expressions 5 Simplify Complex Fractions

Simplify the complex fraction. t + 3t -2 1) 5 - 2t -1 4t A) 5 - 2t2

t3 + 3 5t2 - 2t

Answer: B Type: BI Var: 1 Objective: Simplify Complex Fractions

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t3 + 3 5t3 - 2t2

5 - 2t2 4t

1 1 + 50x 10 1 1 + 5 25x 1 A) 2

1 B) 2x

C) 2

1 25x

Answer: A Type: BI Var: 50+ Objective: Simplify Complex Fractions

m 6m + 5 + 2 2m 1 5 + 2 2m 1 A) 2

m+1

C) m + 5

m+5 2m

Answer: B Type: BI Var: 50+ Objective: Simplify Complex Fractions

2 2 + a b 8 ab b+a 4

16b + 16a a2b2

b + 2a 4

ab D) 2a + 2b

Answer: A Type: BI Var: 1 Objective: Simplify Complex Fractions

12 5)

x+h

12 x

h A) -

1 x+h

x x+h

Answer: C Type: BI Var: 19 Objective: Simplify Complex Fractions

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12 C) -

x x+h

1 x+h

5 1 x-3 x+3 3 x -9 2

4x + 18 x 2 - 9

3 C)

4x + 18 3x2-9

4x + 12 3 4x + 18 3

Answer: D Type: BI Var: 50+ Objective: Simplify Complex Fractions 6 Rationalize the Denominator of a Radical Expression

Rationalize the denominator. -13 1) x A) x x -13 C)

169 x

-13 x x

D) Already rationalized.

Answer: B Type: BI Var: 1 Objective: Rationalize the Denominator of a Radical Expression

9 - 10 2 2+1 A) 19 2 - 29 3

B) 9 - 11 2

C) 19 2 - 29

D) 9 2 - 20 3

Answer: C Type: BI Var: 50+ Objective: Rationalize the Denominator of a Radical Expression

Simply the expression. Assume that all variable expressions represent positive real numbers. y - 33 3) y + 33 A) y + 33

B) y - 33

Answer: D Type: BI Var: 17 Objective: Rationalize the Denominator of a Radical Expression

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C) y + 33

D) y - 33

13x x

13x A)

31 13x 13x

234 13x 13x

234 13x 26x

31 13x 26x

Answer: A Type: BI Var: 50+ Objective: Rationalize the Denominator of a Radical Expression 7 Mixed Exercises

Simplify the expression, if possible. 3 t 1) - 2 t+2 2t +1 2t + 5t + 2 2t + 6 A)

2t +1 t + 2

2t + 6 2t + 1

3t + 6 t+2

3t + 6 2t + 1

Answer: B Type: BI Var: 50+ Objective: Mixed Exercises

1 - 5a -1 - 24a -2 1 - 15a -1 + 56a -2 A)

a+3

a-7

a-8 a-7

C) a - 8

a+3 a-8 a+8 a-7

Answer: A Type: BI Var: 50+ Objective: Mixed Exercises

Write the expression as a single term, factored completely. Do not rationalize the denominator. 3) 1 - 10x -3 + 23x -12 1 - 10x 9 + 23 x 12 - 10x 3 + 23 A) B) x 12 x 12 x 12 - 10x 9 + 23 C)

x 12

Answer: C Type: BI Var: 50+ Objective: Mixed Exercises

Page 39

x 3 - 10x 3 + 23 x3

9 4)

+ 11x

8 11x

9 + 88x A)

1 + 5x B)

8 11x

11x

9 + 88x 2

9 + 5x C)

8 11x

Answer: A Type: BI Var: 48 Objective: Mixed Exercises

19x 2 2

19x + 7 5)

- 19x 2 + 7

19x 2 133x 2

B) -

19x + 7

C) -

133x 2

19x 2 + 7

7 19x

19x 2

19x 2 + 7 7 19x 2 + 7

Answer: B Type: BI Var: 50+ Objective: Mixed Exercises

6) 5 4x 2 + 1 +

20x 4x 2 + 1

5 4x 2 + 4x + 1 4x 2 + 1

5 4x 2 + 4x + 1 5x + 1

20x 2 + 20x + 1 5x + 1

Answer: A Type: BI Var: 14 Objective: Mixed Exercises

0.7 Algebra for Calculus 0 Algebra for Calculus

Write an inequality using absolute value that represents the given condition. 1) The distance between y and L is less than ε (epsilon). A) y + L < ε or L + y < ε B) y < ε + L C) y - L < ε or L - y < ε D) y < ε - L Answer: C Type: BI Var: 1 Objective: Algebra for Calculus

Page 40

20x 2 + 20x + 1 4x 2 + 1

2) The distance between x and c is less than δ (delta). A) x < δ - c C) x + c < δ or c + x < δ

B) x - c < δ or c - x < δ D) x < δ + c

Answer: B Type: BI Var: 1 Objective: Algebra for Calculus

Simplify. 3)

x+7 for x > -7. x+7 A) 0

B) -1

C) 7

D) 1

Answer: D Type: BI Var: 8 Objective: Algebra for Calculus

3 x + h 2 + 4 x + h - 3x 2 + 4x h 6x h+ 3h 2 + 4h + 8x A) C)

h 6x + 3h + 4 h

B) 6x + 3h + 4 D) 6x h+ 3h 2 + 4h + 8x

Answer: B Type: BI Var: 7 Objective: Algebra for Calculus

1 1 7 x + h + 6 7x + 6 5)

h 7 + 12x A)

7x + 7h + 6 7x + 6 -7

7x + 7h + 6 7x + 6

Answer: C Type: BI Var: 47 Objective: Algebra for Calculus

Page 41

-7h 2 B)

7x + 7h + 6 7x + 6 7h 2

7x + 7h + 6 7x + 6

6x + h 3 - 216x 3 h A) 108x 2h + 18xh 2 +h 3 2

C) 108x + 18xh +h

B) 36x 2 + 12xh + h 2

108x 3 + 108x 2h + 18xh 2 +h 3 h

Answer: C Type: BI Var: 6 Objective: Algebra for Calculus

Rationalize the numerator of the expression and simplify. 7x + h + 14 - 7x + 14 7) h A) C)

h 14x + h h 7x + h + 7x

1 7x + h + 7x

1 14x + h

Answer: B Type: BI Var: 50+ Objective: Algebra for Calculus

3 x + h - 3x h 3 A) 6x + 3h

6x + 3h B)

h 3x + 3h + 3x

3h C)

3 D)

3x + 3h + 3x

Answer: D Type: BI Var: 8 Objective: Algebra for Calculus

Factor completely. 5 7 9) x 1/2 + x 5/2 2 2 A)

x 1/2 7x 2 + 5 2

Answer: A Type: BI Var: 8 Objective: Algebra for Calculus

Page 42

B) x 1/2 6x

x 5/2 5x 2 + 7 2

7x 2 + 5 D)

2x 1/2

10) 5 -2x + 5 2 -2 2x 2 - 6 2 + -2x + 5 3 2 2x 2 - 6 8x A) 2 -2x + 5 2x 2 - 6 26x 2 - 40x - 30

B) 2 -2x + 5 2x 2 - 6 -6x 2 + 40x + 30

C) -2 -2x + 5 2 2x 2 - 6 -6x 2 + 40x + 30

D) -2 -2x + 5 2 2x 2 - 6 26x 2 - 40x - 30

Answer: D Type: BI Var: 50+ Objective: Algebra for Calculus

5 t - 1 5 7t + 4 6 - 5 7t + 4 5 7 t - 1 6 11)

7t + 4 6 2 A) C)

70t t - 1 5

7t + 4 7 55 t - 1 5

7t + 4 7

25 t - 1 5 7t + 4 3 5 t - 1 5 14t - 3 7t + 4 7

Answer: C Type: BI Var: 50+ Objective: Algebra for Calculus

12) 8 - x 2 1/2 + x ·

8 - x 2 -1/2 -8x

8 2x-2x+2 A) 8 - x 2 1/2

x2 B) 8 - x 2 1/2

2x2 -4 C) 8 - x 2 1/2

Answer: A Type: BI Var: 36 Objective: Algebra for Calculus

Write the answer as a single term and simplify. 5x 6 x+4 x+4 13) 2 x+4 5x A) 6 -

x+4 x+4

6 - 5x B)

11x + 24 C)

x+4 x+4

Answer: D Type: BI Var: 25 Objective: Algebra for Calculus

Page 43

x+4 x + 24

x+4 x+4

D) -

x2 8 - x 2 1/2

14) x + 3 1/4 +

x 5 x + 3 3/4 x

A) 1 +

5x+3

5+x 3/4

6x + 3

3/4

3x 1/2 B) x + 7 3/2

5x+3

3/4

6x + 15 D)

5 x + 3 3/4

3x 1/2 x + 7 3/2

Answer: D Type: BI Var: 48 Objective: Algebra for Calculus

15)

3x -1/2 x + 7 3 - 3x x+72 3 x+7

7 1/2 3x x + 7 5/2

Answer: D Type: BI Var: 49 Objective: Algebra for Calculus

Page 44

21 3x 1/2 x + 7 3/2

Chapter 1 Equations and Inequalities 1.1 Linear Equations and Rational Equations 0 Concept Connections

Provide the missing information. 1) An equation that can be written in the form ax + b = 0 where a and b are real numbers and a ≠ 0 is called a equation in one variable. Answer: linear Type: SA Var: 1 Objective: Concept Connections

2) A linear equation is also called a

-degree equation because the degree of the variable is 1.

Answer: first Type: SA Var: 1 Objective: Concept Connections

3) A

to an equation is the value of the variable that makes the equation a true statement.

Answer: solution Type: SA Var: 1 Objective: Concept Connections

4) The solution

to an equation is the set of all solutions to the equation.

Answer: set Type: SA Var: 1 Objective: Concept Connections

5) Two equations are

equations if they have the same solution set.

Answer: equivalent Type: SA Var: 1 Objective: Concept Connections

6) The property of equality indicates that adding the same real number to both sides of an equation results in an equivalent equation. Answer: addition Type: SA Var: 1 Objective: Concept Connections

7) The

property of equality indicates that if a = b, then

a c

provided that c ≠ 0.

Answer: division Type: SA Var: 1 Objective: Concept Connections

8) A

equation is one that is true for some values of the variable and false for others.

Answer: conditional Type: SA Var: 1 Objective: Concept Connections

Page 1

9) An is an equation that is true for all values of the variable for which the expressions in the equation are defined. Answer: identity Type: SA Var: 1 Objective: Concept Connections

10) A

is an equation that is false for all values of the variable.

Answer: contradiction Type: SA Var: 1 Objective: Concept Connections

11) A

equation is an equation in which each term contains a rational expression.

Answer: rational Type: SA Var: 1 Objective: Concept Connections

12) If an equation has no solution, then the solution set is the .

set and is denoted by

Answer: empty (or null); { } or ∅ Type: SA Var: 1 Objective: Concept Connections 1 Solve Linear Equations in One Variable

Solve the problem. 1) A train ride is $3.40 per ride. Write a model for the cost C (in $) for x rides on the train. A) C = 3.40 - x B) Cx = 3.40 C) C = 3.40x D) C = 3.40 + x Answer: C Type: BI Var: 50+ Objective: Solve Linear Equations in One Variable

2) A train ride is $2.85 per ride. A commuter can purchase an unlimited-ride card for $45 per month. How many rides are required for a commuter to save money by buying the card? A) 16 rides B) 20 rides C) 22 rides D) 18 rides Answer: A Type: BI Var: 50+ Objective: Solve Linear Equations in One Variable

3) In the mid-nineteenth century, explorers used the boiling point of water to estimate altitude. The boiling temperature of water T (in °F) can be approximated by the model T = -1.83a + 212, where a is the altitude in thousands of feet. Determine the temperature at which water boils at an altitude of 9,000 ft. Round to the nearest degree. A) 214 °F B) 210 °F C) 196 °F D) 228 °F Answer: C Type: BI Var: 26 Objective: Solve Linear Equations in One Variable

Page 2

4) In the mid-nineteenth century, explorers used the boiling point of water to estimate altitude. The boiling temperature of water T (in °F) can be approximated by the model T = -1.83a + 212, where a is the altitude in thousands of feet. Two campers hiking in Colorado boil water for tea. If the water boils at 196°F, approximate the altitude of the campers. Give the result to the nearest hundred feet. A) 8,700 ft B) 2,900 ft C) 8,900 ft D) 1,600 ft Answer: A Type: BI Var: 25 Objective: Solve Linear Equations in One Variable 2 Identify Conditional Equations, Identities, and Contradictions

Determine whether the equation is a conditional equation, an identity, or a contradiction. 1 1) 3(z + 2) - 5z = 4 - z +1 + 2 2 A) Conditional

B) Identity

C) Contradiction

Answer: B Type: BI Var: 27 Objective: Identify Conditional Equations, Identities, and Contradictions

2) 16y + 2(3 - y) = 5 + 14y + 2 A) Conditional Answer: C

B) Identity

C) Contradiction

Type: BI Var: 19 Objective: Identify Conditional Equations, Identities, and Contradictions

3) y - 12 + 3y = 2y + 4 A) Conditional Answer: A

B) Identity

C) Contradiction

Type: BI Var: 50+ Objective: Identify Conditional Equations, Identities, and Contradictions 3 Solve Rational Equations

Solve the rational equation. 11 1 7 1) y + = y 2 3 4 4 A) 45

Answer: A Type: BI Var: 35 Objective: Solve Rational Equations

Page 3

4 87

C) {-4}

D) -

1 4

3 x

4 7 A) 12

B) -

12 7

C) -

7 12

12 7

Answer: B Type: BI Var: 33 Objective: Solve Rational Equations

6 3p - 15 3 = p - 12 p - 12 p A) {6, 2}

B) {-6, -2}

C) {-5, -3}

D) {5, 3}

C) {1}

D) { }

C) {5}

D) {-5, 1}

C) {5, 1}

D) {-8, 8}

Answer: A Type: BI Var: 20 Objective: Solve Rational Equations

3x - 18

x-7

5 1 A) - , 2 3

B) {7, 1}

Answer: C Type: BI Var: 7 Objective: Solve Rational Equations

1 x-4

5 x+1

1 x - 3x - 4 2

A) {5, 1}

B) { }

Answer: C Type: BI Var: 10 Objective: Solve Rational Equations

t - 8 t - 23 1 = 2 t-2 t -4 t+2 A) {5}

B) {-5, -1}

Answer: C Type: BI Var: 50+ Objective: Solve Rational Equations

Page 4

Determine the restrictions on x. 9 6 1 7) = 3x - 5 7x 2 - x 5 A) x ≠ ; x ≠ 0;x ≠ 2 3 3 C) ; x ≠ 0;x ≠ -2 5

3 B) x ≠ ; x ≠ -7;x ≠ 2 5 5 D) x ≠ ; x ≠ -7;x ≠ -2 3

Answer: A Type: BI Var: 50+ Objective: Solve Rational Equations

Solve the rational equation. -21 5 3 8) 2 = x - x - 12 x - 4 x + 3 A) {-4}

B) {-3}

C) { }

D) {3}

Answer: C Type: BI Var: 50+ Objective: Solve Rational Equations 4 Solve Literal Equations for a Specified Variable

Solve for the indicated variable. 1) -8x - 9y = 7 for y 8 7 A) y = - x 9 9

B) y =

8 7 x9 9

8 C) y = -

x+7

D) y =

x+7

Answer: A Type: BI Var: 50+ Objective: Solve Literal Equations for a Specified Variable

2) 3x - y = 2 for y A) y = 3x - 2

B) y = -3x + 2

C) y = 3x + 2

D) y = -3x - 2

Answer: A Type: BI Var: 50+ Objective: Solve Literal Equations for a Specified Variable

3) A = LW for L L A) W = A

B) W =

A L

Answer: D Type: BI Var: 5 Objective: Solve Literal Equations for a Specified Variable

Page 5

C) L =

W A

D) L =

A W

4) H = kx - kx0 for x A) H - kx0 x= k

H + kx0 x0

H + kx0 k

H - kx0 x0

Answer: C Type: BI Var: 39 Objective: Solve Literal Equations for a Specified Variable

5) T = cMN2 for N2 A) N2 = cMT

B) N2 =

T cM

C) N2 =

D) N2 =

Answer: B Type: BI Var: 50+ Objective: Solve Literal Equations for a Specified Variable

6) S = α(T - T0) + S0

for T

1 A) T = (S - S0 + T0) α C) T = S - S + T α 0 0

B) T = α(S - S0) + T0 1 D) T = (S - S0) + T0 α

Answer: D Type: BI Var: 27 Objective: Solve Literal Equations for a Specified Variable

7) q =

c 4

(h + r) for r

A) r =

4q - h c

q B) r =

-h

C) r =

4c q

4q -h

D) r =

-h

Answer: D Type: BI Var: 50+ Objective: Solve Literal Equations for a Specified Variable

1 8) Q =

for D

A) D =

3P Q

B) D =

Q 3P

Answer: D Type: BI Var: 27 Objective: Solve Literal Equations for a Specified Variable

Page 6

C) D =

P 3Q

D) D =

3Q P

9) L =

πq2s for s

3 πq2

3L A) s =

B) s =

πq2

3πq2

L C) s =

3πq2

D) s =

Answer: A Type: BI Var: 27 Objective: Solve Literal Equations for a Specified Variable

10) 9x + ry = tx + 6 6 - ry A) x = 9-t

for x B) x =

tx - ry + 6 9

C) x =

6 - ry t-9

D) x =

t+6 9 + ry

Answer: A Type: BI Var: 50+ Objective: Solve Literal Equations for a Specified Variable 5 Mixed Exercises

Solve the equation. 1) 5 - 2{2 - [-3n - 2(n + 5)]} = -8n + 2(1 + 4n) - 21 A) {5} B) {0}

C) {-2}

D) {1}

Answer: B Type: BI Var: 50+ Objective: Mixed Exercises

Solve the problem. 2) Dema's truck gets 32 mpg on the highway and 18 mpg in the city. The amount of gas he uses A (in 1 1 gal) is given by A = c + h, where c is the number of city miles driven and h is the number of 18 32 highway miles driven. If Dema drove 45 mi in the city and used 8 gal of gas, how many highway miles did he drive? A) 176 miles B) 200 miles C) 192 miles D) 160 miles Answer: A Type: BI Var: 50+ Objective: Mixed Exercises

Solve the equation. 1 1 1 1 3) - x - = - (x + 1) - x 4 6 6 12 A) {0}

B) -

1 3

C) All real numbers Answer: C Type: BI Var: 20 Objective: Mixed Exercises

Page 7

D) { }

6 Expanding Your Skills

Find the value of a so that the equation has the given solution set. 1) ax - 6 = 7x - 26 {5} A) a = 5

B) a = 3

C) a = -

141 5

D) a =

3 5

Answer: B Type: BI Var: 50+ Objective: Expanding Your Skills

1.2 Applications with Linear and Rational Equations 0 Concept Connections

Provide the missing information. 1) The formula for the perimeter P of a rectangle with length l and width w is given by

Answer: P = 2l + 2w Type: SA Var: 1 Objective: Concept Connections

2) The sum of the measures of the angles inscribed inside a triangle is

Answer: 180° Type: SA Var: 1 Objective: Concept Connections

3) If $6000 is borrowed at 7.5% simple interest for 2 yr, then the amount of interest is

Answer: $900 Type: SA Var: 1 Objective: Concept Connections

4) Suppose that 8% of a solution is fertilizer by volume and the remaining 92% is water. How much fertilizer is there in a 2 L bucket of solution? Answer: 0.16 L Type: SA Var: 1 Objective: Concept Connections

5) If d = rt, then t =

? ?

Answer:

d r

Type: SA Var: 1 Objective: Concept Connections

6) If d = rt, then r =

? ?

Answer:

d t

Type: SA Var: 1 Objective: Concept Connections

Page 8

1 Solve Applications Involving Simple Interest

Solve the problem. 1) If $13,000 is borrowed at 5.8% simple interest for 10 years, how much interest will be paid for the loan? A) $7,540.00 B) $9,845.47 C) $20,540.00 D) $22,845.47 Answer: A Type: BI Var: 50+ Objective: Solve Applications Involving Simple Interest

2) Aaron invested a total of $4,100, some in an account earning 8% simple interest, and the rest in an account earning 5% simple interest. How much did he invest in each account if after one year he earned $211 in interest? A) $900 at 8%, $3,200 at 5% B) $200 at 8%, $3,900 at 5% C) $3,200 at 8%, $900 at 5% D) $3,900 at 8%, $200 at 5% Answer: B Type: BI Var: 50+ Objective: Solve Applications Involving Simple Interest 2 Solve Applications Involving Mixtures

Solve the problem. 1) How many gallons of gasoline that is 5% ethanol must be added to 2,000 gallons of gasoline with no ethanol to get a mixture that is 3% ethanol? A) 1,800 B) 4,115 C) 3,000 D) 6,000 Answer: C Type: BI Var: 8 Objective: Solve Applications Involving Mixtures

2) A nurse mixes 90 cc of a 45% saline solution with a 10% saline solution to produce a 20% saline solution. How much of the 10% solution should he use? A) 180 cc B) 225 cc C) 202.5 cc D) 18 cc Answer: B Type: BI Var: 30 Objective: Solve Applications Involving Mixtures 3 Solve Applications Involving Uniform Motion

Solve the problem. 1) Two cars are 261 miles apart and travel toward each other on the same road. They meet in 3 hours. One car travels 3 mph faster than the other. What is the average speed of each car? A) 42 mph; 45 mph B) 41 mph; 44 mph C) 40 mph; 43 mph D) 39 mph; 42 mph Answer: A Type: BI Var: 50+ Objective: Solve Applications Involving Uniform Motion

Page 9

2) A boat can travel 42 miles upstream against the current in the same amount of time it can travel 63 miles downstream with the current. If the boat's average speed in still water is 20 miles per hour, find the speed of the current. Answer: 4 miles per hour Type: SA Var: 50+ Objective: Solve Applications Involving Uniform Motion

3) A consultant traveled 255 miles to attend a meeting, traveling 45 mph hours for the first part of the trip, then increasing to a speed of 60 mph for the second part. If the entire trip took 5 hours, how far did the consultant travel at the faster speed? A) 127.5 mi B) 180 mi C) 120 mi D) 135 mi Answer: C Type: BI Var: 36 Objective: Solve Applications Involving Uniform Motion 4 Solve Applications Involving Rate of Work Done

Solve the problem. 1) It takes Terrell 69 minutes to weed his garden if he does it every 2 weeks, while his wife can get it done in 49 minutes. How long would it take them working together? Round to the nearest tenth of a minute. A) 34.5 minutes B) 28.7 minutes C) 29.5 minutes D) 24.5 minutes Answer: B Type: BI Var: 21 Objective: Solve Applications Involving Rate of Work Done

2) The JUST-SAY-MOW lawn mowing company consists of two people: Marsha and Bob. If Marsha cuts the lawn by herself, she can do it in 3 hours. If Bob cuts the same lawn himself, it takes him an hour longer than Marsha. How long would it take them if they worked together? Round to the nearest hundredth of an hour. A) 1.71 hours B) 1.00 hour C) 4.00 hours D) 3.50 hours Answer: A Type: BI Var: 50+ Objective: Solve Applications Involving Rate of Work Done 5 Solve Applications Involving Proportions

Solve the problem. 1) The property tax on a $160,000.00 house is $2,400.00. At this rate, what is the property tax on a house that is $280,000.00? A) $5,040.00 B) $4,200.00 C) $4,620.00 D) $3,780.00 Answer: B Type: BI Var: 50+ Objective: Solve Applications Involving Proportions

Page 10

2) To estimate the number of bass in a lake, a biologist catches and tags 32 bass. Several weeks later, the biologist catches a new sample of 55 bass and finds that 5 are tagged. How many bass are in the lake? A) 1,760 bass B) 275 bass C) 160 bass D) 352 bass Answer: D Type: BI Var: 50+ Objective: Solve Applications Involving Proportions 6 Mixed Exercises

Solve the problem. 1) The plans for a rectangular deck call for the width to be 4 feet less than the length. Sam wants the deck to have an overall perimeter of 52 feet. What should the length of the deck be? A) 4 feet B) 19 feet C) 28 feet D) 15 feet Answer: D Type: BI Var: 24 Objective: Mixed Exercises

2) The perimeter of a rectangular lot of land is 436 ft. This includes an easement of x feet of uniform width inside the lot on which no building can be done. If the buildable area is 122 ft by 60 ft, determine the width of the easement. A) 9 feet B) 7 feet C) 18 feet D) 4.5 feet Answer: A Type: BI Var: 50+ Objective: Mixed Exercises

3) Suppose that a merchant buys a patio set from the wholesaler for $260. At what price should the merchant mark the patio set so that it may be offered at a discount of 25% but still give the merchant a 20% profit on his $260 investment? A) $312 B) $325 C) $416 D) $377 Answer: C Type: BI Var: 21 Objective: Mixed Exercises

4) Aliyah earned an $6,000 bonus from her sales job for exceeding her sales goals. After paying taxes at a 30% rate, she invested the remaining money in two stocks. One stock returned the equivalent of 10% simple interest after 1 yr, and the other returned 4% at the end of 1 yr. If her investments returned $240.00 (excluding commissions) how much did she invest in each stock A) $3,000 at 4% and $1,200 at 10% B) $1,450 at 4% and $2,750 at 10% C) $1,200 at 4% and $3,000 at 10% D) $2,750 at 4% and $1,450 at 10% Answer: A Type: BI Var: 50+ Objective: Mixed Exercises

Page 11

7 Expanding Your Skills

Solve the problem. 1) One number is 33 more than another number. The quotient of the larger number and smaller number is 5 and the remainder is 1. Find the numbers. A) 10 and 43 B) 5 and 38 C) 11 and 44 D) 8 and 41 Answer: D Type: BI Var: 50+ Objective: Expanding Your Skills

1.3 Complex Numbers 0 Concept Connections

Provide the missing information. 1) The imaginary number i is defined so that i = -1 and i2 =

Answer: -1 Type: SA Var: 1 Objective: Concept Connections

2) For a positive real number, b, the value -b =

Answer: i b Type: SA Var: 1 Objective: Concept Connections

3) Given a complex number a + bi, the value of a is called the called the part.

part and the value of b is

Answer: real; imaginary Type: SA Var: 1 Objective: Concept Connections

4) Given a complex number a + bi, the expression a - bi is called the complex

Answer: conjugate Type: SA Var: 1 Objective: Concept Connections 1 Simplify Imaginary Numbers

Simplify the expression in terms of i: 1) -49 A) i 7 B) 7i

C) 49i

D) -7i

C) 3 2i

D) 9i 2

Answer: B Type: BI Var: 11 Objective: Simplify Imaginary Numbers

2) -18 A) 3i 2

B) -3i 2

Answer: A Type: BI Var: 5 Objective: Simplify Imaginary Numbers

Page 12

Simplify the expression. -144 3) -36 A) -2i

B) 2i

1 2

D) 2

Answer: D Type: BI Var: 5 Objective: Simplify Imaginary Numbers

4) -81 · -3 A) 9 -3

B) -27

C) 9 3

D) -9 3

Answer: D Type: BI Var: 50+ Objective: Simplify Imaginary Numbers

-25

9 A)

B) -

5 3

C) -

5 3

Answer: D Type: BI Var: 50+ Objective: Simplify Imaginary Numbers 2 Write Complex Numbers in the Form a + bi

Identify the real and imaginary parts of the complex number. 1) 11 + 13i A) Real: 11; imaginary: 13i B) Real: 11; imaginary: 13 C) Real: 13; imaginary: 11 D) Real: 24; imaginary: i Answer: B Type: BI Var: 50+ Objective: Write Complex Numbers in the Form a + bi

4 7 ; imaginary: 0

4 B) Real: ; imaginary: i 7

C) Real: 4 ; imaginary: 7

D) Real: 0 ; imaginary:

A) Real:

4 7

Answer: A Type: BI Var: 1 Objective: Write Complex Numbers in the Form a + bi

Page 13

4 7

Simplify and write the result in standard form, a + bi. 14 - -12 3) 2 A)

B) 7 - 2i 3

7-i 3

C) 7 + i 3

D) 7 + 2i 3

Answer: A Type: BI Var: 50+ Objective: Write Complex Numbers in the Form a + bi

4 + -18 6 A)

2 3

Answer: B Type: BI Var: 50+ Objective: Write Complex Numbers in the Form a + bi

-8 - 10i 5)

-2 A) 4 + 5i

B) 4 - 5i

C) 4 - 10i

D) 4 + 10i

C) -2 - 3i 2

D) -2 + 3i 2

C) -1

D) 1

C) -1

D) i

Answer: A Type: BI Var: 50+ Objective: Write Complex Numbers in the Form a + bi

6 - -18 -3 A) -2 - i 2

B) -2 + i 2

Answer: B Type: BI Var: 50+ Objective: Write Complex Numbers in the Form a + bi 3 Perform Operations on Complex Numbers

Simplify. 1) i40 A) i

B) -i

Answer: D Type: BI Var: 8 Objective: Perform Operations on Complex Numbers

2) i15 A) 1 Answer: B

B) -i

Type: BI Var: 8 Objective: Perform Operations on Complex Numbers

Page 14

Perform the indicated operation. Write the answer in the form a + bi. 3) (-12 - 10i) + (17 +14i) A) 29 + 24i B) 5 + 4i C) 9

D) 9i

Answer: B Type: BI Var: 50+ Objective: Perform Operations on Complex Numbers

4) (-4 - 6i) - (9 - 9i) A) -28i

B) -13 + 3i

C) -13 - 15i

D) -10i

C) -30 - 54i

D) 24

C) -70i

D) -30i

Answer: B Type: BI Var: 50+ Objective: Perform Operations on Complex Numbers

5) (-5 - 9i)(6 + 6i) A) 24 - 84i

B) -84 - 84i

Answer: A Type: BI Var: 50+ Objective: Perform Operations on Complex Numbers

6) 7i(-5 + 5i) A) -35 - 35i

B) 35 - 35i

Answer: A Type: BI Var: 50+ Objective: Perform Operations on Complex Numbers

-8 + 3i 5 + 7i A) -

3 - i 5 7

B) -

19 71 i + 74 74

3 + i 5 7

D) -

13 8 - i 5 5

D) 2 - i

C) -

19 71 - i 74 74

Answer: B Type: BI Var: 43 Objective: Perform Operations on Complex Numbers

6-i 2+i A) 2

8 - i 5 5

Answer: B Type: BI Var: 50+ Objective: Perform Operations on Complex Numbers

Page 15

8 + 9i 3-i A)

15 35 - i 8 8

3 7 - i 2 2

15 35 + i 8 8

C) 39 + 2i 9

D) 39

C) 64

D) 64 - 160i

3 7 + i 2 2

Answer: D Type: BI Var: 50+ Objective: Perform Operations on Complex Numbers

10) (6 + -9)(8 - -9) A) 57 - 2i 9

B) 57 + 2i 9

Answer: B Type: BI Var: 50+ Objective: Perform Operations on Complex Numbers

11) (8 - 5i)2 + (8 + 5i)2 A) 78 + 160i

B) 78

Answer: B Type: BI Var: 19 Objective: Perform Operations on Complex Numbers 4 Mixed Exercises 2 - 4ac

Evaluate b for the given values of a, b, and c, and simplify. 1) a = 4, b = -2, and c = 7 A) 6 3 B) -6 3 C) 3i 6

D) 6i 3

Answer: D Type: BI Var: 50+ Objective: Mixed Exercises

1.4 Quadratic Equations 0 Concept Connections

Provide the missing information. 1) A equation is a second degree equation of the form ax2 + bx + c = 0 where a ≠ 0. Answer: quadratic Type: SA Var: 1 Objective: Concept Connections

2) A

equation is a first degree equation of the form ax + b = 0 where a ≠ 0.

Answer: linear Type: SA Var: 1 Objective: Concept Connections

3) The zero product property indicates that if ab = 0, then Answer: a; b Type: SA Var: 1 Objective: Concept Connections

Page 16

= 0 or

= 0.

4) The zero product property indicates that if (5x + 1)(x - 4) = 0, then

= 0 or

= 0.

Answer: (5x + 1); (x - 4) Type: SA Var: 1 Objective: Concept Connections

5) The square root property indicates that if x2 = k, then x =

Answer: ± k Type: SA Var: 1 Objective: Concept Connections

6) The value of n that would make the trinomial x2 + 20x + n a perfect square trinomial is Answer: 100 Type: SA Var: 1 Objective: Concept Connections

7) Given ax2 + bx + c = 0 (a ≠ 0), write the quadratic formula. Answer: x =

-b ± b2 - 4ac 2a

Type: SA Var: 1 Objective: Concept Connections

8) For a quadratic equation ax2 + bx + c = 0, the discriminant is given by the expression . 2 Answer: b - 4ac Type: SA Var: 1 Objective: Concept Connections 1 Solve Quadratic Equations by Using the Zero Product Property

Solve the equation. 1) 5w (5w + 12) = -32 12 A) 0, 5

8 4 , 5 5

C) - 8 , - 4 5 5

D) - 8 , 4 5 5

Answer: C Type: BI Var: 50+ Objective: Solve Quadratic Equations by Using the Zero Product Property

2) t2 - 5t = -4 A) {0, 5}

B) {-4, -1}

C) {0, -5}

Answer: D Type: BI Var: 50+ Objective: Solve Quadratic Equations by Using the Zero Product Property

Page 17

D) {4, 1}

3) y2 - 20y = 0 1 A) 0, 20

B) {0, 20}

C) {20}

D) {0, -20}

Answer: B Type: BI Var: 9 Objective: Solve Quadratic Equations by Using the Zero Product Property

4) 15m(m + 5) = 38m - 20 A) {0, -20}

B) - 4 , - 5 5 3

4 5 ,5 3

4 5 , 5 3

Answer: B Type: BI Var: 50+ Objective: Solve Quadratic Equations by Using the Zero Product Property

5) 9s2 = 4 3 A) 2

2 3

3 3 C) - , 2 2

2 2 D) - , 3 3

Answer: D Type: BI Var: 50+ Objective: Solve Quadratic Equations by Using the Zero Product Property

6) (m + 3)(m - 4) = -6 A) {2, -3}

B) {3, -4}

C) {-2, 3}

D) {-3, 4}

Answer: C Type: BI Var: 12 Objective: Solve Quadratic Equations by Using the Zero Product Property 2 Solve Quadratic Equations by Using the Square Root Property

Solve the equation by using the square root property. 1) f2 = 25 A) {±5} B) {5}

C) {5i}

D) {±5i}

Answer: A Type: BI Var: 50+ Objective: Solve Quadratic Equations by Using the Square Root Property

2) (3x + 10)2 = 81 61 61 A) ,6 6

61 61 i, - i 6 6

C) - 1 , - 19 3 3

Answer: C Type: BI Var: 50+ Objective: Solve Quadratic Equations by Using the Square Root Property

Page 18

D) - 1 3

3) (c + 8)2 = 16 A) {12, 4}

B) {-4, -12}

C) {24, -8}

D) {8, -24}

Answer: B Type: BI Var: 50+ Objective: Solve Quadratic Equations by Using the Square Root Property

4) (3z - 18)2 + 59 = 14 A) {3 5 + 14} C) {6 + i 5, 6 - i 5}

B) {3 5 - 14} D) {14 + 3 5, -14 + 3 5}

Answer: C Type: BI Var: 50+ Objective: Solve Quadratic Equations by Using the Square Root Property

5) 3(x + 8)2 - 15 = 255 A) 8 ± 265

B) -8 ± 3 10

C) -8 ± 265

D) 8 ± 3 10

Answer: B Type: BI Var: 50+ Objective: Solve Quadratic Equations by Using the Square Root Property

6) t -

12 6

17 36

1 - i 17 6

B) -

11 36

17 1 ± 6 6

17 1 i ± 6 6

Answer: D Type: BI Var: 50+ Objective: Solve Quadratic Equations by Using the Square Root Property 3 Complete the Square

Find the value of n so that the expression is a perfect square trinomial and then factor the trinomial. 1) j2 - 4j + n A) n = 2; (j - 2)2 B) n = 4; (j - 2) C) n = 2; (j + 2)2 D) n = 4; (j - 2)2 Answer: D Type: BI Var: 8 Objective: Complete the Square

2) x2 + 20x + n A) n = 400; (x + 20)2 C) n = 100; (x + 10)(x - 10) Answer: B Type: BI Var: 10 Objective: Complete the Square

Page 19

B) n = 100; (x + 10)2 D) n = 100; (x - 10)2

3) Find the value of n so that the expression is a perfect square trinomial and then factor the trinomial. 14 t2 - t + n 3 2 98 2 196 t - 49 98 ; A) n = B) n = ; t 9 9 9 3 49

7 C) n = ; t + 9 3

D) n =

; t-

72 3

Answer: D Type: BI Var: 6 Objective: Complete the Square

Solve the quadratic equation by completing the square and applying the square root property. 4) u2 + 20u + 101 = 0 A) {10 + i} B) {± i} C) {-10 + i} D) {-10 ± i} Answer: D Type: BI Var: 10 Objective: Complete the Square

5) n2 + 18n = -75 A) {9 - 6, 9 + 6}

B) {-9 - 6, -9 + 6}

C) {-9 - 249, -9 + 249}

-18 - 249 -18 + 249 , 2 2

Answer: B Type: BI Var: 50+ Objective: Complete the Square

6) y2 + 53 = 4y A) {4 - i 37, 4 + i 37} C) {-2 - 7i, -2 + 7i}

B) {4 - 37, 4 + 37} D) {2 - 7i, 2 + 7i}

Answer: D Type: BI Var: 28 Objective: Complete the Square

7) 2v2 + 4v + 12 = 0 A) {-2 - i 2, -2 + i 2} C) {-1 - 5, -1 + 5} Answer: D Type: BI Var: 50+ Objective: Complete the Square

Page 20

B) {-2 - 2, -2 + 2} D) {-1 - i 5, -1 + i 5}

8) 2x2 + 6 = 9x A) {9 - 87, 9 + 87} C)

9 - 33 9 + 33 , 4 4

-9 - 33 -9 + 33 , 4 4

D) {-9 - 87, -9 + 87}

Answer: C Type: BI Var: 7 Objective: Complete the Square

9) -5v2 = 5 + 7v 7 51 51 7 A) i, + i 10 10 10 10 7 C) -

7 51 51 i, i + 10 10 10

7 - 69 7 + 69 , 10 10

-7 - 69 -7 + 69 , 10 10

Answer: C Type: BI Var: 36 Objective: Complete the Square

10) 3x2 + 5x - 6 = 0 5 97 A) - ± 6 6

B) -

5 3

47 3

C) -

5 6

47 6

D) -

5 3

Answer: A Type: BI Var: 50+ Objective: Complete the Square 4 Solve Quadratic Equations by Using the Quadratic Formula

Solve the equation by using the quadratic formula. 1) 3x2 + 12x - 15 = 0 1 5 1 5 A) - , B) , 3 3 3 3

C) {-1, 5}

D) {1, -5}

Answer: D Type: BI Var: 27 Objective: Solve Quadratic Equations by Using the Quadratic Formula

2) 2x(x - 2) = 5 A) -1 +

14 , -6 + 2

C) 1 ±

14 2

1± i

14 2

D) -1 +

14 i, -6 + 2

Answer: C Type: BI Var: 7 Objective: Solve Quadratic Equations by Using the Quadratic Formula

Page 21

14 i 2

97 3

3) 6y + 3 = -4y2 3 i 3 3 i 3 A) - + ,- 4 4 4 4

9 1 B) - , 2 3 105 1 1 D) - + , - - 105 12 4 12 4

C) {2 + 3, 2 - 3} Answer: A

Type: BI Var: 1 Objective: Solve Quadratic Equations by Using the Quadratic Formula

4) 5y - 6 + 50y2 = 0 2 3 A) - , 5 10

5 10 ,2 3

3 ± 5

2,791 5

1 2 , 3 5

959 1 i ± 16 16

Answer: A Type: BI Var: 9 Objective: Solve Quadratic Equations by Using the Quadratic Formula

5) 6x(x - 2) = 5 A) 1 ±

B) -1 +

66 66 , -6 + 6 6

D) 1 ±

66 6

6 66 C) -1 +

i, -6 +

66 6

Answer: D Type: BI Var: 7 Objective: Solve Quadratic Equations by Using the Quadratic Formula

6) -

1 = x - 5x2 3 6 9 A) -8, 30

161 10

1 8 C) - , 2 15

Answer: C Type: BI Var: 1 Objective: Solve Quadratic Equations by Using the Quadratic Formula

7) 0.49x2 = 0.28x - 0.04 2 A) ± 7

B) -

2 7

2 C) 7

Answer: C Type: BI Var: 50+ Objective: Solve Quadratic Equations by Using the Quadratic Formula

Page 22

2 D)

8) (3w - 2)(w - 1) = -3 35 35 5 5 i, + i A) 6 6 6 6 C) -

37 i, 5 + 5 6 6 6

37 6

B) i

-5 - 35 -5 + 35 , 6 6

D) - 1 , -2 3

Answer: A Type: BI Var: 28 Objective: Solve Quadratic Equations by Using the Quadratic Formula

9) y2 = 4y - 9 A) {- 2 ± i 5}

B) {-4 ± 2i 5}

C) {4 ± 2i 5}

D) {2 ± i 5}

Answer: D Type: BI Var: 16 Objective: Solve Quadratic Equations by Using the Quadratic Formula

10) t (t - 2) = -2 A) {1 ± 2i }

B) {-1 ± 2i }

C) {1 ± i }

D) {-1 ± i }

Answer: C Type: BI Var: 16 Objective: Solve Quadratic Equations by Using the Quadratic Formula 5 Use the Discriminant

Use the discriminant to determine the type and number of solutions. 1) -2x2 + 5x + 5 = 0 A) Two imaginary solutions B) Two irrational solutions C) Two rational solutions D) One rational solution Answer: B Type: BI Var: 50+ Objective: Use the Discriminant

2) 5x2 + 4x + 5 = 0 A) Two rational solutions C) Two imaginary solutions Answer: C

B) Two irrational solutions D) One rational solution

Type: BI Var: 50+ Objective: Use the Discriminant

3) 6q2 = 1 A) Two rational solutions C) Two irrational solutions Answer: C Type: BI Var: 50+ Objective: Use the Discriminant

Page 23

B) Two imaginary solutions D) One rational solutions

6 Solve an Equation for a Specified Variable

Solve for the indicated variable. 1) c = 9 r for r c2 A) r = 9

B) r =

c2 81

C) r =

D) r =

c 81

Answer: B Type: BI Var: 8 Objective: Solve an Equation for a Specified Variable

2) w =

kr2

for r > 0

3 A) r =

3wk k

B) r =

3w k

C) r =

3 w k

D) r = 3w

C) t =

m h

D) t =

B) t =

a(h - uy) or t = a

a(h + uy) a

Answer: A Type: BI Var: 45 Objective: Solve an Equation for a Specified Variable

3) m = h2kt2x for t > 0 A) t =

mkx hkx

m B) t =

h kx

mhkx hkx

Answer: A Type: BI Var: 50+ Objective: Solve an Equation for a Specified Variable

4) at2 + uy = h

for t

A) t = ±

a(h - uy) a

C) t =

h - uy a

Answer: A Type: BI Var: 50+ Objective: Solve an Equation for a Specified Variable

Page 24

D) t = ± a(h - uy)

5) s = vt +

at2

for t

v±

v2 + 2as

2 A) t =

v±

B) t =

2a C) t =

-v ±

v2 + 2as

v2 + 2as a

-v ± i v2 + 2as

D) t =

Answer: C Type: BI Var: 4 Objective: Solve an Equation for a Specified Variable 7 Mixed Exercises

Solve the equation. 1) y2 + 3y - 11 = (y + 2)(y - 4) 3 3 B) {2, -4} A) , 5 5

D) {-2, 4}

Answer: C Type: BI Var: 50+ Objective: Mixed Exercises

2) 5(x + 2) + x2 = x(x + 5) + 10 A) All real numbers C) No solution

B) 2 5 D) {0}

Answer: A Type: BI Var: 32 Objective: Mixed Exercises

1.5 Applications of Quadratic Equations 0 Concept Connections

Provide the missing information. 1) Write a formula for the area of a triangle of base b and height h. 1 Answer: A = bh 2 Type: SA Var: 1 Objective: Concept Connections

2) Write a formula for the area of a circle of radius r. Answer: A = πr2 Type: SA Var: 1 Objective: Concept Connections

3) Write a formula for the volume of a rectangular solid of length l, width w, and height h. Answer: V = lwh Type: SA Var: 1 Objective: Concept Connections

Page 25

4) Write the Pythagorean theorem for a right triangle with the lengths of the legs given by a and b and the length of the hypotenuse given by c. Answer: a2 + b2 = c2 Type: SA Var: 1 Objective: Concept Connections 1 Solve Applications Involving Quadratic Equations and Geometry

Solve the problem. 1) Ramon wants to fence in a rectangular portion of his back yard against the back of his garage for a vegetable garden. He plans to use 40 feet of fence, and needs fence on only three sides. Find the maximum area he can enclose. (Hint: The lengths of the 3 fenced sides of the rectangle must add up to 40.) A) 225 sq. ft. B) 100 sq. ft. C) 400 sq. ft. D) 200 sq. ft. Answer: D Type: BI Var: 16 Objective: Solve Applications Involving Quadratic Equations and Geometry

2) The product of two consecutive positive even integers is 120. Find the integers. A) 10 and 12 B) 58 and 62 C) 59 and 61 D) 12 and 14 Answer: A Type: BI Var: 8 Objective: Solve Applications Involving Quadratic Equations and Geometry

3) The length of a rectangle is 4 yd more than twice the width x. The area is 390 yd2. Find the dimensions of the given shape. A) 13 yd. by 26 yd. B) 6.5 yd. by 60 yd. C) 26 yd. by 15 yd. D) 13 yd. by 30 yd. Answer: D Type: BI Var: 32 Objective: Solve Applications Involving Quadratic Equations and Geometry

4) The sum of the squares of two consecutive whole numbers is 25. Find the numbers. A) 3 and 4 B) 11 and 12 C) 2 and 3 D) 12 and 13 Answer: A Type: BI Var: 4 Objective: Solve Applications Involving Quadratic Equations and Geometry

5) The sum of an integer and its square is 30. Find the integers. A) 25 and 36 B) 5 and 25 C) 5 and -6 Answer: C Type: BI Var: 8 Objective: Solve Applications Involving Quadratic Equations and Geometry

Page 26

D) -6 and 36

6) An open box is formed from a rectangular piece of cardboard that is 5 in. longer than it is wide, by removing squares of side 4 in. from each corner and folding up the sides. If the volume of the carton is then 336 in3, what were the dimensions of the original piece of cardboard? A) 7 in. by 12 in. B) 11 in. by 16 in. C) 15 in. by 20 in. D) 19 in. by 24 in. Answer: C Type: BI Var: 50+ Objective: Solve Applications Involving Quadratic Equations and Geometry

7) A sprinkler rotates 360° to water a circular region. If the total area watered is approximately 2,200 yd2, determine the radius of the region (the radius is length of the stream of water). Round the answer to the nearest yard. A) 26 yd B) 19 yd C) 6 yd D) 350 yd Answer: A Type: BI Var: 31 Objective: Solve Applications Involving Quadratic Equations and Geometry

8) A patio is configured from a rectangle with two right triangles of equal size attached at the two ends. The length of the rectangle is 38 ft. The base of the right triangle is 4 ft less than the height of the triangle. If the total area of the patio is 1,232 ft2, determine the base and height of the triangular portions. 38 ft

x-4 A) base = 15 ft; height = 19 ft C) base = 19 ft; height = 23 ft

B) base = 18 ft; height = 22 ft D) base = 21 ft; height = 25 ft

Answer: B Type: BI Var: 50+ Objective: Solve Applications Involving Quadratic Equations and Geometry

9) The length of a rectangle is 6 yd more than twice the width x. The area is 416 yd2. Find the dimensions of the rectangle. A) width = 32 yd; length = 13 yd B) width = 16 yd; length = 26 yd C) width = 26 yd; length = 16 yd D) width = 13 yd; length = 32 yd Answer: D Type: BI Var: 44 Objective: Solve Applications Involving Quadratic Equations and Geometry

Page 27

10) The height of a triangle is 4 ft less than the base x. The area is 126 ft2. Find the dimensions of the triangle. A) base = 18 ft; height = 22 B) base = 9 ft; height = 28 C) base = 18 ft; height = 14 D) base = 20 ft; height = 16 Answer: C Type: BI Var: 24 Objective: Solve Applications Involving Quadratic Equations and Geometry

11) The width of a rectangular box is 4 in. The height is one-fifth the length x. The volume is 180 in2. Find the length and the height of the box. A) length = 15 in.; height = 3 in. B) length = 20 in.; height = 4 in. C) length = 3 in.; height = 15 in. D) length = 4 in.; height = 20 in. Answer: A Type: BI Var: 50+ Objective: Solve Applications Involving Quadratic Equations and Geometry

12) The length of the longer leg of a right triangle is 14 ft longer than the length of the shorter leg x. The hypotenuse is 6 ft longer than twice the length of the shorter leg. Find the dimensions of the triangle. A) Short leg = 10, long leg = 24, hypotenuse = 26 B) Short leg = 9, long leg = 23, hypotenuse = 24 C) Short leg = 9, long leg = 23, hypotenuse = 28 D) Short leg = 11, long leg = 25, hypotenuse = 28 Answer: A Type: BI Var: 6 Objective: Solve Applications Involving Quadratic Equations and Geometry

13) A rectangular garden covers 46 yd2. The length is 3 yd longer than the width. Find the length and width. Round to the nearest tenth of a yard. A) length = 9.8; width = 6.8 yd B) length = 6.8; width = 9.8 yd C) length = 5.4; width = 8.4 yd D) length = 8.4; width = 5.4 yd Answer: D Type: BI Var: 50+ Objective: Solve Applications Involving Quadratic Equations and Geometry

14) The height of a triangular truss is 5 ft less than the base. The amount of drywall needed to cover the triangular area is 84 ft2. Find the base and height of the triangle to the nearest tenth of a foot. A) base = 12 ft; height = 7 ft B) base = 21 ft; height = 16 ft C) base = 15.9 ft; height = 10.9 ft D) base = 15.7 ft; height = 10.7 ft Answer: D Type: BI Var: 50+ Objective: Solve Applications Involving Quadratic Equations and Geometry

Page 28

2 Solve Applications Involving Quadratic Models

Solve the problem. 1) A model rocket is launched from a raised platform at a speed of 176 feet per second. Its height in feet is given by h(t) = -16t2 + 176t + 20 (t = seconds after launch). After how many seconds does the object reach its maximum height? A) 2.75 seconds B) 5.5 seconds C) 7.5 seconds D) 20 seconds Answer: B Type: BI Var: 7 Objective: Solve Applications Involving Quadratic Models

2) A model rocket is launched from a raised platform at a speed of 160 feet per second. Its height in feet is given by h(t) = -16t2 + 160t + 20 (t = seconds after launch) What is the maximum height reached by the rocket? A) 840 feet B) 420 feet C) 440 feet D) 210 feet Answer: B Type: BI Var: 7 Objective: Solve Applications Involving Quadratic Models

3) The temperature at a state park for one day in June can be approximated by the function T(x) = 0.289x2 - 5.202x + 83 0 ≤ x ≤ 18 where T is degrees Fahrenheit and x is the number of hours after 5 PM on Friday. At what time is the temperature lowest? Round to the nearest hour. A) 2 AM B) 9 AM C) 11 PM D) 5 PM Answer: A Type: BI Var: 50+ Objective: Solve Applications Involving Quadratic Models

4) The temperature at a state park for one day in June can be approximated by the function T(x) = 0.264x2 - 4.752x + 81 0 ≤ x ≤ 18 where T is degrees Fahrenheit and x is the number of hours after 5 PM on Friday. What was the lowest temperature reached? Round to the nearest whole degree. A) 60 degrees B) 66 degrees C) 64 degrees D) 72 degrees Answer: A Type: BI Var: 50+ Objective: Solve Applications Involving Quadratic Models

5) The gas mileage for a certain vehicle can be approximated by m = -0.05x2 + 3.5x - 49, where x is the speed of the vehicle in mph. Determine the speed(s) at which the car gets 9 mpg. Round to the nearest mph. A) 19 mph and 51 mph B) 35 mph C) 27 mph and 43 mph D) 23 mph and 47 mph Answer: C Type: BI Var: 50+ Objective: Solve Applications Involving Quadratic Models

Page 29

6) The daily profit in dollars made by an automobile manufacturer is P(x) = -45x2 + 2,430x - 15,000 where x is the number of cars produced per shift. How many cars must be produced per shift for the company to maximize its profit? A) 27 B) 32 C) 29 D) 54 Answer: A Type: BI Var: 50+ Objective: Solve Applications Involving Quadratic Models

7) The daily profit in dollars made by an automobile manufacturer is P(x) = -40x2 + 2,240x - 17,000 where x is the number of cars produced per shift. Find the maximum possible daily profit. A) $13,211 B) $13,642 C) $31,360 D) $14,360 Answer: D Type: BI Var: 39 Objective: Solve Applications Involving Quadratic Models

8) A bad punter on a football team kicks a football approximately straight upward with an initial velocity of 89 ft/sec. a. If the ball leaves his foot from a height of 4 ft, write an equation for the vertical height s (in ft) of the ball t seconds after being kicked. b. Find the time(s) at which the ball is at a height of 102.2125 ft. Round to 1 decimal place. A) s = -16t2 + 89t + 4; 1.5 sec and 4 sec B) s = -16t2 + 89t + 4; 2.5 sec and 6.6 sec C) s = -9.8t2 + 89t + 4; 2.5 sec and 6.6 sec D) s = -9.8t2 + 89t + 4; 1.5 sec and 4 sec Answer: A Type: BI Var: 50+ Objective: Solve Applications Involving Quadratic Models

1.6 More Equations and Applications 0 Concept Connections

Provide the missing information. 1) A equation is an equation that has one or more radicals containing a variable. Answer: radical Type: SA Var: 1 Objective: Concept Connections

2) Given an equation of the form um/n = k, raise both sides to the obtain u1 on the left side). n Answer: m Type: SA Var: 1 Objective: Concept Connections

Page 30

power to isolate u (that is, to

3) The equation m2/3 + 10m1/3 + 9 = 0 is said to be in form, because making the substitution u = results in a new equation that is quadratic. Answer: quadratic; m1/3 Type: SA Var: 1 Objective: Concept Connections

4) Consider the equation (4x2 + 1)2 + 4(4x2 + 1) + 4 = 0. If the substitution u=

is made, then the equation becomes u2 + 4u + 4 = 0.

Answer: 4x2 + 1 Type: SA Var: 1 Objective: Concept Connections 1 Solve Polynomial Equations

Solve the equation. 1) x2(x2 + 31) = 180 A) {±5,± 6i}

B) {±5,± 6}

C) {±5i,± 6}

D) {± 5,± 6i}

1 C) 0, , -7 3

1 D) 0, , ±7 3

C) {0, ± 7}

D) {± 7}

C) - 5, ±6

Answer: D Type: BI Var: 36 Objective: Solve Polynomial Equations

2) 2x(3x - 1)(x + 7)2 A) 0, 3, ±7

, -7 3

Answer: C Type: BI Var: 50+ Objective: Solve Polynomial Equations

3) -5(w2 - 7)(w2 + 4) A) {± 7, ±2i}

B) {0, ± 7, ±2i}

Answer: A Type: BI Var: 50+ Objective: Solve Polynomial Equations

4) 180x3 + 36x2 - 5x - 1 = 0 1 A) 5

1 1 B) - , ± 5 6

Answer: B Type: BI Var: 50+ Objective: Solve Polynomial Equations

Page 31

1 ,± i 5 6

5) 100x3 + 25x2 + 4x + 1 = 0 1 1 1 A) A) - , ± 4 4 5

1 1 B) - , ± i 4 5

D) - 4, ±5

9 , -6 2

D) 0, ±i 6

Answer: C Type: BI Var: 50+ Objective: Solve Polynomial Equations

6) 2n2(n2 + 6) = 54 + 9n2 3 2 A) ± , ±i 6 2

3 2 ,i 6 2

Answer: A Type: BI Var: 24 Objective: Solve Polynomial Equations

7) x3 - 8 = x - 2 A) {2, 1 ± i 3}

B) {2, 1 ± 3}

C) {2, -1 ± 2}

D) {2, -1 ± i 2}

Answer: D Type: BI Var: 5 Objective: Solve Polynomial Equations 2 Solve Rational Equations

Solve the equation. 2z 3 1) + =1 z-2 z-4 57 1 ± 2 2

57 i B) 1 ± 2 2

1 2

57 2

7 ± 3

106 i 6

7 ± 3

106 6

C) -

Answer: C Type: BI Var: 50+ Objective: Solve Rational Equations

5z z-5

A) -

1 = -1 z-4 7

3 C)

7 ± 3

106 i 6

106 6

D) -

Answer: C Type: BI Var: 50+ Objective: Solve Rational Equations

Page 32

D) -

1 2

57 i 2

x-4

3x - 9 x-4

A) {4, 1}

5 1 B) - , 2 3

C) {1}

D) { }

C) {1}

D) {-1, 4}

Answer: C Type: BI Var: 7 Objective: Solve Rational Equations

5 8 34 = 2 v - 4 v + 1 v - 3v - 4 A) {-4, 1}

B) ∅

Answer: C Type: BI Var: 50+ Objective: Solve Rational Equations

5) 4x - 5 = A)

3 x

5 ± 37 4

5 ± 73 4

5 ± 73 8

5 ± 37 8

Answer: C Type: BI Var: 50+ Objective: Solve Rational Equations

20 10 + 5 = c2 - 2c c-2 A) { }

B) {±2}

C) {0, 2}

D) {2}

Answer: A Type: BI Var: 16 Objective: Solve Rational Equations

Solve the problem. 7) Fernando's motorboat can travel 35 mi/h in still water. If the boat can travel 7 miles downstream in the same time it takes to travel 3 miles upstream, what is the rate of the river's current? A) 9 mi/h B) 4 mi/h C) 35 mi/h D) 14 mi/h Answer: D Type: BI Var: 50+ Objective: Solve Rational Equations

Page 33

3 Solve Absolute Value Equations

Solve the absolute value equation. 1) 6z - 3 = 7 2 A) 3

B) {7, -6}

5 3

2 3

5 3

Answer: C Type: BI Var: 50+ Objective: Solve Absolute Value Equations

2) 12x - 6 - 15 = -15 1 1 A) , 2 2

B) { }

C) {12, -15}

1 2

Answer: D Type: BI Var: 50+ Objective: Solve Absolute Value Equations

3) b + 4 - 2 = 4 A) {-10, 6}

B) {6, 2}

C) {-10, 2}

D) {10, -10}

C) {-1, -5}

D) {-1}

23 25 , 8 8

D) -3,

Answer: C Type: BI Var: 50+ Objective: Solve Absolute Value Equations

4) 3 - 3w + 9 = 6 A) { }

B) {2, -2}

Answer: A Type: BI Var: 50+ Objective: Solve Absolute Value Equations

5) -

17 2 + |3y - 9| = -4 4 3 A) { }

1 1 B) - , 8 8

4 3

Answer: C Type: BI Var: 50+ Objective: Solve Absolute Value Equations

6) |2v| = -13 - 3v 5 ,0 A) 13

B) {-13}

Answer: C Type: BI Var: 50+ Objective: Solve Absolute Value Equations

Page 34

C) -13, -

13 5

D) {0, 16}

7) -2|x - 4| + 6 = -8 A) {-3, 11}

B) {-6, 14}

C) {3, 5}

D) {2, 6}

Answer: A Type: BI Var: 50+ Objective: Solve Absolute Value Equations

8) |2r + 3| = |5r - 17| 20 20 A) - , 3 3

B) 2,

20 3

69 5

D) ∅

Answer: B Type: BI Var: 50+ Objective: Solve Absolute Value Equations 4 Solve Radical Equations and Equations with Rational Exponents

Solve the equation. 1) -3 + 5x + 5 = 5 59 A) 5

64 5

D) -

1 5

Answer: A Type: BI Var: 33 Objective: Solve Radical Equations and Equations with Rational Exponents

2) m + 55 + 1 = m A) {-12}

B) 8

C) {-12, 9}

D) {9}

Answer: D Type: BI Var: 28 Objective: Solve Radical Equations and Equations with Rational Exponents 4

3) 6 + m = 8 A) {±16}

B) {±4}

C) {16}

D) {4}

Answer: C Type: BI Var: 50+ Objective: Solve Radical Equations and Equations with Rational Exponents

4) -15 = -11 + (q - 2)1/3 A) {62}

B) {66}

C) {-62}

D) { }

Answer: C Type: BI Var: 50+ Objective: Solve Radical Equations and Equations with Rational Exponents

10z + 2 = A) {0}

7z + 11 B) {6}

C) {3}

Answer: C Type: BI Var: 50+ Objective: Solve Radical Equations and Equations with Rational Exponents

Page 35

D) {-3}

6) 4x - 5 + 1 = 4x + 5 141 A) 16

101 16

101 4

61 16

Answer: B Type: BI Var: 50+ Objective: Solve Radical Equations and Equations with Rational Exponents

7) 5 - x + 10 = 7 - x A) {12, -18}

B) {12, -9}

C) {6, -9}

D) {6, -18}

Answer: C Type: BI Var: 36 Objective: Solve Radical Equations and Equations with Rational Exponents

8) 11 - p - 2 + p = -1 A) {±2}

B) {7}

C) {2}

D) {7, 2}

Answer: B Type: BI Var: 50+ Objective: Solve Radical Equations and Equations with Rational Exponents

9) 4d 2/3 - 9d 1/3 - 9 = 0 27 A) ,3 64

B) - 3 , 3 4

, 27

D) -

, 27

Answer: D Type: BI Var: 48 Objective: Solve Radical Equations and Equations with Rational Exponents

10) 3(x - 4)2/3 = 48 A) {-60, 68}

B) {-12, 20}

C) {-68, 60}

D) {-20, 12}

Answer: A Type: BI Var: 50+ Objective: Solve Radical Equations and Equations with Rational Exponents

11) (2x + 4)3/2 = 64 A) 6

B) ±16

C) 16

D) ±6

Answer: A Type: BI Var: 20 Objective: Solve Radical Equations and Equations with Rational Exponents

12) -3 + p = 7 - 32 - p A) {±28}

B) {28}

C) {7}

Answer: D Type: BI Var: 50+ Objective: Solve Radical Equations and Equations with Rational Exponents

Page 36

D) {7, 28}

13) 6(x - 1)6/7 = 12 A) {16/7 + 2}

B) {17/6 + 2}

C) {26/7 + 1}

D) {27/6 + 1}

Answer: D Type: BI Var: 50+ Objective: Solve Radical Equations and Equations with Rational Exponents

14) n4/5 = 3 A) ±

15 4

B) {35/4}

C) {±35/4}

15 4

Answer: B Type: BI Var: 28 Objective: Solve Radical Equations and Equations with Rational Exponents

15) 4p2/3 =

4 1 A) 16

1 B) 64

1 16

1 64

Answer: D Type: BI Var: 12 Objective: Solve Radical Equations and Equations with Rational Exponents

Solve the problem. 16) The amount of time it takes an object dropped from an initial height of h0 feet to reach a height of h feet is given by the formula

How long would it take an object to reach the ground from the top of a building that is 470 feet tall? Round to the nearest tenth of a second. A) 5.4 seconds B) 0.3 seconds C) 29.4 seconds D) 4 seconds Answer: A Type: BI Var: 31 Objective: Solve Radical Equations and Equations with Rational Exponents

17) The amount of time it takes an object dropped from an initial height of h0 feet to reach a height of h feet is given by the formula h0 - h t= . 16 An object dropped from the top of the Sears Tower in Chicago takes 9.7 seconds to reach the ground. Use the above equation to approximate the height of the Sears Tower to the nearest foot. A) 1,219 feet B) 1,032 feet C) 1,584 feet D) 1,505 feet Answer: D Type: BI Var: 5 Objective: Solve Radical Equations and Equations with Rational Exponents

Page 37

18) The yearly depreciation rate for a certain vehicle is modeled by r = 1 -

V 1/n , where V is the value C

of the car after n years, and C is the original cost. a. Determine the depreciation rate for a car that originally cost $18,000 and is worth $11,000 after 3 yr. Round to the nearest tenth of a percent. b. Determine the original cost of a truck that has a yearly depreciation rate of 14% and is worth $12,000 after 5 yr. Round to the nearest $100. A) a. 15.1% per year; b. $14,000 B) a. 15.1% per year; b. $25,500 C) a. 77.2% per year; b. $25,500 D) a. 77.2% per year; b. $14,000 Answer: B Type: BI Var: 50+ Objective: Solve Radical Equations and Equations with Rational Exponents 5 Solve Equations in Quadratic Form

Solve the equation by using substitution. 1) (t + 3)2 - (t + 3) - 12 = 0 A) {1, -6} B) {-1, 6}

C) {4, -3}

D) {-7, 0}

Answer: A Type: BI Var: 9 Objective: Solve Equations in Quadratic Form

2) 3(t2 - 9)2 + 16(t2 - 9) = -5 1 A) ± , ±5 3

B) ± 78 i, ±i2 3

1 C) -

, -5

D) ±

Answer: D Type: BI Var: 39 Objective: Solve Equations in Quadratic Form

Solve the equation. 3) 4z4 + 68z2 + 225 = 0 25 9 A) - , 2 2 C) -

5 2 3 2 3 2 5 2 i i, i, i, 2 2 2 2

Answer: C Type: BI Var: 7 Objective: Solve Equations in Quadratic Form

Page 38

9 25 , 2 2 5 2

D) -

3 2 3 2 5 2 , , 2 2 2

, ±2

4) 30m2 = 216 - m4 A) {-6, 6, - i 6, i 6}

B) {-6, - 6, 6, 6}

C) {- 6, 6, -6i, 6i}

D) {-6i, - i 6, i 6, 6i}

Answer: C Type: BI Var: 50+ Objective: Solve Equations in Quadratic Form

5) 2a4 + 1 = 7a2 A) -

7 + 41 2

i, -

7 - 41 2

7 - 41 7 + 41 , 4 4

-7 - 41 -7 + 41 , 4 4

D) -

7 + 41 2

7 - 41

7 + 41 2

Answer: D Type: BI Var: 8 Objective: Solve Equations in Quadratic Form

6) 9 + 24u-2 = 58u-1 9 A) 10

33 58

4 ,6 9

D) -6, -

4 9

Answer: C Type: BI Var: 50+ Objective: Solve Equations in Quadratic Form

Solve the equation by using substitution. 7) z2/3 + 2z1/3 - 15 = 0 A) {27, -125} B) {6, -6} Answer: A Type: BI Var: 6 Objective: Solve Equations in Quadratic Form

Page 39

C) {9, 25}

D) {27, 125}

8) (4y + 7)2 = 4(4y + 7) + 6 10 5 5 A) - + , - - 10 4 4 4 4 C) {-2 + 10, -2 - 10}

B) {2 + 10, 2 - 10} D)

10 5 , - 10 4 4 4

Answer: A Type: BI Var: 50+ Objective: Solve Equations in Quadratic Form

Solve and express your solution in simplified form. 9) x4 - 3x2 + 2 =0 A) {±1, ± 2} B) {±1, ±2}

C) {1, 2}

D) {1, 2}

C) {±4, ±i 3}

D) {16, 3i}

C) {1, 3}

D) -

Answer: A Type: BI Var: 22 Objective: Solve Equations in Quadratic Form

10) x4 - 13x2 - 48 =0 A) {16, 3}

B) {±4, ± 3}

Answer: C Type: BI Var: 19 Objective: Solve Equations in Quadratic Form

Solve the equation. 92 9 11) 2 + = -3 +42+ y y 9 5 A) - , 4 4

9 B) {-27, -45}

,-3

Answer: D Type: BI Var: 50+ Objective: Solve Equations in Quadratic Form

Make an appropriate substitution and solve the equation. 12) (3x + 7)2 + 2(3x + 7) - 15 = 0 10 2 10 A) - 4, B) - , 3 3 3

C) - 4, -

4 3

2 4 D) - , 3 3

Answer: C Type: BI Var: 50+ Objective: Solve Equations in Quadratic Form

13) (x2 + 4x)2 - 17(x2 + 4x) = -60 A) {12, 5} B) {-5, -6, 2, 1} Answer: B Type: BI Var: 30 Objective: Solve Equations in Quadratic Form

Page 40

C) {-1, -2, 6, 5}

D) {-5, -12}

5 14) -

+1=0

-3 + 14 -3 - 14 , 5 5

B) {-3 + 14, -3 - 14}

C) {3 + 14, 3 - 14}

3 + 14 3 - 14 , 5 5

Answer: B Type: BI Var: 19 Objective: Solve Equations in Quadratic Form

15)

3 1 =4 2 (n + 4) n + 4 A) -1,

B) -1,

C) -5, -

D) -5,

4 3

Answer: C Type: BI Var: 20 Objective: Solve Equations in Quadratic Form

16) m -

12 2 12 - 10 m - 11 = 0 m m

A) {-1, 11}

B) {-12, -4, 1, 3}

C) {1, -11}

D) {-4, -1, 3, 12}

C) {5, -8}

D) {625}

C) {4, 5}

D) {-5, -4, 4, 5}

Answer: D Type: BI Var: 12 Objective: Solve Equations in Quadratic Form

17) n1/2 + 3n1/4 - 40 = 0 A) {25}

B) {625, 4,096}

Answer: D Type: BI Var: 17 Objective: Solve Equations in Quadratic Form

18) 400x -4 - 41x -2 + 1 = 0 1 1 A) , 5 4

B) - 1 , - 1 , 1 , 1 4 5 5 4

Answer: D Type: BI Var: 6 Objective: Solve Equations in Quadratic Form

Page 41

19) 9t - 16 t = 0 3 A) 0, 4

B) 0,

256 81

C) 0,

81 D) 0,

256

4 3

Answer: B Type: BI Var: 12 Objective: Solve Equations in Quadratic Form 6 Mixed Exercises

Solve the equation for the indicated variable. 1) Solve for p: h = 2pq h2 h2q2 A) p = B) p = 2q 4

h2q

h2 C) p =

D) p =

4q2

Answer: A Type: BI Var: 6 Objective: Mixed Exercises

2) Solve for n: M =

Gp1p2 n2

A) n = ± M + Gp1p2 C) n =

B) n = ± M - Gp1p2

± Gp1p2M

D) n =

± Gp1p2 M

Answer: C Type: BI Var: 50+ Objective: Mixed Exercises

3) Solve for p: T = 2π

p n

T2 A) p =

4π2

nT 2 B) p = 2π

C) p = n(T - 2π)

Answer: D Type: BI Var: 27 Objective: Mixed Exercises

4) Solve for x: 25 + x2 - y2 = z A) x = ± z + y2 - 5

B) x = z - y2 - 5

C) x = ± (z - 25)2 + y2

D) x = z2 + y2 - 50z + 625

Answer: C Type: BI Var: 16 Objective: Mixed Exercises

Page 42

T 2 D) p = n 2π

5) Solve for K :

R1Z1

A) K = 2

R2Z2 K2

B) K =

R Z RZ

2 2 1 1

C) K = 2

R2Z2R1Z1

R1Z1

D) K =

R2 Z2K1

K1 R2Z2K1 R1Z1

Answer: D Type: BI Var: 50+ Objective: Mixed Exercises

Solve the equation. 6)

x+ x+2 =4 A)

33 ± 73 2

B) { }

33 + 73 2

33 - 73 2

Answer: D Type: BI Var: 6 Objective: Mixed Exercises

Solve the problem. 7) The equation r = 3

3V gives the radius r of a sphere of volume V. If the radius of a sphere is 6 in., 4π

find the exact volume. A) 144π in.3

B) 96π in.3

C) 3

9 in.3 2π

D) 288π in.3

Answer: D Type: BI Var: 5 Objective: Mixed Exercises

8) The distance d (in miles) that an observer can see on a clear day is approximated by d =

40 where h is the height of the observer in feet. It Rita can see 24.5 mi, how far above ground is her eye level? A) 6 ft B) 20 ft C) 400 ft D) 40 ft Answer: C Type: BI Var: 4 Objective: Mixed Exercises

Page 43

9) Mary is in a boat in the ocean 48 mi from point A, the closest point along a straight shoreline. She needs to dock the boat at a marina x miles farther up the coast, and then drive along the coast to point B, 96 mi from point A. Her boat travels 10 mph, and she drives 60 mph. If the total trip took 7 hr, determine the distance x along the shoreline.

48 mi

A) 36 mi

96 mi B) 43 mi

C) 47 mi

D) 32 mi

Answer: A Type: BI Var: 11 Objective: Mixed Exercises

1.7 Linear, Compound , and Absolute Value Inequalities 0 Concept Connections

Provide the missing information. 1) If a compound inequality consists of two inequalities joined by the word “and,” the solution set is the of the solution sets of the individual inequalities. Answer: intersection Type: SA Var: 1 Objective: Concept Connections

2) The compound inequality a < x and x < b can be written as the three-part inequality

Answer: a < x < b Type: SA Var: 1 Objective: Concept Connections

3) If a compound inequality consists of two inequalities joined by the word “or,” the solution set is the of the solution sets of the individual inequalities. Answer: union Type: SA Var: 1 Objective: Concept Connections

4) If k is a positive real number, then the inequality |x| < k is equivalent to Answer: -k; k Type: SA Var: 1 Objective: Concept Connections

Page 44

<x<

5) If k is a positive real number, then the inequality |x| > k is equivalent to x <

or x

Answer: -k; > Type: SA Var: 1 Objective: Concept Connections

6) If k is a positive real number, then the solution set to the inequality |x| > -k is

Answer: Type: SA Var: 1 Objective: Concept Connections

7) If k is a positive real number, then the solution set to the inequality |x| < -k is

Answer: { } Type: SA Var: 1 Objective: Concept Connections 1 Solve Linear Inequalities in One Variable

Solve the inequality. Write the solution set in interval notation. 1) 9(x - 3) - 8x ≥ -3 A) (24, ∞) B) (-∞, 24] C) [24, ∞)

D) [0, ∞)

Answer: C Type: BI Var: 50+ Objective: Solve Linear Inequalities in One Variable

2) -2(7y - 7) + y > 2y - (-5 + y) 2 9 A) , ∞ B) -∞, 7 14

C) -∞,

9 14

,∞ 14

Answer: B Type: BI Var: 50+ Objective: Solve Linear Inequalities in One Variable

Solve the inequality. Write the solution set in interval notation using fractions. 3) 0.21n - 3 ≤ -0.1(-10 - n) 400 400 97 A) -∞, C) -∞, B) ,∞ 11 11 9

D) -∞,

97 9

Answer: C Type: BI Var: 50+ Objective: Solve Linear Inequalities in One Variable

Solve the inequality. Write the solution set in interval notation. 4) -1 - 2(2x + 1) < x - (-1 - x) 2 2 A) - , ∞ B) - , ∞ C) (-∞, -1) 3 3 Answer: A Type: BI Var: 44 Objective: Solve Linear Inequalities in One Variable

Page 45

D) -∞, -

2 3

2 4 1 5) y - ≥ y + 5 6 5 1 A) -∞, 2

,∞

C) -∞, -

D) -∞,

1 2

Answer: C Type: BI Var: 50+ Objective: Solve Linear Inequalities in One Variable

6) -2(4y - 7) + y ≥ 2y - (-8 + y) 3 3 A) -∞, B) , ∞ 4 4

1 C) - , ∞ 4

D) -∞,

C) (-∞, 0.6]

D) [0.6, ∞)

Answer: A Type: BI Var: 50+ Objective: Solve Linear Inequalities in One Variable

7) 0.31 ≥ 0.04a + 0.07 A) [6, ∞)

B) (-∞, 6]

Answer: B Type: BI Var: 50+ Objective: Solve Linear Inequalities in One Variable

3 1 1 8) (x - 2) - (x - 2) ≥ x + 1 2 4 5 10 10 A) -∞, B) - , ∞ 9 9

,∞

D) -∞,

2 3

Answer: A Type: BI Var: 50+ Objective: Solve Linear Inequalities in One Variable

9) 7 - 5[1 - 2(x - 1)] ≥ 5{1 - [2 - (x + 1)]} 8 2 A) , ∞ B) , ∞ 5 7 Answer: A Type: BI Var: 50+ Objective: Solve Linear Inequalities in One Variable

Page 46

C) - ∞, -

8 5

D) - ∞,

2 7

2 Solve Compound Linear Inequalities

Solve the inequality. Write the solution set in interval notation. 1) 20 > 3x and 11 + 2x ≥ 2 9 20 9 20 A) - , B) -∞, - ∪ ,∞ 2 3 2 3 C) -

20 9 , 3 2

D) { }

Answer: A Type: BI Var: 50+ Objective: Solve Compound Linear Inequalities

9 8

- 5y <

5 4

and

4 7

A) -∞, ∞

y+1<

9 14

5 1 B) - , 8 40

C) { }

D) -∞, -

5 8

Answer: C Type: BI Var: 50+ Objective: Solve Compound Linear Inequalities

3) -2 < -2y + 11 < 6 5 13 A) , 2 2

13 ,6 2

13 5 , 2 2

5 13 , 2 2

Answer: D Type: BI Var: 50+ Objective: Solve Compound Linear Inequalities

Solve the compound inequality. Write the answer in interval notation. 4) 4x ≤ 12 or 9 - x < 0 A) { } B) (-∞, 3] ∪ (9, ∞) C) (-∞, 9)

D) (-∞, ∞)

Answer: B Type: BI Var: 50+ Objective: Solve Compound Linear Inequalities

5) 2x ≤ 4 or 14 - x < 8 A) (-∞, ∞)

B) { }

Answer: C Type: BI Var: 50+ Objective: Solve Compound Linear Inequalities

Page 47

C) (-∞, 2] ∪ (6, ∞)

D) (-∞, 6)

or -8 + 2x ≤ -15 23 7 A) - , 3 2

6) 23 < 3x

C) -∞, -

B) (-∞, ∞) 7 23 D) - , 2 3

23 7 ∪ ,∞ 2 3

Answer: C Type: BI Var: 50+ Objective: Solve Compound Linear Inequalities

Solve the compound inequality. Graph the solution set, and write the solution set in interval notation. 7) -8 < -5x + 2 ≤ 22 A) (-2, 4]

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

9 10 11

B) (-4, 2]

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

C) [-2, 4)

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

D) [-4 , 2)

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

Answer: D Type: BI Var: 50+ Objective: Solve Compound Linear Inequalities

Page 48

8) -1 ≤

2x + 3 3

A) -∞, - 3 ∪

9 ,∞ 2

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

B) - 3,

9 10 11

9 2

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

C) - 3,

9 2

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

D) -∞, - 3 ∪

,∞ 2

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

Answer: B Type: BI Var: 50+ Objective: Solve Compound Linear Inequalities 3 Solve Absolute Value Inequalities

Solve the absolute value inequality. Write the solution in interval notation. 1) y > 13 A) (-∞, -13) ∪ (13, ∞) B) (13, ∞) C) (-∞, -13) D) (-13, 13) Answer: A Type: BI Var: 20 Objective: Solve Absolute Value Inequalities

Page 49

2) x + 6 < 15 A) (-21, 9) C) (-9, 9)

B) (-∞,-21) ∪ (9, ∞) D) (-9, 21)

Answer: A Type: BI Var: 50+ Objective: Solve Absolute Value Inequalities

3) 24 ≤ 2 + -15t + 1 A) {-22, 22}

B) { }

21 23 , 15 15

D) -∞, -

C) -

21 23 ∪ ,∞ 15 15

Answer: D Type: BI Var: 50+ Objective: Solve Absolute Value Inequalities

4) |2b - 23| > -15 A) (-∞, ∞)

B) [4, 19]

C) { }

D) (-∞, 4] ∪ [19, ∞)

C) (-∞, 1) ∪ (17, ∞)

D) (-∞, 7) ∪ (11, ∞)

Answer: A Type: BI Var: 50+ Objective: Solve Absolute Value Inequalities

5) 3|x - 9| + 9 < 15 A) (1, 17)

B) (7, 11)

Answer: B Type: BI Var: 50+ Objective: Solve Absolute Value Inequalities

6) 3|x - 5| + 12 ≥ 15 A) [-4, 14] C) [4, 6]

B) (-∞, 4] ∪ [6, ∞) D) (-∞, -4] ∪ [14, ∞)

Answer: B Type: BI Var: 50+ Objective: Solve Absolute Value Inequalities

m - 12 < 19 4 A) (-64, 88) C) (-∞, -16) ∪ (22, ∞) Answer: A Type: BI Var: 50+ Objective: Solve Absolute Value Inequalities

Page 50

B) (-16, 22) D) (-88, 64)

8) |2x + 7| + 7 > 6 A) (-∞, - 4) ∪ (- 3, ∞) C) (- 4, - 3)

B) { } D) (-∞, ∞)

Answer: D Type: BI Var: 50+ Objective: Solve Absolute Value Inequalities 4 Solve Applications of Inequalities

Solve the problem. 1) In order to ride certain amusement park rides, riders must be at least 46" tall, but no more than 79" tall. Let h represent the height of a prospective rider. Write an inequality that represents the allowable heights. A) h < 46 and h > 79 B) h < 46 or h > 79 C) h < 79 or h > 46 D) h < 79 and h > 46 Answer: D Type: BI Var: 50+ Objective: Solve Applications of Inequalities

2) A skydiving company insists that its customers weigh at least 130 pounds, but no more than 280 pounds, including parachute and other gear. If the total weight of all gear is 25 pounds, write and solve a compound inequality that represents the weight range without gear that is acceptable. A) 155 ≤ w ≤ 305 B) 155 ≤ w ≤ 255 C) 105 ≤ w ≤ 305 D) 105 ≤ w ≤ 255 Answer: D Type: BI Var: 50+ Objective: Solve Applications of Inequalities

3) Sparky has scores of 71, 60, and 69 on his first three Sociology tests. If he needs to keep an average of 70 to stay eligible for lacrosse, what scores on the fourth exam will accomplish this? A) He must score more than 80 B) He must score more than 84 C) He must score 84 or higher. D) He must score 80 or higher. Answer: D Type: BI Var: 50+ Objective: Solve Applications of Inequalities

Write the requested inequality. 4) The cost for a long-distance telephone call is $0.35 for the first minute and $0.10 for each additional minute or a portion thereof. The total cost of the call cannot exceed $3. Write an inequality representing the number of minutes m, a person could talk without exceeding $3. A) m ≤ 26 B) m ≤ 28 C) m ≤ 27 D) m ≤ 29 Answer: C Type: BI Var: 50+ Objective: Solve Applications of Inequalities

Page 51

Solve the problem. 5) The width of a rectangle is fixed at 30 cm, and the perimeter can be no greater than 170 cm. Find the maximum length of the rectangle. A) 140 cm B) 55 cm C) 110 cm D) 70 cm Answer: B Type: BI Var: 35 Objective: Solve Applications of Inequalities

6) Pressure-treated wooden studs can be purchased for $4.88 each. How many studs can be bought if a project's budget allots no more than $200 for studs? A) 40 studs B) 42 studs C) 41 studs D) 43 studs Answer: A Type: BI Var: 50+ Objective: Solve Applications of Inequalities

7) Rita earns scores of 75, 82, 69, 82, and 67 on her five chapter tests for a certain class and a grade of 68 on the class project. The overall average for the course is computed as follows: the average of the five chapter tests makes up 55% of the course grade; the project accounts for 10% of the grade; and the final exam accounts for 35%. What scores can Rita earn on the final exam to earn a “B” in the course if the cut-off for a “B” is an overall score greater than or equal to 80, but less than 90? Assume that 100 is the highest score that can be earned on the final exam and that only whole-number scores are given. A) 96 through 100 inclusive B) 92 through 119 inclusive C) 92 through 100 inclusive D) 96 through 119 inclusive Answer: C Type: BI Var: 50+ Objective: Solve Applications of Inequalities

Write an absolute value inequality equivalent to the expression. 8) "All real numbers whose distance from 0 is more than 82." A) x - 82 ≥ 0 B) x ≥ 82 C) x > 82

D) x - 82 > 0

Answer: C Type: BI Var: 50+ Objective: Solve Applications of Inequalities

9) "All real numbers whose distance from 13 is at most 5" A) y - 13 > 5 B) y - 5 ≤ 13 C) y - 13 < 5 Answer: D Type: BI Var: 50+ Objective: Solve Applications of Inequalities

Page 52

D) y - 13 ≤ 5

10) The results of a political poll indicate that the leading candidate will receive 52% of the votes with a margin of error of no more than 5%. Let x represent the true percentage of votes received by this candidate. Write an absolute value inequality that represents an interval in which to estimate x. A) |x - 52| ≥ 0.05 B) |x - 0.05| ≥ 52 C) |x - 52| ≤ 0.05 D) |x - 0.05| ≤ 52 Answer: C Type: BI Var: 15 Objective: Solve Applications of Inequalities 5 Mixed Exercises

Determine the set of values of x for which the radical expression would produce a real number. 1) 15 - x A) {x | x > 15} B) { } C) {x | x ≤ 15} D) {x | x ≥ 15} Answer: C Type: BI Var: 40 Objective: Mixed Exercises

x + 15 A) all real numbers

B) {x | x > -15}

C) {x | x > 15}

D) {x | x ≥ -15}

Answer: A Type: BI Var: 40 Objective: Mixed Exercises

1.8 Equations and Inequalities For Calculus 0 Equations and Inequalities For Calculus

In Calculus you will see the symbol y'. Treat y' as a variable and solve the equation for y'. 6x 6y 1) + y' = 0 23 7 42x 42x 7x 7x A) y' = B) y' = C) y' = D) y' = 23y 23y 23y 23y Answer: A Type: BI Var: 50+ Objective: Equations and Inequalities For Calculus

2) 3xy3 + 5x2y2y' - y' = 1 1 - 3xy3 A) y' = 2 2 5x y - 1

3y B) y' =

Answer: A Type: BI Var: 38 Objective: Equations and Inequalities For Calculus

Page 53

1 - 3xy3

1 - 3y C) y' =

5x - 1

D) y' =

5x2y2

3) 6y2y' + 30xy + 6x2y' = 5y2 + 25xyy' y(y - 6x) A) y' = 2 6x - 5xy + 6y2 C) y' =

5y(y - x) x - 25xy + y2

B) y' = D) y' =

y - 6x 6x2 - 5x + 6y 5y(y - 6x) 6x - 25xy + 6y2 2

Answer: D Type: BI Var: 22 Objective: Equations and Inequalities For Calculus

4) -5(x + y)2 - 5(x + y)2y' + 5y2y' = -5x2 y(2x + y) x(2x + y) A) B) x(2y + x) y(2y + x)

x2 + y2 (x + y)2

x2 - y2 (x + y)2

Answer: A Type: BI Var: 16 Objective: Equations and Inequalities For Calculus

Simplify the expression. Do not rationalize the denominator. 1 2 1 (4) 5) 2x 3x - 4 + x 4 3x - 4 7x2 - 8 A)

7x2 - 4 B)

3x - 4

x(7x - 8) C)

3x - 4

x(7x - 4) D)

3x - 4

Answer: C Type: BI Var: 50+ Objective: Equations and Inequalities For Calculus

(1)(x2 - 8)1/2 - x 6)

1 2 (x - 8)-1/2 (3x) 3

(x2 - 8)1/2 2 A) -

8 (x2 - 8)3/2

1 - x2 (x2 - 8)5/2

C) -

8 (x2 - 8)5/2

1 - x2 (x2 - 8)3/2

Answer: A Type: BI Var: 50+ Objective: Equations and Inequalities For Calculus

-10x(8x + 1) - (-5x2)(8) (8x + 1)2 10x(4x + 1) A) (8x + 1)2

10x(4x - 1) B) (8x + 1)2

Answer: A Type: BI Var: 26 Objective: Equations and Inequalities For Calculus

Page 54

C) -

40x2 (8x + 1)2

8) 16 - x2 - x

(2x) 2

16 - x

2(8 - x2)

16 - x2

x2 - 8 16 - x2

8 - x2 16 - x2

2(x2 - 6) 16 - x2

Answer: A Type: BI Var: 6 Objective: Equations and Inequalities For Calculus

Find the values of x for which the expression equals zero. -8x(7x + 1) - (-4x2)(7) 9) (7x + 1)2 A) 0,

2 7

B) 0, -

1 7

2 7

C) {0}

D) 0, -

C) {±2}

D) { 2, 2}

Answer: D Type: BI Var: 28 Objective: Equations and Inequalities For Calculus

10) 4 - x2 - x

1 2

A) {± 2}

(2x) 2

4-x

B) {± 2, ±2}

Answer: A Type: BI Var: 6 Objective: Equations and Inequalities For Calculus

Some applications of calculus use a mathematical structure called a power series. To find the interval of convergence of a power series, it is often necessary to solve an absolute value inequality. Solve the absolute value inequality below to find the interval of convergence x+1 11) <1 4 A) (0, 3)

B) (-5, 3)

C) [0, 3]

D) [-5, 3]

Answer: B Type: BI Var: 30 Objective: Equations and Inequalities For Calculus

Solve the problem. 12) A 6-ft person walks away from a lamppost. At the instant the person is 14 ft away from the lamppost, the person's shadow is 10 ft long. Find the height of the lamppost A) 28 ft B) 13 ft C) 52 ft D) 32 ft Answer: B Type: BI Var: 48 Objective: Equations and Inequalities For Calculus

Page 55

13) A water trough has a cross section in the shape of an equilateral triangle with sides of length 1 m. 3 The length is 4 m. Determine the volume of water when the water level is m. 4 3 3 A) 3 2 2 3 m B) m C) 2 3 3 m2 D) 2 m2 8 4 4 8 Answer: B Type: BI Var: 12 Objective: Equations and Inequalities For Calculus

14) A contractor builds a swimming pool with cross section in the shape of a trapezoid. The deep end is 9 ft deep and the shallow end is 3 ft deep. The length of the pool is 60 ft and the width is 25 ft. As the pool is being filled, find the volume of water when the depth is 4 ft. A) 4,000 ft3 B) 2,000 ft3 C) 4,500 ft3 D) 1,620 ft3 Answer: B Type: BI Var: 24 Objective: Equations and Inequalities For Calculus

Page 56

Chapter 2 Functions and Relations 2.1 The Rectangular Coordinate System and Graphing Utilities 0 Concept Connections

Provide the missing information. 1) In a rectangular coordinate system, the point where the x- and y-axes meet is called the . Answer: origin Type: SA Var: 1 Objective: Concept Connections

2) The x- and y-axes divide the coordinate plane into four regions called

Answer: quadrants Type: SA Var: 1 Objective: Concept Connections

3) The distance between two distinct points (x1, y1) and (x2, y2) is given by the formula . Answer: d = (x2 - x1)2 + ( y2 - y1)2 Type: SA Var: 1 Objective: Concept Connections

4) The midpoint of the line segment with endpoints (x1, y1) and (x2, y2) is given by the formula . Answer: M =

x1 + x2 y1 + y2 , 2 2

Type: SA Var: 1 Objective: Concept Connections

5) A to an equation in the variables x and y is an ordered pair (x, y) that makes the equation a true statement. Answer: solution Type: SA Var: 1 Objective: Concept Connections

6) An x-intercept of a graph has a y-coordinate

Answer: 0 Type: SA Var: 1 Objective: Concept Connections

7) A y-intercept of a graph has an x-coordinate of Answer: 0 Type: SA Var: 1 Objective: Concept Connections

Page 1

8) Given an equation in the variables x and y, find the y-intercept by substituting and solving for .

for x

Answer: 0; y Type: SA Var: 1 Objective: Concept Connections 1 Plot Points on a Rectangular Coordinate System

Plot the point on a rectangular coordinate system. 1) A(2, -5) 6 5 4 3 2 1

y A

1 2 3 4 56 x A

-6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6

6 5 4 3 2 1

-6 -5 -4 -3 -2 -1-1

6 5 4 3 2 1

1 2 3 4 5 6 x

-6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6

B) y

6 5 4 3 2 1 1 2 3 4 5 6 x

-2 -3 -4 -5 -6

-6 -5 -4 -3 -2 -1-1

-2 -3 -4 -5 -6

1 2 3 4 5 6 x

Answer: D Type: BI Var: 50+ Objective: Plot Points on a Rectangular Coordinate System 2 Use the Distance and Midpoint Formulas

Find the exact distance between the points. 1) (-9, 4) and (-5, -6) A) 10 2 B) 2 29 Answer: B Type: BI Var: 50+ Objective: Use the Distance and Midpoint Formulas

Page 2

C) 116

D) 200

Find the midpoint of the line segment whose endpoints are the given points. 2) (5.3, 4.7) and (7.2, -9.7) A) (5, -1.25) B) (-0.95, 7.2) C) (6.25, -2.5)

D) (6.25, 7.2)

Answer: C Type: BI Var: 50+ Objective: Use the Distance and Midpoint Formulas

Find the exact distance between the points. 3) ( 5, - 3) and (4 5, -7 3) A) 3 17

B) 2 42

C) 2 58

D) 6 7

Answer: A Type: BI Var: 50+ Objective: Use the Distance and Midpoint Formulas

Find the midpoint of the line segment whose endpoints are the given points. 4) (3, 1) and (9, 4) 3 5 A) (12, 5) B) -3, C) 6, 2 2

D) (-6, 3)

Answer: C Type: BI Var: 50+ Objective: Use the Distance and Midpoint Formulas

5) (-24.6, 38.1) and (-25.7, -17.7) A) (0.55, 27.9) B) (-25.15, 27.9)

C) (6.75, -21.7)

D) (-25.15, 10.2)

5 7 , -2 3 2

D) (5 7, -4 3)

Answer: D Type: BI Var: 50+ Objective: Use the Distance and Midpoint Formulas

6) ( 7, - 3) and (6 7, -5 3) 7 7 A) , -3 3 B) (7 7, -6 3) 2

Answer: A Type: BI Var: 50+ Objective: Use the Distance and Midpoint Formulas

Determine if the given points form the vertices of a right triangle. 7) (-3, 5), (-1, 3), and (-4, 0) A) Yes B) No Answer: B Type: BI Var: 50+ Objective: Use the Distance and Midpoint Formulas

Page 3

3 Graph Equations by Plotting Points

Determine which of the given points are solutions to the given equation. 1) 2x2 + y = 4 I. (3, -14) II. (-3, 14) III. (-3, -14) A) III B) I and III C) I D) I and II Answer: B

E) II

Type: BI Var: 50+ Objective: Graph Equations by Plotting Points

2) |x - 3| - y = 2 I. (1, 0) II. A) I Answer: E

1 3 , 4 4

III. (-5, 10) B) I and III

C) II

D) III

E) I and II

Type: BI Var: 50+ Objective: Graph Equations by Plotting Points

Identify the set of values x for which y will be a real number. 10 3) y = x+7 A) {x | x ≠ 7}

B) {x | x ≠ -7}

C) x | x ≠

10 7

D) {x | x ≠ 10}

Answer: B Type: BI Var: 50+ Objective: Graph Equations by Plotting Points

4) y = x - 8 A) {x | x ≥ 8}

B) {x | x < 8}

Answer: A Type: BI Var: 20 Objective: Graph Equations by Plotting Points

Page 4

C) {x | x > 8}

D) {x | x ≤ 8}

Graph the equation by plotting points. 5) x = |y| - 5 A)

B) 6 y 5 4 3 2 1

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1-1

12 y 11 10 9 8 7 6 5 4 3 2 1

1 2x

-2 -3 -4 -5 -6

-1

D) 2 y 1 -1-1 -2 -3 -4 -5 -6 -7 -8 -9 -10

1 2 3 4 5 6 7 8 9 10 11 x

Answer: A Type: BI Var: 10 Objective: Graph Equations by Plotting Points

Page 5

1 2 3 4 5 6 7 8 9 10 11x

6 y 5 4 3 2 1 -1 -2 -3 -4 -5 -6

1 2 3 4 5 6 7 8 9 10 11 12x

6) y2 - x + 4 = 0 A)

B) 6 5 4 3 2 1

6 5 4 3 2 1 1 2 3 4 5 6 x

-6 -5 -4 -3 -2 -1-1

1 2 3 4 5 6 x

-6 -5 -4 -3 -2 -1-1

-2 -3 -4 -5 -6

D) 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1-1

6 5 4 3 2 1 1 2 3 4 5 6 x

-2 -3 -4 -5 -6

Answer: B Type: BI Var: 8 Objective: Graph Equations by Plotting Points

Page 6

-6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6

1 2 3 4 5 6 x

7) y = |x - 3| A)

B) 6 5 4 3 2 1

6 5 4 3 2 1 1 2 3 4 5 6 x

-6 -5 -4 -3 -2 -1-1

1 2 3 4 5 6 x

-6 -5 -4 -3 -2 -1-1

-2 -3 -4 -5 -6

D) 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1-1

6 5 4 3 2 1 1 2 3 4 5 6 x

-2 -3 -4 -5 -6

Answer: B Type: BI Var: 8 Objective: Graph Equations by Plotting Points

Page 7

-6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6

1 2 3 4 5 6 x

Estimate the x- and y-intercepts from the graph. 8) 8 7 6 5 4 3 2 1 -8 -7 -6 -5 -4 -3 -2 --11 -2 -3 -4 -5 -6 -7 -8

1 2 3 4 5 6 7 8 x

A) x-intercepts: (-2, 0) and (2, 0); y-intercept: (0, 4) B) x-intercept: (4, 0); y-intercepts: (0, -2) and (0, 2) C) x-intercepts: (0, -2) and (0, 2); y-intercept: (4, 0) D) x-intercept: (0, 4); y-intercepts: (-2, 0) and (2, 0) Answer: B Type: BI Var: 10 Objective: Graph Equations by Plotting Points

Page 8

4 Identify x- and y-Intercepts

Estimate the x- and y-intercepts from the graph. 1) 10 y 8 6 4 2 -10 -8

-6

-4

-2

10 x

-2 -4 -6 -8 -10

A) x-int: (0, -3.9), (0, 3.9) y-int: (-3, 0), (5, 0) C) x-int: (0, -3), (0, 5) y-int: (-3.9, 0), (3.9, 0)

B) x-int: (-3.9, 0), (3.9, 0) y-int: (0, -3), (0, 5) D) x-int: (-3, 0), (5, 0) y-int: (0, -3.9), (0, 3.9)

Answer: D Type: BI Var: 24 Objective: Identify x- and y-Intercepts

Determine the x- and y-intercepts of the graph whose points are defined in the table. x -5 0 5 10 15 20 2) y 9 7 2 0 -2 -9 A) x-intercept: (7, 0); y-intercept: (0, 10) B) x-intercept: (10, 0); y-intercept: (0, 7) C) x-intercept: (0, 10); y-intercept: (7, 0) D) x-intercept: (0, 7); y-intercept: (10, 0) Answer: B Type: BI Var: 50+ Objective: Identify x- and y-Intercepts

Find the x- and y-intercepts. 3) y = |x + 1| - 5 A) x-intercept: (4, 0); y-intercept: (0, -4) B) x-intercepts: (4, 0) and (-6, 0); y-intercept: (0, -4) C) x-intercepts: (4, 0) and (-6, 0); y-intercept: (0, -5) D) x-intercept: (-6, 0); y-intercept: (0, -5) Answer: B Type: BI Var: 50 Objective: Identify x- and y-Intercepts

Page 9

(x - 5)2 1

(y - 4)2 4

A) x-intercepts: (-1, 0) and (1, 0); y-intercepts: (0, -2) and (0, 2) B) x-intercept: (6, 0); y-intercept: (0, 6) C) x-intercept: none; y-intercept: none D) x-intercept: (1, 0); y-intercept: (0, 2) Answer: C Type: BI Var: 50+ Objective: Identify x- and y-Intercepts

5) x2 + y = 64 A) x-intercepts: (-8, 0) and (8, 0); y-intercept: (0, 64) B) x-intercept: (8, 0); y-intercept: (0, 64) C) x-intercept: none; y-intercept: (0, 64) D) x-intercept: (0, 64); y-intercepts: (-8, 0) and (8, 0) Answer: A Type: BI Var: 8 Objective: Identify x- and y-Intercepts

Page 10

5 Graph Equations Using a Graphing Utility (Technology Connections)

Graph the equation with a graphing utility on the given viewing window. 1) y = 3x - 5 on [-10, 10, 1] by [-10, 10, 1] A) 10 B)

-10 C)

-10 Answer: A Type: BI Var: 50+ Objective: Graph Equations Using a Graphing Utility (Technology Connections)

Page 11

-10

2) y = 1,000x2 - 1,800x on [-5, 5, 1] by [-1000, 2000, 500] A) 2000 B)

-5

2000

-5

-1000 C)

-1000

2000

-5

2000

-5

-1000 Answer: D Type: BI Var: 26 Objective: Graph Equations Using a Graphing Utility (Technology Connections)

Page 12

-1000

Graph the equations on the standard viewing window. 3) y = x - 3 y = |x + 3| A) 10

-10 C)

-10

-10 Answer: D Type: BI Var: 50+ Objective: Graph Equations Using a Graphing Utility (Technology Connections)

Page 13

-10

6 Mixed Exercises

Solve the problem. 1) A map of a hiking area is drawn so that the Visitor Center is at the origin of a rectangular grid. Two hikers are located at positions (-2, 1) and (1, -3) with respect to the Visitor Center where all units are in miles. A campground is located exactly halfway between the hikers. What are the coordinates of the campground?

3 A) - , 2 2

1 B) - , -1 2

1 C) - , 2 2

3 D) - , -1 2

Answer: B Type: BI Var: 4 Objective: Mixed Exercises

2) The position of an object in a video game is represented by an ordered pair. The coordinates of the ordered pair give the number of pixels horizontally and vertically from the origin. a. Suppose that player A is located at (54, 337) and player B is located at (414, 63). How far apart are the players? Round to the nearest pixel. b. If the two players move directly toward each other at the same speed, where will they meet? A) a. 452 pixels; b. (234, 200) B) a. 452 pixels; b. (59, 376) C) a. 634 pixels; b. (59, 376) D) a. 634 pixels; b. (234, 200) Answer: A Type: BI Var: 50+ Objective: Mixed Exercises

Find the center of a circle if a diameter of the circle has the given endpoints. 3) (-1, -6) and (-7, -2). A) (-4, -2) B) (-4, -4) C) (3, -2) Answer: B Type: BI Var: 50+ Objective: Mixed Exercises

Page 14

D) (3, -4)

Solve the problem. 4) The figure represents the average credit card debt for selected households in Silerville.

Page 15

Assume that the units shown in the grid are in feet. a. Determine the exact length and width of the rectangle shown. b. Determine the perimeter and area. 5) 6 5 4

3 2 1 -6 - 5 - 4 -3 - 2 - 1 -1 -2

1 2

5 6 x

-3 -4 -5 -6

A) a. Length: 4 ft; Width: 3 ft b. Perimeter: 14 ft; Area: 12 ft2

B) a. Length: 4 ft; Width: 3 ft b. Perimeter: 7 ft; Area: 12 ft2

C) a. Length: 4 2 ft; Width: 3 2 ft b. Perimeter: 7 2 ft; Area: 24 ft2

D) a. Length: 4 2 ft; Width: 3 2 ft b. Perimeter: 14 2 ft; Area: 24 ft2

Answer: D Type: BI Var: 48 Objective: Mixed Exercises

Page 16

The endpoints of a diameter of a circle are shown. Find the center and radius of the circle. 6) 7 6 5 4 3

(2, 3)

2 1 1

-7 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 (-2, -5) -5 -6 -7

2 3 4 5

6 7 x

A) center: (0, -1); radius: 2 5 C) center: (0, -1); radius: 20

B) center: (-1, 0); radius: 4 D) center: (-1, 0); radius: 2 5

Answer: A Type: BI Var: 50+ Objective: Mixed Exercises

An isosceles triangle is shown. Find the area of the triangle. Assume that the units shown in the grid are in meters. 7) 7 6 5 4 (-1, 3) 3 2 1 -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 -7

A) Area: 17 m2 Answer: A Type: BI Var: 50+ Objective: Mixed Exercises

Page 17

y (4, 6)

2 3 4 5 6 7 x (2, -2)

B) Area: 17 m2

C) Area: 2 17 m2

D) Area: 34 m2

2.2 Circles 0 Concept Connections

Provide the missing information. 1) A is the set of all points in a plane equidistant from a fixed point called the . Answer: circle; center Type: SA Var: 1 Objective: Concept Connections

2) The distance from the center of a circle to any point on the circle is called the often denoted by r. Answer: radius Type: SA Var: 1 Objective: Concept Connections

3) The standard form of an equation of a circle with center (h, k) and radius r is given by . 2

Answer: (x - h) + (y - k) = r Type: SA Var: 1 Objective: Concept Connections

4) An equation of a circle written in the form x2 + y2 + Ax + By + C = 0 is called the

form of an equation of a circle.

Answer: general Type: SA Var: 1 Objective: Concept Connections 1 Write an Equation of a Circle in Standard Form

Solve the problem. 1) Is the point (-3, -1) on the circle defined by (x + 3)2 + (y + 1)2 = 9 A) Yes

B) No

Answer: B Type: BI Var: 50+ Objective: Write an Equation of a Circle in Standard Form

Determine the center and radius of the circle. 2) (x - 4)2 + (y - 8)2 = 9 A) Center: (-4, -8); Radius: 3 C) Center: (4, 8); Radius: 9 Answer: D Type: BI Var: 50+ Objective: Write an Equation of a Circle in Standard Form

Page 18

B) Center: (-4, -8); Radius: 9 D) Center: (4, 8); Radius: 3

and is

3) (x + 2)2 + (y - 3)2 = 45 A) Center: (2, -3); Radius: 45

B) Center: (2, -3); Radius: 3 5

C) Center: (-2, 3); Radius: 45

D) Center: (-2, 3); Radius: 3 5

Answer: D Type: BI Var: 50+ Objective: Write an Equation of a Circle in Standard Form

Identify the center and radius of the circle. 4) Identify the center and radius of the circle. 42 x+ y2 = 12 7 A) Center =

4 , 0 ; r = 12 7

4 B) Center = - , 0 ; r = 2 3 7

C) Center =

4 ,0;r=2 3 7

D) Center = 0,

Answer: C Type: BI Var: 50+ Objective: Write an Equation of a Circle in Standard Form

Page 19

4 7

; r = 12

Use the given information about a circle to write an equation of the circle in standard form. Graph the circle. 5) Center: (4, 2); Radius: 4 A) (x - 4)2 + (y - 2)2 = 16 B) (x - 4)2 + (y - 2)2 = 4 10 y

10 y

-10 -8 -6 -4 -2

-2

-10 -8 -6 -4 -2

-4

-6

-8

-10

-2

10 y

2 2

-10 -8 -6 -4 -2

-2

-4

-6

-8

-10

Answer: D Type: BI Var: 50+ Objective: Write an Equation of a Circle in Standard Form

Use the given information about a circle to find its equation. 6) Center (-1, -5) and radius 3 B) (x - 1)2 + (y - 5)2 = 3 A) (x - 1)2 + (y - 5)2 = 3 3

Answer: D Type: BI Var: 50+ Objective: Write an Equation of a Circle in Standard Form

Page 20

D) (x + 4) + (y + 2)2 = 16

10 y

C) (x + 1)2 + (y + 5)2 =

C) (x + 4) + (y + 2) = 4

-2

-10

-10 -8 -6 -4 -2

D) (x + 1)2 + (y + 5)2 = 3

7) Center (6, -7) and radius of 5 A) (x + 6)2 + (y - 7)2 = 25 C) (x + 6)2 + (y - 7)2 = 5

B) (x - 6)2 + (y + 7)2 = 5 D) (x - 6)2 + (y + 7)2 = 25

Answer: D Type: BI Var: 50+ Objective: Write an Equation of a Circle in Standard Form

8) Center (-5, -4) and diameter 10 A) (x - 5)2 + (y - 4)2 = 100 C) (x + 5)2 + (y + 4)2 = 25

B) (x + 5)2 + (y + 4)2 = 100 D) (x - 5)2 + (y - 4)2 = 25

Answer: C Type: BI Var: 50+ Objective: Write an Equation of a Circle in Standard Form

Use the given information about a circle to write an equation of the circle in standard form. 9) The endpoints of a diameter are (3, 11) and (-9, -5). A) (x + 3)2 + (y + 11)2 = 20 B) (x + 3)2 + (y - 3)2 = 100 C) (x - 3)2 + (y - 11)2 = 100 D) (x - 3)2 + (y + 3)2 = 20 Answer: B Type: BI Var: 50+ Objective: Write an Equation of a Circle in Standard Form

10) The center is (-5, -4) and another point on the circle is (1, 4) A) (x + 5)2 + (y + 4)2 = 10 B) (x + 5)2 + (y + 4)2 = 100 C) (x - 5)2 + (y - 4)2 = 10 D) (x - 5)2 + (y - 4)2 = 100 Answer: B Type: BI Var: 50+ Objective: Write an Equation of a Circle in Standard Form

Find an equation of the circle with the given characteristics. 11) Center (8, -7) and radius 3 A) (x + 8)2 + (y - 7)2 = 3 B) (x - 8)2 + (y + 7)2 = 3 C) (x - 8)2 + (y + 7)2 = 9 D) (x + 8)2 + (y - 7)2 = 9 Answer: C Type: BI Var: 50+ Objective: Write an Equation of a Circle in Standard Form

12) Center (-10, -2) and radius 6 A) (x - 10)2 + (y - 2)2 = 36 C) (x + 10)2 + (y + 2)2 = 6 Answer: D Type: BI Var: 50+ Objective: Write an Equation of a Circle in Standard Form

Page 21

B) (x - 10)2 + (y - 2)2 = 6 D) (x + 10)2 + (y + 2)2 = 36

13) (-6, 0) and radius 3 A) (x + 6)2 = 9 C) (x + 6)2 + y2 = 9

B) (x - 6)2 + y2 = 3 D) (x - 6)2 = 3

Answer: C Type: BI Var: 50+ Objective: Write an Equation of a Circle in Standard Form

14) (0, 12) and radius 6 A) (y - 12)2 = 6

B) x2 + (y + 12)2 = 6

C) x2 + (y - 12)2 = 6

D) (y + 12)2 = 6

Answer: C Type: BI Var: 50+ Objective: Write an Equation of a Circle in Standard Form

Use the given information about a circle to write an equation of the circle in standard form. 15) The center is (4, -1) and the circle is tangent to the x-axis. A) (x + 4)2 + (y - 1)2 = 1 B) (x - 4)2 + (y + 1)2 = 16 C) (x - 4)2 + (y + 1)2 = 1 D) (x + 4)2 + (y - 1)2 = 16 Answer: C Type: BI Var: 50+ Objective: Write an Equation of a Circle in Standard Form

Solve the problem. 16) Write an equation that represents the set of points that are 5 units from (-5, -7). A) (x + 5)2 + (y + 7)2 = 5 B) |x - 5| + |y - 7| = 5 2 2 C) (x + 5) + (y + 7) = 25 D) |x + 5| + |y + 7| = 5 Answer: C Type: BI Var: 50+ Objective: Write an Equation of a Circle in Standard Form

17) Write an equation of the circle that is tangent to both axes with radius 3 and center in Quadrant IV. 2 2 2 2 A) (x - 3) + (y + 3) = 3 B) (x - 3) + (y + 3) = 3 2

C) (x + 3) + (y - 3) = 3

D) (x + 3) + (y - 3) = 3

Answer: A Type: BI Var: 24 Objective: Write an Equation of a Circle in Standard Form 2 Write the General Form of an Equation of a Circle

Determine the solution set for the equation. 1) (x + 4)2 + (y - 2)2 = 0 A) { } B) {(16, 4)} Answer: D Type: BI Var: 50+ Objective: Write the General Form of an Equation of a Circle

Page 22

C) {(4, -2)}

D) {(-4,2)}

Write the equation in standard form to find the center and radius of the circle. 2) x2 + y2 + 6x + 14y + 37 = 0 A) (x + 3)2 + (y + 7)2 = 21; center (-3, -7), radius 21 B) (x + 3)2 + (y + 7)2 = 21; center (-3, -7), radius 21 C) (x - 3)2 + (y - 7)2 = 21; center (3, 7), radius 21 D) (x + 3)2 + (y + 7)2 = 21; center (3, 7), radius 21 Answer: B Type: BI Var: 50+ Objective: Write the General Form of an Equation of a Circle

Write the equation in standard form to find the center and radius of the circle. Then sketch the graph. 3) x2 + y2 - 6x + 5 = 0 A) (x - 3)2 + y2 = 7 B) (x - 3)2 + y2 = 4

-8

-6

-4

8 y

2 2

-2

8 x

-8

-6

-4

-2

-4

-6

-8

-6

-4

8 x

D) (x - 3) + y = 4 8 y

8 y

2 2

-2

8 x

-8

-6

-4

-2

-4

-6

-8

Answer: D Type: BI Var: 16 Objective: Write the General Form of an Equation of a Circle

Page 23

-8

C) (x - 3) + y = 7

-8

Write the given equation in the form (x - h)2 + (y - k)2 = r2. Identify the center and radius. 4) x2 + y2 + 16x + 8y - 20 = 0 B) (x + 8)2 + ( y + 4)2 = 100 A) (x + 16)2 + ( y + 8)2 = 20 Center: (-16, -8); r = 20 Center: (8, 4); r = 10 2 2 D) (x + 8)2 + ( y + 4)2 = 100 C) (x + 16) + ( y + 8) = 20 Center: (16, 8); r = 20 Center: (-8, -4); r = 10 Answer: D Type: BI Var: 50+ Objective: Write the General Form of an Equation of a Circle

5) x2 + y2 + 8x - 10y + 25 = 0 A) (x + 4)2 + ( y - 5)2 = 16 Center: (4, -5); r = 4 C) (x + 8)2 + ( y - 10)2 = 25 Center: (8, -10); r = 25

B) (x + 8)2 + ( y - 10)2 = 25 Center: (-8, 10); r = 25 D) (x + 4)2 + ( y - 5)2 = 16 Center: (-4, 5); r = 4

Answer: D Type: BI Var: 50+ Objective: Write the General Form of an Equation of a Circle

6) x 2 + y 2 - 6x + 6y - 82 = 0 A) (x - 3)2 + (y + 3)2 = 100 Center: (-3, 3); r = 10 C) (x - 3)2 + (y + 3)2 = 100 Center: (3, -3); r = 10

B) (x + 3)2 + (y - 3)2 = 100 Center: (3, -3); r = 10 D) (x + 3)2 + (y + 3)2 = 100 Center: (3, -3); r = 10

Answer: C Type: BI Var: 50+ Objective: Write the General Form of an Equation of a Circle

7) x2 + y2 + 22x + 21 = 0 A) (x + 11)2 + y2 = 100 Center: (0, 11); r = 10 C) (x + 22)2 + y2 = 21 Center: (0, 22); r = 21

B) (x + 11)2 + y2 = 100 Center: (-11, 0); r = 10 D) (x + 22)2 + y2 = 21 Center: (-22, 0); r = 21

Answer: B Type: BI Var: 50+ Objective: Write the General Form of an Equation of a Circle

Identify the center and radius of the circle. 8) x2 - 10x + y2 = 11 A) Center: (5, 0); C) Center: (10, 0);

r=6 r = 11

Answer: A Type: BI Var: 28 Objective: Write the General Form of an Equation of a Circle

Page 24

B) Center: (0, -5); D) Center: (10, 0);

r=6 r = 36

Write the given equation in the form (x - h)2 + (y - k)2 = r2. Identify the center and radius. 9) x2 + y2 - 8y + 11 = 0 A) Center: (0, 4); r = 5 C) Center: (0, -8); r = 11

B) Center: (0, 8); r = 11 D) Center: (0, -4); r = 5

Answer: A Type: BI Var: 50+ Objective: Write the General Form of an Equation of a Circle

10) 25x2 + 25y2 - 20x + 60y - 360 = 0 A) Center: 4 , - 12 ; r = 360 5 5 2 6 ;r=4 C) Center: - , 5 5

B) Center: - 4 , 12 ; r = 360 5 5 D) Center: 2 , - 6 ; r = 4 5 5

Answer: D Type: BI Var: 50+ Objective: Write the General Form of an Equation of a Circle

Write the equation in standard form: (x - h)2 + (y - k)2 = r2. Then, if possible, identify the center and radius of the circle. If the equation represents a degenerate case, give the solution set. 11) x2 + y2 + 4x + 8y + 11 = 0 A) (x + 2)2 + (y + 4)2 = -9; Degenerate case: { } B) (x + 2)2 + (y + 4)2 = 0; Degenerate case (single point): {(-2, -4)} C) (x + 2)2 + (y + 4)2 = 9; Center: (2, 4); Radius: 3 D) (x + 2)2 + (y + 4)2 = 9; Center: (-2, -4); Radius: 3 Answer: D Type: BI Var: 50+ Objective: Write the General Form of an Equation of a Circle

12) x2 + y2 - 4x + 2y + 5 = 0 A) (x - 2)2 + (y + 1)2 = -5; Degenerate case: { } B) (x - 2)2 + (y + 1)2 = 25; Center: (2, -1); Radius: 5 C) (x - 2)2 + (y + 1)2 = 0; Degenerate case (single point): {(2, -1)} D) (x - 2)2 + (y + 1)2 = 25; Center: (-2, 1); Radius: 5 Answer: C Type: BI Var: 24 Objective: Write the General Form of an Equation of a Circle

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Solve the problem. 13) A cell tower is a site where antennas, transmitters, and receivers are placed to create a cellular network. Suppose that a cell tower is located at a point A(-7, -6) on a map and its range is 2.5 mi. Write an equation that represents the boundary of the area that can receive a signal from the tower. Assume that all distances are in miles. A) (x + 7)2 + (y + 6)2 = 2.5 B) |x + 7| + |(y + 6)| = 2.5 2 2 C) (x + 7) + (y + 6) = 6.25 D) |x - 7| + |(y - 6)| = 2.5 Answer: C Type: BI Var: 50+ Objective: Write the General Form of an Equation of a Circle

14) Find the shortest distance from the origin to a point on the circle defined by x2 + y2 - 14x - 18y + 121 = 0. A) 2 130 -+ 3

B) 130 - 3

C) 130 + 3

D) 2 130 - 3

Answer: B Type: BI Var: 50+ Objective: Write the General Form of an Equation of a Circle

2.3 Functions and Relations 0 Concept Connections

Provide the missing information. 1) A set of ordered pairs (x, y) is called a in x and y. The set of x values in the relation is called the of the relation. The set of values is called the range of the relation. Answer: relation; domain; y Type: SA Var: 1 Objective: Concept Connections

2) Explain what it means for a relation to define y as a function of x. Answer: A relation defines y as a function of x if for each value of x in the domain, there is exactly one value of y in the range. Type: SA Var: 1 Objective: Concept Connections

3) If the graph of a set of points (x, y) has two points aligned vertically then the relation (does/does not) define y as a function of x. Answer: does not Type: SA Var: 1 Objective: Concept Connections

4) Given a function defined by y = f (x), the statement f (2) = 4 is equivalent to what ordered pair? Answer: (2, 4) Type: SA Var: 1 Objective: Concept Connections

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5) Given a function defined by y = f (x), to find the

-intercept, evaluate f (0).

Answer: y Type: SA Var: 1 Objective: Concept Connections

6) Given a function defined by y = f (x), to find the x-intercept(s), substitute 0 for solve for x. Answer: f (x) Type: SA Var: 1 Objective: Concept Connections

7) Given, f (x) =

x+1

the domain is restricted so that x ≠

x+5 Answer: -5 Type: SA Var: 1 Objective: Concept Connections

8) Given g(x) = x - 5, the domain is restricted so that x ≥ Answer: 5 Type: SA Var: 1 Objective: Concept Connections 1 Determine Whether a Relation is a Function

Write the domain and range of the relation. 1) 5 y 4 3 2 1 -5 -4 -3 -2 -1 -1

-2 -3 -4 -5

A) Domain: {-3, -2, 1, 2}; Range: {-5, -1, 1, 2, 5} B) Domain: {-3, -2, 1, 2, 2}; Range: {-5, -1, 1, 2, 5} C) Domain: {-5, -1, 1, 2, 5}; Range: {-3, -2, 1, 2} D) Domain: {-5, -1, 1, 2, 5}; Range: {-3, -2, 1, 2, 2} Answer: A Type: BI Var: 50+ Objective: Determine Whether a Relation is a Function

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and

For the given relation, write the domain, write the range, and determine if the relation defines y as a function of x. 2) 5 y 4 3 2 1 -5

-4

-3

-2

-1

-1 -2 -3 -4 -5

A) Domain: {-1, 1, 3}; Range: {-4, -2, 4}; not a function B) Domain: {-1, 1, 3}; Range: {-4, -2, 4}; function C) Domain: {-4, -2, 4}; Range: {-1, 1, 3}; function D) Domain: {-4, -2, 4}; Range: {-1, 1, 3}; not a function Answer: D Type: BI Var: 50+ Objective: Determine Whether a Relation is a Function

Identify the domain and range of the relation, and determine whether the relation is a function. 3) {(-7, -12), (-3, -5), (1, 16), (8, 18)} A) Domain: {-12, -5, 16, 18}; Range: {-7, -3, 1, 8}; Function B) Domain: {-7, -3, 1, 8}; Range: {-12, -5, 16, 18}; Function C) Domain: {-12, -5, 16, 18}; Range: {-7, -3, 1, 8}; Not a function D) Domain: {-7, -3, 1, 8}; Range: {-12, -5, 16, 18}; Not a function Answer: B Type: BI Var: 50+ Objective: Determine Whether a Relation is a Function

4) {(-9, 4), (-1, 2), (18, -2), (-1, -4)} A) Domain: {-9, -1, 18}; Range: {-4, -2, 2, 4}; Not a function B) Domain: {-9, -1, -1, 18}; Range: {-4, -2, 2, 4}; Function C) Domain: {-4, -2, 2, 4}; Range: {-9, -1, 18}; Function D) Domain: {-4, -2, 2, 4}; Range: {-9, -1, 18}; Not a function Answer: A Type: BI Var: 50+ Objective: Determine Whether a Relation is a Function

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Determine whether the relation defines y as a function of x. 5)

A) Not a function

B) Function

Answer: B Type: BI Var: 6 Objective: Determine Whether a Relation is a Function

6) 6

5 4 3 2 1 -6 -5 -4 -3 -2 -1 -1

6 x

-2 -3 -4 -5 -6

A) Function Answer: A Type: BI Var: 50+ Objective: Determine Whether a Relation is a Function

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B) Not a function

7) 5 y 4 3 2 1 -5 -4 -3 -2 -1 -1

-2 -3 -4 -5

A) Function

B) Not a function

Answer: A Type: BI Var: 50+ Objective: Determine Whether a Relation is a Function

8) 5 y 4 3 2 1 -5 -4 -3 -2 -1 -1

-2 -3 -4 -5

A) Function Answer: B

B) Not a function

Type: BI Var: 50+ Objective: Determine Whether a Relation is a Function

9) x = |y + 1| A) Not a function Answer: A Type: BI Var: 16 Objective: Determine Whether a Relation is a Function

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B) Function

10)

A) Function

B) Not a function

Answer: B Type: BI Var: 6 Objective: Determine Whether a Relation is a Function

11) (x + 6)2 + (y - 7)2 = 25 A) Not a function Answer: A

B) Function

Type: BI Var: 50+ Objective: Determine Whether a Relation is a Function 2 Apply Function Notation

Solve the problem. 1) Find f(5) for the given function. f (x) = 2 A) 7 B) 5

C) 2

D) 10

Answer: C Type: BI Var: 50+ Objective: Apply Function Notation

Evaluate the function for the indicated value, then simplify. 2) f (x) = -2x - 2; find f (a - 1), then simplify as much as possible. A) a B) a - 5 C) -2a

D) -2a - 3

Answer: C Type: BI Var: 50+ Objective: Apply Function Notation

3) f (x) = 2x2 - 4x; find f (8) A) 224 B) 124

C) 96

D) -16

Answer: C Type: BI Var: 50+ Objective: Apply Function Notation

4) f (x ) = x 2 + 5x; find f (a + 1), then simplify as much as possible. A) a 2 + 3a + 6 B) a 2 + 7a + 6 C) a 2 + 3a + 2 Answer: B Type: BI Var: 40 Objective: Apply Function Notation

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D) a 2 + 7a + 7

5) f (t ) = t 2 + 5t; find f (t - 5), then simplify as much as possible. A) t 2 + 15t B) t 2 - 15t C) t 2 - 20t

D) t 2 - 5t

Answer: D Type: BI Var: 11 Objective: Apply Function Notation

Evaluate as indicated. 6) Find f (a + 2) for the given function. f(x) = x - 5 A) a - 10 B) x + a - 3

C) x + a - 10

D) a - 3

C) 23

D) -23

C) 25 + x

D) 3x2 + 8x - 5

Answer: D Type: BI Var: 50+ Objective: Apply Function Notation

7) If K(x) = x - 9 + x, find K(-7). A) 9 B) -9 Answer: A Type: BI Var: 50+ Objective: Apply Function Notation

8) If f (x) = 3x2 + 8x - 3, find and simplify f (2 + x). A) 3x2 + 2x + 25 B) 3x2 + 20x + 25 Answer: B Type: BI Var: 50+ Objective: Apply Function Notation

9) If z(t) = 2t2 + 7t - 4, find z(-1) and z(4). A) z(-1) = -13; z(4) = 56 C) z(-1) = -9; z(4) = 56

B) z(-1) = 5; z(4) = 35 D) z(-1) = -7; z(4) = 88

Answer: C Type: BI Var: 49 Objective: Apply Function Notation

10) If f (x) = 4x2 + 4x - 5, find and simplify f (2 + x). A) 4x2 + 4x - 7 B) 4x2 + 20x + 19

C) 4x2 + 2x + 19

Answer: B Type: BI Var: 50+ Objective: Apply Function Notation

11) Let f (x) = 7x - 2. Find f (m - 6) and simplify. A) 7x - 8 C) 7m - 8 Answer: B Type: BI Var: 50+ Objective: Apply Function Notation

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B) 7m - 44 D) 7xm - 42x - 2m + 12

D) 19 + x

Find and simplify f (x + h). 12) f (x) = -3x2 + 2x + 1 A) -3x2 2xh + h2 + 2x + 2h + 1 C) -3x2 - 3h2 + 2x h + 1

B) -3x2 - 6xh - 3h2 + 2x + 2h + 1 D) -3x2 - 3h2 + 2x + 2h + 1

Answer: B Type: BI Var: 50+ Objective: Apply Function Notation

13) f (x) = 3 - 4x2 A) -4x2 + h2 + 3 C) -4x2 - 8xh - 4h2 + 3

B) -4x2 + 2xh + h2 + 3 D) f (x) = -4x2 + h + 3

Answer: C Type: BI Var: 50+ Objective: Apply Function Notation

14) f (x) = x3 - 5x + 8 A) x3 + 3x2h + 3xh2 + h3 - 5x - 5h + 8 C) x3 - 5x - 5h + 8

B) x3 + h3 - 5x - 5h + 8 D) x3 - 5x + h + 8

Answer: A Type: BI Var: 50+ Objective: Apply Function Notation

Solve the problem. 15) Consider the function f = {(-6, -3), (-2, 3), (2, -1), (4, 2)} Determine f (-2) A) 4 B) 1 C) 2

D) 3

Answer: D Type: BI Var: 50+ Objective: Apply Function Notation

16) Consider the function z = {(9, -5),(-7, 3),(-1, 9)}. Find the function value z(-7). Answer: 3 Type: SA Var: 50+ Objective: Apply Function Notation

17) Sarita needs to drive 350 mi. After having driven x miles, the distance remaining r(x) (in mi) is given by r(x) = 350 - x. Evaluate r(250) and interpret the meaning. A) r(250) = 250; Sarita still has 250 miles remaining. B) r(250) = 100; Sarita still has 100 miles remaining. C) r(250) = 100; Sarita has traveled 100 miles. D) r(250) = 250; Sarita has traveled 250 miles. Answer: B Type: BI Var: 40 Objective: Apply Function Notation

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18) If bottled water costs $0.63 per bottle, then the cost, C (in dollars), for b bottles is defined by C(b) = 0.63b. How many bottles of water can be purchased for $7.56? A) 5 bottles B) 12 bottles C) 16 bottles D) 8 bottles Answer: B Type: BI Var: 50+ Objective: Apply Function Notation

19) The number of accidents in 1 month involving drivers x years of age in a certain country can be approximated by the function f (x) = 3x2 - 150x + 2,000. Find the number of accidents in 1 month that involved 47-year olds. A) 7,191 B) 2,000 C) 1,577 D) 6,627 Answer: C Type: BI Var: 50+ Objective: Apply Function Notation

20) Suppose the weight (in pounds) of a baby boy x months old, for his first 10 months, can be approximated by the function f (x) = 1.3x + 7.7. Find the predicted weight at the age of 1 months. A) 8.7 lb B) 9 lb C) 7.7 lb D) 10 lb Answer: B Type: BI Var: 50+ Objective: Apply Function Notation

21) For over 20 years, the population of Tressel, Ohio has been increasing linearly according tothe function P(t) = 225t + 7,000 where P is the number of residents, and t is years after 1980. Compute P(0) and interpret its meaning in the context of this problem. A) P(0) = 7,225; This was the population of the town in 1981. B) P(0) = 7,000; This was the population of the town in 1981. C) P(0) = 7,225; This was the population of the town in 1980. D) P(0) = 7,000; This was the population of the town in 1980. Answer: D Type: BI Var: 50+ Objective: Apply Function Notation

Page 34

22) At one college, a study found that the average grade point average decreased linearlyaccording to the function g(h) = 3.00 - 0.10h where h is the number of hours per week spent watching reality shows on television. Compute g(5) and interpret its meaning. A) g(5) = 2.50. On average, watching 5 hours of reality programming per week will decrease your GPA by 2.50. B) g(5) = 3.50. This tells us that the average GPA of students that watch 5 hours of reality programming per week is 3.50. C) g(5) = 3.50. On average, watching 5 hours of reality programming per week will increase your GPA by 3.50. D) g(5) = 2.50. This tells us that the average GPA of students that watch 5 hours of reality programming per week is 2.50. Answer: D Type: BI Var: 50+ Objective: Apply Function Notation 3 Determine x- and y-Intercepts of a Function Defined by y=f(x)

Determine the x- and y-intercepts for the given function. 1) f (x) = x - 2 A) x-intercept: (-2, 0); y-intercept: (0, 4) B) x-intercept: (4, 0); y-intercept: (0, -2) C) x-intercepts: (-4, 0) and (4, 0); y-intercept: (0, -2) D) x-intercept: (-2, 0); y-intercepts: (0, -4) and (0, 4) Answer: B Type: BI Var: 18 Objective: Determine x- and y-Intercepts of a Function Defined by y=f(x)

2) g(x) = -7x - 15 A) x-intercept: -

15 , 0 ; y-intercept: (0, -15) 7

C) x-intercept: 0, -

15 ; y-intercept: (-15, 0) 7

B) x-intercept: (0, -15); y-intercept: (22, 0) D) x- and y-intercept: (0, 0)

Answer: A Type: BI Var: 50+ Objective: Determine x- and y-Intercepts of a Function Defined by y=f(x)

Page 35

3) B(x) = 7x + 3 A) x-intercept: -

7 3 3

, 0 ; y-intercept: (3, 0)

C) x-intercept: - , 0 ; y-intercept: (0, -3) 7

B) x-intercept: -

3 7

, 0 ; y-intercept: (0, 3)

D) x- and y-intercept: (0, 0)

Answer: B Type: BI Var: 43 Objective: Determine x- and y-Intercepts of a Function Defined by y=f(x)

4) q(x) = -x2 + 18 A) x-intercepts: (18, 0), (-18, 0); y-intercept: (0, 3 2) C) x-intercepts: (3 2, 0), (-3 2, 0); y-intercept: (0, 18)

B) x-intercept: none; y-intercept: (0, 18) D) x-intercept: (0, 18); y-intercepts: (3 2, 0), (-3 2, 0)

Answer: C Type: BI Var: 8 Objective: Determine x- and y-Intercepts of a Function Defined by y=f(x)

5) r(x) = |x - 8| A) x-intercept: none; y-intercept: (0, 8) B) x-intercept: (8, 0); y-intercept: (0, 8) C) x-intercept: (0, 8); y-intercept: (0, 0) D) x-intercepts: (0, 8), (0, -8); y-intercept: (0, 8) Answer: B Type: BI Var: 18 Objective: Determine x- and y-Intercepts of a Function Defined by y=f(x)

Page 36

Solve the problem. 6) A student decides to finance a used car over a 5-yr (60-month) period. After making a down payment of $5,000, the remaining cost of the car including tax and interest is $13,140. The amount owed y = A(t) (in $) is given by A(t) = 13,140 - 219t, where t is the number of months after purchase and 0 ≤ t ≤ 60. Determine the t-intercept and y-intercept. Amount Owed on Vehicles after t Months

Amount Owed ($)

15000 12000 9000 6000 3000

Number of Months

A) t-intercept: (60,0); y-intercept: (0,13,140) B) t-intercept: (0,13,140); y-intercept: (60,0) C) t-intercept: (13,140, 0); y-intercept: (0, 60) D) t-intercept: (0, 60); y-intercept: (13,140, 0) Answer: A Type: BI Var: 50+ Objective: Determine x- and y-Intercepts of a Function Defined by y=f(x)

Page 37

4 Determine Domain and Range of a Function

Determine the domain and range of the function. 1) 5 y 4 3 2 1 -5

-4

-3

-2

-1

-1 -2 -3 -4 -5

A) Domain: (-∞, 2]; Range: (-∞, ∞) C) Domain: [2, ∞); Range: (-∞, ∞)

B) Domain: (-∞, ∞); Range: [2, ∞) D) Domain: (-∞, ∞); Range: (-∞, 2]

Answer: B Type: MC Var: 50+ Objective: Determine Domain and Range of a Function

2) 5 y 4 3 2 1 -5 -4 -3 -2 -1 -1

-2 -3 -4 -5

A) Domain: [-2, ∞]); Range [-3, -∞) C) Domain: [-2, ∞); Range: (-∞, -1] Answer: C Type: MC Var: 20 Objective: Determine Domain and Range of a Function

Page 38

B) Domain: (-∞, -1]; Range: [-2, ∞) D) Domain: (-2, ∞); Range: (-∞, -1)

3) 5 y 4 3 2 1 -5 -4 -3 -2 -1 -1

-2 -3 -4 -5

A) Domain: (-∞, 1]; Range [3, ∞) C) Domain: (-∞, ∞); Range (-∞, 1]

B) Domain: (-∞, 1]; Range (-∞, ∞) D) Domain: (-∞, ∞); Range (-∞, ∞)

Answer: C Type: MC Var: 50+ Objective: Determine Domain and Range of a Function

Write the domain in interval notation. x+7 4) f(x) = x+3 A) (-∞, -7) ∪ (-7, ∞) C) (-∞, 7) ∪ (7, ∞)

B) (-∞, 3) ∪ (3, ∞) D) (-∞, -3) ∪ (-3, ∞)

Answer: D Type: BI Var: 50+ Objective: Determine Domain and Range of a Function

5) k (x) =

x+9 x -4

A) (-∞, -9) ∪ (-9, ∞) C) (-∞, -4) ∪ (-4, ∞)

B) (-∞, 9) ∪ (9, ∞) D) (-∞, 4) ∪ (4, ∞)

Answer: D Type: BI Var: 50+ Objective: Determine Domain and Range of a Function

4 6) f(x) =

3-x

A) (-∞, 3)

B) (-∞, 3]

Answer: A Type: BI Var: 50+ Objective: Determine Domain and Range of a Function

Page 39

C) (3, ∞)

D) [3, ∞)

7) z(a) = a + 9 A) [-9, ∞)

B) (-∞, ∞)

C) [0, ∞)

D) (-9, ∞)

C) (- ∞, 8]

D) [0, 16]

Answer: A Type: BI Var: 9 Objective: Determine Domain and Range of a Function

8) y (t ) = 8 - t A) (-∞, ∞)

B) (- ∞, 8)

Answer: C Type: BI Var: 20 Objective: Determine Domain and Range of a Function

9) m(x) =

5 |x| + 1

A) (-∞, -5) ∪ (-5, ∞) C) (∞, ∞)

B) (-∞, -1) ∪ (-1, 1) ∪ (1, ∞) D) (-∞, -1) ∪ (-1, ∞)

Answer: C Type: BI Var: 50+ Objective: Determine Domain and Range of a Function

-14 10) r(x) =

x2 + 81

A) (∞, 0) ∪ (0, ∞) C) (∞, ∞)

B) (∞, -9) ∪ (9, ∞) D) (-∞, -9) ∪ (-9, 9) ∪ (9, ∞)

Answer: C Type: BI Var: 50+ Objective: Determine Domain and Range of a Function

11) f (x) =

x-4 x-5

A) ((-∞, - 5) ∪ (- 5, 5) ∪ ( 5, ∞) C) (-∞, ∞)

B) (-∞, 5) ∪ (5, ∞) D) (-∞, 4) ∪ (4, ∞)

Answer: B Type: BI Var: 50+ Objective: Determine Domain and Range of a Function

12) a(x) = 8 - x A) [-8, ∞) C) (-∞, -8) ∪ (-8, ∞) Answer: D Type: BI Var: 50+ Objective: Determine Domain and Range of a Function

Page 40

B) (-8, ∞) D) (-∞, 8]

13) g(t) =

5-t

A) (-∞, ∞)

B) (-∞, 5]

D) (-∞, 5) ∪ (5, ∞)

C) [5, ∞)

Answer: A Type: BI Var: 42 Objective: Determine Domain and Range of a Function

14) r(x) = x2 + 2x - 35 A) (-∞, -5) ∪ (-5, 7) ∪ (7, ∞) C) (-∞, -7) ∪ (-7, 5) ∪ (5, ∞)

B) (-∞, -8) ∪ (-8, ∞) D) (-∞, ∞)

Answer: D Type: BI Var: 50+ Objective: Determine Domain and Range of a Function

15) w(a) =

6 - |a - 3| A) (-∞, -3) ∪ (-3, 9) ∪ (9, ∞) C) (-∞, ∞)

B) (-∞, 9) ∪ (9, ∞) D) (-∞, -3) ∪ (-3, ∞)

Answer: A Type: BI Var: 50+ Objective: Determine Domain and Range of a Function

16) h(c) =

c + 11 - 1 A) (-11, ∞) C) [-11, -10) ∪ (-10, ∞)

B) [-11, ∞) D) (-11, -10) ∪ (-10, ∞)

Answer: C Type: BI Var: 50+ Objective: Determine Domain and Range of a Function

17) p(x) = 7x + 1; x ≥ 0 B) ( -∞, -

C) [0, 4)

D) (-∞, ∞)

Answer: A Type: BI Var: 50+ Objective: Determine Domain and Range of a Function

Page 41

1 )∪ - ,∞ 7 7

A) [0, ∞)

5 Interpret a Function Graphically

Use the graph of y = f (x) to answer the questions. 1) a. Determine f (-1) b. Find all x for which f (x) = -4 5 y 4 3 2 1 -5

-4

-3

-2

-1

-1 -2 -3 -4 -5

A) f (-1) = 1; f (x) = -4 for x = 4 B) f (-1) = -4; f (x) = -4 for x = 4 C) f (-1) = -4; f (x) = -4 for all x on the interval (-1,2) D) f (-1) = 1; f (x) = -4 for all x on the interval (-1, 2) Answer: D Type: BI Var: 50+ Objective: Interpret a Function Graphically

Page 42

a. Determine f (-1) b. Find the range of f. 5 y 4 3 2 1 -5

-4

-3

-2

-1

-1 -2 -3 -4 -5

A) f (-1) = 1; Range: {-2} ∪[0, ∞) C) f (-1) = -2; Range: [-2, ∞)

B) f (-1) = -2; Range: {-2} ∪[0, ∞) D) f (-1) = 1; Range: [-2, ∞)

Answer: A Type: MC Var: 50+ Objective: Interpret a Function Graphically

a. Determine the y-intercept b. Find f (-4) 5

4 3 2 1 -5 -4 -3 -2 -1 -1

5 x

-2 -3 -4 -5

A) y-intercept: (1, 0); f (-4) = 2 C) y-intercept: (1, 0); f (-4) = -2 Answer: B Type: BI Var: 9 Objective: Interpret a Function Graphically

Page 43

B) y-intercept: (0, -1); f (-4) = 2 D) y-intercept: (0, -1); f (-4) = -2

a. Determine the domain b. Determine the x-intercept 5

4 3 2 1 -5 -4 -3 -2 -1 -1

5 x

-2 -3 -4 -5

A) Domain: [-5, 0]; x-intercept (0, -1) C) Domain: (-∞, ∞); x-intercept (0, -1)

B) Domain: (-∞, 4); x-intercept (-1, 0) D) Domain: [-5, 0] ∪ {2}; x-intercept (-1, 0)

Answer: B Type: BI Var: 3 Objective: Interpret a Function Graphically 6 Mixed Exercises

Write a function defined by y = f (x) subject to the following conditions. 1) The value of f (x) is four more than eight times x. A) f (x) = 8x - 4 B) f (x) = 8x + 4 C) f (x) = 4x + 8

D) f (x) = 4x - 8

Answer: B Type: BI Var: 50+ Objective: Mixed Exercises

2) The value of f (x) is nine times the square root of x. A) f (x) = 9x2 B) f (x) = (9x)2

C) f (x) = 9x

D) f (x) = 9 x

Answer: D Type: BI Var: 18 Objective: Mixed Exercises

2.4 Linear Equations in Two Variables and Linear Functions 0 Concept Connections

Provide the missing information. 1) A diagram is a visual representation of a set of data points represented as ordered pairs. Answer: scatter Type: SA Var: 1 Objective: Concept Connections

Page 44

2) A equation in the variables x and y can be written in the form Ax + By = C, where A and B are not both zero. Answer: linear Type: SA Var: 1 Objective: Concept Connections

3) An equation of the form x = k where k is a constant represents the graph of a

line.

Answer: vertical Type: SA Var: 1 Objective: Concept Connections

4) An equation of the form y = k where k is a constant represents the graph of a

line.

Answer: horizontal Type: SA Var: 1 Objective: Concept Connections

5) True or false: The slope between any two distinct points on a nonvertical line is a constant. Answer: True Type: SA Var: 1 Objective: Concept Connections

6) Write the formula for the slope of a line between the two distinct points (x1, y1) and (x2, y2). Answer: m =

y2 - y1 x2 - x1

Type: SA Var: 1 Objective: Concept Connections

7) The slope of a horizontal line is

Answer: zero Type: SA Var: 1 Objective: Concept Connections

8) The slope of a vertical line is

Answer: undefined Type: SA Var: 1 Objective: Concept Connections

9) An equation written in the form y = mx + b is said to be written in form. Answer: slope-intercept Type: SA Var: 1 Objective: Concept Connections

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10) A function f is a linear function if f(x) = represents the y-intercept.

where m represents the slope and (0, b)

Answer: mx + b Type: SA Var: 1 Objective: Concept Connections

11) If f is defined on the interval [x1, x2], then the average rate of change of f on the interval [x1, x2] is given by the formula Answer: m =

f (x2) - f (x1) x2 - x1

Type: SA Var: 1 Objective: Concept Connections

12) The graph of a constant function defined by f(x) = b is a (horizontal/vertical) line. Answer: horizontal Type: SA Var: 1 Objective: Concept Connections

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1 Graph Linear Equations in Two Variables

Graph the equation and identify the x- and y-intercepts. 1) -5x - 2y = 10 A) x-intercept: (-5, 0); y-intercept: (0, -2) 5 y

-5 -4 -3 -2 -1

-1

-3

-2

-3

-2

5 x

-1

-3

-4

-5

D) x-intercept: (-5, 0); y-intercept: (0, 2) y

5 y

1 1

-1

-3

5 x

-5 -4 -3 -2 -1

-1

-2

-3

-4

-5

Answer: B Type: BI Var: 39 Objective: Graph Linear Equations in Two Variables

Page 47

-4

-2

-4

-5

-2

C) x-intercept: (2, 0); y-intercept: (0, -5)

-5

B) x-intercept: (-2, 0); y-intercept: (0, -5)

2) 5y + 1 = 21 A) x-intercept: none; y-intercept: (0, 4) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

C) x-intercept: none; y-intercept: (0, -4)

D) x-intercept: (-4,0); y-intercept: none

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: A Type: BI Var: 50+ Objective: Graph Linear Equations in Two Variables

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B) x-intercept: (4,0); y-intercept: none

3) 5x + 4 = 24 A) x-intercept: (-4,0); y-intercept: none 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

C) x-intercept: none; y-intercept: (0, -4)

D) x-intercept: (4,0); y-intercept: none

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: D Type: BI Var: 50+ Objective: Graph Linear Equations in Two Variables

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B) x-intercept: none; y-intercept: (0, 4)

4) 3x = -5y A) x-intercept: (0, 0); y-intercept: (0, 0) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

C) x-intercept: (0, 0); y-intercept: (0, 0)

D) x-intercept: none; y-intercept: none

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: B Type: BI Var: 8 Objective: Graph Linear Equations in Two Variables

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B) x-intercept: (0, 0); y-intercept: (0, 0)

5) 0.03x + 0.01y = 0.03 A) x-intercept: (0, 3); y-intercept: (1, 0)

B) x-intercept: (-1, 0); y-intercept: (0, 3)

5 y

-5 -4 -3 -2 -1

-1

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

C) x-intercept: (0, 3); y-intercept: (-1, 0)

D) x-intercept: (1, 0); y-intercept: (0, 3)

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: D Type: BI Var: 18 Objective: Graph Linear Equations in Two Variables 2 Determine the Slope of a Line

Find the average slope of the ramp. 1) 250 in.

1,000 in. 1 A) m = 4

B) m = 4

Answer: D Type: BI Var: 15 Objective: Determine the Slope of a Line

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C) m = -4

D) m =

1 4

Find the slope of the ramp pictured below. 2)

A) -88

C) 88

25 3

Answer: B Type: BI Var: 1 Objective: Determine the Slope of a Line

Solve the problem. 3) If a plane loses 800 feet in altitude over a horizontal distance of 14,000 feet, what is the slope? 2 2 2 35 A) B) 37 D) 35 C) 2 35 Answer: A Type: BI Var: 50+ Objective: Determine the Slope of a Line

Determine the slope of the line passing through the given points. 4) (1, -1) and (-8, -3) 2 2 9 A) m = B) m = C) m = 9 9 2

D) m =

9 2

Answer: B Type: BI Var: 50+ Objective: Determine the Slope of a Line

5) (-2, -2) and (-2, 3) A) 0

B) Undefined

C) 1

D) 5

C) 1

D) 0

Answer: B Type: BI Var: 50+ Objective: Determine the Slope of a Line

6) (-7, -10) and (5, -10) A) 12

B) Undefined

Answer: D Type: BI Var: 50+ Objective: Determine the Slope of a Line

7) (8, -1) and (10, -7) 1 A) m = 3

B) m = 3

Answer: D Type: BI Var: 50+ Objective: Determine the Slope of a Line

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C) m =

1 3

D) m = -3

3 2 and 1 8 , - ,5 3 9 7 285 A) m = 77

B) m = -

75 77

C) m =

285 112

D) m = -

21 325

Answer: C Type: BI Var: 50+ Objective: Determine the Slope of a Line

1 1 5 5 , and - , 3 2 4 2 23 A) m = 24

B) m =

C) m = -

D) m =

B) m = 0.77

C) m = -1.3

D) m = 1.3

C) m = - 2

D) m =

Answer: B Type: BI Var: 50+ Objective: Determine the Slope of a Line

10) (-3.4, 3.4) and (-4.4, 2.1). A) m = -0.77 Answer: D

Type: BI Var: 50+ Objective: Determine the Slope of a Line

11) (3 11, 6 6) and ( 11, 6) 5 66 A) m = B) m = 2 22 Answer: D Type: BI Var: 50+ Objective: Determine the Slope of a Line

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5 66 22

Determine the slope of the line. 12) 5 y 4 3 2 1 -5

-4

-3

-2

-1

-1 -2 -3 -4 -5

A) m =

1 B) m = 3

C) m = -

1 3

D) m = -3

Answer: B Type: BI Var: 36 Objective: Determine the Slope of a Line

13) 5 y 4 3 2 1 -5

-4

-3

-2

-1

-1 -2 -3 -4 -5

A) m =

B) m =

3 Answer: B Type: BI Var: 4 Objective: Determine the Slope of a Line

Page 54

3 5

C) m = -

5 3

D) m = -

3 5

Solve the problem. 14) If the slope of a line is

7 10

, how much vertical change will be present for a horizontal change of

63 ft? A) 6.3 ft

B) 90 ft

C) 44.1 ft

D) 441 ft

Answer: C Type: BI Var: 50+ Objective: Determine the Slope of a Line 3 Apply the Slope-Intercept Form of a Line

Write the equation in slope-intercept form and determine the slope and y-intercept. 1) -2x = -3y - 6 2 2 2 2 A) y = x + 6; slope: ; y-intercept: (0, 6) B) y = x + 6; slope: ; y-intercept: (0, 6) 3 3 3 3 2 2 2 2 C) y = x - 2; slope: ; y-intercept: (0, -2) D) y = - x - 2; slope: - ; y-intercept: (0, -2) 3 3 3 3 Answer: C Type: BI Var: 40 Objective: Apply the Slope-Intercept Form of a Line

Determine the slope and the y-intercept of the line. 2) 6 = -6y A) Slope: 0; y-intercept: (-1, 0) C) Slope: undefined; y-intercept: (-1, 0)

B) Slope: 0; y-intercept: (0, -1) D) Slope: 1; y-intercept: (0, -1)

Answer: B Type: BI Var: 50+ Objective: Apply the Slope-Intercept Form of a Line

3) 6x - 5y = 4 6 A) Slope: -

; y-intercept: 0, -

C) Slope: Slope:

5 6 5

; y-intercept: 0, -

; y-intercept: 0, - 5 4 6

D) Slope: Slope:

6 ; y-intercept: 0, 4 5

4 5

Answer: C Type: BI Var: 50+ Objective: Apply the Slope-Intercept Form of a Line

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B) Slope: Slope:

Write the equation in slope-intercept form. Then, graph the line using the slope and y-intercept. 4) 3x = 3 - y 5 y 4 3 2 1 -5 -4 -3 -2 -1 -1

-2 -3 -4 -5

A) y = 3x + 3

B) y = 3x - 3 5 y

5 y

-5 -4 -3 -2 -1 -1

-1

-2

-3

-4

-5

C) y = -3x - 3

D) y = -3x + 3 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: D Type: BI Var: 32 Objective: Apply the Slope-Intercept Form of a Line

Page 56

-5 -4 -3 -2 -1

5) -3x + 5y = -15 5 y 4 3 2 1 -5 -4 -3 -2 -1 -1

-2 -3 -4 -5

3 A) y = x + 3 5

5 B) y = x - 3 3 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

3 C) y = -

-5

x-3

D) y = x - 3 5

5 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: D Type: BI Var: 40 Objective: Apply the Slope-Intercept Form of a Line

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6) -5x - 2y = 0 5 y 4 3 2 1 -5 -4 -3 -2 -1 -1

-2 -3 -4 -5

5 A) y = - x 2

2 B) y = x 5 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

C) y = x 2

D) y = - x 5 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: A Type: BI Var: 50+ Objective: Apply the Slope-Intercept Form of a Line

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Determine if the function is linear, constant, or neither. 5 7) f(x) = 3 A) linear

B) constant

C) neither

Answer: B Type: BI Var: 33 Objective: Apply the Slope-Intercept Form of a Line

Use the slope-intercept form to write an equation of the line that passes through the given point and has the given slope. Use function notation where y = f(x). 8) (-4, -5); m = 4 A) f(x) = 4x - 4 B) f(x) = 4x + 11 C) f(x) = 4x - 25 D) f(x) = 4x - 5 Answer: B Type: BI Var: 50+ Objective: Apply the Slope-Intercept Form of a Line

9) (-3, 1); m = -

3 2 A) f(x) = - x - 3 3

B) f(x) = -

x-1

C) f(x) = -

x+5

D) f(x) =

x-1

Answer: B Type: BI Var: 50+ Objective: Apply the Slope-Intercept Form of a Line

Use the slope-intercept form to write an equation of the line that passes through the given points. Use function notation where y = f(x). 10) (2, -3) and (-7, -11) 8 1 9 1 9 8 43 43 A) f(x) = - x B) f(x) = - x C) f(x) = x D) f(x) = x 9 9 8 9 8 9 9 9 Answer: D Type: BI Var: 50+ Objective: Apply the Slope-Intercept Form of a Line

11) (10, 2) and (8, 10) A) f(x) = 4x + 38

B) f(x) = -4x + 38

Answer: C Type: BI Var: 50+ Objective: Apply the Slope-Intercept Form of a Line

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C) f(x) = -4x + 42

D) f(x) = -4x - 42

Use the slope-intercept form to write an equation of the line that passes through the given point and has the given slope. Use function notation where y = f(x). 12) (9, -3); m = 0 A) f(x) = -3 B) f(x) = 9x C) f(x) = 0 D) f(x) = 9 Answer: A Type: BI Var: 50+ Objective: Apply the Slope-Intercept Form of a Line

13) (-2.5, 4.3); m = -5.2 A) f(x) = -8.7x - 5.2 C) f(x) = -5.2x - 8.7

B) f(x) = -5.2x + 8.7 D) f(x) = -5.2x + 4.3

Answer: C Type: BI Var: 50+ Objective: Apply the Slope-Intercept Form of a Line 4 Compute Average Rate of Change

Find the slope of the secant line indicated with a dashed line. 1) 16 14

12 10 8 6 4 2 -16 -14 -12 -10 -8 -6 -4 -2 -2

4 6 8 10 12 14 16 x

(-4,2 -5) -6 -8

(-2, -14)

-10 -12 -14 -16

A) m =

4 9

B) m = -

Answer: C Type: BI Var: 50+ Objective: Compute Average Rate of Change

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9 4

C) m =

9 4

D) m = -

4 9

Solve the problem. 2) The population of a certain country since 1990 can be approximated by f(t) = 0.008t2 + 2.1t + 175 where f(t) is the population in millions and t represents the number of years since 1990. Find the average rate of change in the country's population between 1990 and 2010. Round to 1 decimal place. A) 2.0 million/yr B) 2.3 million/yr C) 2.9 million/yr D) 2.5 million/yr Answer: B Type: BI Var: 50+ Objective: Compute Average Rate of Change

3) The function given by y = f (x) shows the value of $6,000 invested at 5% interest compounded continuously, x years after the money was originally invested. Find the average amount earned per year between the 25th year and 30th year. y 32000 28000

(30, 26,890)

Value ($)

24000 (25, 20,942)

20000

(20, 16,310)

16000 12000

(15, 12,702)

8000

(10, 9,892) (5, 7,704)

4000

10 15 20 Number of Years

A) $1,189.60/year

B) $5,948.00/year

C) $20,942.00/year

D) $26,890.00/year

Answer: A Type: BI Var: 40 Objective: Compute Average Rate of Change

Determine the average rate of change of the function on the given interval. 4) f (x) = x + 1 on [0, 3] 1 3 3 A) B) C) 3 3 3 Answer: D Type: BI Var: 20 Objective: Compute Average Rate of Change

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1 3

5) f (x) = 3x2 + 3 on [3, 5] A) -24

C) 24

5 2

Answer: C Type: BI Var: 50+ Objective: Compute Average Rate of Change

6) f (x) = x3 + 3 on [2, 3] A) 19

B) -19

C) -

19 2

Answer: A Type: BI Var: 40 Objective: Compute Average Rate of Change 5 Solve Equations and Inequalities Graphically

Use the graph to solve the equation and inequality. Write the solution to the inequality in interval notation. 1) a. 3x - 3 = 2x - 1 b. 3x - 3 > 2x - 1 5 y 4 3

(2, 3)

2 1 -5

-4

-3

-2

-1

-1 -2 -3 -4 -5

A) a. {2}; b. (-∞, 2) C) a. {3}; b. (3, ∞) Answer: B Type: BI Var: 24 Objective: Solve Equations and Inequalities Graphically

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B) a. {2}; b. (2, ∞) D) a. {3} b. (-∞, 3}

a. 3x - 1 = 2x - 2 b. 3x - 1 < 2x - 2 5 y 4 3 2 1 -5

-4

-3

-2

-1

-1 -2 -3 (-1, -4)

-4 -5

A) a. {-1}; b. (-∞, -1) C) a. {-4} b. (-∞, -4}

B) a. {-4}; b. (-4, ∞) D) a. {-1}; b. (-1, ∞)

Answer: A Type: BI Var: 50+ Objective: Solve Equations and Inequalities Graphically

2.5 Applications of Linear Equations and Modeling 0 Concept Connections

Provide the missing information. 1) Given a point (x1, y1) on a line with slope m, the point-slope formula is given by Answer: y - y1 = m(x - x1) Type: SA Var: 1 Objective: Concept Connections

2) If two nonvertical lines have the same slope but different y-intercepts, then the lines are (parallel/perpendicular). Answer: parallel Type: SA Var: 1 Objective: Concept Connections

3) If m1 and m2 represent the slopes of two nonvertical perpendicular lines, then m1m2 = . Answer: -1 Type: SA Var: 1 Objective: Concept Connections

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4) Suppose that y = C (x) represents the cost to produce x items, and that y = R (x) represents the revenue for selling x items. The profit P (x) of producing and selling x items is defined by P (x) = . Answer: R (x) - C (x) Type: SA Var: 1 Objective: Concept Connections 1 Apply the Point-Slope Formula

Use the point-slope formula to write an equation of the line that passes through the given points. Write the answer in slope-intercept form (if possible). 1) (3, -3) and (-4, -5) 7 3 2 3 2 27 7 27 D) y = - x A) y = x B) y = - x C) y = x 7 7 2 7 2 7 7 7 Answer: A Type: BI Var: 50+ Objective: Apply the Point-Slope Formula

2) Passes through (4, 1) and the slope is undefined. A) y = 1 B) y = x + 1

C) y = x + 4

D) x = 4

Answer: D Type: BI Var: 50+ Objective: Apply the Point-Slope Formula

Write an equation of the line satisfying the given conditions. Write the answer in standard form. 6 3) The line has a slope of - and contains the point (-5, -8). 7 6 6 86 83 B) y = - x C) x + y = A) 6x + 7y = 86 D) 6x + 7y = -86 7 7 7 7 Answer: D Type: BI Var: 50+ Objective: Apply the Point-Slope Formula

Write an equation of the line satisfying the given conditions. Write the answer in slope-intercept form. 2 4) The line passes through the point (15, -2) and has a slope of . 5 2 2 2 2 B) y = x + 15 C) y = x - 8 D) y = x + 22 A) y = - x - 8 5 5 5 5 Answer: C Type: BI Var: 50+ Objective: Apply the Point-Slope Formula

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5) The line passes through the point (2, 13) and has a slope of 4. A) y = 4x + 2 B) y = -4x + 13 C) y = 4x + 13

D) y = 4x + 5

Answer: D Type: BI Var: 50+ Objective: Apply the Point-Slope Formula

3 6) The line passes through the point (-4, 1) and has a slope of . 2 3 3 3 B) y = x + 7 C) y = x - 5 A) y = - x + 7 2 2 2

D) y =

x-4

Answer: B Type: BI Var: 50+ Objective: Apply the Point-Slope Formula

7) The line passes through (12, 9) and (9, 9). 1 21 A) y = - x + B) y = 9 6 2

1 21 x6 2

C) x = 9

D) y = -

C) y = -12

D) y = -12x

Answer: B Type: BI Var: 50+ Objective: Apply the Point-Slope Formula

8) The line passes through (-12, -12) and (-12, -4). A) y = 12 B) x = -12 Answer: B Type: BI Var: 50+ Objective: Apply the Point-Slope Formula 2 Determine the Slopes of Parallel and Perpendicular Lines

The slope of a line is given. a. Determine the slope of a line parallel to the given line, if possible. b. Determine the slope of a line perpendicular to the given line, if possible. 8 1) m = 5 8 5 8 B) a. m = 0; b. m = A) a. m = ; b. m = 5 8 5 5 8 5 C) a. m = 0; b. m = D) a. m = ; b. m = 8 5 8 Answer: D Type: BI Var: 44 Objective: Determine the Slopes of Parallel and Perpendicular Lines

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The slope of a line is given. Find the slope of a line parallel to the given line. 11 2) m = 10 11 10 10 A) B) C) 10 11 11

D) -

11 10

Answer: A Type: BI Var: 50+ Objective: Determine the Slopes of Parallel and Perpendicular Lines

The slope of a line is given. Find the slope of a line perpendicular to the given line. 14 3) m = 5 5 14 5 A) B) C) 14 5 14

D) -

14 5

Answer: C Type: BI Var: 50+ Objective: Determine the Slopes of Parallel and Perpendicular Lines

The slope of a line is given. a. Determine the slope of a line parallel to the given line, if possible. b. Determine the slope of a line perpendicular to the given line, if possible. 4) m is undefined A) a. m is undefined; b. m = 0 B) a. m = 0; b. m = -1 C) a. m = 0; b. m is undefined D) a. m = 0; b. m = 1 Answer: A Type: BI Var: 2 Objective: Determine the Slopes of Parallel and Perpendicular Lines

Determine if the lines defined by the given equations are parallel, perpendicular, or neither. 7 5) y= x- 2 5 7 y=- x-4 5 A) perpendicular

B) neither

C) parallel

Answer: B Type: BI Var: 50+ Objective: Determine the Slopes of Parallel and Perpendicular Lines

6) -4y = 2x + 5 -4x = 8y + 3 A) perpendicular Answer: B

B) parallel

Type: BI Var: 50+ Objective: Determine the Slopes of Parallel and Perpendicular Lines

Page 66

C) neither

7) -4x - 9y = -4 2 3 x + y = -9 3 2 A) perpendicular Answer: C

B) neither

C) parallel

Type: BI Var: 50+ Objective: Determine the Slopes of Parallel and Perpendicular Lines

8) -4y = -3x - 3 -12x = 9y + 8 A) perpendicular Answer: A

B) neither

C) parallel

Type: BI Var: 50+ Objective: Determine the Slopes of Parallel and Perpendicular Lines

9) -4x - 4y = -1 1 1 - x+ y=3 2 2 A) parallel Answer: C

B) neither

C) perpendicular

Type: BI Var: 50+ Objective: Determine the Slopes of Parallel and Perpendicular Lines

10) -5x - 4y = 6 1 2 x+ y=8 2 3 A) perpendicular Answer: B

B) neither

C) parallel

Type: BI Var: 50+ Objective: Determine the Slopes of Parallel and Perpendicular Lines

Write an equation of the line satisfying the given conditions. Write the answer in standard form with no fractional coefficients. 11) Passes through (-1, -4) and is parallel to the line defined by 5x + 3y = -8 A) 5x + 3y = -4 B) 5x + 3y = -1 C) 5x + 3y = -17 D) 5x + 3y = -5 Answer: C Type: BI Var: 50+ Objective: Determine the Slopes of Parallel and Perpendicular Lines

12) Passes through (4, 4) and is perpendicular to the line defined by -5x + 4y = -6 A) 4x - 5y = -4 B) 5x - 4y = 4 C) -5x - 4y = -36 Answer: D Type: BI Var: 50+ Objective: Determine the Slopes of Parallel and Perpendicular Lines

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D) 4x + 5y = 36

Write an equation of the line satisfying the given conditions. 13) Passes through (1, 4) and is parallel to the y-axis. A) y = 1 B) x = 4 C) x = 1

D) y = 4

Answer: C Type: BI Var: 50+ Objective: Determine the Slopes of Parallel and Perpendicular Lines

14) The line passes through (42, 17) and is parallel to y = 6. 1 1 A) x = B) y = C) y = 17 6 6

D) x = 42

Answer: C Type: BI Var: 50+ Objective: Determine the Slopes of Parallel and Perpendicular Lines

15) The line passes through (25, 24) and is perpendicular to y = 6. 1 1 C) x = A) x = 25 B) y = 6 6

D) y = 24

Answer: A Type: BI Var: 50+ Objective: Determine the Slopes of Parallel and Perpendicular Lines 3 Create Linear Functions to Model Data

Solve the problem. 1) Joey borrows $2,400 from his grandfather and pays the money back in monthly payments of $200. a. Write a linear function that represents the remaining money owed L(x) after x months. b. Evaluate L(10) and interpret the meaning in the context of this problem. A) L(x) = -200x + 2,400; L(10) = 400, This represents the amount Joey has paid his grandfather after 10 months. B) L(x) = 200x + 2,400; L(10) = 4,400, This represents the amount Joey still owes his grandfather after 10 months. C) L(x) = -200x + 2,400; L(10) = 400, This represents the amount Joey still owes his grandfather after 10 months. D) L(x) = 200x + 2,400; L(10) = 4,400, This represents the amount Joey has paid his grandfather after 10 months. Answer: C Type: BI Var: 50+ Objective: Create Linear Functions to Model Data

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2) A bakery makes and sells pastries. The fixed monthly cost to the bakery is $770. The cost for labor, taxes, and ingredients for the pastries amounts to $0.90 per pastry. The pastries sell for $1.60 each. a. Write a linear profit function representing the profit for producing and selling x pastries. b. Determine the break-even point for the bakery. A) a. P(x) = 1.60x - 770; b. 481 pastries B) a. P(x) = 0.7x - 770; b. 1,100 pastries C) a. P(x) = 1.60x + 770; b. 481 pastries D) a. P(x) = 0.7x + 770; b. 1,100 pastries Answer: B Type: BI Var: 50+ Objective: Create Linear Functions to Model Data

3) The graph shows the number of organ transplants y in a certain country for the years 2005 to 2010 where x represents the number of years since 2005. y 200 190 180 170 160

(4, 162)

150 140 130 120 110

(1, 114)

100 90 1

a. Use the points (1, 114) and (4, 162) to write a linear model for these data. Round the slope to one decimal place and the y-intercept to the nearest whole unit. b. Use the model to approximate the number of organ transplants performed in 2012. A) a. y = 16.0x + 98 b. approximately 210 B) a. y = 16.0x + 98 b. approximately 290 C) a. y = 16.0x + 114 b. approximately 306 D) a. y = 16.0x + 114 b. approximately 226 Answer: A Type: BI Var: 50+ Objective: Create Linear Functions to Model Data

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Plot the point on a rectangular coordinate system. 4) The figure represents the average credit card debt for selected households in Silerville.

Let y represent the credit card debt in dollars. Let x represent the year, where x = 0 corresponds to the year 1990, x = 4 represents 1994, and so on. a. Use the ordered pairs given in the graph, (0, 3581) and (16, 8106) to find a linear equation to estimate the average credit card debt versus the year. Round the slope to the nearest tenth. b. Use the model from (a) to estimate the average debt in 2003. Round to the nearest dollar. c. Interpret the slope of the model in the context of this problem. A) a. y = 282.8x + 3,581 b. $7,257 in 2003 c. The average household credit card debt is increasing at approximately $283 per year. B) a. y = 282.8x + 3,581 b. $7,257 in 2003 c. The average household credit card debt is increasing at approximately $3,581 per year. C) a. y = -282.8x + 3,581 b. $95.4 in 2003 c. The average household credit card debt is decreasing at approximately $283 per year. D) a. y = 282.8x + 3,581 b. -$95.4 in 2003 c. The average household credit card debt is decreasing at approximately $3,581 per year. Answer: A Type: BI Var: 4 Objective: Create Linear Functions to Model Data

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Solve the problem. 5) At the Jumping Jack cookie factory, quality assurance inspectors remove broken or otherwise defective cookies from a moving conveyor belt prior to packaging. Based on past studies, the plant manager knows that the inspectors eliminate 98% of the defective cookies at a conveyor belt speed of 9 feet per minute. As the belt speed increases, the factory can produce more cookies per hour, but at the cost of lower quality of the packaged product. Inspectors collect only 60% of defective cookies at a belt speed of 18 feet per minute.

a. Find an equation of the line through the given points. Write the equation in slope-intercept form. Round the slope and y-intercept each to 1 decimal place. b. Use the equation from part (b) to predict the efficiency of the inspectors at a belt speed of 15 feet per minute. Round to the nearest percent. A) a. y = 136.0 - 4.2x b. 66.5% at a belt speed of 15 feet per minute. B) a. y = 136.0 + 4.2x b. 73% at a belt speed of 15 feet per minute. C) a. y = 136.0 + 4.2x b. 66.5% at a belt speed of 15 feet per minute. D) a. y = 136.0 - 4.2x b. 73% at a belt speed of 15 feet per minute. Answer: D Type: BI Var: 12 Objective: Create Linear Functions to Model Data

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6) The following figure represents the average hourly wage for employees of MegaMart from 1980 to 2005.

Let y represent the hourly wage and let x represent the year, where x = 0 corresponds to the year 1980, x = 1 represents 1981, and so on. Then the average wage can be approximated by the equation y = 0.17x + 3.02, where 0 ≤ x ≤ 25. Use the linear equation to approximate the average wage for the year 1,985 and compare it to the actual wage of $3.75 per hour. A) $4.72 per hour; the calculated wage was $0.97 more than the actual wage. B) $3.02 per hour; the calculated wage was $0.17 more than the actual wage. C) $3.87 per hour; the calculated wage was $0.12 more than the actual wage. D) $3.02 per hour; the calculated wage was $0.73 less than the actual wage. Answer: C Type: BI Var: 4 Objective: Create Linear Functions to Model Data

7) In 2005, a special mixed-nut blend at a store cost $1.35 per lb, and in 2010 the blend cost $1.83 per lb. Let y represent the cost of a pound of the mixed-nut blend x years after 2005. Use a linear equation model to estimate the cost of a pound of the mixed-nut blend in 2007. A) $1.59 B) $1.64 C) $1.54 D) $1.68 Answer: C Type: BI Var: 50+ Objective: Create Linear Functions to Model Data

8) A certain medicine is administered to animals based on the weight. The recommended dose is 4 mg per lb of the animal's weight. Construct a linear function and use it to determine what size animal would require a 1,300-mg dose of the medicine. A) 325 lb B) 2,600 lb C) 5,200 lb D) 650 lb Answer: A Type: BI Var: 17 Objective: Create Linear Functions to Model Data

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9) A tumor originally weighed 34 g. Every day, chemotherapy treatment reduces the size of the tumor by 2.49 g. Express the size of the tumor as a linear function of the number of days spent in chemotherapy then determine how much the tumor weighs after 5 days of treatment. A) W(x) = -2.49x + 34; 21.55 g B) W(x) = 34x - 2.49; 21.55 g C) W(x) = -2.49x + 34; 12.45 g D) W(x) = 34x - 2.49; 12.45 g Answer: A Type: BI Var: 50+ Objective: Create Linear Functions to Model Data

10) The fixed and variable costs to produce an item are given along with the price at which an item is sold. Determine the break-even point. Fixed cost: $2,093 Variable cost per items: $34.50 Price at which the item is sold: $80.00 A) 26 items

B) 18 items

C) 61 items

D) 46 items

Answer: D Type: BI Var: 50+ Objective: Create Linear Functions to Model Data

11) A pediatrician records the age x (in yr) and average height y (in inches) for girls between the ages of 2 and 10. a. Use the points (2, 37) and (6, 50) to write a linear model for these data. b. Use the model to forecast the average height of 11-yr-old girls. y 70 60

Height (in.)

(6, 50)

40 (2, 37)

30 20 10 2

Age (yr)

A) a. y = 3x + 29.5; b. 62.5 in. C) a. y = 3.25x + 30.5; b. 66.25 in. Answer: C Type: BI Var: 50+ Objective: Create Linear Functions to Model Data

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B) a. y = 3x + 29.5; b. 63.25 in. D) a. y = 3.25x + 30.5; b. 67.25 in.

4 Create Models Using Linear Regression

Solve the problem. 1) The graph shows the number of organ transplants y in a certain country for the years 2005 to 2010 where x represents the number of years since 2005. y 200 190 180 170 160

(4, 161)

150 140 130 120 110

(1, 110)

100 90 1

The linear model y = 17.0x + 93 was created using the points (1, 110) and (4, 161). a. Interpret the meaning of the slope in the context of this problem. b. Interpret the meaning of the y-intercept in the context of this problem. A) a. m = 17.0 means that 17.0 more organ transplants were performed in 2009 than in 2006. b. The y-intercept is (0, 93) and means that health officials predicted that approximately 93 organ transplants would be performed in the year 2005. B) a. m = 17.0 means that 17.0 more organ transplants were performed in 2009 than in 2006. b. The y-intercept is (0, 96) and means that health officials predicted that approximately 96 organ transplants would be performed in the year 2005. C) a. m = 17.0 means that the number of organ transplants increased at an average rate of 17.0 per yr during this time period. b. The y-intercept is (0, 93) and means that approximately 93 organ transplants were performed in the year 2005. D) a. m = 17.0 means that the number of organ transplants increased at an average rate of 17.0 per yr during this time period. b. The y-intercept is (0, 96) and means that approximately 96 organ transplants were performed in the year 2005. Answer: C Type: BI Var: 50+ Objective: Create Models Using Linear Regression

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Compute the least-squares regression line for the given data set. 2) x 2 3 4 5 6 7 y 3.4 -1.7 -3 -4.3 -10.9 -13.9 A) y = 3.46x + 3.4 C) y = 3.46x + 9.77

B) y = 9.77x - 3.30 D) y = -3.30x + 9.77

Answer: D Type: BI Var: 50+ Objective: Create Models Using Linear Regression

Compute the least-squares regression line for predicting the ribeye price from the corn price. 3) One of the primary feeds for beef cattle is corn. The following table presents the average price in dollars for a bushel of corn and a pound of ribeye steak for 10 consecutive months. Corn Price ($/bu) 6.67 5.75 6.06 5.92 6.36 6.13

Ribeye Price ($/lb) 14.42 11.90 12.35 12.62 13.01 13.27

A) y = -2.22 + 0.41x C) y = 2.22 + 0.41x Answer: D Type: BI Var: 50+ Objective: Create Models Using Linear Regression

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B) y = 2.46 - 2.22x D) y = -2.22 + 2.46x

Compute the least-squares regression line for predicting the temperature from the chirp rate. 4) The common cricket can be used as a crude thermometer. The colder the temperature, the slower the rate of chirping. The table below shows the average chirp rate of a cricket at various temperatures. Chirp Rate (chirps/second) 3.8 2.3 2.2 1.1 1.5 1.2

Temperature (°F) 71.2 52 48.9 34.5 53.8 42.9

A) y ≈ 13.59 + 28.18x C) y ≈ 11.09 + 28.18x

B) y ≈ 28.18 + 13.59x D) y ≈ 28.18 + 11.09x

Answer: D Type: BI Var: 50+ Objective: Create Models Using Linear Regression

Solve the problem. 5) The data in the table shows the number of violent crimes in a certain city in the even years since 2000. Years Since Number of 2000 (x) Violent Crimes (y) 0 19 2 33 4 43 6 46 8 57 10 63 12 78 Use the data to find the least-squares regression line. Round the slope and y-intercept to 1 decimal place. Use the model to approximate the number of violent crimes in the city in 2005. A) y = 4.8x + 22.1; approximately 46.1 B) y = 4.5x + 21.5; approximately 44.0 C) y = 4.1x + 20.9; approximately 44.5 D) y = 4.4x + 22.3; approximately 44.3 Answer: B Type: BI Var: 50+ Objective: Create Models Using Linear Regression

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Use the scatter plot to determine if a linear regression model appears to be appropriate. 6) y

A) No Answer: B Type: BI Var: 24 Objective: Create Models Using Linear Regression

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B) Yes

Solve the problem. 7) The table below shows the wind speed y (in mph) of a hurricane versus the barometric pressure x (in mb). Barometric Pressure (mb) (x) 1,005 1,004 1,000 996 984 970 949 929 904

Wind Speed (mph) (y) 35 45 50 65 80 100 110 145 160

a. Use the data in the table to find the least-squares regression line. Round the slope to 2 decimal places and the y-intercept to the nearest whole unit. b. Use the model in part (a) to approximate the wind speed of a hurricane with a barometric pressu of 900 mb. A) a. y = -1.38x + 1,395; b. 155 mph C) a. y = -1.2x + 1,251; b. 155 mph

B) a. y = -1.38x + 1,395; b. 153 mph D) a. y = -1.2x + 1,251; b. 171 mph

Answer: D Type: BI Var: 50+ Objective: Create Models Using Linear Regression

2.6 Transformations of Graphs 0 Concept Connections

Provide the missing information. 1) A function defined by f (x) = mx + b is a rectangular coordinate system. Answer: linear

function and its graph is a line in a

Type: SA Var: 1 Objective: Concept Connections

2) Let c represent a positive real number. The graph of y = f (x) + c is the graph of y = f (x) shifted (up/down/left/right) c units. Answer: up Type: SA Var: 1 Objective: Concept Connections

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3) Let c represent a positive real number. The graph of y = f (x + c) is the graph of y = f (x) shifted (up/down/left/right) c units. Answer: left Type: SA Var: 1 Objective: Concept Connections

4) Let c represent a positive real number. The graph of y = f (x - c) is the graph of y = f (x) shifted (up/down/left/right) c units. Answer: right Type: SA Var: 1 Objective: Concept Connections

5) Let c represent a positive real number. The graph of y = f (x) - c is the graph of y = f (x) shifted (up/down/left/right) c units. Answer: down Type: SA Var: 1 Objective: Concept Connections

6) The graph of y = 3f (x) is the graph of y = f (x) with a (choose one: vertical stretch, vertical shrink, horizontal stretch, horizontal shrink). Answer: vertical stretch Type: SA Var: 1 Objective: Concept Connections

7) The graph of y = f (3x) is the graph of y = f (x) with a (choose one: vertical stretch, vertical shrink, horizontal stretch, horizontal shrink). Answer: horizontal shrink Type: SA Var: 1 Objective: Concept Connections

1 8) The graph of y = f ( x) is the graph of y = f (x) with a (choose one: vertical stretch, vertical shrink, 3 horizontal stretch, horizontal shrink). Answer: horizontal stretch Type: SA Var: 1 Objective: Concept Connections

9) The graph of y =

1 f (x) is the graph of y = f (x) with a (choose one: vertical stretch, vertical shrink, 3

horizontal stretch, horizontal shrink). Answer: vertical shrink Type: SA Var: 1 Objective: Concept Connections

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10) The graph of y = -f (x) is the graph of y = f (x) reflected across the

axis.

Answer: x Type: SA Var: 1 Objective: Concept Connections 1 Recognize Basic Functions

From memory match each equation with its graph. 1)

f(x) =

x g(x) = x2 I

h(x) = |x| II

III y

-3 -2 -1

-1

-3 -2 -1

-1

-3 -2 -1

-2

-3

A) f(x), III; g(x), II; h(x), I C) f(x), I; g(x), III; h(x), II Answer: D

-1

B) f(x), II; g(x), III; h(x), I D) f(x), III; g(x), I; h(x), II

Type: BI Var: 4 Objective: Recognize Basic Functions

g(x) =

f(x) = x

1 x

h(x) = x2

II y

-3 -2 -1 -1

-2

-3

A) f(x), III; g(x), I; h(x), II C) f(x), I; g(x), III; h(x), II Answer: C Type: BI Var: 4 Objective: Recognize Basic Functions

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III

B) f(x), III; g(x), II; h(x), I D) f(x), II; g(x), III; h(x), I

2 Apply Vertical and Horizontal Translations (Shifts)

Use translations to graph the given function. 1) f (x) = |x + 1| A)

5 y

-5 -4 -3 -2 -1 -1

-1

-2

-3

-4

-5

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: B Type: BI Var: 12 Objective: Apply Vertical and Horizontal Translations (Shifts)

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-5 -4 -3 -2 -1

Sketch the graph using transformations of a parent function (without a table of values). 2) r(x) = x + 2 A)

B) 7 y 6 5 4 3 2 1

-7 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 -7

7 y 6 5 4 3 2 1 1 2 3 4 5 6

1 2 3 4 5 6

D) 7 y 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 -7

7 y 6 5 4 3 2 1 1 2 3 4 5 6

Answer: D Type: BI Var: 16 Objective: Apply Vertical and Horizontal Translations (Shifts)

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-7 -6 -5 -4 -3 -2 --11 -2 -3 -4 -5 -6 -7

3) m(x) =

x A)

B) 7 y 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 -7

7 y 6 5 4 3 2 1 1 2 3 4 5 6

1 2 3 4 5 6

D) 7 y 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 -7

7 y 6 5 4 3 2 1 1 2 3 4 5 6

Answer: C Type: BI Var: 12 Objective: Apply Vertical and Horizontal Translations (Shifts)

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-7 -6 -5 -4 -3 -2 --11 -2 -3 -4 -5 -6 -7

4) a(x) = x2 - 3 A)

B) 7 y 6 5 4 3 2 1

-7 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 -7

7 y 6 5 4 3 2 1 1 2 3 4 5 6

1 2 3 4 5 6

D) 7 y 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 -7

7 y 6 5 4 3 2 1 1 2 3 4 5 6

Answer: B Type: BI Var: 12 Objective: Apply Vertical and Horizontal Translations (Shifts)

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-7 -6 -5 -4 -3 -2 --11 -2 -3 -4 -5 -6 -7

5) h(x) = A)

x-2 B) 7 y 6 5 4 3 2 1

-7 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 -7

7 y 6 5 4 3 2 1 1 2 3 4 5 6

1 2 3 4 5 6

D) 7 y 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 -7

7 y 6 5 4 3 2 1 1 2 3 4 5 6

Answer: C Type: BI Var: 16 Objective: Apply Vertical and Horizontal Translations (Shifts)

Page 85

-7 -6 -5 -4 -3 -2 --11 -2 -3 -4 -5 -6 -7

Use translations to graph the given function. 1 6) g(x) = +3 x-1 A)

B) 5 y

5 y

-5 -4 -3 -2 -1 -1

-1

-2

-3

-4

-5

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: A Type: BI Var: 24 Objective: Apply Vertical and Horizontal Translations (Shifts)

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-5 -4 -3 -2 -1

7) a(x) = x - 2 - 1 A)

B) 5 y

5 y

-5 -4 -3 -2 -1 -1

-1

-2

-3

-4

-5

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: D Type: BI Var: 24 Objective: Apply Vertical and Horizontal Translations (Shifts)

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-5 -4 -3 -2 -1

8) b(x) = |x - 2| + 3 A)

B) 5 y

5 y

-5 -4 -3 -2 -1 -1

-1

-2

-3

-4

-5

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: B Type: BI Var: 24 Objective: Apply Vertical and Horizontal Translations (Shifts)

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-5 -4 -3 -2 -1

3 Apply Vertical and Horizontal Shrinking and Stretching

Use transformations to graph the given function. 1) f(x) = 3x2 A)

5 y

-5 -4 -3 -2 -1 -1

-5 -4 -3 -2 -1

-1

-2

-3

-4

-5

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: C Type: BI Var: 8 Objective: Apply Vertical and Horizontal Shrinking and Stretching

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2) f (x) = 4x A)

B) 5 y

5 y

-5 -4 -3 -2 -1 -1

-5 -4 -3 -2 -1

-1

-2

-3

-4

-5

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: A Type: BI Var: 6 Objective: Apply Vertical and Horizontal Shrinking and Stretching

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3) q(x) = 5x A)

B) 5 y

5 y

-5 -4 -3 -2 -1 -1

-5 -4 -3 -2 -1

-1

-2

-3

-4

-5

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-1 -2

-3

-4

-5

Type: BI Var: 6 Objective: Apply Vertical and Horizontal Shrinking and Stretching

Solve the problem. 4) Use the graph of y = f (x) below to graph y = 5f (x) y 10 9 8 7 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1

-5 -4 -3 -2 -1

-2

Answer: A

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1 2 3 4 5 6 7x

B) y

10 9 8 7 6 5 4 3 2 1

10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7x

-6 -5 -4 -3 -2 -1

D) y

10 9 8 7 6 5 4 3 2 1

10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7x

-6 -5 -4 -3 -2 -1

Answer: A Type: BI Var: 16 Objective: Apply Vertical and Horizontal Shrinking and Stretching

1 5) Use the graph of y = b(x) below to graph y = b( x). 2 y 9 6 3

-9

-6

-3 -3 -6 -9

Page 92

1 2 3 4 5 6 7x

9 x

-6 -5 -4 -3 -2 -1

1 2 3 4 5 6 7x

B) y

-9

-6

-3

9 x

-9

-6

-3

-6

-9

9 x

D) y

-9

-6

-3

9 x

-9

-6

-3

-6

-9

Answer: B Type: BI Var: 4 Objective: Apply Vertical and Horizontal Shrinking and Stretching

Page 93

4 Apply Reflections Across the x- and y-Axes

Graph the function by applying an appropriate reflection. 1) f(x) = A)

x B) 5 y

5 y

-5 -4 -3 -2 -1 -1

-1

-2

-3

-4

-5

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: D Type: BI Var: 4 Objective: Apply Reflections Across the x- and y-Axes

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-5 -4 -3 -2 -1

Graph the equation by plotting points. 2) p(x) = -x3 A)

5 y

-5 -4 -3 -2 -1 -1

-1

-2

-3

-4

-5

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: B Type: BI Var: 3 Objective: Apply Reflections Across the x- and y-Axes

Page 95

-5 -4 -3 -2 -1

Use the graph of y = f(x) below to graph y = -f(x) 3) 5 y 4 3 2 1 -5

-4

-3

-2

-1

-1 -2 -3 -4 -5

B) 5 y

5 y

-5 -4 -3 -2 -1 -1

-1

-2

-3

-4

-5

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: C Type: BI Var: 12 Objective: Apply Reflections Across the x- and y-Axes

Page 96

-5 -4 -3 -2 -1

Use the graph of y = f (x) below to graph y = f (-x) 4) 5 y 4 3 2 1 -5

-4

-3

-2

-1

-1 -2 -3 -4 -5

B) 5 y

5 y

-5 -4 -3 -2 -1 -1

-1

-2

-3

-4

-5

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: A Type: BI Var: 12 Objective: Apply Reflections Across the x- and y-Axes

Page 97

-5 -4 -3 -2 -1

5) 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1-1

1 2 3 4 5 6 x

-2 -3 -4 -5 -6

B) 6 5 4 3 2 1

6 5 4 3 2 1 1 2 3 4 5 6 x

-6 -5 -4 -3 -2 -1-1

1 2 3 4 5 6 x

-6 -5 -4 -3 -2 -1-1

-2 -3 -4 -5 -6

D) 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1-1

6 5 4 3 2 1 1 2 3 4 5 6 x

-2 -3 -4 -5 -6

Answer: B Type: BI Var: 36 Objective: Apply Reflections Across the x- and y-Axes

Page 98

-6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6

1 2 3 4 5 6 x

5 Summarize Transformations of Graphs

Use transformations to graph the given function. 1) f(x) = -(x + 3)2 + 2 A)

7 y 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 -7

7 y 6 5 4 3 2 1 1 2 3 4 5 6

1 2 3 4 5 6

D) 7 y 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 -7

7 y 6 5 4 3 2 1 1 2 3 4 5 6

Answer: C Type: BI Var: 24 Objective: Summarize Transformations of Graphs

Page 99

-7 -6 -5 -4 -3 -2 --11 -2 -3 -4 -5 -6 -7

2) f(x) =

|x + 2| + 3

2 A)

B) 7 y 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 -7

7 y 6 5 4 3 2 1 1 2 3 4 5 6

1 2 3 4 5 6

D) 7 y 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 -7

7 y 6 5 4 3 2 1 1 2 3 4 5 6

Answer: A Type: BI Var: 50+ Objective: Summarize Transformations of Graphs

Page 100

-7 -6 -5 -4 -3 -2 --11 -2 -3 -4 -5 -6 -7

3) f(x) = -x - 4 A)

B) 5 y

5 y

-5 -4 -3 -2 -1 -1

-1

-2

-3

-4

-5

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: C Type: BI Var: 8 Objective: Summarize Transformations of Graphs

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-5 -4 -3 -2 -1

A function g is given. Identify the parent function. Then use the steps for graphing multiple transformations of functions to list, in order, the transformations applied to the parent function to obtain the graph of g. 4) g(x) =

4 x+5

-3 1

A) Parent function: f (x) =

; Shift the graph of f to the right 5 units, shrink the graph vertically

1 by a factor of , and shift the graph upward by 3 units. 4 1 B) Parent function: f (x) = ; Shift the graph of f to the left 5 units, stretch the graph vertically x by a factor of 4, and shift the graph downward by 3 units. 1 C) Parent function: f (x) = ; Shift the graph of f to the right 5 units, stretch the graph vertically x by a factor of 4, and shift the graph upward by 3 units. 1 D) Parent function: f (x) = ; Shift the graph of f to the left 5 units, shrink the graph vertically by x 1 a factor of , and shift the graph downward by 3 units. 4 Answer: B Type: BI Var: 50+ Objective: Summarize Transformations of Graphs

1 5) g(x) =

(x + 1.3)2 - 2.5

A) Parent function: f (x) = x2; Shift the graph of f to the left 1.3 units, strech the graph vertically by a factor of 5, and shift the graph downward by 2.5 units. B) Parent function: f (x) = x2; Shift the graph of f to the right 1.3 units, stretch the graph vertically by a factor of 5, and shift the graph upward by 2.5 units. C) Parent function: f (x) = x2; Shift the graph of f to the right 1.3 units, shrink the graph 1 vertically by a factor of , and shift the graph upward by 2.5 units. 5 D) Parent function: f (x) = x2; Shift the graph of f to the left 1.3 units, shrink the graph vertically 1 by a factor of , and shift the graph downward by 2.5 units. 5 Answer: D Type: BI Var: 50+ Objective: Summarize Transformations of Graphs

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6) g(x) =

1 5

x + 2.7 + 2.6

A) Parent function: f (x) = x; Shift the graph of f to the right 2.7 units, shrink the graph 1 vertically by a factor of , and shift the graph downward by 2.6 units. 5 B) Parent function: f (x) = x; Shift the graph of f to the left 2.7 units, strech the graph vertically by a factor of 5, and shift the graph upward by 2.6 units. C) Parent function: f (x) = x; Shift the graph of f to the left 2.7 units, shrink the graph vertically 1 by a factor of , and shift the graph upward by 2.6 units. 5 D) Parent function: f (x) = x; Shift the graph of f to the right 2.7 units, strech the graph vertically by a factor of 5, and shift the graph downward by 2.6 units. Answer: C Type: BI Var: 50+ Objective: Summarize Transformations of Graphs

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Use transformations to graph the given function. 7) p(x) = -3|x + 3| - 1 A)

7 y 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 -7

7 y 6 5 4 3 2 1 1 2 3 4 5 6

1 2 3 4 5 6

D) 7 y 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 -7

7 y 6 5 4 3 2 1 1 2 3 4 5 6

Answer: D Type: BI Var: 50+ Objective: Summarize Transformations of Graphs

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-7 -6 -5 -4 -3 -2 --11 -2 -3 -4 -5 -6 -7

6 Mixed Exercises

The graph of y = f (x) is given. Graph the indicated function. 1) Graph y = -f (x - 2) - 2 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1-1

1 2 3 4 5 6 x

-2 -3 -4 -5 -6

B) 6 5 4 3 2 1

6 5 4 3 2 1 1 2 3 4 5 6 x

-6 -5 -4 -3 -2 -1-1

1 2 3 4 5 6 x

-6 -5 -4 -3 -2 -1-1

-2 -3 -4 -5 -6

D) 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6

Answer: A Type: BI Var: 24 Objective: Mixed Exercises

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6 5 4 3 2 1 1 2 3 4 5 6 x

-6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6

1 2 3 4 5 6 x

2) Graph y = 2f (x + 1) + 4 y

6 5 4 3 2 1

1 2 3 4 5 6 x

-6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6

B) 6 5 4 3 2 1

6 5 4 3 2 1 1 2 3 4 5 6 x

-6 -5 -4 -3 -2 -1-1

1 2 3 4 5 6 x

-6 -5 -4 -3 -2 -1-1

-2 -3 -4 -5 -6

D) 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6

Answer: A Type: BI Var: 15 Objective: Mixed Exercises

Page 106

6 5 4 3 2 1 1 2 3 4 5 6 x

-6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6

1 2 3 4 5 6 x

3) Graph y = f (-x) - 4 y

6 5 4 3 2 1

1 2 3 4 5 6 x

-6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6

B) 6 5 4 3 2 1

6 5 4 3 2 1 1 2 3 4 5 6 x

-6 -5 -4 -3 -2 -1-1

1 2 3 4 5 6 x

-6 -5 -4 -3 -2 -1-1

-2 -3 -4 -5 -6

D) 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6

Answer: A Type: BI Var: 7 Objective: Mixed Exercises

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6 5 4 3 2 1 1 2 3 4 5 6 x

-6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6

1 2 3 4 5 6 x

Write a function based on the given parent function and the transformations in the given order. 4) Parent function: y = x3 1. Shift 6.8 units to the right. 2. Reflect across the y-axis. 3. Shift downward 5.2 units. A) y = (-x + 6.8)3 + 5.2 B) y = (-x - 6.8)3 - 5.2 C) y = - (x + 6.8)3 + 5.2 D) y = - (x - 6.8)3 - 5.2 Answer: B Type: BI Var: 50+ Objective: Mixed Exercises 3

5) Parent function y = x 1. Shift 9 units to the left. 2. Shift horizontally by a factor of 5. 3. Reflect across the x-axis. 1 A) y = - 3 x + 9 B) y = - 3 5x + 9 5

1 C) y = 3 - x - 9 5

D) y = -

5x - 9

Answer: A Type: BI Var: 50+ Objective: Mixed Exercises

6) Parent function y =

1 x

1. Stretch vertically by a factor of 6. 2. Reflect across the x-axis. 3. Shift downward 9 units. 6 6 B) y = - 9 A) y = + 9 x x

C) y = -

6 x

-9

D) y = -

x - 4.5|

D) y = |-5x - 4.5|

Answer: C Type: BI Var: 50+ Objective: Mixed Exercises

7) Parent function y = |x| 1. Shift 4.5 units to the right. 2. Shrink horizontally by a factor of

5 3. Reflect across the y-axis. 1 B) y = |-5x + 4.5| A) y = |- x + 4.5| 5 Answer: D Type: BI Var: 50+ Objective: Mixed Exercises

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C) y = |-

1 5

7 Expanding Your skills

Use transformations on the basic functions to write a rule y = f (x) that would produce the given graph. 1) 5 y 4 3 2 1 -5 -4 -3 -2 -1 -1

-2 -3 -4 -5

A) f (x) = -(x + 1)2 + 2 C) f (x) = -(x - 1)2 + 2

B) f (x) = (x - 1)2 - 2 D) f (x) = (x + 1)2 - 2

Answer: C Type: BI Var: 50+ Objective: Expanding Your skills

2) 5 y 4 3 2 1 -5 -4 -3 -2 -1 -1

-2 -3 -4 -5

A) f (x) = C) f (x) =

1 x-1 1

-2 -2

x+1 Answer: C Type: BI Var: 29 Objective: Expanding Your skills

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B) f (x) =

+2 x+1 1 D) f (x) = +2 x-1

3) 6 y 5 4 3 2 1 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5

1 2 3 4 5 6x

-6

A) T y = |x + 1| - 2

B) y = |x - 2| + 1

C) y = |x - 1| + 2

Answer: D Type: BI Var: 50+ Objective: Expanding Your skills

2.7 Analyzing Graphs of Functions and Piecewise-Defined Functions 0 Concept Connections

Provide the missing information. 1) A graph of an equation is symmetric with respect to the if replacing x by -x results in an equivalent equation.

-axis

Answer: y Type: SA Var: 1 Objective: Concept Connections

2) A graph of an equation is symmetric with respect to the if replacing y by -y results in an equivalent equation.

-axis

Answer: x Type: SA Var: 1 Objective: Concept Connections

3) A graph of an equation is symmetric with respect to the if replacing x by -x and y by -y results in an equivalent equation. Answer: origin Type: SA Var: 1 Objective: Concept Connections

4) An even function is symmetric with respect to the Answer: y -axis Type: SA Var: 1 Objective: Concept Connections

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D) y = |x + 2| - 1

5) An odd function is symmetric with respect to the

Answer: origin Type: SA Var: 1 Objective: Concept Connections

6) The expression

represents the greatest integer, less than or equal to x.

Answer: x or int(x) or floor(x) Type: SA Var: 1 Objective: Concept Connections 1 Test for Symmetry

Determine whether the graph of the equation is symmetric with respect to the x-axis, y-axis, origin, or none of these. 1) x = y4 - y6 A) origin B) y-axis C) x-axis D) none of these Answer: C Type: BI Var: 24 Objective: Test for Symmetry

2) x = y2 + 6 A) x-axis C) y-axis Answer: A

B) x-axis, y-axis, and origin D) none of these

Type: BI Var: 30 Objective: Test for Symmetry

3) |x| + |y| = 7.3 A) x-axis, y-axis, and origin C) x-axis

B) y-axis D) none of these

Answer: A Type: BI Var: 50+ Objective: Test for Symmetry

5 4) y = -

x+5

A) x-axis, y-axis, and origin C) x-axis Answer: D Type: BI Var: 50+ Objective: Test for Symmetry

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B) y-axis D) none of these

2 Identify Even and Odd Functions

Use the graph to determine if the function is even, odd, or neither. 1) y

-3

-2

-1

3 x

-1

A) even

B) odd

C) neither

Answer: A Type: BI Var: 6 Objective: Identify Even and Odd Functions

Find f(-x) and determine whether f is odd, even, or neither. 2) f (x) = 4x5 - 5x4 A) f (-x) = -4x5 + 5x4; f is odd. B) f (-x) = 4x5 + 5x4; f is neither odd nor even. C) f (-x) = -4x5 - 5x4; f is even. D) f (-x) = -4x5 - 5x4; f is neither odd nor even. Answer: D Type: BI Var: 50+ Objective: Identify Even and Odd Functions

Determine if the function is odd, even, or neither. 3) f (x) = 3x4 + 3|x3| - 3 A) neither B) even

C) odd

Answer: B Type: BI Var: 50+ Objective: Identify Even and Odd Functions

4) f (x) =

x3 7x 2

A) Odd Answer: A Type: BI Var: 50+ Objective: Identify Even and Odd Functions

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B) Neither

C) Even

5) f (x) =

x3 |x + 5|

A) Odd

B) Even

C) Neither

B) Even

C) Odd

Answer: C Type: BI Var: 50+ Objective: Identify Even and Odd Functions

6) f (x) =

- 3x 8x 2

A) Neither Answer: C Type: BI Var: 50+ Objective: Identify Even and Odd Functions 3 Graph Piecewise-Defined Functions

Evaluate the function for the given values of x. 1) -5x + 4, for x < -1 2 f(x) = x + 3, for -1 ≤ x < 2 1, for x ≥ 2 (a) f(-1); (b) f(3) A) (a) 9; (b) 12 B) (a) 4; (b) 1

C) (a) 4; (b) 12

D) (a) 9; (b) 1

C) -12

D) -5

Answer: B Type: BI Var: 50+ Objective: Graph Piecewise-Defined Functions

Evaluate the function for the indicated value. 2) Evaluate f (-12). 7 f (x) = x + 1 -5 A) 11

x ≤ -2 -2 < x ≤ 4 x>4 B) 7

Answer: B Type: BI Var: 50+ Objective: Graph Piecewise-Defined Functions

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3) Evaluate f (-1). 9 f (x) = x + 1 -5 A) -1

x ≤ -1 -1 < x ≤ 5 x>5 B) -5

C) 0

D) 9

C) 0

D) 7

C) -5

D) 7

C) 9

D) -2

Answer: D Type: BI Var: 50+ Objective: Graph Piecewise-Defined Functions

4) Evaluate f (-1). 7 f (x) = x + 1 -5 A) -5

x ≤ -2 -2 < x ≤ 2 x>2 B) -1

Answer: C Type: BI Var: 50+ Objective: Graph Piecewise-Defined Functions

5) Evaluate f (3). 7 f (x) = x + 1 -5 A) 3

x ≤ -1 -1 < x ≤ 3 x>3 B) 4

Answer: B Type: BI Var: 50+ Objective: Graph Piecewise-Defined Functions

6) Evaluate f (10). 9 f (x) = x + 1 -2 A) 11

x ≤ -1 -1 < x ≤ 2 x>2 B) 10

Answer: D Type: BI Var: 50+ Objective: Graph Piecewise-Defined Functions

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Match the function with the graph. 7) 5 y 4 3 2 1 -5

-4

-3

-2

-1

-1 -2 -3 -4 -5

A) f(x) = x + 2 for x ≤ 3 C) f(x) = x + 2 for x ≤ 1 Answer: C Type: MC Var: 46 Objective: Graph Piecewise-Defined Functions

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B) f(x) = x + 2 for x ≥ 3 D) f(x) = x + 2 for x ≥ 1

Graph the function. x - 1, t(x) = -x2, 8)

for x > 0 for x ≤ 0

B) 5 y

5 y

-5 -4 -3 -2 -1 -1

-1

-2

-3

-4

-5

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: D Type: BI Var: 14 Objective: Graph Piecewise-Defined Functions

Page 116

-5 -4 -3 -2 -1

-3 for -4 ≤ x < 0 r(x) = 2 for 0 ≤ x < 2 -2 for x ≥ 2 9) A)

5 y

-5 -4 -3 -2 -1 -1

-1

-2

-3

-4

-5

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: D Type: BI Var: 50+ Objective: Graph Piecewise-Defined Functions

Page 117

-5 -4 -3 -2 -1

10) s(x) =

-x + 3 for x ≤ 3 x - 3 for x > 3 B)

A) 5 y

5 y

-5 -4 -3 -2 -1 -1

-1

-2

-3

-4

-5

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: B Type: BI Var: 12 Objective: Graph Piecewise-Defined Functions

Page 118

-5 -4 -3 -2 -1

5 for -4 < x < -2 for -2 ≤ x < 3 11) m(x) = -x x - 3 for x ≥ 3 A)

B) 5 y

5 y

-5 -4 -3 -2 -1 -1

-5 -4 -3 -2 -1

-1

-2

-3

-4

-5

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: A Type: BI Var: 18 Objective: Graph Piecewise-Defined Functions

Evaluate the step function defined by f(x) = int(x) for the given value of x. 12) f (-1.9) A) 1.9 B) -2 C) -1.9 Answer: B Type: BI Var: 38 Objective: Graph Piecewise-Defined Functions

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D) -1

Graph the function. 13) f(x) = int(x + 3) A)

-4

-3

-2

B) 4 y

4 y

1 1

-1

-3

-2

-1 -1

-2

-3

-4

-3

-2

4 y

1 1

-1

-4

-3

-2

-1

-2

-3

-4

Answer: D Type: BI Var: 8 Objective: Graph Piecewise-Defined Functions

Page 120

-4

-1

Write a piecewise-defined function to model the monthly cost C(x) (in $) as a function of the number of minutes used x for the month. 14) A cell phone plan charges $45.75 per month, plus $9.55 in taxes, plus $0.35 per minute for calls beyond the 500-min monthly limit. 55.3, for x ≤ 500 C(x) = A) 55.3 + 0.35(x - 500), for x > 500 B) C) D)

C(x) = 55.3, 0.35(x - 500),

for x ≤ 500 for x > 500

C(x) =

55.3, 0.35x,

for x ≤ 500 for x > 500

55.3, 55.3 + 0.35x,

for x ≤ 500 for x > 500

C(x) =

Answer: A Type: BI Var: 50+ Objective: Graph Piecewise-Defined Functions 4 Investigate Increasing, Decreasing, and Constant Behavior of a Function

Solve the problem. 1) Use interval notation to write the intervals over which f is constant 5 y 4 3 2 1 -5 -4 -3 -2 -1 -1

-2 -3 -4 -5

A) (2, 5)

B) (-4, -1)

C) (2, 2)

Answer: D Type: BI Var: 50+ Objective: Investigate Increasing, Decreasing, and Constant Behavior of a Function

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D) (-1, 2)

Use interval notation to write the intervals over which f is (a) increasing, (b) decreasing, and (c) constant. 2) y 5 4 3 2 1 -5 -4 -3 -2 -1 -1

5 x

-2 -3 -4 -5

A) a. (-∞, 2) ∪ (2, ∞) b. never decreasing c. (-2, 2) C) a. (-4, ∞) b. (-∞, -4) c. never constant

B) a. (-∞, -2) ∪ (2, ∞) b. never decreasing c. (-2, 2) D) a. never increasing b. (-∞, -2) ∪ (2, ∞) c. (-2, 2)

Answer: B Type: BI Var: 10 Objective: Investigate Increasing, Decreasing, and Constant Behavior of a Function

Page 122

3) 5 y 4 3 2 1 -5 -4 -3 -2 -1

-1

-2 -3 -4 -5

A) a. (-∞, -2) b. (-2, ∞) c. never constant C) a. (-∞, 2) b. (2, ∞) c. x = 2

B) a. (-∞, 2) b. (2, ∞) c. never constant D) a. (-∞, -1) b. (-1, ∞) c. never constant

Answer: B Type: BI Var: 20 Objective: Investigate Increasing, Decreasing, and Constant Behavior of a Function

4) y 5 4 3 2 1 -5 -4 -3 -2 -1 -1

5 x

-2 -3 -4 -5

A) a. (-∞, -2) ∪ (2, ∞) b. never decreasing c. (-2, 2) C) a. (5, ∞) b. (-∞, 5) c. never constant

B) a. never increasing b. (-∞, -2) ∪ (2, ∞) c. (-2, 2) D) a. (-∞, -3) ∪ (-3, ∞) b. never decreasing c. (-2, 2)

Answer: B Type: BI Var: 10 Objective: Investigate Increasing, Decreasing, and Constant Behavior of a Function

Page 123

5 Determine Relative Minima and Maxima of a Function

Identify the location and value of any relative maxima or minima of the function. 1) 5 y 4 3 2 1 -5 -4 -3 -2 -1 -1

-2 -3 -4 -5

A) At x = 2, the function has a relative minimum of 0. At x = 4, the function has a relative maximum of 2. B) At x = 1, the function has a relative maximum of -2. At x = 0, the function has a relative minimum of 2. At x = 2, the function has a relative maximum of 4. C) At x = 0, the function has a relative minimum of 2. At x = 2, the function has a relative maximum of 4. D) At x = -2, the function has a relative maximum of 1. At x = 2 the function has a relative minimum of 0. At x = 4 the function has a relative maximum of 2. Answer: D Type: BI Var: 24 Objective: Determine Relative Minima and Maxima of a Function

Page 124

2) 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1-1

1 2 3 4 5 6 x

-2 -3 -4 -5 -6

A) At x = -3, the function has a relative minimum of -5. At x = 3, the function has a relative minimum of -5. B) At x = -3, the function has a relative minimum of -5. At x = 0, the function has a relative maximum of 0. At x = 3, the function has a relative minimum of -5. C) At x = -4.2, the function has a relative minimum of 0. At x = 0, the function has a relative maximum of 0. At x = 4.2, the function has a relative minimum of 0. D) At x = -3, the function has a relative minimum of 0. At x = 0, the function has a relative maximum of 0. At x = 3, the function has a relative minimum of 0. Answer: B Type: BI Var: 10 Objective: Determine Relative Minima and Maxima of a Function

Page 125

Solve the problem. 3) The graph shows the height h (in meters) of a roller coaster t seconds after the ride starts.

a. Over what interval(s) does the height increase? b. Over what interval(s) does the height decrease? A) a. (0, 10) and (30, 50) b. (10, 20) and (50, 70) C) a. (10, 50) b. (50, 70) Answer: A Type: BI Var: 4 Objective: Determine Relative Minima and Maxima of a Function

Page 126

B) a. (50, 70) b. (10, 50) D) a. (10, 20) and (30, 40) b. (50, 70)

6 Mixed Exercises

Produce a rule for the function whose graph is shown. 1) y 5 4 3 2 1 -5 -4 -3 -2 -1

-1

5 x

-2 -3 -4 -5

3 3 1 C) g(x) = 3

A) g(x) =

for x ≥ 1 for x < 3 for x < 3 for x ≥ 3

1 3 1 D) g(x) = 3 B) g(x) =

for x ≤ 3 for x > 3 for x ≥ 3 for x < 3

Answer: D Type: BI Var: 50+ Objective: Mixed Exercises

2) 6 5 4

3 2 1 1 2

-6 -5 -4 -3 -2 -1 -1 -2

5 6 x

-3 -4 -5 -6

A) m(x) =

x2 - 3 for x < 1 for x ≥ 1

-2

C) m(x) =

(x + 3) -2

Answer: A Type: BI Var: 10 Objective: Mixed Exercises

Page 127

for x < 1 for x ≥ 1

B) m(x) =

x2 + 3 for x < 1 -2 for x ≥ 1

D) m(x) =

x2 - 3 for x > 1 -2 for x ≤ 1

3) A math tutor makes $15.00 per hour in the tutoring lab at her school. During final exam week, she earns overtime at $22.50 per hour for the work exceeding her normal 40-hr work week. Which graph best depicts her total salary for the week as a function of the number of hours worked? A) B) $

$ 1200

1200

1000

800

600

400

200

60 time

D) $

1200

1000

800

600

400

200

200 10

Answer: A Type: BI Var: 8 Objective: Mixed Exercises

Use the given information to a. Graph the function. b. Write the domain in interval notation. c. Write the range in interval notation. -x2 - 2 for x ≤ 1 1 x for x > 1 4) f (x) = 2

Page 128

60 time

A) domain: (-∞, ∞) range: (-∞, ∞) 6

B) domain: (-∞, ∞) 1 range: (-∞, ) 2

2 -6

-4

-2

6 x

-2

-6

-4

-2

-4

-2

-6

-4

6 x

-6

C) domain: (-∞, ∞)

D) domain: (-∞, ∞) 1

1 range: (-∞, -2) ∪ ( , ∞) 2

range: (-∞, -2) ∪ ( , ∞) 2 6

-6

-4

-2

6 x

-6

-4

-2

-4

-6

Type: BI Var: 40 Objective: Mixed Exercises

-2

Answer: D

Page 129

|x| for x < 4 -x for x ≥ 4 A) domain: (-∞, ∞) range: (-4, ∞)

5) f (x) =

-6

-4

B) domain: (-∞, ∞) range: (-∞, -4] ∪ [0, ∞) y

-2

6 x

-2 -2

-4

-6

6 x

-2

6 x

-6

-4

6 x

-2

-4

-6

Answer: B Type: BI Var: 4 Objective: Mixed Exercises

2.8 Algebra of Functions and Function Composition 0 Concept Connections

Provide the missing information. 1) The function f + g is defined by (f + g)(x) = Answer: f (x) + g(x) Type: SA Var: 1 Objective: Concept Connections

Page 130

D) domain: (-∞, ∞) range: (-∞, -4) ∪ (0, ∞) 6

-4

-2

C) domain: (-∞, ∞) range: [-4, ∞)

-6

f 2) The function Answer:

f (x)

(x) is defined by

provided that

= 0.

g ; g(x)

g(x) Type: SA Var: 1 Objective: Concept Connections

3) Let h represent a positive real number. Given a function defined by y = f (x), the difference quotient is given by . Answer:

f (x + h) - f (x) h

Type: SA Var: 1 Objective: Concept Connections

4) The composition of f and g, denoted by (f ∘ g)(x) is defined by = Answer: f (g(x)) Type: SA Var: 1 Objective: Concept Connections

Page 131

1 Perform Operations on Functions

Find (f + g)(x) and identify the graph of f + g. 1) f (x) = x2 and g(x) = 1 A) (f + g)(x) = x2 + 1 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

C) (f + g)(x) = x2 + 1

5 y

-1

D) (f + g)(x) = (x + 1)2

5 y

-5 -4 -3 -2 -1

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: A Type: BI Var: 6 Objective: Perform Operations on Functions

Page 132

B) (f + g)(x) = (x + 1)2

2) f (x) = |x| and g(x) = -3 A) (f + g)(x) = |x - 3|

B) (f + g)(x) = |x| - 3

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

C) (f + g)(x) = |x| - 3

D) (f + g)(x) = |x - 3|

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: B Type: BI Var: 6 Objective: Perform Operations on Functions

Evaluate the function for the given value of x. 3) f (x) = -3x, g(x) = |x - 6|, (f · g)(-4) = ? A) (f · g)(-4) = -30 C) (f · g)(-4) = 120

B) (f · g)(-4) = -120 D) (f · g)(-4) = 6

Answer: C Type: BI Var: 50+ Objective: Perform Operations on Functions

4) f (x) = -2x, g(x) =

, (f - g)(1) = ? x+2 5 7 A) (f - g)(1) = B) (f - g)(1) = 3 3

Answer: D Type: BI Var: 50+ Objective: Perform Operations on Functions

Page 133

C) (f - g)(1) =

5 3

D) (f - g)(1) = -

7 3

f 5) f (x) = -2x, g(x) = |x + 2|, 3 f A) g (-5) = 10

(-5) = ?

g 10 f B) g (-5) = 3

10 f C) g (-5) = 3

3 f D) g (-5) = 10

Answer: B Type: BI Var: 50+ Objective: Perform Operations on Functions

6) f (x) = 8x, g(x) = -5x2 - 5, (g + f )(x) = ? A) (g + f )(x) = -5x2 + 8x - 5 C) (g + f )(x) = -5x2 - 8x - 5

B) (g + f )(x) = -5x2 - 8x + 5 D) (g + f )(x) = -5x2 + 8x + 5

Answer: A Type: BI Var: 50+ Objective: Perform Operations on Functions

7) p(x) = x2 + 7x, q(x) = x + 3, (p · q)(x) = ? A) x + 3 + 7 x + 3 C) (p · q)(x) = (x2 + 7x) x + 3

B) (p · q)(x) = x2 + 7x ·

D) (p · q)(x) = (x2 + 7x)(x + 3)

Answer: C Type: BI Var: 50+ Objective: Perform Operations on Functions

8) r (x) = 3x, p (x) = x 2+ 6x, (p - r)(x) A) (p - r)(x) = x 2 + 18x C) (p - r)(x) = x 2 - 3x Answer: D Type: BI Var: 50+ Objective: Perform Operations on Functions

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x+3

B) (p - r)(x) = x 2 + 9x D) (p - r)(x) = x 2 + 3x

Find the indicated function and write its domain in interval notation. q 9) p(x) = x2 + 6x, q(x) = 4 - x, (x) = ? p 4-x q A) (x) = ; (-∞, -6) ∪ (-6, 0) ∪ (0, 4) p x2 + 6x 4-x q B) (x) = ; (-∞, -6) ∪ (-6, 0) ∪ (0, ∞) p x2 + 6x 4-x q C) (x) = ; (-∞, -6) (-6, 0) ∪ (0, ∞) 2 p x + 6x 4-x q D) (x) = ; (-∞, -6) ∪ (-6, 0) ∪ (0, 4] 2 p x + 6x Answer: D Type: BI Var: 14 Objective: Perform Operations on Functions

x-5 10) s(x) =

, t(x) =

x-7 , (s · t)(x) = ? 5-x

x2 - 49 1 A) ; (-∞, 7) ∪ (7, ∞) 7-x 1 C) ; (-∞, -7) ∪ (-7, 5) ∪ (5, 7)∪ (7, ∞) x+7

Answer: C Type: BI Var: 15 Objective: Perform Operations on Functions

11) s(x) = x - 4 , t(x) = x - 6 s , ( )(x) = ? x2 - 36 4-x t 1 A) ; (-∞, -6) ∪ (-6, 4) ∪ (4, 6)∪ (6, ∞) x+6 (x - 4)2 B) ; (-∞, -6) ∪ (-6, 4) ∪ (4, 6)∪ (6, ∞) (x - 6)2(x + 6) 1 C) ; (-∞, -6) ∪ (-6, ∞) x+6 (x - 4)2 D) ; (-∞, -6) ∪ (6, ∞) (x - 6)2(x + 6) Answer: B Type: BI Var: 15 Objective: Perform Operations on Functions

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; (-∞, -7) ∪ (-7, 5) ∪ (5, 7)∪ (7, ∞) 7-x 1 D) ; (-∞, -7) ∪ (7, ∞) x+7 B)

x-2

x-3 , (s + t)(x) = ? x2 - 9 2-x x3 - 2x2 - 13x + 31 A) ; (-∞, -3) ∪ (-3, 2) ∪ (2, 3)∪ (3, ∞) (x + 3)(x - 3)(x - 2)

12) s(x) =

, t(x) =

x3 - 4x2 - 5x + 23 B) ; (-∞, -3) ∪ (-3, 2) ∪ (2, 3)∪ (3, ∞) (x + 3)(x - 3)(x - 2) C) D)

x3 - 4x2 - 5x + 23 ; (-∞, -3) ∪ (3, ∞) (x + 3)(x - 3)(x - 2)

x3 - 2x2 - 13x + 31 (x + 3)(x - 3)(x - 2)

; (-∞, -3) ∪ (3, ∞)

Answer: B Type: BI Var: 15 Objective: Perform Operations on Functions

x-5

x-6 , (s + t)(x) = ? x - 36 5-x x3 - 7x2 - 26x + 191 A) ; (-∞, -6) ∪ (-6, 5) ∪ (5, 6)∪ (6, ∞) (x + 6)(x - 6)(x - 5)

13) s(x) =

, t(x) =

x3 - 5x2 - 46x + 241 (x + 6)(x - 6)(x - 5)

; (-∞, -6) ∪ (6, ∞)

x3 - 7x2 - 26x + 191 C) ; (-∞, -6) ∪ (6, ∞) (x + 6)(x - 6)(x - 5) D)

x3 - 5x2 - 46x + 241 (x + 6)(x - 6)(x - 5)

; (-∞, -6) ∪ (-6, 5) ∪ (5, 6)∪ (6, ∞)

Answer: D Type: BI Var: 15 Objective: Perform Operations on Functions

Page 136

x-2 14) s(x) =

, t(x) = x + 8, (s · t)(x) = ? x2 - 64 (x - 2) A) ; (-8, 2) ∪ (2, 8)∪ (8, ∞) (x + 8)(x - 8) x + 8 B)

(x - 2) x + 8 ; (-8, 2) ∪ (2, 8)∪ (8, ∞) (x + 8)(x - 8)

(x - 2) x + 8 ; (-8, 8) ∪ (8, ∞) (x + 8)(x - 8) (x - 2) ; (-8, 8) ∪ (8, ∞) D) (x + 8)(x - 8) x + 8 C)

Answer: C Type: BI Var: 28 Objective: Perform Operations on Functions 2 Evaluate a Difference Quotient

Find

f(x + h) - f(x)

for the given function.

h 1) f (x) = x2 + 8x. A) 2x + h + 8

B) 2x + 8

C) 2xh + h2 + 8

D) 1

C) -3h - 12

D) -3h

Answer: A Type: BI Var: 14 Objective: Evaluate a Difference Quotient

Find the difference quotient and simplify. 2) f (x) = -3x - 6 A) -3 B) -6 Answer: A Type: BI Var: 50+ Objective: Evaluate a Difference Quotient

Find

f(x + h) - f(x)

for the given function.

h 3) f (x) =

1 x-4

A) C)

1 (x - 4)(x + h - 4)

1 2x + h - 8

Answer: A Type: BI Var: 16 Objective: Evaluate a Difference Quotient

Page 137

1 (x - 4)(x + h - 4) 1 D) 2x + h - 8

4) f (x) = 5x2 - 6x + 4 A) 5h2 + 10xh - 6h

B) 10x - 6

C) -2x + 5h + 2

D) 10x + 5h - 6

Answer: D Type: BI Var: 50+ Objective: Evaluate a Difference Quotient

5) f (x) = x3 - 11 A) 3x2 + 3x + h2 C) 3x2 + 3xh + h2 - 22

B) 3x2 + 3xh + h2 D) 3x2 + 3x + h2 - 22

Answer: B Type: BI Var: 24 Objective: Evaluate a Difference Quotient 3 Compose and Decompose Functions

Evaluate the function for the given value of x. 1) f (x) = x2 + 3x, g(x) = 5x + 2, (f ∘ g)(3) = ? A) ( f ∘ g)(3) = 272 C) ( f ∘ g)(3) = 340 Answer: C

B) ( f ∘ g)(3) = 306 D) ( f ∘ g)(3) = 92

Type: BI Var: 50+ Objective: Compose and Decompose Functions

2) f (x) = x2 + 3x, g(x) = 5x - 4, (g ∘ f )(-3) = ? A) (g ∘ f )(-3) = -0 C) (g ∘ f )(-3) = -4 Answer: C

B) (g ∘ f )(-3) = 0 D) (g ∘ f )(-3) = 304

Type: BI Var: 50+ Objective: Compose and Decompose Functions

3) g(x) = 3x, h(x) = x3 + 5x, (g ∘ h)(2) = ? A) 3 2 B) 3 6

C) 18 6

D) 5 3

C) 12 6 - 3

D) Undefined

C) 45

D) 91,395

Answer: B Type: BI Var: 50+ Objective: Compose and Decompose Functions

4) g(x) = 4x - 3, h(x) = 3 x - 7, (h ∘ g)(1) = ? A) 24 3 - 3

B) 3 6

Answer: D Type: BI Var: 50+ Objective: Compose and Decompose Functions

5) f (x) = x3 + 6x, ( f ∘ f )(3) = ? A) 90 B) 2,025 Answer: D Type: BI Var: 50+ Objective: Compose and Decompose Functions

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6) f (x) = x3 - 4x, g(x) = 5x , h(x) = 4x + 5, ( f ∘ h ∘ g)(5) = ? A) 15,525 B) 135 C) 13,125

D) 425

Answer: A Type: BI Var: 50+ Objective: Compose and Decompose Functions

Find the indicated function and write its domain in interval notation. 7) m(x) = x + 2, n(x) = x - 3, (m ∘ n)(x) = ? A) (m ∘ n)(x) = (x - 3) x + 2; domain: [-2, ∞) B) (m ∘ n)(x) = x2 - 6; domain: [-6, ∞) C) (m ∘ n)(x) = x - 1; domain: [1, ∞) D) (m ∘ n)(x) = x + 2 - 3; domain: [-2, ∞) Answer: C Type: BI Var: 48 Objective: Compose and Decompose Functions

, (q ∘ n)(x) = ? x+6 1 A) (q ∘ n)(x) = + 3; domain: (-∞, -6) ∪ (-6, ∞) x+6 1 B) (q ∘ n)(x) = ; domain: (-∞, 9) ∪ (9, ∞) x+9 1 C) (q ∘ n)(x) = ; domain: (-∞, -9) ∪ (-9, ∞) x+9 1 D) (q ∘ n)(x) = + 3; domain: (-∞, 6) ∪ (6, ∞) x+6

8) n(x) = x + 3, q(x) =

Answer: C Type: BI Var: 50+ Objective: Compose and Decompose Functions

9) r(x) = |-3x - 7|, n(x) = x + 4, (r ∘ n)(x) = ? A) (r ∘ n)(x) = |-3x - 7| + 4; domain: (-∞, ∞) C) (r ∘ n)(x) = |-3x - 19|; domain: (-∞, ∞) Answer: C Type: BI Var: 50+ Objective: Compose and Decompose Functions

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7 B) (r ∘ n)(x) = |-3x - 7| + 4; domain: (-∞, - ) 3 19 D) (r ∘ n)(x) = |-3x - 19|; domain: (-∞, - ) 3

10) n(x) = x - 5, p(x) = x2 + 6x, (p ∘ n)(x) = ? A) (p ∘ n)(x) = x2 + 6x - 5; domain (-∞, ∞) B) (p ∘ n)(x) = x2 - 4x - 5; domain (-∞, ∞) C) (p ∘ n)(x) = x2 - 4x - 5; domain (-∞, -6) ∪ (-6, ∞) D) (p ∘ n)(x) = x2 + 6x - 5; domain [0, ∞) Answer: B Type: BI Var: 50+ Objective: Compose and Decompose Functions

, g(x) = 4 - x, ( f ∘ g)(x) = ? x2 - 11 9 4-x A) 2 ; (-∞, 11) ∪ ( 11, ∞) x - 11 9 C) ; (-∞, -7) ∪ (-7, ∞) x+7

11) f (x) =

9 4-x ; (-∞, 11) ∪ ( 11, 4] x2 - 11 9 D) ; (-∞, -7) ∪ (-7, 4] x+7 B)

Answer: D Type: BI Var: 50+ Objective: Compose and Decompose Functions

12) f (x) =

, g(x) =

x-1 36 A)

100 - x2 36

36 2

, ( f ∘ g)(x) = ?

x - 64 ; (-∞, -10) ∪ (-10, -8) ∪ (-8, 8) ∪ (8, 10) ∪ (10, ∞)

; (-∞, -8) ∪ (-8, 8) ∪ (8, ∞) x2 + 100 36 C) ; (-∞, -10) ∪ (-10, 10) ∪ (10, ∞) 100 - x2 36 D) 2 ; (-∞, ∞) x + 100 B)

Answer: A Type: BI Var: 4 Objective: Compose and Decompose Functions

Page 140

Solve the problem. 13) The cost to buy tickets online for a dance show is $60 per ticket, that is, the cost function is C(x) = 60x for x tickets to the show. There is a sales tax of 4.5% and a processing fee of $7 for a group of tickets, that is, T(x) = 1.045x + 7 is the total cost for x dollars spent on tickets. (a) Find (T ∘ C)(x) (b) Find (T ∘ C)(7) and interpret its meaning in the context of the problem. A) (a) (T ∘ C)(x) = 62.7x + 420; (b) (T ∘ C)(7) = 858.9; The average cost per ticket. B) (a) (T ∘ C)(x) = 62.7x + 7; (b) (T ∘ C)(7) = 445.9; The average cost per ticket. C) (a) (T ∘ C)(x) = 62.7x + 420; (b) (T ∘ C)(7) = 858.9; The total cost to purchase 7 tickets. D) (a) (T ∘ C)(x) = 62.7x + 7; (b) (T ∘ C)(7) = 445.9; The total cost to purchase 7 tickets. Answer: D Type: BI Var: 50+ Objective: Compose and Decompose Functions

Find two functions f and g such that h(x) = (f ∘ g)(x). 14) h(x) = (x + 5)6 A) f (x) = x + 5 and g(x) = x6 C) f (x) = x and g(x) = x6 + 5

B) f (x) = (x + 5) and g(x) = (x + 5)5 D) f (x) = x6 and g(x) = x + 5

Answer: D Type: MC Var: 50+ Objective: Compose and Decompose Functions

15) h(x) =

8x - 3

A) f (x) = 8x - 3 and g(x) = C) f(x) = x - 3 and g(x) =

B) f (x) =

D) f (x) =

5 5

x and g(x) = 8x - 3 8x and g(x) = x - 3

Answer: B Type: BI Var: 50+ Objective: Compose and Decompose Functions

16) h(x) =

6 x-6

A) f (x) = 6 and g(x) =

1 x-6

C) f (x) =

and g(x) = x - 6

x Answer: C Type: MC Var: 50+ Objective: Compose and Decompose Functions

Page 141

B) f (x) = -6 and g(x) = D) f (x) =

1 x-6

6 x

and g(x) = 6

4 Mixed Exercises

The graphs of f and g are shown. Find the values for the given values of x, if possible. g 1) a. ( f + g)(-2); b. (1); c. (g ∘ f )(3) f y = f(x)

5 y 4 3 2 1

-5 -4 -3 -2 -1 -1

-2 y = g(x)

-3 -4 -5

3 A) a. 7; b. - ; c. -2 2

2 B) a. 1; b. - ; c. -2 3

2 C) a. 1; b. - ; c. 3 3

3 D) a. 7; b. - ; c. 3 2

Answer: B Type: BI Var: 50+ Objective: Mixed Exercises

Page 142

2) a. (h · k)(0); b. 5

k (-4); h

c. (k - h)(1)

4 3 2 1 -5 -4 -3 -2 -1 -1

y = h(x) 1

5 x

-2 y = k(x)

-3 -4 -5

A) a. undefined; b.

2 ; c. -2 3

C) a. -4; b. undefined; c. 2

B) a. -4; b. undefined; c. -2 2 D) a. -4; b. ; c. -2 3

Answer: B Type: BI Var: 50+ Objective: Mixed Exercises

Refer to the values of k(x) and p(x) in the table, and evaluate the function for the given value of x. 3) x k(x) p(x) -3 -6 -7 -2 6 1 1 2 4 6 -2 5 (p ∘ k)(-2) A) 4 Answer: B Type: BI Var: 50+ Objective: Mixed Exercises

Page 143

B) 5

C) -3

D) 2

Solve the problem. 4) A chess master creates instructional videos about various opening strategies. She sells the videos in 3-hr packages for $35 each. Her one-time initial cost to produce each 3-hr video package is $4,600 (this includes labor and the cost of computer supplies). The cost to package and ship each DVD is $2.95. a. Write a linear cost function that represents the cost C(x) to produce, package, and ship x 3-hr video packages. b. Write a linear revenue function to represent the revenue R(x) for selling x 3-hr video packages. c. Evaluate (R - C)(x) and interpret its meaning in the context of this problem. A) a. C(x) = 8.85x + 4,600; B) a. C(x) = 8.85x + 4,600; b. R(x) = 35x b. R(x) = 35x c. (R - C)(x) = 4,600 - 26.15x represents c. (R - C)(x) = 26.15x - 4,600 represents the profit for selling x DVDs. the profit for selling x DVDs. C) a. C(x) = 35x D) a. C(x) = 2.95x + 4,600; b. R(x) = 2.95x + 4,600; b. R(x) = 35x c. (R - C)(x) = 32.05x - 4,600 represents c. (R - C)(x) = 32.05x - 4,600 represents the profit for selling x DVDs. the profit for selling x DVDs. Answer: D Type: BI Var: 50+ Objective: Mixed Exercises

5) f (x) = {(-6, 6), (-2, -3), (5, -8), (1, 9)}, g(x) = (2, 5), (1, 8), (9, -6), (-2, -3)} Find ( f ∘ g)(1) A) undefined B) 6 C) -3

D) 9

Answer: A Type: BI Var: 50+ Objective: Mixed Exercises 5 Expanding Your skills

Solve the problem. 1) A car accelerates from 0 to 91.2 ft/sec in 8 sec. The distance d(t) (in ft) that the car travels t seconds after motion begins is given by d(t) = 5.7t2, where 0 ≤ t ≤ 8. d(t + h) - d(t) a. Find the difference quotient . h b. Use the difference quotient to determine the average rate of speed on the interval 4 ≤ t ≤ 6. A) a. 5.7(t + h); B) a. 11.4t + 5.7h; C) a. 11.4t + 5.7h; D) a. 5.7(t + h); b. 34.2 ft/sec b. 57 ft/sec b. 79.8 ft/sec b. 79.8 ft/sec Answer: B Type: BI Var: 50+ Objective: Expanding Your skills

Page 144

2) If a is b plus nine, and c is the square of a, write c as a function of b. A) c(b) = b + 81 B) c(b) = 2b + 18 C) c(b) = (b + 9)2

D) c(b) = b2 + 81

Answer: C Type: BI Var: 18 Objective: Expanding Your skills

3) If q is r minus six, and s is the square root of q, write s as a function of r. s(r) = r - 6 B) s(r) = 6 - r C) s(r) = r - 6 A) Answer: A

D) s(r) = 6 - r

Type: BI Var: 9 Objective: Expanding Your skills

4) If x is twice y, and z is two less than x, write z as a function of y. A) z(y) = 2y - 2 B) z(y) = 2(2 - y) C) z(y) = 2 - 2y

D) z(y) = 2(y - 2)

Answer: A Type: BI Var: 9 Objective: Expanding Your skills

5) If m is one-sixth of n, and p is five less than m, write p as a function of n. 1 1 1 A) p(n) = 5n B) p(n) = (n - 5) C) p(n) = n - 5 6 6 6

D) p(n) = n -

Answer: C Type: BI Var: 48 Objective: Expanding Your skills

6) Given f (x) = |-5x3 + 8|, define m, n, h, and k such that f (x) = (m ∘ n ∘ h ∘ k)(x). A) m(x) = |x|, n(x) = x + 8, h(x) = x3, k(x) = -5x B) m(x) = x3, n(x) = -5x, h(x) = x + 8, k(x) = |x| C) m(x) = |x|, n(x) = x + 8, h(x) = -5x, k(x) = x3 D) m(x) = x3, n(x) = x + 8, h(x) = -5x, k(x) = |x| Answer: C Type: BI Var: 50+ Objective: Expanding Your skills

Page 145

5 6

Chapter 3 Polynomial and Rational Functions 3.1 Quadratic Functions and Applications 0 Concept Connections

Provide the missing information. 1) A function defined byf (x) = ax2 + bx + c (a ≠ 0) is called a

function.

Answer: quadratic Type: SA Var: 1 Objective: Concept Connections

2) The vertical line drawn through the vertex of a quadratic function is called the symmetry.

Answer: axis Type: SA Var: 1 Objective: Concept Connections

3) Given f (x) = a(x - h)2 + k (a ≠ 0), the vertex of the parabola is the point

Answer: (h, k) Type: SA Var: 1 Objective: Concept Connections

4) Given f (x) = a(x - h)2 + k, if a < 0 , the parabola opened (upward/downward) and the (minimum/maximum) value is . Answer: upward; maximum; k Type: SA Var: 1 Objective: Concept Connections

5) Given f (x) = a(x - h)2 + k, if a > 0 , the parabola opened (upward/downward) and the (minimum/maximum) value is . Answer: downward; minimum;k Type: SA Var: 1 Objective: Concept Connections

6) The graph of f (x) = a(x - h)2 + k, a ≠ 0, is a parabola and the axis of symmetry is a line given by . Answer: x = h Type: SA Var: 1 Objective: Concept Connections

Page 1

1 Graph a Quadratic Function in Vertex Form

Determine whether the graph of the parabola opens upward or downward and determine the range. 1) f (x) = -3(x - 2)2 - 2 A) Downward B) Upward C) Upward D) Downward Range: (-∞, 2] Range: [-2, ∞) Range: [2, ∞) Range: (-∞, -2] Answer: D Type: BI Var: 50+ Objective: Graph a Quadratic Function in Vertex Form

Identify the vertex and determine the minimum or maximum value of the function. 2) f (x) = -2(x - 4)2 + 7 A) Vertex: (4, -7) B) Vertex: (4, 7) C) Vertex: (4, 7) D) Vertex: (4, -7) Maximum: -7 Maximum: 7 Minimum: 7 Minimum: -7 Answer: B Type: BI Var: 50+ Objective: Graph a Quadratic Function in Vertex Form

Determine the x- and y-intercepts for the given function. 3) f (x) = -(x - 2)2 - 4 A) x-intercept: none y-intercept: (0, -8) C) x-intercept: (2, 0) y-intercept: (0, -4) Answer: A Type: BI Var: 50+ Objective: Graph a Quadratic Function in Vertex Form

Page 2

B) x-intercept: (2, 0) y-intercept: (0, 4) D) x-intercepts: (0, 0) and (4, 0) y- intercept: (0, -8)

Sketch the function and determine the axis of symmetry. 4) f (x) = 3(x - 1)2 + 6 A) Axis of symmetry: x = 1 10 y

10 y

-10 -8 -6 -4 -2 -2

8 10x

-10 -8 -6 -4 -2 -2

-4

-6

-8

-10

C) Axis of symmetry: x = 6

10 y

2 2

8 10x

D) Axis of symmetry: x = -6

10 y

-10 -8 -6 -4 -2 -2

8 10x

-10 -8 -6 -4 -2 -2

-4

-6

-8

-10

Answer: A Type: BI Var: 50+ Objective: Graph a Quadratic Function in Vertex Form

Page 3

B) Axis of symmetry: x = -1

Graph the function and determine the minimum or maximum value of the function. 1 2 5) m(x) = (x + 1) 4 A) maximum value= 0 B) maximum value= 0 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

C) minimum value= 0

D) minimum value= 0

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: A Type: BI Var: 50+ Objective: Graph a Quadratic Function in Vertex Form

Page 4

-5 -4 -3 -2 -1 -1

Graph the parabola and the axis of symmetry. Label the coordinates of the vertex, and write the equation of the axis of symmetry. 6) y = - (x - 5)2 + 2 A) B) y

2 2

-8 -6 -4 -2

-2

-4

-6

-8

Vertex (5, 2) axis of symmetry:x = 5

Vertex (-5, -2) axis of symmetry:x = -5

D) y

2 2

-8 -6 -4 -2

-2

-4

-6

-8

Vertex (5, -2) axis of symmetry:x = 5 Answer: A Type: BI Var: 50+ Objective: Graph a Quadratic Function in Vertex Form

Page 5

Vertex (-5, 2) axis of symmetry:x = -5

Graph the function. 1 2 7) g(x) = - (x - 1) + 2 2 A)

B) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: B Type: BI Var: 50+ Objective: Graph a Quadratic Function in Vertex Form

Page 6

-5 -4 -3 -2 -1 -1

8) h(x) = -

1 3

(x - 3) + 6

B) y

-6

-4

2 2

-2

-6

-4

-2

-4

-6

D) y

-6

-4

2 2

-2

-6

-4

-2

-4

-6

Answer: A Type: BI Var: 50+ Objective: Graph a Quadratic Function in Vertex Form

Page 7

Identify the graph of the function. 9) h(x) = (x + 1)2 - 2 A)

-6

-4

2 2

-2

-6

-4

-2

-4

-6

D) y

-6

-4

2 2

-2

-6

-4

-2

-4

-6

Answer: A Type: BI Var: 47 Objective: Graph a Quadratic Function in Vertex Form

Find the vertex. 10) f (x) = -2(x - 19)2 + 13 A) (-19, 13)

B) (13, 19)

C) (13, -19)

D) (19, 13)

C) 8, 1

D) 8, -1

Answer: D Type: BI Var: 50+ Objective: Graph a Quadratic Function in Vertex Form

11) p x = -

x+82+1

7 A) -8, -1

B) -8, 1

Answer: B Type: BI Var: 50+ Objective: Graph a Quadratic Function in Vertex Form

Page 8

2 Write f(x) = ax2 + bx + c (a ≠ 0) in Ve rtex Form

Find the vertex of the parabola. 1) f (x) = x2 + 16x - 8 A) (-8, -72)

B) (16, -8)

C) -8,

11 4

D) (-16, 8)

Answer: A Type: BI Var: 50+ Objective: Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form

2) q(x) = 2x2 - 12x - 1 A) (3, -19)

B) (-3, -19)

C) (-3, -31)

D) (3, -31)

C) (3, -1)

D) (-1, -3)

Answer: A Type: BI

Var: 24

Objective: Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form

3) h(x) = -2x2 + 12x - 19 A) (0, -19)

B) (-3, -1)

Answer: C Type: BI

Var: 50+ Objective: Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form

Identify the vertex, axis of symmetry, and intercepts for the graph of the function. 4) g(x) = x2 - 2x - 15 A) Vertex at (-1, -12); axis: y = -12; x-intercepts: (-3, 0) and (5, 0); y-intercept: (0, -15) B) Vertex at (1, -16); axis: y = -16; x-intercepts: none; y-intercept: (-3, 0) C) Vertex at (1, -16); axis: x = 1; x-intercepts: (-3, 0) and (5, 0); y-intercept: (0, -15) D) Vertex at (-1, -12); axis: x = -1; x-intercepts: none; y-intercept: (-3, 0) Answer: C Type: BI Var: 12 Objective: Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form

5) y = x2 + 8x + 14 A) Vertex at (4, 62); axis: x = 4; x-intercepts: none; y-intercept: (-4 + 2, 0) B) Vertex at (-4, -2); axis: x = -4; x-intercepts: (-4 - 2, 0) and (-4 + 2, 0) ; y-intercept: (0, 14) C) Vertex at (-4, -2); axis: y = -2; x-intercepts: none; y-intercept: (-4 - 2, 0) D) Vertex at (4, 62); axis: y = 62; x-intercepts: (-4 Answer: B Type: BI Var: 12 Objective: Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form

Page 9

2, 0) and (-4 + 2, 0) ; y-intercept: (0, 14)

6) g(x) = -x2 + 6x - 11 A) Vertex at (-3, -38); axis: y = -38; x-intercepts: none; y-intercept: (0, 11) B) Vertex at (3, -2); axis: y = -2; x-intercepts: (3 - 2, 0) and (3 + 2, 0); y-intercept: (0, 11) C) Vertex at (3, -2); axis: x = 3; x-intercepts: none; y-intercept: (0, -11) D) Vertex at (-3, -38); axis: x = -3; x-intercepts: (3 - 2, 0) and (3 + 2, 0); y-intercept: (0, -11) Answer: C Type: BI Var: 6 Objective: Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form

7) y = -x2 - 2x + 8 A) Vertex at (1, 5); axis: x = 1; x-intercepts: none; y-intercept: (2, 0) B) Vertex at (-1, 9); axis: x = -1; x-intercepts: (-4, 0) and (2, 0); y-intercept: (0, 8) C) Vertex at (-1, 9); axis: y = 9; x-intercepts: (-4, 0) and (2, 0); y-intercept: (0, 8) D) Vertex at (1, 5); axis: y = 5; x-intercepts: none; y-intercept: (2, 0) Answer: B Type: BI Var: 12 Objective: Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form

Write the function in vertex form and determine the range. 8) f (x) = -3x2 + 18x - 33 A) f (x) = -3(x + 3)2 - 6 B) f (x) = -3(x + 3)2 - 6 Range: [3, ∞) Range: (-∞, -6] 2 C) f (x) = -3(x - 3) - 6 D) f (x) = -3(x - 3)2 - 6 Range: (-∞, -6] Range: [3, ∞) Answer: C Type: BI

Var: 50+

Objective: Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form

Write the domain and range of the function in interval notation. 9) f (x) = x 2 + 10x + 6 A) Domain: (0, ∞) B) Domain: (-∞, ∞) Range: (-∞, -19] Range: [-19, ∞) C) Domain: (-∞, ∞) D) Domain: (-∞, ∞) Range: [6, ∞) Range: (-∞, 6] Answer: B Type: BI

Var: 41

Objective: Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form

Page 10

10) q(x) = 2x2 - 12x + 2 A) Domain : [0, ∞) Range: (∞, ∞) C) Domain: [-16, ∞) Range: (-∞, ∞) Answer: B

B) Domain: (-∞, ∞) Range: [-16, ∞) D) Domain: (-∞, ∞) Range: (-∞, -16]

Type: BI Var: 24 Objective: Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form

Determine the minimum or maximum value of the function. 11) k(x) = 3x2 - 4x 4 4 4 A) Maximum: B) Maximum: C) Minimum: 3 3 3

D) Minimum:

4 3

Answer: C Type: BI

Var: 18

Objective: Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form

Identify the vertex and determine the minimum or maximum value of the function. 12) f (x) = -2x2 + 8x - 2 A) Vertex: (2, -6) B) Vertex: (2, 6) C) Vertex: (2, -6) D) Vertex: (2, 6) Minimum: -6 Maximum: 6 Maximum: -6 Minimum: 6 Answer: B Type: BI

Var: 50+

Objective: Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form

Determine the x- and y-intercepts for the given function. 13) f (x) = 2x2 + 8x - 10 A) x-intercept: (-2, 0) y-intercept: (0, 18) C) x-intercept: (-2, 0) y-intercept: (0, -18) Answer: D Type: BI Var: 50+ Objective: Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form

Page 11

B) x-intercept: none y-intercept: (0, -10) D) x-intercepts: (-5, 0) and (1, 0) y- intercept: (0, -10)

Write the function in vertex form and determine the x-intercept(s) 14) h x = -5x 2 + 11x A) h x = -5 x -

11 2

10 B) h x = 5 x +

11 2

C) h x = -5 x -

10 D) h x = 5 x +

11 2 10

; 0,0 and -

10 11 2

121

; 0,0 and 11 , 0 10 5 121

; 0,0 and 11 , 0 20 5

121

; 0,0 and - 11 , 0 10 5

Answer: C Type: BI Var: 7 Objective: Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form

15) p x = x 2 +9x + 11 2 9 37 ; 0, 0 A) p x = x + 2 4 9 2 37 -9 + 37 , 0 and -9 - 37 , 0 - ; 2 4 2 2 2 9 37 ; 0, 0 C) p x = x 2 4 B) p x = x +

D) p x = x +

9 2 37 -9 + 37 , 0 and -9 - 37 , 0 + ; 2 4 2 2

Answer: B Type: BI

Var: 12

Objective: Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form

Page 12

Sketch the function and determine the axis of symmetry. 16) f (x) = 2x2 - 4x + 6 A) Axis of symmetry: x = 1

-20

20 y

-10

-20

-10

-20

C) Axis of symmetry: x = -4

-20

B) Axis of symmetry: x = 4

D) Axis of symmetry: x = -1

20 y

-10

-20

-10

-20

Answer: A Type: BI Var: 50+ Objective: Write f(x) = ax2 + bx + c (a ≠ 0) in Vertex Form 3 Find the Vertex of a Parabola by Using the Vertex Formula

Find the vertex of the parabola by applying the vertex formula. 12 - 6x - 7 1) f (x) = - x 5 3 256 3 76 A) - , B) (15, -52) C) , 25 5 25 5 Answer: D Type: BI Var: 50+ Objective: Find the Vertex of a Parabola by Using the Vertex Formula

Page 13

D) (-15, 38)

2) f x = 2x 2 - 48x - 87 A) 24, - 87

B) 8, -343

C) - 8, 425

D) - 24, -343

Answer: B Type: BI Var: 50+ Objective: Find the Vertex of a Parabola by Using the Vertex Formula

3) h a = 6a 2 + 16 1 385 A) , 12 24

B) 0, 16

C) 0, - 16

1 385 ,12 24

Answer: B Type: BI Var: 50+ Objective: Find the Vertex of a Parabola by Using the Vertex Formula

4) P x = 1.4x 2 + 1.3x - 5.1 A) -0.46, -5.402 B) -0.46, -5.994

C) 0.46, 5.994

Answer: A Type: BI Var: 50+ Objective: Find the Vertex of a Parabola by Using the Vertex Formula

Page 14

D) 0.46, -4.206

Identify the vertex and sketch the graph. 5) f x = 2x 2 + 4 A)

-4 -3 -2 -1 -1

-5 -4 -3 -2 -1 -1

Vertex: 4, 0

Vertex: 0, - 4

D) y

Vertex: 0, 4

-4 -3 -2 -1 -1

Vertex: 0, 4

Answer: C Type: BI Var: 12 Objective: Find the Vertex of a Parabola by Using the Vertex Formula

Page 15

6) f x = -2x 2 - 24x - 72 A)

8 y

-2 -1 -1

-8 -7 -6 -5 -4 -3 -2 -1 -1

-2

Vertex: 6, 0

Vertex: -6, 0

D) 2 y

2 y

-8 -7 -6 -5 -4 -3 -2 -1 -1

-2 -1 -1

-2

-3

-4

-5

-6

-7

-8

Vertex: -6, 0

Vertex: 6, 0

Answer: C Type: BI Var: 18 Objective: Find the Vertex of a Parabola by Using the Vertex Formula 4 Solve Applications Involving Quadratic Functions

Solve the problem. 1) The population P(t) of a culture of bacteria is given byP(t) = -1,840t2 + 81,000t + 10,000, where t is the time in hours since the culture was started. Determine the time at which the population is at a maximum. Round to the nearest hour. A) 44 hr B) 9 hr C) 18 hr D) 22 hr Answer: D Type: BI Var: 50+ Objective: Solve Applications Involving Quadratic Functions

Page 16

2) A fireworks mortar is launched straight upward from a pool deck platform4 m off the ground at an initial velocity of61 m/sec. The height of the mortar can be modeled byh(t) = -4.9t2 + 61t + 4, where h(t) is the height in meters and t is the time in seconds after launch. What is the maximum height? Round to the nearest meter. A) 4 m B) 237 m C) 6 m D) 194 m Answer: D Type: BI Var: 50+ Objective: Solve Applications Involving Quadratic Functions

3) A model rocket is launched from a raised platform at a speed of112 feet per second. Its height in feet is given byh(t) = -16t 2 + 112t + 20 (t = seconds after launch). After how many seconds does the object reach its maximum height? A) 3.5 seconds B) 5.5 seconds C) 20 seconds D) 1.75 seconds Answer: A Type: BI Var: 7 Objective: Solve Applications Involving Quadratic Functions

4) The daily profit in dollars made by an automobile manufacturer is P(x) = -30x2 + 1,560x - 1,470 where x is the number of cars produced per shift. Find the maximum possible daily profit. A) $15,989 B) $20,315 C) $17,305 D) $18,810 Answer: D Type: BI Var: 50+ Objective: Solve Applications Involving Quadratic Functions

5) A family wants to fence in a rectangular area for a garden. One side of the garden will border their house and will not be fenced. Find the dimensions of the garden with greatest area that can be enclosed with 80 ft of fencing. A) 20 ft by 40 ft B) 40 ft by 40 ft C) 9 ft by 9 ft D) 20 ft by 20 ft Answer: A Type: BI Var: 21 Objective: Solve Applications Involving Quadratic Functions

6) An automobile manufacturer can produce up to 300 cars per day. The profit made from the sale of these vehicles can be modeled by the functionP(x) = -10x2 + 1,900x - 34,000, where P(x) is the profit in dollars and x is the number of automobiles made and sold. How many cars should be made and sold to maximize profit? A) 1,900 B) 45 C) 95 D) 190 Answer: C Type: BI Var: 50+ Objective: Solve Applications Involving Quadratic Functions

Page 17

7) The sum of two positive numbers is 26. What two numbers will maximizethe product? A) 13 and 13 B) -13 and 13 C) 11 and 15 D) 1 and 25 Answer: A Type: BI Var: 50+ Objective: Solve Applications Involving Quadratic Functions

8) The monthly profit for a small company that makes long-sleeve T-shirts depends on the price per shirt. If the price is too high, sales will drop. If the price is too low, the revenue brought in may not cover the cost to produce the shirts. After months of data collection, the sales team determines that the monthly profit is approximated byf p = -33p 2 + 1,254p - 10,763, where p is the price per shirt and f p is the monthly profit based on that price. Find the price that generates the maximum profit and find the maximum profit. A) p = $19 B) p = $19 C) p = $38 D) p = $38 f p = $1,150 f p = $24,976 f p = $1,150 f p = $10,763 Answer: A Type: BI Var: 50+ Objective: Solve Applications Involving Quadratic Functions

9) A trough at the end of a gutter spout is meant to direct water away from a house. The homeowner makes the trough from a rectangular piece of aluminum that is24 in. long and 8 in. wide. He makes a fold along the two long sides a distance of x inches from the edge. 24 in. x 8 - 2x a. Write a function to represent the volume in terms of x. b. What value of x will maximize the volume of water that can be carried by the gutter? A) a. V(x) = -48x2 + 192x; b. x = 1 B) a. V(x) = -48x2 + 192x; b. x = 2 C) a. V(x) = -2x2 + 8x; b. x = 2 D) a. V(x) = -2x2 + 8x; b. x = 1 Answer: B Type: BI Var: 50+ Objective: Solve Applications Involving Quadratic Functions

Page 18

5 Create Quadratic Models Using Regression

Solve the problem. 1) Gas mileage is tested for a car under different driving conditions. At lower speeds, the car is driven in stop and go traffic. At higher speeds, the car must overcome more wind resistance. The variable x given in the table represents the speed (in mph) for a compact car, andm(x) represents the gas mileage (in mpg). Use regression to find a quadratic function to model the data. x 25 30 35 40 45 50 55 60 65 m (x) 20.7 24.4 27 28.4 28.7 27.8 25.8 22.6 18.3 A) m(x) = -0.023x2 + 2.01x - 15.2 C) m(x) = -0.0600x2 + 0.03x + 6.0

B) m(x) = 0.0260x2 - 0.06x + 6.0 D) m(x) = 2.0100x2 - 0.02x - 15.2

Answer: A Type: BI Var: 50+ Objective: Create Quadratic Models Using Regression

2) Gas mileage is tested for a car under different driving conditions. At lower speeds, the car is driven in stop and go traffic. At higher speeds, the car must overcome more wind resistance. The variable x given in the table represents the speed (in mph) for a compact car, andm(x) represents the gas mileage (in mpg). Use regression to find a quadratic function to model the data and then use the function to determine the speed with the greatest gas mileage. Round to the nearest mile per hour. Carry out all computations up to the final answer without rounding. x 25 30 35 40 45 50 55 60 65 m (x) 21.4 25.4 28.2 29.9 30.4 29.8 28 25.1 21 A) 45 mph

B) 41 mph

Answer: A Type: BI Var: 50+ Objective: Create Quadratic Models Using Regression

Page 19

C) 47 mph

D) 43 mph

3) Tetanus bacillus bacteria are cultured to produce tetanus toxin used in an inactive form for the tetanus vaccine. The amount of toxin produced per batch increases with time and then decreases as the culture becomes unstable. The variablet is the time in hours after the culture has started, and y t is the yield of toxin in grams. t 8 16 24 32 40 48 56 64 72 80 88 96 y t 0.55 1.13 1.62 1.80 1.88 1.98 1.92 1.78 1.51 1.33 0.69 0.13 a. Use regression to find a quadratic function to model the data. b. At what time is the yield the greatest? Round to the nearest hour. A) a. y = -0.0008148t2 + 0.08155t + 0.646 B) a. y = -0.0008148t2 + 0.08155t + 0.646 b. 49 hr b. 50 hr 2 C) a. y = -0.000831t + 0.081t + 0.028 D) a. y = -0.000831t2 + 0.081t + 0.028 b. 51 hr b. 49 hr Answer: D Type: BI Var: 4 Objective: Create Quadratic Models Using Regression 6 Mixed Exercises

Determine the number of x-intercepts of the graph of f (x) = ax2 + bx + c (a ≠ 0) based on the discriminant of the related equation f (x) = 0. (Hint: Recall that the discriminant is b2 - 4ac.) 1) f (x) = -9x2 + 4x + 6 A) One B) Two C) None Answer: B Type: BI Var: 50+ Objective: Mixed Exercises 7 Expanding Your Skills

Find the value of c that gives the function the given minimum value. 1) f (x) = 5x2 - 10x + c; minimum value1 A) c = 6 B) -5 C) 5

D) c = -4

Answer: A Type: BI Var: 50+ Objective: Expanding Your Skills

Find the value of b that gives the function the given maximum value. 2) f (x) = -x2 + bx + 2; maximum value6 A) b = -4 or b = 4 B) b = -2 or b = 2 C) b = 2 Answer: A Type: BI Var: 50+ Objective: Expanding Your Skills

Page 20

D) b = 4

3.2 Introduction to Polynomial Functions 0 Concept Connections

Provide the missing information. 1) A function defined byf (x) = anxn + an-1xn-1 + an-2xn-2 + ... + a1x + a0 where an, an-1, an-2, ... a1, a0 are real numbers and an ≠ 0 is called a function. Answer: polynomial Type: SA Var: 1 Objective: Concept Connections

1 2) The function given by f (x) = -3x5 + 2x + x (is/is not) a polynomial function. 2 Answer: is Type: SA Var: 1 Objective: Concept Connections

3) The function given by f (x) = -3x5+ 2 x +

(is/is not) a polynomial function.

Answer: is not Type: SA Var: 1 Objective: Concept Connections

4) A quadratic function is a polynomial function of degree

Answer: 2 Type: SA Var: 1 Objective: Concept Connections

5) A linear function is a polynomial function of degree

Answer: 1 Type: SA Var: 1 Objective: Concept Connections

6) Determine if the statement is true or false. The graph of the function defined byf (x) = |x| is smooth. Answer: False Type: SA Var: 1 Objective: Concept Connections

7) Determine if the statement is true or false. The function defined byf (x) =

1 x

Answer: False Type: SA Var: 1 Objective: Concept Connections

Page 21

is continuous.

8) The end behavior of the graph of f (x) = -6x2 is (up/down) to the left and (up/down) to the right. Answer: down; down Type: SA Var: 1 Objective: Concept Connections

9) The end behavior of the graph of f (x) = -8x3 is (up/down) to the left and (up/down) to the right. Answer: up; down Type: SA Var: 1 Objective: Concept Connections

10) The values of x in the domain of a polynomial functionf for which f (x) = 0 are called the of the function. Answer: zeros Type: SA Var: 1 Objective: Concept Connections

11) Given the function defined byg(x) = -3(x - 1)3(x + 5)4, the value 1 is a zero with multiplicity , and the value -5 is a zero with multiplicity . Answer: 3; 4 Type: SA Var: 1 Objective: Concept Connections

1 5 12) Given the function defined byh(x) =

, the value 0 is a zero with multiplicity

x (x + 0.6)

2 , and the value -0.6 is a zero with multiplicity

Answer: 5; 3 Type: SA Var: 1 Objective: Concept Connections

13) What is the maximum number of turning points of the graph of f (x) = -3x6 - 4x5 - 5x4 + 2x2 + 6? Answer: 5 Type: SA Var: 1 Objective: Concept Connections

14) If the graph of a polynomial function has 3 turning points, what is the minimum degree of the function? Answer: 4 Type: SA Var: 1 Objective: Concept Connections

Page 22

15) If c is a real zero of a polynomial function and the multiplicity is 3, does the graph of the function cross the x-axis or touch the x-axis (without crossing) at (c, 0)? Answer: cross Type: SA Var: 1 Objective: Concept Connections

16) If c is a real zero of a polynomial function and the multiplicity is 6, does the graph of the function cross the x-axis or touch the x-axis (without crossing) at (c, 0)? Answer: touch (without crossing) Type: SA Var: 1 Objective: Concept Connections

17) Suppose that f is a polynomial function and thata < b. If f(a) and f(b) have opposite signs, then what conclusion can be drawn from the intermediate value theorem? Answer: f has at least one zero on the interval [a, b]. Type: SA Var: 1 Objective: Concept Connections

18) An even function is symmetric with respect to the

axis.

Answer: y Type: SA Var: 1 Objective: Concept Connections

19) An odd function is symmetric with respect to the

Answer: origin Type: SA Var: 1 Objective: Concept Connections

1 20) What is the leading term of f (x) = -

(x - 3) (3x + 5) ?

Answer: -3x6 Type: SA Var: 1 Objective: Concept Connections

21) What is the leading term of f (x) = -0.6x2(10x + 1)3(x - 1)4? Answer: -600x9 Type: SA Var: 1 Objective: Concept Connections

22) If the leading term of a polynomial function isaxn where a is negative and n is odd, determine the end behavior. Answer: up to the left and down to the right Type: SA Var: 1 Objective: Concept Connections

Page 23

1 Determine the End Behavior of a Polynomial Function

Determine the end behavior of the graph of the function. 1) f (x) = 8x6 + 3x5 + 3x4 + 7 A) Up left and down right C) Down left and down right

B) Up left and up right D) Down left and up right

Answer: B Type: BI Var: 50+ Objective: Determine the End Behavior of a Polynomial Functio

2) f (x) = 8x3(4 - x)(7x - 6)4 A) Down left and up right C) Up left and up right

B) Up left and down right D) Down left and down right

Answer: D Type: BI Var: 50+ Objective: Determine the End Behavior of a Polynomial Functio

3) 4(x + 2)(8x + 7)3(x + 2)5 A) As x → -∞, f (x) → ∞; As x → ∞, f (x) → -∞ B) As x → -∞, f (x) → -∞; As x → ∞, f (x) → -∞ C) As x → -∞, f (x) → -∞; As x → ∞, f (x) → ∞ D) As x → -∞, f (x) → ∞; As x → ∞, f (x) → ∞ Answer: C Type: BI Var: 50+ Objective: Determine the End Behavior of a Polynomial Functio 2 Identify Zeros and Multiplicity of Zeros

Find the zeros of the function and state the multiplicities. 1) f (x) = 2x5 + 9x4 + 7x3 7 A) 1 (multiplicity2.5), (multiplicity 2.5) 2 7 B) -1 (multiplicity 2.5), - (multiplicity 2.5) 2 7 C) 0 (multiplicity3), -1 (multiplicity 1), - (multiplicity 1) 2 7 D) 0 (multiplicity3), 1 (multiplicity 1), (multiplicity 1) 2 Answer: C Type: BI Var: 50+ Objective: Identify Zeros and Multiplicity of Zero

Page 24

2) f (x) = -2x6(x + 7)2(x - 5)6 A) -7 (multiplicity 2), 5 (multiplicity6) B) 0 (multiplicity6), 7 (multiplicity 2), -5 (multiplicity 6) C) 7 (multiplicity2), -5 (multiplicity6) D) 0 (multiplicity6), -7 (multiplicity2), 5 (multiplicity 6) Answer: D Type: BI Var: 50+ Objective: Identify Zeros and Multiplicity of Zero

3) f (x) = 4x(9x + 8)(2x + 5)(x + 6)(x - 6) 8 5 A) 0, , ; each of multiplicity 1; and6 of multiplicity 2 9 2 8 5 , , ±6; each of multiplicity 1 9 2 8 5 C) 0, - , - , ± 6 ; each of multiplicity 1 9 2 8 5 D) 0, - , - ; each of multiplicity 1; and 6 of multiplicity 2 9 2 B) 0,

Answer: C Type: BI Var: 50+ Objective: Identify Zeros and Multiplicity of Zero

4) f (x) = [x - (4 - 7)][x - (4 + 7)] A) -4 ± 7; each of multiplicity 1 C) 4 + 7; multiplicity 2

B) 4 - 7; multiplicity 2 D) 4 ± 7; each of multiplicity 1

Answer: D Type: BI Var: 40 Objective: Identify Zeros and Multiplicity of Zero

5) f x = 4x 4 - 101x 2 + 25 1 A) , 25; each of multiplicity 1 4 1 C) - , - 25; each of multiplicity 1 4 Answer: D Type: BI Var: 20 Objective: Identify Zeros and Multiplicity of Zero

Page 25

1 B) , 5; each of multiplicity 1 2 1 1 D) -5, - 2 , 2 , 5; each of multiplicity 1

6) m x = x 5 - 3x 3 A) - 3, 0, 3; each of multiplicity 1 B) - 3, 3 each of multiplicity 1; 0 of multiplicity3 C) - 3, 0, 3; each of multiplicity3 D) - 3, 3 each of multiplicity3; 0 of multiplicity 1 Answer: B Type: BI Var: 50+ Objective: Identify Zeros and Multiplicity of Zero 3 Apply the Intermediate Value Theorem

Determine whether the intermediate value theorem guarantees that the function has a zero on the given interval. 1) f (x) = x3 - 8x2 + 14x + 9; [1, 2] A) Yes B) No Answer: B Type: BI Var: 50+ Objective: Apply the Intermediate Value Theorem

A table of values is given for Y1 = f (x). Determine whether the intermediate value theorem guarantees that the function has a zero on the given interval. 2) Y1 = 17x4 - 39x3 + 186x2 + 298x + 398 [-2, -1] X

-4 -3 -2 -1 0 1 2 3

9,030 3,608 1,130 342 398 860 1,698 3,290

X=-4

A) No Answer: A Type: BI Var: 50+ Objective: Apply the Intermediate Value Theorem

Page 26

B) Yes

Solve the problem. 3) Given f (x) = 2x3 + 7x2 + 14x - 9. Use the intermediate value theorem to determine whether f (x) has a zero on the interval [0, 1], find the zero if it exists. A) 1 C)

B) -1

1 2

D) There is no zero on the interval.

Answer: C Type: BI Var: 50+ Objective: Apply the Intermediate Value Theorem 4 Sketch a Polynomial Function

Determine if the graph can represent a polynomial function. If so, assume the end behavior and all turning points are represented on the graph. 1) y 5 4 3 2 1 -5 -4 -3 -2 -1 -1

-2 -3 -4 -5

a. Determine the minimum degree of the polynomial based on the number of turning points. b. Determine whether the leading coefficient is positive or negative based on the end behavior and whether the degree of the polynomial is odd or even. c. Approximate the real zeros of the function, and determine if their multiplicity is odd or even. A) a. Minimum degree 2 b. Leading coefficient negative degree even c. -2 (odd multiplicity), 1 (even multiplicity),2 (even multiplicity) B) a. Minimum degree 3 b. Leading coefficient negative degree odd c. -2 (odd multiplicity), 1 (even multiplicity),2 (even multiplicity) C) a. Minimum degree 3 b. Leading coefficient positive degree odd c. -2, 1, and 2 (each with odd multiplicity) D) Not a polynomial function. Answer: C Type: BI Var: 50+ Objective: Sketch a Polynomial Function

Page 27

2) y 5 4 3 2 1 -5 -4 -3 -2 -1 -1

-2 -3 -4 -5

a. Determine the minimum degree of the polynomial based on the number of turning points. b. Determine whether the leading coefficient is positive or negative based on the end behavior and whether the degree of the polynomial is odd or even. c. Approximate the real zeros of the function, and determine if their multiplicity is odd or even. A) a. Minimum degree 4 b. Leading coefficient positive degree even c. -3 (odd multiplicity), -2 (odd multiplicity), 2 (even multiplicity) B) a. Minimum degree 3 b. Leading coefficient negative degree odd c. -3 (even multiplicity),-2 (even multiplicity),2 (odd multiplicity) C) a. Minimum degree 3 b. Leading coefficient positive degree odd c. -3, -2, and 2 (each with odd multiplicity) D) Not a polynomial function. Answer: A Type: BI Var: 48 Objective: Sketch a Polynomial Function

Page 28

3) 10 y 9 8 7 6 5 4 3 2 1 -10-9 -8 -7 -6 -5 -4 -3 -2 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10

1 2 3 4 5 6 7 8 9 10 x

a. Determine the minimum degree of the polynomial based on the number of turning points. b. Determine whether the leading coefficient is positive or negative based on the end behavior and whether the degree of the polynomial is odd or even. c. Approximate the real zeros of the function, and determine if their multiplicity is odd or even. A) a. Minimum degree 3 b. Leading coefficient negative degree odd c. -2 (even multiplicity) B) a. Minimum degree 2 b. Leading coefficient positive degree even c. -2 (even multiplicity) C) Not a polynomial function. D) a. Minimum degree 2 b. Leading coefficient positive degree even c. -1.6 and -4.4 (each of odd multiplicity) Answer: C Type: BI Var: 42 Objective: Sketch a Polynomial Function

Page 29

Sketch the function. 4) f (x) = -x3 - 3x2 A)

B) 10 y

10 y

-10 -8 -6 -4 -2

-2

-4

-6

-8

-10

D) 10 y

10 y

-10 -8 -6 -4 -2

-2

-10 -8 -6 -4 -2

-2

-4

-6

-8

-10

Answer: D Type: BI Var: 12 Objective: Sketch a Polynomial Function

Page 30

-10 -8 -6 -4 -2

1 5) g(x) =

(x - 5)(x + 4)(x + 1)

B) 10 y

10 y

-10 -8 -6 -4 -2

-2

-4

-6

-8

-10

D) 10 y

10 y

-10 -8 -6 -4 -2

-2

-10 -8 -6 -4 -2

-2

-4

-6

-8

-10

Answer: B Type: BI Var: 50+ Objective: Sketch a Polynomial Function

Page 31

-10 -8 -6 -4 -2

6) f (x) = -0.4(x - 1)2(x - 4)3 A)

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Answer: D Type: BI Var: 50+ Objective: Sketch a Polynomial Function

Page 32

-5 -4 -3 -2 -1 -1

7) m(x) = 25x5 - 25x4 - 9x3 + 9x2 A)

-2

-4

-6

D) y

-2

-4

-6

Answer: B Type: BI Var: 10 Objective: Sketch a Polynomial Function

Page 33

8) f (x) = -x4 + 15x2 - 44 A)

B) y

60 50 40 30 20 10

-5

5 x

-10 -20 -30 -40 -50 -60

-5

D) 60 50 40 30 20 10 -5

-10 -20 -30 -40 -50 -60

y 5

5 x

-5

Answer: B Type: BI Var: 12 Objective: Sketch a Polynomial Function

Solve the problem. 1 6 9) g x = - x - 3 5 a. Identify the power function of the form y = x n that is the parent function to the given graph. b. In order, outline the transformations that would be required on the graph of y = x n to make the graph of the given function. c. Match the function with the graph. i.

Page 34

ii.

1 y -5 -4 -3 -2 -1 -1

4 y 1

-2

-3 -4

-5 -4 -3 -2 -1 -1

-5

-2

-6

-3

-7

-4

-8

-5

-9

-6

iii.

iv. 7 y

7 y

-5 -4 -3 -2 -1 -1

-2

-3

A) a. y = x b. Expand y = x 6 vertically by a factor of 5. Reflect across the x - axis. Shift upward 3 units. c. Graph iv. B) a. y = x 6 1 b. Shrink y = x 6 horizontally by a factor of . Reflect across the x - axis. Shift downward 3 5 units. c. Graph iii. C) a. y = x 6 b. Expand y = x 6 horizontally by a factor of 5. Reflect across the x - axis. Shift downward 3 units. c. Graph iv. D) a. y = x 6 1 b. Shrink y = x 6 vertically by a factor of . Reflect across the x - axis. Shift downward 3 5 units. c. Graph i. Answer: D Type: MC Var: 32 Objective: Sketch a Polynomial Function

Page 35

10) f x = -

x-14

8 a. Identify the power function of the form y = x n that is the parent function to the given graph. b. In order, outline the transformations that would be required on the graph of y = x n to make the graph of the given function. c. Match the function with the graph. i.

ii. 9 y

5 y

4 3

-4 -3 -2 -1 -1

-2

-3 1

-7 -6 -5 -4 -3 -2 -1 -1

-4

iii.

-5

iv. 1 y

-3 -2 -1 -1

5 y 1

4 3

-2

-3

-4 -5

-6 -5 -4 -3 -2 -1 -1

-6

-2

-7

-3

-8

-4

-9

A) a. y = x

-5

1 b. Shift y = x 4 to the left 1 units. Shrink vertically by a factor of . Reflect across the x-axis. 4 c. Graph iii. B) a. y = x 4 1 b. Shift y = x 4 to the right 1 units. Shrink vertically by a factor of . Reflect across the x-axis. 8 c. Graph ii. C) a. y = x 4 1 b. Shift y = x 4 to the right 1 units. Shrink vertically by a factor of . Reflect across the x-axis. 8 c. Graph iii. Page 36

D) a. y = x 4 1 b. Shift y = x 4 to the left 1 units. Shrink vertically by a factor of . 4 c. Graph i. Answer: C Type: MC Var: 28 Objective: Sketch a Polynomial Function

11) p x = 3 x + 3 3 - 3 a. Identify the power function of the form y = x n that is the parent function to the given graph. b. In order, outline the transformations that would be required on the graph of y = x n to make the graph of the given function. c. Match the function with the graph. i.

ii. 4 y

2 y

-7 -6 -5 -4 -3 -2 -1 -1

1 -7 -6 -5 -4 -3 -2 -1 -1

-2

-3

-2

-4

-3

-5

-4

-6

-5

-7

-6

-8

iii.

iv. 2 y

1 y

1 -7 -6 -5 -4 -3 -2 -1 -1

-7 -6 -5 -4 -3 -2 -1 -1 -2

-2

-3

-4

-5

-6

-7

-8

-9

A) a. y = x 3 1 b. Shift y = x 3 to the left 3 units. Shrink vertically by a factor of . Shift downward 3 units. 3 c. Graph iv. Page 37

B) a. y = x 3 b. Shift y = x 3 to the left 3 units. Stretch vertically by a factor of 3. Shift downward 3 units. c. Graph ii. C) a. y = x 3 b. Shift y = x 3 to the left 3 units. Stretch vertically by a factor of 3. Shift downward 3 units. c. Graph iii. D) a. y = x 3 b. Shift y = x 3 to the left 3 units. Stretch vertically by a factor of 3. Shift downward 3 units. c. Graph i. Answer: B Type: MC Var: 50+ Objective: Sketch a Polynomial Function

Sketch the function. 1 12) m x = x+2 x- 2 x+3 3 10 A)

B) 10 y 8 6 4 2 -5 -4 -3 -2 -1-2

18 y 16 14 12 10 8 6 4 2

-4 -6 -8 -10 -12 -14 -16 -18

-5 -4 -3 -2 -1-2 -4 -6 -8 -10

D) 6 y 4 2 -5 -4 -3 -2 -1-2 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22

Answer: B Type: MC Var: 50+ Objective: Sketch a Polynomial Function

Page 38

14 y 12 10 8 6 4 2 -5 -4 -3 -2 -1-2 -4 -6 -8 -10 -12 -14

5 Mixed Exercises

Determine if the statement is true or false. 1) If c is a real zero of an even polynomial function, then -c is also a zero of the function. A) False B) True Answer: B Type: BI Var: 7 Objective: Mixed Exercises

3.3 Division of Polynomials and the Remainder and Factor Theorems 0 Concept Connections

Provide the missing information. 1) Given the division algorithm identify the polynomials representing the dividend, divisor, quotient, and remainder. f (x) = d(x) ·q(x) + r(x) Answer: dividend: f (x); divisor: d(x); quotient: q(x); and remainder: r(x) Type: SA Var: 1 Objective: Concept Connections

2) Given

2x3 - 5x2 - 6x + 1

-8

= 2x + x - 3 + , use the division algorithm to check the result. x-3 x-3 Answer: (x - 3)(2x2 + x - 3) + (-8) = 2x3 - 5x2 - 6x + 1 Type: SA Var: 1 Objective: Concept Connections

3) The remainder theorem indicates that if a polynomialf (x) is divided by x - c, then the remainder is . Answer: f (c) Type: SA Var: 1 Objective: Concept Connections

4) Given a polynomial f (x), the factor theorem indicates that if f (c) = 0, then x - c is a (x). Furthermore, if x - c is a factor of f (x), then f (c) = .

of f

Answer: factor; 0 Type: SA Var: 1 Objective: Concept Connections

5) Answer true or false. If 5 is a zero of a polynomial, then (x - 5) is a factor of the polynomial. Answer: True Type: SA Var: 1 Objective: Concept Connections

6) Answer true or false. If (x + 3) is a factor of a polynomial, then 3 is a zero of the polynomial. Answer: False Type: SA Var: 1 Objective: Concept Connections

Page 39

1 Divide Polynomials Using Long Division

Use long division to divide. 1) (4x3 - 17x2 - 1x + 29) ÷ (x - 4) 495 A) 4x2 - 33x + 131 x-4 9 C) 4x2 - 1x - 5 + x-4

B) 4x2 - 1x + 4 D) 4x2 - 33x - 364

Answer: C Type: BI Var: 50+ Objective: Divide Polynomials Using Long Division

2) (-150x2 + 15x4 + 135) ÷ (3x + 9) 240 A) 5x3 - 65x + x+3 420 C) 155x3 - 50x x+3

B) 5x3 - 15x2 - 5x + 15 D) 5x3 + 15x2 - 5x - 15

Answer: B Type: BI Var: 50+ Objective: Divide Polynomials Using Long Division

3) (x5 + 2x4 - 2x2 - 5x - 12) ÷ (x2 - 3) A) x3 + 2x2 + 7x + 4 C) x3 + 2x2 - 3x - 8 +

B) x3 + 2x2 + 3x + 4 + 4x + 12 2

D) x3 + 5x2 + 13 +

x -3

x2 - 3 34x - 12 x2 - 3

Answer: B Type: BI Var: 50+ Objective: Divide Polynomials Using Long Division

12x4 - 6x3 + 25x2 - 11x - 6 4x2 - 2x -1 A) 3x2 + 3x + 8 C) 3x2 + 7x +

B) 3x2 + 10x + 1 3x + 1

4x2 - 2x -1 Answer: D Type: BI Var: 50+ Objective: Divide Polynomials Using Long Division

Page 40

D) 3x2 + 7 +

3x + 1 4x2 - 2x -1

64x3 + 1 4x + 1 A) 16x2 + 1

B) 16x2 - 1

C) 16x2 - 4x + 1

D) 16x2 + 4x + 1

Answer: C Type: BI Var: 28 Objective: Divide Polynomials Using Long Division

6) 3x 3 - 4x 2 + 8 ÷ 2x - 1 3 A)

3 B) x 2 - 2x - 1 2

2 x + 2x - 1

59 8

C) x - x - + 2 4 8 2x - 1

D) x + x + + 2 4 8 2x - 1

Answer: C Type: BI Var: 50+ Objective: Divide Polynomials Using Long Division 2 Divide Polynomials Using Synthetic Division

Solve the problem. 1) The following table represents the result of a synthetic division. 1 9 9 4 5 9

Use x as the variable. Identify the divisor. A) 27 C) x + 1

B) x - 1 D) 9x3 + 9x2 + 4x + 5

Answer: B Type: BI Var: 50+ Objective: Divide Polynomials Using Synthetic Division

2) The following table represents the result of a synthetic division. -2 -9 -8 5 -8 -9

-20

-15

Use x as the variable. Identify the quotient. A) x - 2 C) x + 2 Answer: D Type: BI Var: 50+ Objective: Divide Polynomials Using Synthetic Division

Page 41

59 8

B) -9x3 - 8x2 + 5x - 8 D) -9x2 + 10x - 15

3) The following table represents the result of a synthetic division. 2 8 -6 -7 -7 8

Use x as the variable. Identify the remainder. A) 8x3 - 6x2 - 7x - 7 C) 19

B) x - 2 D) 8x2 + 10x + 13

Answer: C Type: BI Var: 50+ Objective: Divide Polynomials Using Synthetic Division

4) The following table represents the result of a synthetic division. -3 5 9 -4 -5 5

-15

-42

-6

-47

Use x as the variable. Identify the dividend. A) 5x3 + 9x2 - 4x - 5 C) x + 3

B) 5x2 - 6x + 14 D) -47

Answer: A Type: BI Var: 50+ Objective: Divide Polynomials Using Synthetic Division

Use synthetic division to divide the polynomials. 5) (s2 + 2s - 8) ÷ (s + 4) A) s + 2

B) s + 2 -

C) s - 2

D) s + 2 +

s+4

Answer: C Type: BI Var: 50+ Objective: Divide Polynomials Using Synthetic Division

6) (7w3 + 2w2 - 3w - 14) ÷ (w - 1) A) 7w2 + 9w - 6 C) 7w2 + 9w + 6 -

B) 7w2 + 9w + 6 8 w-1

Answer: C Type: BI Var: 50+ Objective: Divide Polynomials Using Synthetic Division

Page 42

D) 7w2 + 7w + 2 -

16 w+1

7) (s4 + 5s3 + 2s2 - 17s + 7) ÷ (s - 1) A) s3 + 6s2 + 8s - 9 C) s3 + 6s2 + 8s +

B) s3 + 6s2 + 8s - 9 -

s-1 22 D) s3 + 4s2 - 2s - 15 + s+1

-9s + 2 s-1

Answer: B Type: BI Var: 50+ Objective: Divide Polynomials Using Synthetic Division

6x5 + 16x4 + 13x3 + 23x2 + 24x - 4 x+2 A) 6x4 + 4x3 + 5x2 + 13x - 2

B) 6x4 + 28x3 + 69x2 + 161x +

688 x+2

C) 6x3 + 28x2 + 69x + 161 +

346x + 688 x+2

D) 6x3 + 4x2 + 5x + 13 -

x+2

Answer: A Type: BI Var: 50+ Objective: Divide Polynomials Using Synthetic Division

x5 - 1,024 x-4 A) x4 + 256

B) x4 - 4x3 + 16x2 - 64x + 256 -

2,048 x-4

C) x4 - 256

D) x4 + 4x3 + 16x2 + 64x + 256

Answer: D Type: BI Var: 6 Objective: Divide Polynomials Using Synthetic Division

10) -3x 4 - 10x 3+ 38x 2 + 58x - 52 ÷ x -

2 3

A) -3x 3 - 4x 2 + 10x - 26 C) -x 3 + 4x 2 - 10x - 26 Answer: B Type: BI Var: 50+ Objective: Divide Polynomials Using Synthetic Division

Page 43

B) -3x 3 - 12x 2 + 30x + 78 D) -3x 3 + 12x 2 - 30x - 78

3 Apply the Remainder and Factor Theorems

Use the remainder theorem to evaluate the polynomial for the given value of x. 1) f (x) = 3x4 - 7x3 - 7x2 + 42x - 19; f (3) A) 98 B) 161 C) -224

D) 224

Answer: A Type: BI Var: 50+ Objective: Apply the Remainder and Factor Theorems

Use the remainder theorem to determine if the given number c is a zero of the polynomial. 2) x4 + 9x3 + 22x2 + 19x + 45; c = -3 A) No B) Yes Answer: A Type: BI Var: 50+ Objective: Apply the Remainder and Factor Theorems

Use the factor theorem to determine if the given binomial is a factor of f (x). 3) f (x) = x4 + 8x3 + 11x2 - 11x + 3; x + 3 A) Yes B) No Answer: A Type: BI Var: 50+ Objective: Apply the Remainder and Factor Theorems

4) f (x) = 2x3 + x2 - 9x - 6; x - 1 A) Yes Answer: B

B) No

Type: BI Var: 50+ Objective: Apply the Remainder and Factor Theorems

5) f x = x 3 + 343 a. x - 7 b. x + 7 A) a. No b. No Answer: C

B) a. Yes b. No

C) a. No b. Yes

Type: BI Var: 7 Objective: Apply the Remainder and Factor Theorems

Use the remainder theorem to determine if the given binomial is a factor of f (x). 6) f (x) = x3 + 3x2 - 28x - 84; x - 5 7 A) Yes B) No Answer: B Type: BI Var: 50+ Objective: Apply the Remainder and Factor Theorems

Page 44

D) a. Yes b. Yes

Solve the problem. 7) Use synthetic division and the remainder theorem to determine if[x - (3 - 2i)] is a factor of f (x) = x2 - 6x + 13. A) Yes

B) No

Answer: A Type: BI Var: 50+ Objective: Apply the Remainder and Factor Theorems

8) Factor f (x) = 3x3 - 2x2 - 53x - 60 given that -3 is a zero. A) (x + 3)(3x2 + 7x - 32) B) (x - 3)(3x - 4)(x - 5) 2 C) (x - 3)(3x + 7x - 32) D) (x + 3)(3x + 4)(x - 5) Answer: D Type: BI Var: 50+ Objective: Apply the Remainder and Factor Theorems

Use the factor theorem to factor the polynomial. 9) Factor f (x) = 6x3 + 31x2 - 29x + 6 given that

is a zero.

3 A) f (x) = (3x + 1)(2x + 1)(x + 6) C) f (x) = (3x - 1)(2x - 1)(x + 6)

B) f (x) = (3x - 1)(2x + 1)(x - 6) D) f (x) = (3x + 1)(2x - 1)(x - 6)

Answer: C Type: BI Var: 50+ Objective: Apply the Remainder and Factor Theorems

Write a polynomial f (x) that meets the given conditions. Answers may vary. 10) Degree 3 polynomial with zeros 1, 7, and 6 A) f (x) = x3 + 14x2 + 43x + 42 B) f (x) = x3 + 14x2 + 55x + 42 C) f (x) = x3 - 14x2 + 55x - 42 D) f (x) = x3 - 14x2 + 43x - 42 Answer: C Type: BI Var: 50+ Objective: Apply the Remainder and Factor Theorems

11) Degree 4 polynomial with zeros4 and -

(each with multiplicity 1) and 0 (withmultiplicity2).

5 A) f (x) = 5x4 - 26x3 + 24x2 C) f (x) = 6x4 - 19x3 - 20x2 Answer: B Type: BI Var: 50+ Objective: Apply the Remainder and Factor Theorems

Page 45

B) f (x) = 5x4 - 14x3 - 24x2 D) f (x) = 6x4 - 29x3 - 20x2

12) Degree 2 polynomial with zeros 3 5 and -3 5 A) f (x) = x2 - 6 5x + 45 C) f (x) = x2 - 45

B) f (x) = x2 + 45 D) f (x) = x2 + 6 5x + 45

Answer: C Type: BI Var: 50+ Objective: Apply the Remainder and Factor Theorems

13) Degree 3 polynomial with zeros 4, 5i, and -5i A) f (x) = x3 + 4x2 - 25x - 100 C) f (x) = x3 - 4x2 + 25x - 100

B) f (x) = x3 + 4x2 + 25x + 100 D) f (x) = x3 - 4x2 - 25x + 100

Answer: C Type: BI Var: 50+ Objective: Apply the Remainder and Factor Theorems 4 Mixed Exercises

Solve the problem. 1) Find m so that x + 4 is a factor of 5x3 + 18x2 + mx + 16 A) m = 16 B) m = 4 C) m = -4

D) m = -16

Answer: C Type: BI Var: 50+ Objective: Mixed Exercises

3.4 Zeros of Polynomials 0 Concept Connections

Provide the missing information. 1) The of a polynomial f (x) are the solutions (or roots) of the equation f (x) = 0. Answer: zeros Type: SA Var: 1 Objective: Concept Connections

2) If f (x) is a polynomial of degree n ≥ 1 with complex coefficients, thenf (x) has exactly complex zeros, provided that each zero is counted by its multiplicity. Answer: n Type: SA Var: 1 Objective: Concept Connections

3) The conjugate zeros theorem states that if f (x) is a polynomial with real coefficients, and ifa + bi is a zero of f (x), then is also a zero of f (x). Answer: a - bi Type: SA Var: 1 Objective: Concept Connections

Page 46

4) A real number b is called a(n) bound of the real zeros of a polynomialf (x) if all real zeros of f (x) are less than or equal to b. Answer: upper Type: SA Var: 1 Objective: Concept Connections

5) A real number a is called a lower bound of the real zeros of a polynomialf (x) if all real zeros of f (x) are or equal to a. Answer: greater than Type: SA Var: 1 Objective: Concept Connections

6) Explain why the number 7 cannot be a rational zero of the polynomial f (x) = 2x3 + 5x2 - x + 6. Answer: 7 is not the ratio of any of the factors of 6 over the factors of 2. Type: SA Var: 1 Objective: Concept Connections 1 Apply the Rational Zero Theorem

List the possible rational zeros. 1) f (x) = x6 - 2x4 + 7x2 + 25 A) 1, 5, 25 B) -1, -5, -25

C) ±1, ±5, ±25

1 1 D) ±1, ± , ± 5 25

Answer: C Type: BI Var: 50+ Objective: Apply the Rational Zero Theorem

2) f (x) = 25x4 - 7x3 + 2x + 10 2 2 1 1 A) ±1, ± , ± , ±5, ±2, ±10, ± , ± 5 25 5 25 5 25 1 1 C) ±1, ± , ± , ±5, ± , ±25, ± 5 25 2 2

2 2 1 1 B) 1, , , 5, 2, , 5 25 5 25 2 2 1 1 D) -1, - , , -5, -2, - , 5 25 5 25

Answer: A Type: BI Var: 50+ Objective: Apply the Rational Zero Theorem

Find all the rational zeros. 3) f (x) = x4 - 6x3 - 5x2 - 12x - 14 A) -1, 7 B) -1, 7, -2 Answer: A Type: BI Var: 50+ Objective: Apply the Rational Zero Theorem

Page 47

C) ±1, ±7

D) ±1, ±7, ±2

4) f (x) = 4x4 + 9x3 - 34x2 - 25x - 4 1 1 A) , 4 B) - , -4 4 4

1 C)

1 , 4, ±1

D) -

, -4, 1

Answer: B Type: BI Var: 50+ Objective: Apply the Rational Zero Theorem

Find all the zeros. 5) f (x) = x3 + 10x2 + 25x + 18 A) 2, -4 ± 7

B) -2, -4 ± 7

C) 2, -4 ± 7i

D) -2, -4 ± 7i

Answer: B Type: BI Var: 50+ Objective: Apply the Rational Zero Theorem

6) f (x) = 3x3 - 5x2 - 18x + 30 5 5 A) - , ± 6 B) , ± 6 3 3

5 C)

5 , ±6

D) -

, ±6

Answer: B Type: BI Var: 50+ Objective: Apply the Rational Zero Theorem

7) f (x) = x3 - 10x2 + 57x - 82 A) 2, 4 ± 5i B) -2, -4 ± 5i

C) -2, 4 ± 5i

D) 2, -4 ± 5i

C) ±14, ±1i

D) ± 14, ±1i

Answer: A Type: BI Var: 50+ Objective: Apply the Rational Zero Theorem

8) f (x) = x4 - 13x2 - 14 A) ±14, ±1

B) ± 14, ±1

Answer: D Type: BI Var: 29 Objective: Apply the Rational Zero Theorem 2 Apply the Fundamental Theorem of Algebra

A polynomial f (x) and one of its zeros are given. Find all the zeros. 1) f (x) = x4 - 6x3 + 8x2 + 30x - 65; 3 - 2i is a zero A) ± 5, 3 ± 2i

B) ± 5, ±3 - 2i

Answer: A Type: BI Var: 50+ Objective: Apply the Fundamental Theorem of Algebra

Page 48

C) ±5, 3 ± 2i

D) ±5, ±3 - 2i

A polynomial f (x) and one of its zeros are given. Factor f (x) as a product of linear factors. 2) f (x) = 4x3 - 23x2 + 46x + 13; 3 + 2i is a zero A) (4x + 1)(x - (3 + 2i))(x - (3 - 2i)) B) (4x - 1)(x - (3 + 2i))(x - (3 - 2i)) C) (4x + 1)(x - (3 + 2i))(x + (3 + 2i)) D) (4x - 1)(x - (3 + 2i))(x + (3 + 2i)) Answer: A Type: BI Var: 50+ Objective: Apply the Fundamental Theorem of Algebra

A polynomial f (x) and two of its zeros are given. Solve the equation f (x) = 0. 3 3) f (x) = 2x5 + 19x4 + 75x3 + 90x2 - 162x - 324; -3 - 3i and are zeros 2 3 3 A) -3, ± , -3 ± 3i B) -2, ± , -3 ± 3i 2 2 3 3 C) -2, -3, , -3 ± 3i D) -2, -3, , ±3 - 3i 2 2 Answer: C Type: BI Var: 50+ Objective: Apply the Fundamental Theorem of Algebra

Write a polynomial f (x) that meets the given conditions. 4) Degree 3 polynomial with integer coefficients with zeros-7i and

4 5

B) f (x) = 25x3 - 75x2 + 296x - 112 D) f (x) = 25x3 - 5x2 - 264x + 112

A) f (x) = 5x - 4x + 245x - 196 C) f (x) = 5x3 - 4x2 - 245x + 196 Answer: A Type: BI Var: 50+ Objective: Apply the Fundamental Theorem of Algebra

5) Polynomial of lowest degree with zeros of-2 (multiplicity 2) and3 (multiplicity 2) and with f (0) = -108 A) f (x) = x4 - 2x3 - 11x2 + 12x - 108 B) f (x) = -3x4 + 6x3 - 111x2 - 36x - 108 C) f (x) = -3x4 + 6x3 + 33x2 - 18x - 108 D) f (x) = -3x4 + 6x3 + 33x2 - 36x - 108 Answer: D Type: BI Var: 50+ Objective: Apply the Fundamental Theorem of Algebra

3 6) Polynomial of lowest degree with zeros of

1 (multiplicity 2) and

f (0) = -18 A) f (x) = 96x3 - 176x2 + 102x - 18 C) f (x) = 96x3 - 176x2 - 6x - 18 Answer: A Type: BI Var: 50+ Objective: Apply the Fundamental Theorem of Algebra

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(multiplicity 1) and with

B) f (x) = 48x3 - 88x2 + 3x - 18 D) f (x) = 48x3 - 88x2 + 51x - 18

7) Polynomial of lowest degree with real coefficients and with zeros-8 + 4i (multiplicity 1)and 0 (multiplicity 2) A) f (x) = x4 + 16x + 80 B) f (x) = x4 + 16x3 + 80x2 C) f (x) = x4 + 16x + 48 D) f (x) = x4 + 16x3 + 48x2 Answer: B Type: BI Var: 50+ Objective: Apply the Fundamental Theorem of Algebra

8) Polynomial of lowest degree with real coefficients and zeros4i and 2 - i. A) f (x) = x 4 - 4x 3 + 12x 2 + 64x - 64 B) f (x) = x 4 + 4x 3 - 12x 2 - 64x + 64 C) f (x) = x 4 - 4x 3 + 21x 2 - 64x + 80 D) f (x) = x 4 - 4x 3 + 19x 2 - 64x + 48 Answer: C Type: BI Var: 42 Objective: Apply the Fundamental Theorem of Algebra 3 Apply Descartes' Rule of Signs

Determine the number of possible positive and negative real zeros for the given function. 1) f (x) = -8x7 - 7x4 - 6x3 + 4x2 + 6x + 5 A) Positive: 1; Negative: 4 or 2 B) Positive: 4; Negative: 1 C) Positive: 1; Negative: 4 D) Positive: 4 or 2; Negative: 1 Answer: A Type: BI Var: 50+ Objective: Apply Descartes' Rule of Signs

Use Descartes' rule of signs to determine the total number of real zeros and the number of positive and negative real zeros. (Hint: First factor out x to its lowest power.) 2) f (x) = 5x11 + 2x9 + 9x7 - 9x6 A) 11 real zeros; f (x) has 1 positive real zero, 4 negative real zeros and the number 0 is a zero of multiplicity6. B) 2 real zeros; f (x) has 1 positive real zero, no negative real zeros and the number 0 is a zero of multiplicity 1. C) 6 real zeros; f (x) has 1 positive real zero, no negative real zeros and the number 0 is a zero of multiplicity5. D) 7 real zeros; f (x) has 1 positive real zero, no negative real zeros and the number 0 is a zero of multiplicity6. Answer: D Type: BI Var: 50+ Objective: Apply Descartes' Rule of Signs

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4 Find Upper and Lower Bounds

Solve the problem. 1) Determine if the upper bound theorem identifies4 as an upper bound for the real zeros of f (x). f (x) = 8x3 - 8x2 + 3x - 9 A) No B) Yes Answer: B Type: BI Var: 50+ Objective: Find Upper and Lower Bounds

2) Determine if the lower bound theorem identifies-4 as a lower bound for the real zeros of f (x). f (x) = 3x3 + 2x2 - 9x + 4 A) Yes B) No Answer: A Type: BI Var: 50+ Objective: Find Upper and Lower Bounds

Find the zeros and their multiplicities. Consider using Descartes' rule of signs and the upper and lower bound theorem to limit your search for rational zeros. 3) f (x) = 20x3 - 47x2 + 11x + 6 1 3 1 3 B) - , , and 2 (each with multiplicity 1) A) , - 5 , and -2 (each with multiplicity 1) 4 5 4 C) -1, 3, and 2 (each with multiplicity 1)

D) 1, -3, and -2 (each with multiplicity 1)

Answer: B Type: BI Var: 50+ Objective: Find Upper and Lower Bounds

4) f (x) = x9 + 10x8 + 27x7 + 20x6 + 50x5 A) 0 (multiplicity5), 5 (multiplicity 2) and ±2i (each multiplicity 1) B) 0 (multiplicity5), 5 (multiplicity 2) and ± 2 (each multiplicity 1) C) 0 (multiplicity5), 5 (multiplicity 2) and ±i 2 (each multiplicity 1) D) 0 (multiplicity5), -5 (multiplicity 2) and ±i 2 (each multiplicity 1) Answer: D Type: BI Var: 50+ Objective: Find Upper and Lower Bounds

5) f (x) = 3x4 - 32x3 + 122x2 - 188x + 80 2 A) -4, - , 3 ± i (each multiplicity 1) 3 2 C) 4, , ±3i (each multiplicity 1) 3 Answer: D Type: BI Var: 50+ Objective: Find Upper and Lower Bounds

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2 B) -4, -

, ±3i (each multiplicity 1)

2 D) 4,

, 3 ± i (each multiplicity 1)

5 Mixed Exercises

Determine if the statement is true or false. 1) A polynomial with real coefficients of degree 11 must have at least one real zero. A) False B) True Answer: B Type: BI Var: 50+ Objective: Mixed Exercises 6 Expanding Your Skills

Solve the problem. 1) A food company originally sells cereal in boxes with dimensions25 cm by 14 cm by 10 cm. To make more profit, the company decreases each dimension of the box byx centimeters but keeps the price the same. If the new volume is 2,208 cm3 by how much was each dimension decreased? A) 1 cm B) 2 cm C) 4 cm D) 3 cm Answer: B Type: BI Var: 50+ Objective: Expanding Your Skills

2) A rectangle is bounded by the x-axis and a parabola defined byy = 4 - x 2. What are the dimensions of the rectangle if the area is 6 cm2? Assume that all units of length are in centimeters. y

(x, y)

A) 2 cm by 3 cm 13 - 1 B) 1 cm by 6 cm or

1 + 13 by

C) 2 cm by 3 cm or 13 - 1 cm by D) 13 - 1 cm by

1 + 13 cm 2

Answer: C Type: BI Var: 4 Objective: Expanding Your Skills

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1 + 13 cm 2

3) f x = x 4 - 4x 2 - 221 a. Factor the polynomial over the set of real numbers. b. Factor the polynomial over the set of complex numbers. A) a. f x = x - 13 x + 13 x 2 + 17 b. f x = x - 13 x + 13 x + 17i x - 17i B) a. f x = x - 17 x + 17 x - 13 x + 13 b. f x = x - 17 x + 17 x + 13 x - 13 C) a. f x = x - 17 x + 17 x 2 + 13 b. f x = x - 17 x + 17 x + 13i x - 13i D) a. f x = x 2 + 17 x 2 + 13 b. f x = x -

17i x + 17i x + 13i x - 13i

Answer: C Type: BI Var: 50+ Objective: Expanding Your Skills

3.5 Rational Functions 0 Concept Connections

Provide the missing information. 1) A

function is a function that can be written in the formf (x) =

p(x)

where p(x) and

q(x) q(x) are polynomials, and q(x) ≠ 0. Answer: rational Type: SA Var: 1 Objective: Concept Connections

2) The domain of a rational function defined byf (x) =

p(x) q(x)

. Answer: q(x) Type: SA Var: 1 Objective: Concept Connections

3) The notation x → ∞ is read as

Answer: x approaches infinity Type: SA Var: 1 Objective: Concept Connections

4) The notation x → 5⁻ is read as Answer: x approaches 5 from the left. Type: SA Var: 1 Objective: Concept Connections

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is all real numbers excluding the zeros of

5) The line x = c is a asymptote of the graph of a function f if f (x) approaches infinity or negative infinity asx approaches from either the left or right. Answer: vertical; c Type: SA Var: 1 Objective: Concept Connections

6) To locate the vertical asymptotes of a function, determine the real numbersx where the denominator is zero, but the numerator is . Answer: nonzero Type: SA Var: 1 Objective: Concept Connections

2x3 + 7 7) Given f (x) =

5x3

, the graph of f will behave like the graph of which of the following functions

for large2 values of |x|? 2x a. y = b. y = 5x

c. y =

d. y = x 5

Answer: c Type: SA Var: 1 Objective: Concept Connections

8) Consider a rational function in which the degree of the numerator isn and the degree of the denominator is m. If n m, then the x-axis is the horizontal asymptote. If n m, then the function has no horizontal asymptote. Answer: <; > Type: SA Var: 1 Objective: Concept Connections

9) The graph of f (x) =

- 3 is the graph of y =

x+6

shifted (left/right) 6 units and (up/down) 3

units. Answer: left; down Type: SA Var: 1 Objective: Concept Connections

10) A rational function will have a slant asymptote if the degree of the numerator is exactly greater than the degree of the denominator. Answer: one Type: SA Var: 1 Objective: Concept Connections

Page 54

11) To find an equation for the slant asymptote of a rational function, begin by dividing the numerator by the . Answer: denominator Type: SA Var: 1 Objective: Concept Connections

12) If C (x) represents the cost to manufacture x items, then C (x) = cost per item. C (x) Answer: x

represents the average

Type: SA Var: 1 Objective: Concept Connections 1 Apply Notation Describing Infinite Behavior of a Function

Write the domain in interval notation. x2 - 81 1) f (x) = x+9 A) (-∞, 9) ∪ (9, ∞) C) (-∞, ∞)

B) (-∞, -9) ∪ (-9, ∞) D) (-∞, -9] ∪ [-9, ∞)

Answer: B Type: BI Var: 18 Objective: Apply Notation Describing Infinite Behavior of a Functio

7x - 1

2) f (x) =

4x2 - 27x - 81 9 ∪ - 9, 1 ∪ 1, 9 ∪ (9, ∞) A) -∞, 4 4 7 7 9 9 C) -∞, ∪ - , 9 ∪ (9, ∞) 4 4

9 9 ∪ ,∞ 4 4 1 9 9 1 D) -∞, - 9 ∪ -9, ∪ - , ∪ ,∞ 7 7 4 4 B) -∞, - 9 ∪ -9, -

Answer: C Type: BI Var: 50+ Objective: Apply Notation Describing Infinite Behavior of a Functio

3) f (x) =

16x x + 100 2

A) (-∞, -10) ∪ (-10, 10) ∪ (10, ∞) C) (-∞, 10) ∪ (10, ∞)

B) (-∞, 0) ∪ (0, ∞) D) (-∞, ∞)

Answer: D Type: BI Var: 50+ Objective: Apply Notation Describing Infinite Behavior of a Functio

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Refer to the graph of the function and complete the statements. 4) As x → -∞, f (x) → . As x → ∞, f (x) → . y 15 10 5

-15

-10

-5

-5 -10 -15

B) ∞; -∞

A) 3; 3

C) -∞; ∞

D) -5; -5

Answer: A Type: BI Var: 50+ Objective: Apply Notation Describing Infinite Behavior of a Functio

5) As x → -∞, f (x) → As x → ∞, f (x) →

. . y 10 8 6 4 2

-10 -8

-6

-4

-2

-2 -4 -6 -8 -10

A) -∞; ∞

B) 5, 5

C) ∞; ∞

Answer: D Type: BI Var: 50+ Objective: Apply Notation Describing Infinite Behavior of a Functio

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D) 1; 1

The domain is The range is

. . y 15 10 5

-15

-10

-5

-5 -10 -15

A) (-∞, -7) ∪ (-7, ∞); (-∞, -5) ∪ (-5, ∞) C) (-∞, -5) ∪ (-5, ∞); (-∞, -7) ∪ (-7, ∞)

B) (-∞, ∞); (-∞, -5) ∪ (-5, ∞) D) (-∞, -5) ∪ (-5, ∞); (-∞, ∞)

Answer: C Type: BI Var: 50+ Objective: Apply Notation Describing Infinite Behavior of a Functio

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7) The domain is The range is

. . y 10 8 6 4 2

-10 -8

-6

-4

-2

-2 -4 -6 -8 -10

A) (-9, ∞); (-∞, 6) ∪ (6, ∞);

C) (-∞, ∞); (-∞, ∞)

B) (-∞, ∞); (-9, ∞)

D) (-∞, 6) ∪ (6, ∞); (-9, ∞)

Answer: D Type: BI Var: 50+ Objective: Apply Notation Describing Infinite Behavior of a Functio

The graph is decreasing over the interval The graph is increasing over the interval

. .

y 15 10 5

-15

-10

-5

-5 -10 -15

A) (-∞, -2); (-2, ∞) Answer: A

B) ∅; (-∞, ∞)

C) (-∞, ∞); ∅

Type: BI Var: 50+ Objective: Apply Notation Describing Infinite Behavior of a Functio

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D) (-2, ∞); (-∞, -2)

9) The graph is increasing over the interval(s) The graph is decreasing over the interval(s)

. .

y 10 8 6 4 2 -10 -8

-6

-4

-2

10 x

-2 -4 -6 -8 -10

A) (-∞, -3) ∪ (-3, ∞); Never decreasing C) (-∞, 3) ∪ (3, ∞); Never decreasing

B) Never increasing; (-∞, -3) ∪ (-3, ∞) D) Never increasing; (-∞, 3) ∪ (3, ∞)

Answer: B Type: BI Var: 50+ Objective: Apply Notation Describing Infinite Behavior of a Functio

Page 59

10)

As x → -3-, f (x) → As x → -3+, f (x) →

. . y

15 10 5

-15

- 10

-5

-5 -10 -15

A) -∞; ∞

B) -2; -2

C) -∞; -∞

D) ∞; ∞

Answer: C Type: BI Var: 50+ Objective: Apply Notation Describing Infinite Behavior of a Functio -

11) As x → -5 , f (x) → + As x → -5 , f (x) →

. . y 10 8 6 4 2

-10 -8

-6

-4

-2

10 x

-2 -4 -6 -8 -10

A) -5, -5

B) ∞; -∞

C) 6; 6

Answer: B Type: BI Var: 50+ Objective: Apply Notation Describing Infinite Behavior of a Functio

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D) -∞; ∞

2 Identify Vertical Asymptotes

Determine the vertical asymptote(s) of the graph of the function. x + 9 1) f (x) = 2 7x + 9x - 10 5 5 A) x = -2 and x = B) x = 2 and x = 7 7 C) x = 9

D) None

Answer: A Type: BI Var: 50+ Objective: Identify Vertical Asymptote

2) f (x) =

x + 81 A) x = 9

B) x = 9 and x = 0

C) x = 9 and x = -9

Answer: D Type: BI Var: 9 Objective: Identify Vertical Asymptote

x+4

3) h (x) =

2x2 + 11x + 5 1 5 A) x = and x = 2 2 C) x =

1 and x = -5 2

B) x = -

and x = -5

2 D) x = 4 and x = 5

Answer: B Type: BI Var: 50+ Objective: Identify Vertical Asymptote

a2 + 4 4) f (a) =

a2 + 3a - 5 5 A) a = 3 -3 + 29 -3 - 29 C) a = and a = 2 2

Answer: C Type: BI Var: 50+ Objective: Identify Vertical Asymptote

Page 61

B) a = 2 and a = -2 D) None

D) None

5) m(x) =

x 2

x +4 A) x = 4 and x = 0

B) x = -4 and x =

1 4

C) None

D) x = 4 and x = -4

Answer: C Type: BI Var: 9 Objective: Identify Vertical Asymptote

Identify Horizontal Asymptotes

a. Identify the horizontal asymptote (if any). b. If the graph of the function has a horizontal asymptote, determine the point where the graph crosses th horizontal asymptote. 7 1) f (x) = x2 - 7x + 9 A) a. y = 0 b. (0, 0) C) a. y = 7 b. (0, 7)

B) a. y = 0 b. Graph does not cross y = 0. D) a. No horizontal asymptote b. Not applicable

Answer: B Type: BI Var: 50+ Objective: Identify Horizontal Asymptote

2x2 + 2x - 5 2) f (x) =

x2 + 2

A) a. y = 0 5 b. - , 0 2

B) a. y = 2 9 b. , 2 2

C) a. y = 0 b. Graph does not cross y = 0.

D) a. No horizontal asymptote b. Not applicable

Answer: B Type: BI Var: 50+ Objective: Identify Horizontal Asymptote

Page 62

5x - 10

3) s(x) =

x2 + 5x - 4 A) a. y = 2 b. (0, 2)

B) a. y = 0 5 b. (2

C) a. y = 0 b. (2, 0)

D) a. No horizontal asymptote b. Not applicable

, 0)

Answer: C Type: BI Var: 50+ Objective: Identify Horizontal Asymptote

x2 - 7x2 + 1 4) f (x) =

8x + 7

A) a. y = 1 1 b. , 1 7

B) a. y = 0 1 b. , 0 7

1 D) a. No horizontal asymptote b. Not applicable

C) a. y = 8 b.

1 1 , 7 8

Answer: D Type: BI Var: 50+ Objective: Identify Horizontal Asymptote

Solve the problem.

10x 3 + 8

5) The graph of f x = A) y =

10x

13x 3

will behave like which function for large values ofx ? B) y =

13 Answer: B Type: BI Var: 50+ Objective: Identify Horizontal Asymptote

Page 63

10 13

C) y =

10 13

D) y =

10 13x

Identify Slant Asymptotes

Identify the asymptotes. x3 - 4x2 - 9x + 8 1) f (x) = x2 - 6 A) Vertical asymptotes: x = 6 and x = Horizontal asymptote: y = 1 C) Vertical asymptotes: x = 6 and x = Slant asymptote: y = -3x - 16

6 6

B) Vertical asymptotes: x = 6 and x = Slant asymptote: y = x - 4 D) Vertical asymptotes: x = 6 and x = -6 Horizontal asymptote: y = 1

Answer: B Type: BI Var: 50+ Objective: Identify Slant Asymptote

4x3 + 3x + 3 2) f (x) =

2x2 - 4x + 3

A) Horizontal asymptote: y = 1 C) Slant asymptote: y = 13x - 9

B) Slant asymptote: y = 2x + 4 D) Horizontal asymptote: y = 2

Answer: B Type: BI Var: 50+ Objective: Identify Slant Asymptote

2x2 + 7 3) f (x) =

A) Vertical asymptote: x = 2; C) Vertical asymptote: x = 0; Slant asymptote: y = 2x

B) Vertical asymptote: x = 0; Slant asymptote: y = 2x + 7 D) Vertical asymptote: x = 0;

Answer: C Type: BI Var: 44 Objective: Identify Slant Asymptote

-2x2 + 5x - 4 4) f (x) =

x+3

A) Horizontal asymptote: x = 0 Vertical asymptote: x = 3 C) Vertical asymptote: x = -3 Answer: B Type: BI Var: 50+ Objective: Identify Slant Asymptote

Page 64

B) Vertical asymptote: x = -3 Slant asymptote: y = -2x + 11 D) Horizontal asymptote: x = 0 Slant asymptote: y = -2x + 11

4x + 5

5) r(x) =

x3 - 5x2 - 9x + 45 A) Vertical asymptotes: x = -3, x = 3, and x = 5; horizontal asymptote: y = 0 B) Vertical asymptotes: x = -3, x = 3, and x = 5; horizontal asymptote: y = -

5 4

C) horizontal asymptote: y = 0 D) Vertical asymptotes: x = -3, x = 3, and x = -5; horizontal asymptote: y = 0 Answer: A Type: BI Var: 50+ Objective: Identify Slant Asymptote Graph Rational Functions

1 Graph the function by using a transformation of the graph of y = . x 1 1) f (x) = x+2

B) y

-5

D) y

-5

Answer: C Type: BI Var: 10 Objective: Graph Rational Functions

Page 65

-5

Graph the function by using transformations of the graph of y = 2) m(x) =

1 x2

. x2

B) 10 y

10 y

-10 -8 -6 -4 -2

-2

-10 -8 -6 -4 -2

-2

-4

-6

-8

-10

D) 10 y

10 y

-10 -8 -6 -4 -2

-2

-10 -8 -6 -4 -2

-2

-4

-6

-8

-10

Answer: B Type: BI Var: 13 Objective: Graph Rational Functions

Page 66

3) f (x) =

2 -2

(x - 1) A)

-10

10 y

-5

-10

10 y

-5

-10

-5

-10

Answer: D Type: BI Var: 50+ Objective: Graph Rational Functions

Page 67

-10

Graph the function. 5 4) f (x) = 3x + 8 A)

B) y

-5

D) y

-5

Answer: B Type: BI Var: 50+ Objective: Graph Rational Functions

Page 68

-5

5) f (x) =

4x x2 - 4x - 5

B) y

-5

D) y

-5

Answer: A Type: BI Var: 50+ Objective: Graph Rational Functions

Page 69

-5

6) f (x) =

x2 + 6x + 9 x

B) 15 y

15 y

15x

-15

D) 15 y

15 y

15x

-15

Answer: D Type: BI Var: 20 Objective: Graph Rational Functions

Page 70

15x

-15

15x

-15

-4x2 7) f (x) =

x2 + 4

B) y

-5

D) y

-5

Answer: D Type: BI Var: 32 Objective: Graph Rational Functions

Page 71

-5

8) p(x) =

6 x2 - 16

B) 10 y

10 y

-10 -8 -6 -4 -2

-2

-10 -8 -6 -4 -2

-4

-6

-8

-10

D) 10 y

10 y

-10 -8 -6 -4 -2

-2

-10 -8 -6 -4 -2

-2

-4

-6

-8

-10

Answer: C Type: BI Var: 20 Objective: Graph Rational Functions

Page 72

-2

9) g(x) =

x3 + 5x2 - 4x - 20 x2 + 3x B)

A) 10 y

10 y

-10 -8 -6 -4 -2

-2

-4

-6

-8

-10

D) 10 y

10 y

-10 -8 -6 -4 -2

-2

-10 -8 -6 -4 -2

-2

-4

-6

-8

-10

Answer: B Type: BI Var: 6 Objective: Graph Rational Functions

Page 73

-10 -8 -6 -4 -2

10) d(x) =

x2 + 3x - 4 x+3

B) 10 y

10 y

-10 -8 -6 -4 -2

-2

-4

-6

-8

-10

D) 10 y

10 y

-10 -8 -6 -4 -2

-2

-10 -8 -6 -4 -2

-2

-4

-6

-8

-10

Answer: C Type: BI Var: 50+ Objective: Graph Rational Functions

Page 74

-10 -8 -6 -4 -2

11) f x =

x-2 x-3

B) 10 y

10 y

-10 -8 -6 -4 -2 -2

8 10 x

-4

-6

-8

-10

8 10 x

D) 10 y

10 y

-10 -8 -6 -4 -2 -2

8 10 x

-10 -8 -6 -4 -2 -2

-4

-6

-8

-10

Answer: D Type: BI Var: 16 Objective: Graph Rational Functions

Page 75

-10 -8 -6 -4 -2 -2

Use Rational Functions in Applications

Solve the problem. 1) A sports trainer has monthly costs of $50.51 for phone service and $43.53 for his website and advertising. In addition he pays a $10 fee to the gym for each session in which he trains a client. His monthly costs can be represented by the function C(x) = 94.04 + 10x, where x is the number of training sessions. Write a function representing the average costC (x) for x sessions. 94.04 A) C (x) = 94.04 + 10x2 B) C (x) = + 10x x C) C (x) =

94.04 + 10x x

D) C (x) = 94.04x + 10x2

Answer: C Type: BI Var: 50+ Objective: Use Rational Functions in Applications

2) An on-demand printing company has monthly overhead costs of$1,900 in rent, $450 in electricity, $80 for phone service, and $230 for advertising and marketing. The printing cost is$40 per thousand pages for paper and ink. The average cost for printing x thousand pages can be represented by the function 2,660 + 40x C (x) = . x For a given month, if the printing company could print an unlimited number of pages, what value would the average cost per thousand pages approach? What does this mean in the context of the problem? A) The average cost would approach $40 per thousand pages or equivalently $0.04 per page. This is the cost per page in the absence of fixed costs. B) The average cost would approach infinity. The more pages the company prints, the higher the average cost. C) The average cost would approach $2,660 per thousand pages. This is the total of the fixed monthly costs. D) The average cost would approach $0 per thousand pages. The more pages the company prints, the lower the average cost. Answer: A Type: BI Var: 50+ Objective: Use Rational Functions in Applications

Page 76

3) A power company burns coal to generate electricity. The cost C(x) (in $1000) to remove x% of the air pollutants is given by 800x C (x) = . 100 - x If the power company budgets $1.2 million for pollution control, what percentage of the air pollutants can be removed? Round to two decimal places, if necessary. A) 0.6%

B) 0.15%

C) 14.98%

D) 60%

Answer: D Type: BI Var: 8 Objective: Use Rational Functions in Applications

4) A power company burns coal to generate electricity. The cost C x (in $1000) to remove x% of the air pollutants is given by: 500x C x = 140 - x a. Compute the cost to remove 25% of the air pollutants. b. If the power company budgets $1.4 million for pollution control, what percentage of the air pollutants can be removed? 2500000 2500000 2500 2500 A) a. B) a. C) a. D) a. 23 23 23 23 b. 103.16%

b. 135.17%

b. 103.16%

Answer: C Type: BI Var: 50+ Objective: Use Rational Functions in Applications 7

Expanding Your Skills

Write the domain in interval notation and identify any vertical asymptotes. x 2 + x - 56 1) f x = x-7 A) -∞, 7 ∪ 7, ∞ ; x = 7 C) -∞, ∞ ; None Answer: B Type: BI Var: 9 Objective: Expanding Your Skills

Page 77

B) -∞, 7 ∪ 7, ∞ ; None D) -∞, ∞ ; x = 7

b. 135.17%

2x - 4

2) f x =

x 2 + 5x - 14 5 y 4 3 2 1 -5 -4 -3 -2 -1 -1

-2 -3 -4 -5

A) -∞, -7 ∪ -7, 2 ∪ 2, ∞ ; x = -7 C) -∞, -2 ∪ -2, 7 ∪ 7, ∞ ; x = -7, x = 2

B) -∞, -7 ∪ -7, 2 ∪ 2, ∞ ; x = -7, x = 2 D) -∞, -2 ∪ -2, 7 ∪ 7, ∞ ; x = -7

Answer: A Type: BI Var: 50+ Objective: Expanding Your Skills

3.6 Polynomial and Rational Inequalities 0 Concept Connections

Provide the missing information. 1) Let f (x) be a polynomial. An inequality of the formf (x) < 0, f (x) > 0, f (x) ≥ 0, or f (x) ≤ 0 is called a inequality. If the polynomial is of degree , then the inequality is also called a quadratic inequality. Answer: polynomial; 2 Type: SA Var: 1 Objective: Concept Connections

2) Let f (x) be a rational expression. An inequality of the formf (x) < 0, f (x) > 0, f (x) ≥ 0, or f (x) ≤ 0 is called a inequality. Answer: rational Type: SA Var: 1 Objective: Concept Connections

3) The solutions to an inequality f (x) < 0 are the values of x on the intervals where f (x) is (positive/negative). Answer: negative Type: SA Var: 1 Objective: Concept Connections

Page 78

4) The solution set for the inequality (x + 10)2 ≥ -4 is inequality (x + 10)2 ≤ -4 is .

, whereas the solution set for the

Answer: (-∞, ∞); { } Type: SA Var: 1 Objective: Concept Connections 1 Solve Polynomial Inequalities

The graph of y = f (x) is given. Solve the inequality. 1) f (x) < 0

A) (-∞, 1] ∪ [2, ∞)

B) (1, 2)

C) (-∞, 1) ∪ (2, ∞)

D) [1, 2]

C) (-∞, 3)

D) (-∞, 0) ∪ (0, 3)

Answer: C Type: BI Var: 6 Objective: Solve Polynomial Inequalities

2) f (x) < 0

A) {0} ∪ (3, ∞)

B) (3, ∞)

Answer: B Type: BI Var: 6 Objective: Solve Polynomial Inequalities

Page 79

3) f (x) ≥ 0 8 y 7 6 5 4 3 2 1 -8 -7 -6 -5 -4 -3 -2 -1-1

1 2 3 4 5 6 7 8x

-2 -3 -4 -5 -6 -7 -8

A) (-∞, -10]

B) (-∞, ∞)

C) [-10, ∞)

D) { }

Answer: D Type: BI Var: 50+ Objective: Solve Polynomial Inequalities

Solve the inequality. Write the solution set in interval notation. 4) (7x - 9)(5x - 9) < 0 9 9 9 9 9, 9 -∞, ,∞ -∞, ,∞ B) C) ∪ ∪ A) 7 5 7 5 7 5 Answer: D Type: BI Var: 50+ Objective: Solve Polynomial Inequalities

5) -x2 - 15x - 54 ≥ 0 A) (-∞, -9) ∪ (-6, ∞) C) [-9, -6]

B) (-9, -6) D) (-∞, -9] ∪ [-6, ∞)

Answer: C Type: BI Var: 50+ Objective: Solve Polynomial Inequalities

6) 7y2 + 5y ≤ -9(7y + 5) 5 A) (-∞, -9] ∪ - , ∞ 7 C) (-9, ∞) Answer: B Type: BI Var: 50+ Objective: Solve Polynomial Inequalities

Page 80

B) -9, -

5 7

D) (-∞, -9]

D) 9 , 9 7 5

7) d2 > 9d A) (0, 9)

B) (9, ∞)

C) (-∞, 0) ∪ (9, ∞)

D) (-∞, 9]

Answer: C Type: BI Var: 16 Objective: Solve Polynomial Inequalities

8) - x2 - 10x - 24 ≤ 0 A) (-∞, -4] ∪ [6, ∞) C) (-∞, 4] ∪ [6, ∞)

B) (-∞, ∞) D) (-∞, -6] ∪ [-4, ∞)

Answer: D Type: BI Var: 20 Objective: Solve Polynomial Inequalities

9) 2x3 + 9x2 < 32x + 144 9

A) (-∞, -4) ∪ (4, ∞)

B) -

C) (-4, 4)

D) -∞, -

, -4 ∪ (4, ∞) 9 2

∪ (-4, 4)

Answer: D Type: BI Var: 15 Objective: Solve Polynomial Inequalities

10) a2 + 6a + 9 > 0 A) (-∞, 3) ∪ (3, ∞) C) (-∞, ∞)

B) (-∞, -3) ∪ (3, ∞) D) (-∞, -3) ∪ (-3, ∞)

Answer: D Type: BI Var: 9 Objective: Solve Polynomial Inequalities

11) -16y - 64 ≥ y2 A) (-∞, -8) ∪ (-8, ∞) C) (-∞, ∞)

B) {-8} D) { }

Answer: B Type: BI Var: 9 Objective: Solve Polynomial Inequalities

12) - x2 - 6x + 16 < 0 A) (-∞, ∞)

B) (-∞, -2) ∪ (8, ∞)

Answer: D Type: BI Var: 42 Objective: Solve Polynomial Inequalities

Page 81

C) (-8, 2) ∪ (2, ∞)

D) (-∞, -8) ∪ (2, ∞)

13) 4x2 - 19x - 30 ≥ 0 5 A) (-∞, -6] ∪ , ∞ 4 C) -6,

5 B) -∞, -

∪ [6, ∞)

5 4

D) -

Answer: B Type: BI Var: 50+ Objective: Solve Polynomial Inequalities

14) t 2 - 8t + 16 ≥ 0 A) (-∞, ∞) C) (-∞, -4) ∪ (4, ∞)

B) (-∞, 4) ∪ (4, ∞) D) (-∞, -4) ∪ (-4, ∞)

Answer: A Type: BI Var: 9 Objective: Solve Polynomial Inequalities

15) (k - 8)(k - 3)(k + 5) < 0 A) (-∞, -5) ∪ (3, 8) C) (-∞, -8) ∪ (-3, 5)

B) (-5, 3) ∪ (8, ∞) D) (-8, -3) ∪ (5, ∞)

Answer: A Type: BI Var: 50+ Objective: Solve Polynomial Inequalities

16) -4u(u + 2)2(5 - u) > 0 A) (-∞, -2) ∪ (-2, 0) ∪ (5, ∞) C) (-5, ∞) ∪ (-∞, 2) ∪ (2, 0)

B) (-2, 0) ∪ (5, ∞) D) (-∞, -2) ∪ (0, 5)

Answer: A Type: BI Var: 45 Objective: Solve Polynomial Inequalities

17) w4 - 29w2 + 100 ≤ 0 A) (-5, -2) ∪ (2, 5)

B) (-∞, -5) ∪ (5, ∞)

C) [-5, -2] ∪ [2, 5]

D) (-∞, -5] ∪ [5, ∞)

C) (-∞, ∞)

D) -

Answer: C Type: BI Var: 10 Objective: Solve Polynomial Inequalities

18) -2 > (5x + 8)2 8 A) { }

B) -

Answer: A Type: BI Var: 50+ Objective: Solve Polynomial Inequalities

Page 82

,∞

8 ,∞ 5

19) 4w2 > 4w - 1 1 ∪ - 1, ∞ A) -∞, 4 4

B) -∞, 1 ∪ 1 , ∞ 4 4 1 1 D) -∞, ∪ ,∞ 2 2

C) (-∞, ∞) Answer: D Type: BI Var: 16 Objective: Solve Polynomial Inequalities

20) (x + 7)(x + 11) ≤ -4 A) {-9}

B) (-∞, ∞)

C) [-11, -7]

D) { }

C) (3, 4)

D) (3, ∞)

Answer: A Type: BI Var: 36 Objective: Solve Polynomial Inequalities 2 Solve Rational Inequalities

The graph of y = f (x) is given. Solve the inequality. 1) f (x) > 0 8 y 7 6 5 4 3 2 1 -8 -7 -6 -5 -4 -3 -2 -1-1

1 2 3 4 5 6 7 8x

-2 -3 -4 -5 -6 -7 -8

A) [3, 4]

B) (-∞, 3)

Answer: C Type: BI Var: 50+ Objective: Solve Rational Inequalities

Page 83

2) f (x) ≤ 0 8 y 7 6 5 4 3 2 1 -8 -7 -6 -5 -4 -3 -2 -1-1

1 2 3 4 5 6 7 8x

-2 -3 -4 -5 -6 -7 -8

A) (-∞, ∞) C) (-∞, -3] ∪ [-3, ∞)

B) { } D) {-3}

Answer: B Type: BI Var: 50+ Objective: Solve Rational Inequalities

Solve the inequality. Write the solution set in interval notation. x+1 3) ≤0 x-2 A) (-2, 1]

B) (-1, 2)

C) (-2, 1)

D) [-1, 2)

C) (-∞, 1] ∪ [6, ∞)

D) (-∞, -6) ∪ (1, ∞)

Answer: D Type: BI Var: 50+ Objective: Solve Rational Inequalities

x+6

x-1 A) (-∞, -6] ∪ [1, ∞)

B) (-∞, 1) ∪ (6, ∞)

Answer: D Type: BI Var: 50+ Objective: Solve Rational Inequalities

Page 84

3x - 4 <0 3 + 2x A) -∞, C) -

3 4 ∪ ,∞ 3 2

B) -∞, -

4 3 , 3 2

D) -

3 4 3 ∪ , 2 3 2

3 4 , 2 3

Answer: D Type: BI Var: 50+ Objective: Solve Rational Inequalities

9-x

≥0

x + 11 A) (-∞, -11) ∪ (9, ∞) C) (-∞, -11] ∪ [9, ∞)

B) [-11, 9) D) (-11, 9]

Answer: D Type: BI Var: 50+ Objective: Solve Rational Inequalities

40 - 8x ≤0 x2 A) [5, ∞)

B) (-∞, 5)

C) (5, ∞)

D) (-∞, 5]

C) (-∞, 0) ∪ (0, ∞)

D) (-∞, 5] ∪ [5, ∞)

C) (-∞, -2]

D) (-∞, -2)

Answer: A Type: BI Var: 50+ Objective: Solve Rational Inequalities

x2 x2 + 25

≥0

A) (-∞, ∞)

B) {0}

Answer: A Type: BI Var: 9 Objective: Solve Rational Inequalities

≥7

x+2 A) (-∞, 0)

B) (-∞, 0]

Answer: D Type: BI Var: 50+ Objective: Solve Rational Inequalities

Page 85

10)

y-4 41 A) (-∞, 4) ∪

41 ,∞ 9

,∞

∪ (4, ∞)

D) 4,

31 C) -∞, -

41 9

Answer: A Type: BI Var: 50+ Objective: Solve Rational Inequalities

11)

≥3

m-9 35 3

A) 9,

B) -

C) (-∞, 9) ∪

35 ,∞ 3

D) -∞,

35 3

Answer: A Type: BI Var: 50+ Objective: Solve Rational Inequalities

12)

9h > -7 h -1 7 A) -∞, 7

∪ (1, ∞)

B) -∞,

7 ∪ (1, ∞) 16

7 ,∞ 16

Answer: B Type: BI Var: 50+ Objective: Solve Rational Inequalities

13)

t- 6 t2+4

A) (-4, 6)

B) (-∞, 6)

Answer: B Type: BI Var: 50+ Objective: Solve Rational Inequalities

Page 86

C) { }

D) (-∞, ∞)

14)

r+7 r2 + 2

>0 B) (-7, ∞) D) (-∞, -7) ∪ (-2, ∞)

A) { } C) (-∞, ∞) Answer: B Type: BI Var: 50+ Objective: Solve Rational Inequalities

15)

2 7 ≤ 1-x 3- x A) (1, 3)

B) -∞,

19 5

C) (-∞, 1) ∪ (3, ∞)

D) (1, 3) ∪

19 ,∞ 5

Answer: D Type: BI Var: 50+ Objective: Solve Rational Inequalities 3 Solve Applications Involving Polynomial and Rational Inequalities

Solve the problem. 1) The population P(t) of a bacteria culture is given by P(t) = -1,500t2 + 48,000t + 18,000, where t is the time in hours after the culture is started. Determine the time(s) at which the population will be greater than 388,500 organisms. A) The population will be greater than 388,500 organisms between 13 hr and 25 hr after the culture is started. B) The population will be greater than 388,500 organisms between 13 hr and 19 hr after the culture is started. C) The population will be greater than 388,500 organisms from 19 hr after the culture is started. D) The population will be greater than 388,500 organisms from 25 hr after the culture is started. Answer: B Type: BI Var: 50+ Objective: Solve Applications Involving Polynomial and Rational Inequalitie

Page 87

2) The average round trip speed S (in mph) of a vehicle traveling a distance ofd miles in each direction 2d is given by S = where r and r are the rates of speed for the initial trip and the return trip, 1 2 d d + r1 r 2 respectively. Suppose a motorist travels 100 miles to an event and averages 50 mph for the trip to the event. Determine the average speed necessary for the return trip if the motorist wants the average speed for the round trip to be at least 60 mph. Round to one decimal place, if necessary. A) The motorist must average a speed of at least 70 mph on the return trip. B) The motorist must average a speed of at least 75.0 mph on the return trip. C) The motorist must average a speed of at least 55 mph on the return trip. D) The motorist must average a speed of at least 65 mph on the return trip. Answer: B Type: BI Var: 32 Objective: Solve Applications Involving Polynomial and Rational Inequalitie

3) A landscaping team plans to build a rectangular garden that is between240 yd2 and 360 yd2 in area. For aesthetic reasons, they also want the length to be 1.2 times the width. Determine the restrictions on the width so that the dimensions of the garden will meet the required area. Give exact values and the approximated values to the nearest tenth of a yard. A) The width should be between 20 yd and 10 6 yd. This is between approximately20.0 yd and 24.5 yd. B) The width should be between 10 2 yd and 10 3 yd. This is between approximately14.1 yd and 17.3 yd. C) The width should be between 4 30 yd and 12 5 yd. This is between approximately21.9 yd and 26.8 yd. D) The width should be between 4 15 yd and 6 10 yd. This is between approximately15.5 yd and 19.0 yd. Answer: B Type: BI Var: 19 Objective: Solve Applications Involving Polynomial and Rational Inequalitie

Solve the inequality. Write the solution set in interval notation. 4) 36z 2 - 9 < 0 1 1 1 1 A) - , B) - , 2 2 2 2 1 1 1 1 ∪ ∪ C) - ∞, ,∞ D) - ∞,,∞ 2 2 2 2 Answer: B Type: BI Var: 50+ Objective: Solve Applications Involving Polynomial and Rational Inequalitie

Page 88

5) 4p 2 ≥ 17 17 A) -

17 2

B) - ∞,-

17 ∪ 2

17 2

D) - ∞, -

17 ∪ 2

17 ,∞ 2

17 C) -

,∞

Answer: B Type: BI Var: 50+ Objective: Solve Applications Involving Polynomial and Rational Inequalitie

Solve the problem. 6) A professional fireworks team shoots an 8-in. mortar straight upwards from ground level with an initial speed of 256 ft/sec. a. Write a function modeling the vertical positions t in ft of the shell at a time t seconds after launch. b. The spectators can see the shell rising once it clears a768-ft tree line. From what period of time after launch is the shell visible? A) a. s t = -32t 2 + 256t b. The spectators can see the shell between 4 and 12 sec after launch. B) a. s t = -32t 2 + 256t b. The spectators can see the shell between 4 and 24 sec after launch. C) a. s t = -16t 2 + 256t b. The spectators can see the shell between 4 and 12 sec after launch. D) a. s t = -16t 2 + 256t + 768 b. The spectators can see the shell between 4 and 24 sec after launch. Answer: C Type: BI Var: 50+ Objective: Solve Applications Involving Polynomial and Rational Inequalitie 4 Mixed Exercises

Write the domain of the function in interval notation. 1) h(a) = 64 - a2 A) (-∞, -8] ∪ [8, ∞) B) [8, ∞)

C) [-8, 8]

D) (-8, 8)

Answer: C Type: BI Var: 11 Objective: Mixed Exercises

2) p(x) = x2 - 14 A) [- 14, 14] C) (-∞, ∞) Answer: D Type: BI Var: 10 Objective: Mixed Exercises

Page 89

B) [ 14, ∞) D) (-∞, - 14] ∪ [ 14, ∞)

3) k(x) =

1 2x + 3x - 54

A) -6,

9 2

B) { } 9

C) (-∞, -6) ∪

,∞

D) (-∞, ∞)

Answer: C Type: BI Var: 50+ Objective: Mixed Exercises

4) h x =

3x x+5

A) -∞, -5 ∪ -5, ∞

B) 0, ∞

C) -5, ∞

D) -∞, -5 ∪ 0, ∞

Answer: D Type: BI Var: 50+ Objective: Mixed Exercises

3.7 Variation 0 Concept Connections

Provide the missing information. 1) If k is a nonzero constant real number, then the statement y = kx implies that y varies

as x.

Answer: directly Type: SA Var: 1 Objective: Concept Connections

2) If k is a nonzero constant real number, then the statement y =

implies that y varies

as x.

x Answer: inversely Type: SA Var: 1 Objective: Concept Connections

3) The value of k in the variation models y = kx and y =

is called the

x Answer: constant; variation Type: SA Var: 1 Objective: Concept Connections

4) If y varies directly as two or more other variables such as x and w, then y = kxw, and we say that y varies as x and w. Answer: jointly Type: SA Var: 1 Objective: Concept Connections

Page 90

5) a. Given y = 2x, evaluate y for the given values of x. x = 1, x = 2, x = 3, x = 4, and x = 5. b. How does y change when x is doubled? c. How does y change when x is tripled? d. Complete the statement. Given y = 2x, when x increases, y (increases/decreases) proportionally. e. Complete the statement. Given y = 2x, when x decreases, y (increases/decreases) proportionally. Answer: a. 2; 4; 6; 8; 10 b. y is also doubled. c. y is also tripled. d. increases e. decreases Type: SA Var: 1 Objective: Concept Connections

6) a. Given y =

24 , evaluate y for the given values of x. x

x = 1, x = 2, x = 3, x = 4, and x = 6. b. How does y change when x is doubled? c. How does y change when x is tripled? d. Complete the statement. Given y =

, when x

x increases, y (increases/decreases) proportionally. e. Complete the statement. Given y =

, when x

x decreases, y (increases/decreases) proportionally. Answer: a. 24; 12; 8; 6; 4 b. y is one-half its original value. c. y is one-third its original value. d. decreases e. increases Type: SA Var: 1 Objective: Concept Connections

Page 91

7) The time required to drive from Atlanta, Georgia, to Nashville, Tennessee, varies the average speed at which a vehicle travels.

Answer: inversely Type: SA Var: 1 Objective: Concept Connections

8) The amount of a person’s paycheck varies

as the number of hours worked.

Answer: directly Type: SA Var: 1 Objective: Concept Connections

9) The volume of a right circular cone varies and as the height of the cylinder.

as the square of the radius of the cylinder

Answer: jointly Type: SA Var: 1 Objective: Concept Connections

10) A student’s grade on a test varies as the number of hours the student spends studying for the test. Answer: directly Type: SA Var: 1 Objective: Concept Connections 1 Write Models Involving Direct, Inverse, and Joint Variation

Write a variation model using k as the constant of variation. 1) The variable m is directly proportional to n. k k A) m = B) m = k + n C) n = m n

D) m = kn

Answer: D Type: BI Var: 2 Objective: Write Models Involving Direct, Inverse, and Joint Variation

2) The variable s varies indirectly ast. A) s = kt

B) s =

k t

C) s = k + s

D) t = ks

Answer: B Type: BI Var: 2 Objective: Write Models Involving Direct, Inverse, and Joint Variation

3) The variable v varies jointly as w and x and inversely as the fourth root of y. A) v =

4 kwx

kwx B) v = 4 y

C) v = wx

Answer: B Type: BI Var: 4 Objective: Write Models Involving Direct, Inverse, and Joint Variation

Page 92

4 y

4 D) v = wx

4) The variable n is directly proportional to the fifth power of α and inversely proportional to the fifth power of z. kα5 z5 kz5 z5 5 A) n = B) n = 5 C) n = α D) n = 5 z5 kα k α Answer: A Type: BI Var: 50+ Objective: Write Models Involving Direct, Inverse, and Joint Variation 2 Solve Applications Involving Variation

Find the constant of variation k. 1) y varies directly as x. When x is 16, y is 20. A) k = 80

B) k = 320

C) k =

16 5

D) k =

C) k =

8 3

D) k = 72

Answer: D Type: BI Var: 39 Objective: Solve Applications Involving Variation

2) p is inversely proportional to q. When q is 8, p is 3. 3 A) k = B) k = 24 8 Answer: B Type: BI Var: 13 Objective: Solve Applications Involving Variation

3) N varies jointly as t and p. When t is 6 and p is 3, N is 36. A) k = 648 B) k = 2 C) k = 18

D) k = 72

Answer: B Type: BI Var: 50+ Objective: Solve Applications Involving Variation

Solve the problem. 4) R varies directly as x2 and inversely ast when R = 30 when x = 6 and t = 12. Find R when x = 7 and t = 9. 490 140 9 3 A) R = D) R = B) R = C) R = 490 140 9 3 Answer: A Type: BI Var: 50+ Objective: Solve Applications Involving Variation

5) The yield on a bond varies inversely as the price. The yield on a particular bond is3% when the price is $150. What price is necessary for a yield of4.5%? Round to the nearest cent, if necessary. A) $225.00 B) $111.11 C) $100.00 D) $9.00 Answer: C Type: BI Var: 50+ Objective: Solve Applications Involving Variation

Page 93

6) The resistance of a wire varies directly as its length and inversely as the square of its diameter. A 50-ft wire with a 0.4-in. diameter has a resistance of 0.0125 Ω. Find the resistance of a 20-ft wire with a diameter of 0.1-in. Round to 4 decimal places if necessary. A) 0.0800 Ω B) 0.0025 Ω C) 0.0100 Ω D) 0.3162 Ω Answer: A Type: BI Var: 50+ Objective: Solve Applications Involving Variation

7) The cost, in dollars, of filling your gas tank is directly proportional to the amount purchased. If11 gallons of gas costs $13.20, how much would 14 gallons cost? A) $9.17 B) $16.80 C) $16.20 D) $18.20 Answer: B Type: BI Var: 50+ Objective: Solve Applications Involving Variation

8) The number of hours it takes to paint a house is inversely proportional to the number of people painting. If it takes 3 workers 17.7 hours to paint a certain house, how long would it take 5 workers? Round to one tenth of an hour. A) 10.6 hours B) 15.7 hours C) 53.0 hours D) 19.7 hours Answer: A Type: BI Var: 50+ Objective: Solve Applications Involving Variation

9) The area of a picture projected on a wall varies directly as the square of the distance from the projector to the wall. a. If a 12-ft distance produces a 81 ft2 picture, what is the area of the picture when the projection unit is moved to a distance of 16 ft from the wall? b. If the projected image is 324 ft2, how far is the projector from the wall? A) a. 108 ft2; B) a. 169 ft2; C) a. 196 ft2; D) a. 144 ft2; b. 48 ft b. 36 ft b. 54 ft b. 24 ft Answer: D Type: BI Var: 50+ Objective: Solve Applications Involving Variation

10) The body mass index (BMI) of an individual varies directly as the weight of the individual and inversely as the square of the height of the individual. The body mass index for a154-lb person who is 71 in. tall is 21.48. Determine the BMI for an individual who is77 in. tall and 164 lb. Round to 2 decimal places. A) 20.42 B) 18.67 C) 19.45 D) 19.06 Answer: C Type: BI Var: 50+ Objective: Solve Applications Involving Variation

Page 94

11) The strength of a wooden beam varies jointly as the width of the beam and the square of the thickness of the beam, and inversely as the length of the beam. A beam that is48 in. long, 6 in. wide, and 4 in. thick can support a load of 1,668 lb. Find the maximum load that can be safely supported by a board that is 8 in. wide, 120 in. long, and 5 in. thick. A) 1,390 lb B) 1,676 lb C) 1,529 lb D) 1,533 lb Answer: A Type: BI Var: 50+ Objective: Solve Applications Involving Variation

12) The period of a pendulum is the length of time required to complete one swing back and forth. The period varies directly as the square root of the length of the pendulum. If it takes 1.6 sec for a 0.64 m pendulum to complete one period, what is the period of a 1.21 m pendulum? A) 2.2 sec B) 3 sec C) 2.6 sec D) 0.5 sec Answer: A Type: BI Var: 50+ Objective: Solve Applications Involving Variation

13) The amount of a pain reliever that a physician prescribes for a child varies directly as the weight of the child. A physician prescribes170 mg of the medicine for a 20-lb child. a. How much medicine would be prescribed for a40-b child? b. If 140 mg of medicine was prescribed, what is the weight of the child? A) a. 190 mg B) a. 340 mg C) a. 340 mg D) a. 190 mg b. 16.47 lb b. -5 lb b. 16.47 lb b. -5 lb Answer: C Type: BI Var: 50+ Objective: Solve Applications Involving Variation

Page 95

Chapter 4 Exponential and Logarithmic Functions 4.1 Inverse Functions 0 Concept Connections

Provide the missing information. 1) Given the function f : = {(1, 2), (2, 3), (3, 4)} write the set of ordered pairs representing f -1 Answer: {(2, 1), (3, 2), (4, 3)} Type: SA Var: 1 Objective: Concept Connections

2) A function f is a a ≠ b, then f (a) ≠ f (b).

function if for a and b in the domain of f, if

Answer: one-to-one Type: SA Var: 1 Objective: Concept Connections

3) If no horizontal line intersects the graph of a function f in more than one point then f is a function. Answer: one-to-one Type: SA Var: 1 Objective: Concept Connections

4) The graph of a function and its inverse are symmetric with respect to the line

Answer: y = x Type: SA Var: 1 Objective: Concept Connections

5) The function f : = {(1, 5), (-2, 3), (-4, 2), (2, 5)} (is/is not) a one-to-one function. Answer: is not Type: SA Var: 1 Objective: Concept Connections

6) A function defined by y = f (x) (is/is not) a one-to-one function if no horizontal line intersects the graph of f in more than one point. Answer: is Type: SA Var: 1 Objective: Concept Connections

7) Given a one-to-one function defined by y = f (x), if f (a) = f (b), then a Answer: = Type: SA Var: 1 Objective: Concept Connections

Page 341

8) Let f be a one-to-one function and let g be the inverse of f. Then (f ∘ g)(x) = (g ∘ f ) (x) =

and

Answer: x ; x Type: SA Var: 1 Objective: Concept Connections

9) The notation of f.

is often used to represent the inverse of a function f and not the reciprocal

-1

Answer: f

Type: SA Var: 1 Objective: Concept Connections

10) If (a, b) is a point on the graph of a one-to-one function f, then the corresponding ordered pair -1

is a point on the graph of f

Answer: (b, a) Type: SA Var: 1 Objective: Concept Connections

11) The function defined by f (x) = x2 - 9 (is/is not) a one-to-one function, whereas g(x) = x2 - 9; x ≥ 0 (is/is not) a one-to-one function. Answer: is not; is Type: SA Var: 1 Objective: Concept Connections 1 Identify One-to-One Functions

A relation in x and y is given. Determine if the relation defines y as a one-to-one function of x. 1) {(-6, 8), -5, -8), (4, -6), (1, 3)} A) Yes B) No Answer: A Type: BI Var: 50+ Objective: Identify One-to-One Functions

A) Yes Answer: A Type: BI Var: 8 Objective: Identify One-to-One Functions

Page 342

B) No

3) x y 1.5 1.12 5.8 -0.34 -6.2 -2.56 4.3 -0.66 A) Yes

B) No

Answer: A Type: BI Var: 50+ Objective: Identify One-to-One Functions

Determine if the relation defines y as a one-to-one function of x. 4) 5 y 4 3 2 1 -5 -4 -3 -2 -1

-1

-2 -3 -4 -5

A) No

B) Yes

Answer: B Type: BI Var: 50+ Objective: Identify One-to-One Functions

5) 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1-1

1 2 3 4 5 6 x

-2 -3 -4 -5 -6

A) No Answer: A Type: BI Var: 32 Objective: Identify One-to-One Functions

Page 343

B) Yes

6) 6 5 4 3 2 1

-6 -5 -4 -3 -2 -1-1

1 2 3 4 5 6 x

-2 -3 -4 -5 -6

A) Yes

B) No

Answer: A Type: BI Var: 5 Objective: Identify One-to-One Functions

7) 5 y 4 3 2 1 -2

-1

-2 -3 -4 -5

A) No Answer: A Type: BI Var: 12 Objective: Identify One-to-One Functions

Page 344

B) Yes

8) 5 y 4 3 2 1 -5 -4 -3 -2 -1 -1

-2 -3 -4 -5

A) No

B) Yes

Answer: A Type: BI Var: 20 Objective: Identify One-to-One Functions

Use the definition of a one-to-one function to determine if the function is one-to-one. 9) f (x) = -2x2 - 13 A) Yes B) No Answer: B Type: BI Var: 50+ Objective: Identify One-to-One Functions

10) f (x) = |x - 4| A) Yes Answer: B

B) No

Type: BI Var: 18 Objective: Identify One-to-One Functions 2 Determine Whether Two Functions Are Inverses

Determine whether the two functions are inverses. 6+x 1) f (x) = 6x + 6 and g (x) = 6 A) No

B) Yes

Answer: A Type: BI Var: 50+ Objective: Determine Whether Two Functions Are Inverses

2) f (x) =

2 2 + 3x and g (x) = x+3 x

A) No Answer: A Type: BI Var: 50+ Objective: Determine Whether Two Functions Are Inverses

Page 345

B) Yes

Solve the problem. 3) There were 2,300 applicants for enrollment to the freshman class at a small college in the year 2010. The number of applicants has risen linearly by roughly 170 per year. The number of applications f (x) is given by f (x) = 2,300 + 170x, where x is the number of years since 2010. a. Determine if the function g(x) =

x - 2,300 is the inverse of f. 170

b. Interpret the meaning of function g in the context of the problem. A) a. No b. The value g(x) represents the number of years since the year 2010 based on the number of applicants to the freshman class, x. B) a. No b. The value g(x) represents the number of applicants to the freshman class based on the number of years since 2010, x. C) a. Yes b. The value g(x) represents the number of years since the year 2010 based on the number of applicants to the freshman class, x. D) a. Yes b. The value g(x) represents the number of applicants to the freshman class based on the number of years since 2010, x. Answer: C Type: BI Var: 50+ Objective: Determine Whether Two Functions Are Inverses 3 Find the Inverse of a Function

A one-to-one function is given. Write an expression for the inverse function. 6-x 1) f (x) = 2 6 - 2x 6 + 2x A) f -1(x) = B) f -1(x) = 6 + 2x C) f -1(x) = x x Answer: D Type: BI Var: 48 Objective: Find the Inverse of a Function

2) f (x) = 3 x + 7 A) f -1(x) = x3 + 7 C) f -1(x) = (x - 7)3 Answer: D Type: BI Var: 18 Objective: Find the Inverse of a Function

Page 346

B) f -1(x) = (x + 7)3 D) f -1(x) = x3 - 7

D) f -1(x) = 6 - 2x

3) f (x) = 4x3 - 9 -1

A) f

x+9 (x) = 4

C) f -1(x) =

-1

B) f

x-9

(x) =

x+9 4

D) f -1(x) = 3

x+9 4

Answer: D Type: BI Var: 50+ Objective: Find the Inverse of a Function

4) m(x) = 7x3 + 2 x+2 B) m (x) = 7

x+2 A) m (x) = 7 C) m-1(x) =

-1

x-2

D) m-1(x) = 3

x-2 7

Answer: D Type: BI Var: 50+ Objective: Find the Inverse of a Function

5) f (x) =

x+8 x+6

A) f -1(x) =

x-6

B) f -1(x) =

x-8

x+6 x+8

C) f -1(x) =

8 - 6x x-1

Answer: C Type: BI Var: 50+ Objective: Find the Inverse of a Function

6) g (x) = r (x + q)7 + s x-s -q A) g -1(x) = 7 r x C) g -1(x) = 7 - s - q r Answer: A Type: BI Var: 12 Objective: Find the Inverse of a Function

Page 347

B) g -1(x) = 7

-s-q r x-s D) g -1(x) = 7 -q r

D) f -1(x) =

8 + 6x x+1

Given f (x), write an equation for f -1(x) and then graph f (x) and f -1(x) on the same coordinate system. 7) f (x) = x2 - 9; x ≤ 0 B) f -1(x) = x + 9 A) f -1(x) = - x + 9 10 y

10 y

10x

-10

10x

-10

C) f -1(x) = x + 9

D) f -1(x) = - x + 9 10 y

10 y

10x

-10

10x

-10

Answer: D Type: BI Var: 8 Objective: Find the Inverse of a Function

Solve the problem. 8) Given that the domain of a one-to-one function f is [-8, -1) and the range of f is (8, ∞), state the domain and range of f -1. A) Domain: (1, 8] B) Domain: [-8, -1) Range: (-∞, -8) Range: (8, ∞) C) Domain: (-∞, -8) D) Domain: (8, ∞) Range: (1, 8] Range: [-8, -1) Answer: D Type: BI Var: 50+ Objective: Find the Inverse of a Function

Page 348

9) Given f (x) = |x| - 3; x ≥ 0, write an equation for f -1(x). (Hint: Sketch f (x) and note the domain and range.) A) f -1(x) = |x | + 3; x ≥ -3 B) f -1(x) = |x | + 3; x ≥ 0 C) f -1(x) = x + 3; x ≥ -3 D) f -1(x) = |x + 3|; x ≥ 0 Answer: C Type: BI Var: 9 Objective: Find the Inverse of a Function

Find the inverse mentally. 10) f (x) = 7x - 8 A) f -1(x) = 8 -

B) f -1(x) =

x 7

C) f -1(x) =

x+8 7

D) f -1(x) =

Answer: C Type: BI Var: 50+ Objective: Find the Inverse of a Function

11) f (x) =

8x + 5

-1

A) f (x) = (8x) - 5 x-5 C) f (x) = 8

B) f (x) =

-1

Answer: D Type: BI Var: 50+ Objective: Find the Inverse of a Function

Solve the problem. 12) f (x) = x + 3 a. Graph f (x) b. Write an equation for f -1(x) c. Write the domain of f -1 in interval notation

Page 349

-1

D) f (x) =

x 7 -5

(x - 5)7 8

8-x

A) a.

B) a. 8

-8

-6

-4

2 2

-2

8 x

-4

-2

-4

-6

-8

2 2

-2

8 x

b. f -1(x) = x2 - 3; x ≥ 0; c. [0, ∞) D) a.

8 x

-8

-6

-4

-2

-4

-6

-8

Answer: B Type: BI Var: 8 Objective: Find the Inverse of a Function

-2

b. f -1(x) = x2 - 3; x ≥ -3; c. [-3, ∞)

Page 350

-4

-2

-6

-2

b. f -1(x) = x2 + 3; x ≥ 0; c. [0, ∞) C) a.

-8

b. f -1(x) = x2 + 3; x ≥ 3; c. [3, ∞)

4 Mixed Exercises

The graph of a function is given. Graph the inverse function. 1) 5

4 3 2 1 -5

-4

-3

-2

-1

5 x

-1 -2 -3 -4 -5

B) 5

-5

-4

-3

-2

1 1

-1

5 x

-4

-3

-2

-1

-2

-3

-4

-5

5 x

D) 5

-5

-4

-3

-2

1 1

-1

5 x

-5

-4

-3

-2

-1

-2

-3

-4

-5

Answer: D Type: BI Var: 50+ Objective: Mixed Exercises

Page 351

-5

2) y 4 3 2 1 -4

-3

-2

-1

-1 -2 -3 -4

B) y

-4

-3

-2

1 1

-1

-4

-3

-2

-1

-2

-3

-4

D) y

-4

-3

-2

1 1

-1

-4

-3

-2

-1

-2

-3

-4

Answer: B Type: BI Var: 6 Objective: Mixed Exercises

Page 352

3) y 4 3 2 1 -4

-3

-2

-1

-1 -2 -3 -4

B) y

-4

-3

-2

1 1

-1

-4

-3

-2

-1

-2

-3

-4

D) y

-4

-3

-2

1 1

-1

-4

-3

-2

-1

-2

-3

-4

Answer: C Type: BI Var: 7 Objective: Mixed Exercises

Page 353

Solve the problem. 4) The millage rate is the amount of property tax per $1000 of the taxable value of a home. For a certain county the millage rate is 29 mil. A city within the county also imposes a flat fee of $101 per home. a. Write a function representing the total amount of property tax T(x) for a home with a taxable valu of x thousand dollars. b. Write an equation for T-1(x). c. What does the inverse function represent in the context of this problem? A) a. T(x) = 29x + 101 x - 101 b. T-1(x) = 29 c. T-1(x) represents the taxable value of a home (in $1000) based on x dollars of property tax paid on the home. B) a. T(x) = 101x + 29 x - 29 b. T-1(x) = 101 c. T-1(x) represents the amount of property tax paid based on x thousand dollars of taxable value of the home. C) a. T(x) = 29x + 101 x - 101 b. T-1(x) = 29 c. T-1(x) represents the amount of property tax paid based on x thousand dollars of taxable value of the home. D) a. T(x) = 101x + 29 x - 29 b. T-1(x) = 101 c. T-1(x) represents the taxable value of a home (in $1000) based on x dollars of property tax paid on the home. Answer: A Type: BI Var: 50+ Objective: Mixed Exercises

Page 354

5) Based on data from a hurricane, the function defined by w(x) = -1.17x + 1,225 gives the wind speed w(x) (in mph) based on the barometric pressure x (in millibars, mb). Write a function representing the inverse of w and interpret its meaning in context. 1,225 - x A) w -1(x) = ; The inverse gives the barometric pressure w -1(x) for a given wind speed. 1.17 B) w -1(x) =

x - 1,225 ; The inverse gives the barometric pressure w -1(x) for a given wind speed. 1.17

C) w -1(x) =

x - 1,225 ; The inverse gives the wind speed w -1(x) for a given barometric pressure. 1.17

D) w -1(x) =

1,225 - x 1.17

Answer: A Type: BI Var: 50+ Objective: Mixed Exercises

4.2 Exponential Functions 0 Concept Connections

Provide the missing information. 1) Given a real number b where b > 0 and b ≠ 1, a function defined by f (x) = exponential function. Answer: b

is called an

Type: SA Var: 1 Objective: Concept Connections

2) The function defined by y = x3 (is/is not) an exponential function, whereas the function defined by y = 3x (is/is not) an exponential function. Answer: is not; is Type: SA Var: 1 Objective: Concept Connections

3) The graph of f (x) =

5x 3

is (increasing/decreasing) over its domain.

Answer: increasing Type: SA Var: 1 Objective: Concept Connections

4) The graph of f (x) =

3x 5

is (increasing/decreasing) over its domain.

Answer: decreasing Type: SA Var: 1 Objective: Concept Connections

Page 355

5) The domain of an exponential function f (x) = bx is

Answer: (-∞, ∞) Type: SA Var: 1 Objective: Concept Connections

6) The range of an exponential function f (x) = bx is

Answer: (0, ∞) Type: SA Var: 1 Objective: Concept Connections

7) All exponential functions f (x) = bx pass through the point

Answer: (0, 1) Type: SA Var: 1 Objective: Concept Connections

8) The horizontal asymptote of an exponential function f (x) = bx is the line

Answer: y = 0 Type: SA Var: 1 Objective: Concept Connections

9) The function defined by f (x) = 1x (is/is not) an exponential function. Answer: is not Type: SA Var: 1 Objective: Concept Connections

10) As x → ∞, the value of 1 +

1x x

approaches

Answer: e Type: SA Var: 1 Objective: Concept Connections

11) The function f (x) = ex is the exponential function base exponential function.

and is also called the

Answer: e ; natural Type: SA Var: 1 Objective: Concept Connections

12) The formula A = Pert gives the amount A in an account after t years at an interest rate r under the assumption that interest is compounded . Answer: continuously Type: SA Var: 1 Objective: Concept Connections

Page 356

1 Graph Exponential Functions

Evaluate the function at the given value of x. Round to 4 decimal places if necessary. 1) f (x) = 4x; f (4.2) A) 337.794 B) 16.8 C) 311.1696 D) Undefined Answer: A Type: BI Var: 50+ Objective: Graph Exponential Functions

2) f (x) = 2x; f ( 6) A) 4.899

B) 6

C) 5.4622

D) Undefined

Answer: C Type: BI Var: 29 Objective: Graph Exponential Functions

3) h (x) = A)

; h (-3)

1 27

B) 9

1 9

D) 27

Answer: D Type: BI Var: 15 Objective: Graph Exponential Functions

4) f (x) =

1x 4

;

f (-5)

A) -1.25

B) 0.0016

C) 1,024

D) Undefined

C) 0.1205

D) Undefined

Answer: C Type: BI Var: 24 Objective: Graph Exponential Functions

5) f (x) =

1x 7

;

A) 0.5566

f (0.4e) B) 0.1553

Answer: C Type: BI Var: 50+ Objective: Graph Exponential Functions

Page 357

Graph the function and write the domain and range in interval notation. 6) f (x) = 5x A) B) y

-5

Domain: (0, ∞) Range: (-∞, ∞)

Domain: (-∞, ∞) Range: (0, ∞)

D) y

-5

Domain: (0, ∞) Range: (-∞, ∞) Answer: D Type: BI Var: 6 Objective: Graph Exponential Functions

Page 358

-5

Domain: (-∞, ∞) Range: (0, ∞)

7) f (x) =

2x 5

B) y

-5

Domain: (0, ∞) Range: (-∞, ∞)

D) y

-5

Domain: (-∞, ∞) Range: (0, ∞) Answer: D Type: BI Var: 8 Objective: Graph Exponential Functions

Page 359

-5

Domain: (-∞, ∞) Range: (0, ∞)

8) f (x) =

1x 2 y

-5

Domain: (0, ∞) Range: (-∞, ∞)

-5

Domain: (0, ∞) Range: (-∞, ∞) Answer: B Type: BI Var: 5 Objective: Graph Exponential Functions

Page 360

Domain: (-∞, ∞) Range: (0, ∞) y

-5

Domain: (-∞, ∞) Range: (0, ∞)

9) f (x) =

3x 2 y

-5

Domain: (-∞, ∞) Range: (0, ∞)

-5

Domain: (0, ∞) Range: (-∞, ∞) Answer: D Type: BI Var: 21 Objective: Graph Exponential Functions

Page 361

Domain: (0, ∞) Range: (-∞, ∞) y

-5

Domain: (-∞, ∞) Range: (0, ∞)

Solve the problem. 10) Use the graph of y = 3x to graph the function. Write the domain and range in interval notation. f (x) = 3x + 5 + 2 A)

-10

10 y

-5

-10

Domain: (-∞, ∞) Range: (-5, ∞)

Domain: (-∞, ∞) Range: (-2, ∞)

-10

10 y

-5

-10

-5

-10

Domain: (-∞, ∞) Range: (5, ∞)

Domain: (-∞, ∞) Range: (2, ∞)

Answer: D Type: BI Var: 50+ Objective: Graph Exponential Functions

Page 362

-10

11) Use the graph of y = 2- x to graph the function .Write the domain and range in interval notation. f (x) = 2- x A) B)

-10

10 y

-5

-10

Domain: (-∞, ∞) Range: (-∞, 0)

-10

10 y

-5

-10

-5

-10

Domain: (-∞, ∞) Range: (0, ∞) Answer: C

Domain: (-∞, ∞) Range: (0, ∞)

Type: BI Var: 5 Objective: Graph Exponential Functions

Page 363

-10

12) Use the graph of y = 3x to graph the function. Write the domain and range in interval notation. f (x) = -3x A) B)

-10

10 y

-5

-10

Domain: (-∞, ∞) Range: (0, ∞)

Domain: (-∞, ∞) Range: (-∞, 0)

-10

10 y

-5

-10

-5

-10

Domain: (-∞, ∞) Range: (0, ∞)

Domain: (-∞, ∞) Range: (-∞, 0)

Answer: B Type: BI Var: 50+ Objective: Graph Exponential Functions

Page 364

-10

13) Use the graph of y = 5x to graph the function. Write the domain and range in interval notation. f (x) = 5x - 5 A) B)

-10

10 y

-5

-10

-5

-10

Domain: (-∞, ∞) Range: (-5, ∞)

-10

10 y

-5

-10

-5

-10

Domain: (-∞, ∞) Range: (-∞, -5)

Answer: B Type: BI Var: 20 Objective: Graph Exponential Functions

14) Use the graph of y = f (x) =

Page 365

1x 3

1 x+1 +3 3

to graph the function. Write the domain and range in interval notation.

-10

10 y

-5

-10

-5

-10

Domain: (-∞, ∞) Range: (-∞, -3)

Domain: (-∞, ∞) Range: (-3, ∞)

-10

10 y

-5

5 -5

-10

-5 -5

-10

Domain: (-∞, ∞) Range: (3, ∞)

Domain: (-∞, ∞) Range: (-∞, 3)

Answer: C Type: BI Var: 50+ Objective: Graph Exponential Functions

15) Use the graph of y =

1x 3

1x f (x) = -

Page 366

to graph the function. Write the domain and range in interval notation.

-10

10 y

-5

-10

-5

-10

Domain: (-∞, ∞) Range: (2, ∞)

Domain: (-∞, ∞) Range: (-2, ∞)

-10

10 y

-5

-10

-5 -5

-10

Domain: (-∞, ∞) Range: (-∞, 2)

Domain: (-∞, ∞) Range: (-∞, -2)

Answer: C Type: BI Var: 12 Objective: Graph Exponential Functions 2 Evaluate the Exponential Function Base e

Evaluate the function at the given value of x. Round to 4 decimal places if necessary. 1) f (x) = ex; f (-2) A) -5.4366 B) 6.5809 C) 0.1353 D) 0.6931 Answer: C Type: BI Var: 50+ Objective: Evaluate the Exponential Function Base e

Page 367

Use transformations of the graph y = ex to graph the function. Write the domain and range in interval notation. 2) f (x) = ex + 2 A) B) 5 y

5 y

-5 -4 -3 -2 -1 -1

-5 -4 -3 -2 -1

-1

-2

-3

-4

-5

Domain: (-∞,∞) Range: (0,∞)

Domain: (-∞,∞) Range: (2,∞)

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Domain: (-∞,∞) Range: (0,∞) Answer: A Type: BI Var: 6 Objective: Evaluate the Exponential Function Base e

Page 368

Domain: (-∞,∞) Range: (-2,∞)

3) f (x) = ex + 1 A)

B) 5 y

5 y

-5 -4 -3 -2 -1 -1

-1

-2

-3

-4

-5

Domain: (-∞, ∞) Range: (0, ∞)

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Domain: (-∞, ∞) Range: (1, ∞)

Domain: (-∞, ∞) Range: (-1, ∞)

Answer: C Type: BI Var: 6 Objective: Evaluate the Exponential Function Base e

Page 369

-5 -4 -3 -2 -1

4) f (x) = -ex + 1 A)

B) 5 y

5 y

-5 -4 -3 -2 -1 -1

-1

-2

-3

-4

-5

Domain: (-∞, ∞) Range: (-∞, 0)

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

Domain: (-∞, ∞) Range: (-∞, 1)

Domain: (-∞, ∞) Range: (-∞, -1)

Answer: C Type: BI Var: 6 Objective: Evaluate the Exponential Function Base e

Page 370

-5 -4 -3 -2 -1

3 Use Exponential Functions to Compute Compound Interest

Solve the problem. 1) Suppose $14,000 is invested with 4% interest for 5 yr under the following compounding options. Complete the table. Compounding Option n Value Result a. Daily b. Continuously A) Compounding Option n Value Result a. Daily 365 $17,099.45 b. Continuously n/a $17,783.62 B) Compounding Option n Value Result a. Daily 1 $17,033.14 b. Continuously n/a $17,783.62 C) Compounding Option n Value Result a. Daily 365 $17,099.45 b. Continuously n/a $17,099.64 D) Compounding Option n Value Result a. Daily 1 $17,033.14 b. Continuously n/a $17,099.64 Answer: C Type: BI Var: 50+ Objective: Use Exponential Functions to Compute Compound Interest

2) Bethany needs to borrow $8,000. She can borrow money at 6.9% simple interest for 3 yr or she can borrow at 6.5% with interest compounded continuously for 3 yr. Which option results in less total interest? A) 6.9% simple interest results in less total interest B) 6.5% compounded continuously results in less total interest Answer: A Type: BI Var: 50+ Objective: Use Exponential Functions to Compute Compound Interest

3) James wants to invest $8,000. He can invest the money at 7.2% simple interest for 30 yr or he can invest at 7% with interest compounded continuously for 30 yr. Which option results in more total interest? A) 7% compounded continuously results in more total interest B) 7.2% simple interest results in more total interest Answer: A Type: BI Var: 50+ Objective: Use Exponential Functions to Compute Compound Interest

Page 371

4 Use Exponential Functions in Applications

Solve the problem. 1) The atmospheric pressure on an object decreases as altitude increases. If a is the height (in km) above sea level, then the pressure P(a) (in mmHg) is approximated by P(a) = 760e-0.13a. Determine the atmospheric pressure at 7.176 km. Round to the nearest whole unit. A) 299 mmHg B) 4,789 mmHg C) 261 mmHg D) 379 mmHg Answer: A Type: BI Var: 50+ Objective: Use Exponential Functions in Applications

2) Newton's law of cooling indicates that the temperature of a warm object will decrease exponentially with time and will approach the temperature of the surrounding air. The temperature T(t) is modeled by T(t) = Ta + (T0 - Ta)e-kt. In this model, Ta represents the temperature of the surrounding air, T0 represents the initial temperature of the object and t is the time after the object starts cooling. The value of k is the cooling rate and is a constant related to the physical properties o the object. A cake comes out of the oven at 335°F and is placed on a cooling rack in a 70°F kitchen. After checking the temperature several minutes later, it is determined that the cooling rate k is 0.050. Write a function that models the temperature T(t) (in °F) of the cake t minutes after being removed from the oven. A) T(t) = 70 + 335e0.050t C) T(t) = 70 + 265e-0.050t Answer: C Type: BI Var: 50+ Objective: Use Exponential Functions in Applications

Page 372

B) T(t) = 70 + 265e0.050t D) T(t) = 335 + 70e0.050t

3) Newton's law of cooling indicates that the temperature of a warm object will decrease exponentially with time and will approach the temperature of the surrounding air. The temperature T(t) is modeled by T(t) = Ta + (T0 - Ta)e-kt. In this model, Ta represents the temperature of the surrounding air, T0 represents the initial temperature of the object and t is the time after the object starts cooling. The value of k is the cooling rate and is a constant related to the physical properties o the object. Water in a water heater is originally 131°F. The water heater is shut off and the water cools to the temperature of the surrounding air, which is 80°F. The water cools slowly because of the insulation inside the heater, and the rate of cooling is 0.00348. Dominic does not like to shower with water less than 116°F. If Dominic waits 24 hr after the heater is shut off, will the water still be warm enough for a shower? How warm will the water be? A) No; the temperature after 24 hr will be approximately 106°F B) Yes; the temperature after 24 hr will be approximately 128°F C) No; the temperature after 24 hr will be approximately 80°F D) Yes; the temperature after 24 hr will be approximately 127°F Answer: D Type: BI Var: 50+ Objective: Use Exponential Functions in Applications

4) Scientists often use a process called carbon dating to estimate the age of archaeological finds. The process measures the amount of carbon-14, a radioactive isotope with a half-life of 5,730 years. If a sample of wood from an ancient artifact had 20 grams of carbon-14 initially, the amount remaining, in grams, is given by t/5,730 A(t) = 20 1 2 where t is the number of years since the tree died. How many grams would be present after 5,730 years? A) 5 g

B) 40 g

Answer: D Type: BI Var: 19 Objective: Use Exponential Functions in Applications

Page 373

C) 0.27 g

D) 10 g

5) The population of bacteria culture was 2000 at noon, and was increasing at a rate of 10% per hour. The number can be found using the function P(t) = 2,000(1.1)t where t is the number of hours past noon. Predict the population 11 hours later, at 11 PM to the nearest whole number. A) 24,200

B) 3,706

C) 5,706

D) 7,726

Answer: C Type: BI Var: 10 Objective: Use Exponential Functions in Applications

6) From 1995-2005, the hourly pay for lifeguards at an outdoor swimming pool increased by 2% per year. The hourly pay, P(t), in dollars, t yr after 1995 is given by P(t) = 5.60(1.02)t. What was the hourly pay for the lifeguards in 2,001? A) $6.18

B) $6.27

C) $6.31

D) $6.43

Answer: C Type: BI Var: 50+ Objective: Use Exponential Functions in Applications

7) After taking a certain antibiotic, the amount of amoxicillin A(t), in milligrams, remaining in the patient's system t hr after taking 1,000 mg of amoxicillin is A(t) = 1,000e-0.49t. How much amoxicillin is in the patient's system 3 hr after taking the medication? Round to the nearest tenth of a mg. A) 229.9 mg

B) 689.8 mg

Answer: A Type: BI Var: 50+ Objective: Use Exponential Functions in Applications

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C) 361.0 mg

D) 33.9 mg

8) A radioactive isotope X is a by-product of nuclear fission with a half-life of 28.7 yr. After a nuclear accident, large areas surrounding the site were contaminated with isotope X. If 10 micrograms of X 1 t / 28.7 is present in a sample, the function A(t) = 10 gives the amount A(t), in micrograms, 2 present after t years. Evaluate the function for the given values of t and interpret the meaning in context. a. A(57.4) b. A(100) A) a. After 57.4 years the amount of isotope X is 5 µg. b. After 100 years the amount of isotope X is approximately 0.894 µg. B) a. After 57.4 years the amount of isotope X is 2.5 µg. b. After 100 years the amount of isotope X is approximately 1.138 µg. C) a. After 57.4 years the amount of isotope X is 5 µg. b. After 100 years the amount of isotope X is approximately 1.138 µg. D) a. After 57.4 years the amount of isotope X is 2.5 µg. b. After 100 years the amount of isotope X is approximately 0.894 µg. Answer: D Type: BI Var: 50+ Objective: Use Exponential Functions in Applications

9) The population of a country in 2010 was approximately 35 million with an annual growth rate of 0.804%. At this rate, the population P(t) (in millions) can be approximated by P(t) = 35(1.00809)t where t is the time in years since 2010. a. Evaluate P(0) and interpret its meaning in the context of this problem. b. Evaluate P(6) and interpret its meaning in the context of this problem. Round the population value to the nearest million. A) a. The initial population in 2010 was approximately 35 million. b. The population in 2,016 will be approximately 37 million more people. B) a. The population increases by approximately 35 million in the first year. b. The population in 2,016 will be approximately 38 million. C) a. The initial population in 2010 was approximately 35 million. b. The population in 2,016 will be approximately 37 million. D) a. The population increases by approximately 35 million in the first year. b. The population in 2,016 will be approximately 38 million more people. Answer: C Type: BI Var: 50+ Objective: Use Exponential Functions in Applications

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10) A veterinarian depreciates a $10,000 X-ray machine. He estimates that the resale value V(t) (in $) after t years is 90% of its value from the previous year. Therefore, the resale value can be approximated by V(t) = 10,000(0.9)t. a. Find the resale value after 3 yr. b. If the veterinarian wants to sell his practice 10 yr after the X-ray machine was purchased, how much is the machine worth? Round to the nearest $100. A) a. $2,710; b. $3,500 B) a. $2,710; b. $6,500 C) a. $7,290; b. $6,500 D) a. $7,290; b. $3,500 Answer: D Type: BI Var: 48 Objective: Use Exponential Functions in Applications

4.3 Logarithmic Functions 0 Concept Connections

Provide the missing information. 1) Given positive real numbers x and b such that b ≠ 1, y = logb x is the is equivalent to b y = x.

function base b and

Answer: logarithmic Type: SA Var: 1 Objective: Concept Connections

2) Given y = logb x, the value y is called the .

, b is called the

, and x is called the

Answer: logarithm, base, argument Type: SA Var: 1 Objective: Concept Connections

3) The logarithmic function base 10 is called the logarithmic function, and the logarithmic function base e is called the logarithmic function. Answer: common; natural Type: SA Var: 1 Objective: Concept Connections

4) Given y = log x, the base is understood to be . Answer: 10; e Type: SA Var: 1 Objective: Concept Connections

5) logb 1 =

because b□= 1.

Answer: 0; 0 Type: SA Var: 1 Objective: Concept Connections

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. Given y = ln x, the base is understood to be

6) logb b =

because b□ = b.

Answer: 1; 1 Type: SA Var: 1 Objective: Concept Connections

7) f (x) = logb x and g(x) = bx are inverse functions. Therefore, logb bx = log x b b =

and

Answer: x ; x Type: SA Var: 1 Objective: Concept Connections

8) Given y = logb x, if b > 1, then the graph of the function is a(n) (increasing/decreasing) logarithmic function. If 0 < b < 1, then the graph is (increasing/decreasing). Answer: increasing; decreasing Type: SA Var: 1 Objective: Concept Connections

9) The graph of y = logb x passes through the point (1, 0) and the line (horizontal/vertical) asymptote.

is a

Answer: x = 0; vertical Type: SA Var: 1 Objective: Concept Connections

10) The graph of f (x) = log4 (x + 6) - 8 is the graph of y = log4 x shifted 6 units (left/right/upward/downward) and shifted 8 units (left/right/upward/downward). Answer: left; downward Type: SA Var: 1 Objective: Concept Connections 1 Convert Between Logarithmic and Exponential Forms

Write the equation in exponential form. 1) log8 512 = b b

512 b=8

512

= 512

b = 512

Answer: D Type: BI Var: 19 Objective: Convert Between Logarithmic and Exponential Forms

2) log 4 16 = 2 A) 162 = 4

B) 24 = 16

Answer: D Type: BI Var: 14 Objective: Convert Between Logarithmic and Exponential Forms

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C) 216 = 4

D) 42 = 16

3) log

= -6

3 729

Answer: 3-6 =

729 Type: SA Var: 8 Objective: Convert Between Logarithmic and Exponential Forms

4) log

= -3

8 512

1 A) 512 = -38

1 B) 512 = 3-8

1 -3 =8 512

D) 8-3 =

D) 61/2 = 36

C) logb 2 = 8

D) log8 b = 2

C) log3 243 = 5

D) log5 243 = 3

1 512

Answer: D Type: BI Var: 17 Objective: Convert Between Logarithmic and Exponential Forms

5) log36 6 =

1 2

A) 2-6 =

B) 361/2 = 6

Answer: B Type: BI Var: 19 Objective: Convert Between Logarithmic and Exponential Forms

Write the equation in logarithmic form. 6) 82 = b A) log2 b = 8

B) logb 8 = 2

Answer: D Type: BI Var: 19 Objective: Convert Between Logarithmic and Exponential Forms

7) 35 = 243 A) log243 3 = 5

B) log243 5 = 3

Answer: C Type: BI Var: 23 Objective: Convert Between Logarithmic and Exponential Forms

8) 105 = 100,000 A) log100,000 5 = 10 C) log 100,000 = 5 Answer: C Type: BI Var: 8 Objective: Convert Between Logarithmic and Exponential Forms

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B) 100,000 = log 5 D) log5 100,000 = 10

9) 5-4 =

1 625

A) log 5 = 625

B) log

-4

= -4

C) log (-4) = 625

5 625

D) -log 625 = -4

Answer: B Type: BI Var: 25 Objective: Convert Between Logarithmic and Exponential Forms 2 Evaluate Logarithmic Expressions

Simplify the expression. 1) log 2 16 A) 8

B) - 8

C) -4

D) 4

Answer: D Type: BI Var: 6 Objective: Evaluate Logarithmic Expressions

2) log

7 49

A) 2

B) -2

C) -

1 2

Answer: B Var: 19 Type: BI Objective: Evaluate Logarithmic Expressions

3) ln

1 e2 A) -2

C) 2

D) -

1 2

Answer: A Type: BI Var: 8 Objective: Evaluate Logarithmic Expressions

4) log1/5 625 A) 1 4

B) -4

Answer: B Type: BI Var: 19 Objective: Evaluate Logarithmic Expressions

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C) 4

D) -

1 4

5) log 0.00001 1 A) 5

B) -5

C) -4

1 4

Answer: B Type: BI Var: 4 Objective: Evaluate Logarithmic Expressions

Simplify the expression without using a calculator. 6) log 10,000,000,000 A) 11 B) 10

C) 9

D) 12

C) -4

D) 4

C) 4

D) -4

C) 5

Answer: B Type: BI Var: 5 Objective: Evaluate Logarithmic Expressions

1 7) log1/2

1 A) 4

B) -

1 4

Answer: D Type: BI Var: 13 Objective: Evaluate Logarithmic Expressions

8) log5/4 A) -

256 625 1 4

1 B) 4

Answer: D Type: BI Var: 12 Objective: Evaluate Logarithmic Expressions

9) log2

A) -5

B) -

1 5

Answer: D Type: BI Var: 13 Objective: Evaluate Logarithmic Expressions

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1 5

10) log6

1 216

2 A) 3

B) -

3 2

3 C) 2

D) -

2 3

Answer: B Type: BI Var: 6 Objective: Evaluate Logarithmic Expressions

Approximate the value of the logarithm to four decimal places. 11) log 512 A) 2.6016 B) 2.7093 C) 6.2383

D) 2.8392

Answer: B Type: BI Var: 50+ Objective: Evaluate Logarithmic Expressions

12) log 38,018,195 A) 7.5800

B) 3.8018

C) 17.4536

D) 10.8018

C) -9.2000

D) -3.2840

C) -20.1675

D) -19.1675

Answer: A Type: BI Var: 50+ Objective: Evaluate Logarithmic Expressions

13) log 0.00052 A) -7.5617

B) -5.2000

Answer: D Type: BI Var: 50+ Objective: Evaluate Logarithmic Expressions

14) log(6.8 ·10 -19) A) -17.1675

B) -18.1675

Answer: B Type: BI Var: 50+ Objective: Evaluate Logarithmic Expressions

Approximate f (x) = ln x for the given value of x. Round to four decimal places. 15) f (460) A) 6.6000 B) 4.6000 C) 2.6628

D) 6.1312

Answer: D Type: BI Var: 50+ Objective: Evaluate Logarithmic Expressions

16) f ( 229) A) 2.7169

B) 1.5133

Answer: A Type: BI Var: 25 Objective: Evaluate Logarithmic Expressions

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C) 2.5133

D) 1.1799

17) f (9π) A) 2.8274

B) 3.8274

C) 3.3420

D) 1.4514

C) 5

D) x5

Answer: C Type: BI Var: 6 Objective: Evaluate Logarithmic Expressions 3 Apply Basic Properties of Logarithms

Simplify the expression. log (x5 - 7) 1) 4 4

A) -7

B) x5 - 7

Answer: B Type: BI Var: 50+ Objective: Apply Basic Properties of Logarithms

2) 3

A) 4x + 4y 3

B) 12x + 12y

8 3

C) x5

D) 4

C) 1

D) π

4x + 4y

Answer: D Type: BI Var: 50+ Objective: Apply Basic Properties of Logarithms 5

3) ln ex + 4 A) x5 + 4

B) 5

Answer: A Type: BI Var: 50+ Objective: Apply Basic Properties of Logarithms

4) log π 1 A) 0

1 π

Answer: A Type: BI Var: 2 Objective: Apply Basic Properties of Logarithms

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4 Graph Logarithmic Functions

Graph the function. 1) y = log7x A)

-8

-6

-4

8 y

2 2

-2

8 x

-6

-4

-2 -2

-4

-6

-8

8 x

-8

-6

-4

8 y

2 2

-2

8 x

-8

-6

-4

-2

-4

-6

-8

Answer: A Type: BI Var: 8 Objective: Graph Logarithmic Functions

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-8

-2

2) y = log1/3x A)

-8

-6

-4

8 y

2 2

-2

8 x

-6

-4

-2 -2

-4

-6

-8

8 x

-8

-6

-4

8 y

2 2

-2

8 x

-8

-6

-4

-2

-4

-6

-8

Answer: A Type: BI Var: 8 Objective: Graph Logarithmic Functions

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-8

-2

3) y = ln x A)

-8

-6

-4

8 y

2 2

-2

8 x

-6

-4

-2 -2

-4

-6

-8

8 x

-8

-6

-4

8 y

2 2

-2

8 x

-8

-6

-4

-2

-4

-6

-8

Answer: C Type: BI Var: 18 Objective: Graph Logarithmic Functions

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-8

-2

a. Use transformations to graph the function. b. Write the domain and range in interval notation. c. Determine the vertical asymptote. 4) y = log5(x + 3) A) a.

-8

B) a.

-6

-4

8 y

2 2

-2

8 x

-6

-4

-2

-4

-6

-8

8 y

2 2

-2

8 x

-8

-6

-4

-2

-4

-6

-8

b. domain: (3, ∞), range (-∞, ∞) c. vertical asymptote: x = 3

b. domain: (0, ∞), range (-∞, ∞) c. vertical asymptote: x = 0

Type: BI Var: 20 Objective: Graph Logarithmic Functions

8 x

b. domain: (0, ∞), range (-∞, ∞) c. vertical asymptote: x = 0 D) a.

8 y

Answer: A

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-6

-2

b. domain: (-3, ∞), range (-∞, ∞) c. vertical asymptote: x = -3 C) a.

-8

-2

8 x

5) y = log 4(x - 1) A)

-8

-6

-4

8 y

2 2

-2

8 x

-6

-4

-2

-4

-6

-8

b. domain: (1, ∞), range (-∞, ∞) c. vertical asymptote: x = 1

b. domain: (-1, ∞), range (-∞, ∞) c. vertical asymptote: x = -1

8 x

-8

-6

-4

8 y

2 2

-2

8 x

-8

-6

-4

-2

-4

-6

-8

b. domain: (0, ∞), range (-∞, ∞) c. vertical asymptote: x = 0

Answer: A Type: BI Var: 18 Objective: Graph Logarithmic Functions

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-8

-2

8 x

6) y = 1 + log5(x) A) a.

-8

B) a.

-6

-4

8 y

2 2

-2

8 x

-4

-2 -4

-6

-8

2 2

-2

8 x

-8

-6

-4

-2

-4

-6

-8

b. domain: (0, ∞), range (-∞, ∞) c. vertical asymptote: x = 0

b. domain: (1, ∞), range (-∞, ∞) c. vertical asymptote: x = 1

Type: BI Var: 20 Objective: Graph Logarithmic Functions

8 x

b. domain: (-1, ∞), range (-∞, ∞) c. vertical asymptote: x = -1 D) a.

Answer: C

Page 388

-4

-6

-2

b. domain: (0, ∞), range (-∞, ∞) c. vertical asymptote: x = 0 C) a.

-8

-2

8 x

Solve the problem. 7) Use transformations of the graph of y = log x to graph the function. 4

y = log4(x - 3) - 4 A)

-10

10 y

-5

-10

-5

-10

10 y

-5

-10

-5

-10

Answer: C Type: BI Var: 50+ Objective: Graph Logarithmic Functions

Write the domain in interval notation. 8) f (x) = log(2 - x) A) (0, ∞) B) (-∞, 2)

C) [2, ∞)

Answer: B Type: BI Var: 9 Objective: Graph Logarithmic Functions

9) f (x) = ln(x2 + 8x + 12) A) (-∞, 2) ∪ (6, ∞) C) (-∞, 2] ∪ [6, ∞) Answer: B Type: BI Var: 50 Objective: Graph Logarithmic Functions

Page 389

B) (-∞, -6) ∪ (-2, ∞) D) [-6, -2]

D) (-∞, 2]

10) f (x) = ln(x2 + 7) A) (-∞, ∞)

B) (7, ∞)

C) [10, ∞)

D) (-∞, 7)

C) (-∞, 3) ∪ (3, ∞)

D) (-3, 3)

C) (-3, 3)

D) (-∞, -3) ∪ (3, ∞)

C) [-15, ∞)

D) (-∞, 15)

Answer: A Type: BI Var: 10 Objective: Graph Logarithmic Functions

11) f (x) = log5 (3 - x)2 A) (-∞, ∞)

B) (-∞, -3) ∪ (3, ∞)

Answer: C Type: BI Var: 40 Objective: Graph Logarithmic Functions

12) f (x) = log3 (x2 - 9) A) (-∞, ∞)

B) (-∞, 3) ∪ (3, ∞)

Answer: D Type: BI Var: 11 Objective: Graph Logarithmic Functions

13) f (x) = 14 A) (-∞, 15]

1 15 - x B) (-15, ∞)

Answer: D Type: BI Var: 50+ Objective: Graph Logarithmic Functions 5 Use Logarithmic Functions in Applications

Solve the problem. 1) The local magnitude ML (on the Richter scale) of an earthquake of intensity I is given by I M = log where I0 is a minimum reference intensity of a “zero-level” earthquake against which L I0 the intensities of other earthquakes may be compared. How many times more intense is an earthquake of magnitude 5.8 than an earthquake of magnitude 2.9? Round to the nearest whole number. A) Approximately 2 times more intense. C) Approximately 794 times more intense. Answer: C Type: BI Var: 50+ Objective: Use Logarithmic Functions in Applications

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B) Approximately 481 times more intense. D) Approximately 18 times more intense.

2) Scientists use the pH scale to represent the level of acidity or alkalinity of a liquid. This is based on the molar concentration of hydronium ions, [H+]. Since the values of [H+] vary over a large range, 1 × 100 mole per liter to 1 × 10-14 mole per liter (mol/L), a logarithmic scale is used to compute the pH. The formula pH = -log[H+] represents the pH of a liquid as a function of its concentration of hydronium ions, [H+]. The pH scale ranges from 0 to 14. Pure water is taken as neutral having a pH of 7. A pH less than 7 is acidic. A pH greater than 7 is alkaline (or basic). Two acids have [H+] values of 3.2 × 10-3 mol/L and 2.4 × 10-6 mol/L respectively. Find the pH for each acid. Which substance is more acidic? Round pH values to 1 decimal place. A) 2.5; 5.6; The first acid is more acidic. B) 5.7; 12.9; The second acid is more acidic C) 5.7; 12.9; The first acid is more acidic. D) 2.5; 5.6; The second acid is more acidic. Answer: A Type: BI Var: 50+ Objective: Use Logarithmic Functions in Applications

3) The level of a sound in decibels is calculated using the formula D = 10 · log(I × 1012) where I is the intensity of the sound waves in watts per square meter. A electric shaver puts out 0.0001 watt per square meter. How many decibels is that? A) 800 B) 80 C) 8 D) -40 Answer: B Type: BI Var: 6 Objective: Use Logarithmic Functions in Applications 6 Mixed Exercises

Solve the problem. 1) The number n of monthly payments of P dollars each required to pay off a loan of A dollars in its entirety at interest rate r is given by log 1 -

Ar 12P

log 1+

r 12

n=-

A college student wants to buy a car and realizes that he can only afford payments of $180 per month. If he borrows $5,000 at 6% interest and pays it off, how many months will it take him to retire the loan? Round to the nearest month. A) 30 months B) 28 months C) 31 months D) 29 months Answer: A Type: BI Var: 50+ Objective: Mixed Exercises

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7 Expanding Your Skills

Write the domain in interval notation. x-5 1) f (x) = log9 x+8 A) (-∞, -8) ∪ (5, ∞)

B) (-8, 5)

C) (-8, 5]

D) (-∞, ∞)

B) (-5, ∞)

C) [-4, ∞)

D) [-5, ∞)

B) [-15, ∞)

C) (-∞, -15)

D) (-∞, -15]

Answer: A Type: BI Var: 50+ Objective: Expanding Your Skills

2) f (x) = ln( x + 5 - 1) A) (-4, ∞) Answer: A Type: BI Var: 16 Objective: Expanding Your Skills

3) f (x) = log

1 x + 15

A) (-15, ∞) Answer: A Type: BI Var: 30 Objective: Expanding Your Skills

4.4 Properties of Logarithms 0 Concept Connections

Provide the missing information. 1) The product property of logarithms indicates that logb (xy) = numbers b, x, and y where b ≠ 1. Answer: logb x + logb y

for positive real

Type: SA Var: 1 Objective: Concept Connections

x 2) The

property of logarithms indicates that logb

numbers b, x, and y where b ≠ 1. Answer: quotient; logb x - logb y Type: SA Var: 1 Objective: Concept Connections

3) The power property of logarithms indicates that for any real number p, logb x p = for positive real numbers b, x, and y where b ≠ 1. Answer: p logb x Type: SA Var: 1 Objective: Concept Connections

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for positive real

4) The change-of-base formula indicates that logb x can be written as a ratio of logarithms with base a □ as: logb x = . □ Answer:

loga x loga b

Type: SA Var: 1 Objective: Concept Connections

5) The change-of-base formula is often used to convert a logarithm to a ratio of logarithms with base or base so that a calculator can be used to approximate the logarithm. Answer: 10; e Type: SA Var: 1 Objective: Concept Connections

6) To use a graphing utility to graph the function defined by y = log5 x, use the change-of-base formula to write the function as y = or y = . log x ln x Answer: ; log 5 ln 5 Type: SA Var: 1 Objective: Concept Connections 1 Apply the Product, Quotient, and Power Properties of Logarithms

Use the product property of logarithms to write the logarithm as a sum of logarithms. Then simplify if possible. 1) log9 (xy) A) log9 x × log9 y

B) log9 x - log9 y

C) log9 x + log9 y

D) (log9 x) · y

Answer: C Type: BI Var: 19 Objective: Apply the Product, Quotient, and Power Properties of Logarithms

2) log1/2 (abc) 1 1 1 A) log a + log b + log c 2 2 2 C) (log1/2 a)(log1/2 b)(log1/2 c)

B) log

a - log 1/2

1/2

D) log1/2 a + log1/2 b + log1/2 c

Answer: D Type: BI Var: 8 Objective: Apply the Product, Quotient, and Power Properties of Logarithms

Page 393

b - log 1/2

3) log2 8x A) log2 x + log2 8

B) 3 log2 x

C) 3 + log2 x

D) (log2 8)(log2 x)

Answer: C Type: BI Var: 15 Objective: Apply the Product, Quotient, and Power Properties of Logarithms

4) log(10,000y) A) 10,000 log y C) log y + log 10,000

B) 4 + log y D) 4 log y

Answer: B Type: BI Var: 4 Objective: Apply the Product, Quotient, and Power Properties of Logarithms

5) log (c (c + 4)) A) logc2 + log 4c C) 3log c + log 4

B) 2log c + log 4c D) log c + log (c + 4)

Answer: D Type: BI Var: 11 Objective: Apply the Product, Quotient, and Power Properties of Logarithms

6) log9[(17 + t) · u] A) log9(17 + t) + log9u)]

B) (log917 + log9t)· log9u

C) log917 · log9t + log9u

D) log917 + log9t · log9u

Answer: A Type: BI Var: 50+ Objective: Apply the Product, Quotient, and Power Properties of Logarithms

Use the quotient property of logarithms to write the logarithm as a difference of logarithms. Then simplify if possible. t 7) log 8 64 A) log8 t - 2

B) log8 t - log8 64

log8 t log8 64

Answer: A Type: BI Var: 15 Objective: Apply the Product, Quotient, and Power Properties of Logarithms

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log8 t 2

8) ln

e 18 A) 1 - ln(18)

ln(18)

ln(e) C) ln(18)

ln(e) - ln(18)

Answer: A Type: BI Var: 16 Objective: Apply the Product, Quotient, and Power Properties of Logarithms

m4 + n 9) log

1,000

A) log(m4 + n) - 3 C) 4 log m · log n - 4

B) 4 - 4 log m · log n D) 4 log m + log n - 3

Answer: A Type: BI Var: 16 Objective: Apply the Product, Quotient, and Power Properties of Logarithms

Apply the power property of logarithms. 10) log(2t - 7)3 A) 3 log 2t - 3 log 7 C) 3 log(2t - 7)

B) 3 log 2 · 3 log t - 3 log 7 D) 3 log 2 + 3 log t - 3 log 7

Answer: C Type: BI Var: 50+ Objective: Apply the Product, Quotient, and Power Properties of Logarithms 8 9

11) ln x

ln x · 9ln x

9 B) ln x 8

9ln x

D) 9ln x - 8ln x

8ln x

Answer: B Type: BI Var: 19 Objective: Apply the Product, Quotient, and Power Properties of Logarithms

12) ln 10kt A) kt + ln 10

B) 10 ln k · 10 ln t

C) kt ln 10

Answer: C Type: BI Var: 16 Objective: Apply the Product, Quotient, and Power Properties of Logarithms

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D) 10 ln kt

2 Write a Logarithmic Expression in Expanded Form

Write the logarithm as a sum or difference of logarithms. Simplify each term as much as possible. 1) log 1 p3q 2 32 1 A) + 3log p + log q B) -5 + 3log p + log q 2 2 2 2 5 3 D) log p + log q C) -15log p + log q 2 2 2 5 2 Answer: B Type: BI Var: 50+ Objective: Write a Logarithmic Expression in Expanded Form

2) log6 x4y5 A) 4log6 x + 5log6 y C) 20log6 x + 20log6 y

B) 9(log6 x + log6 y) 4 5 D) log x + log y 6

Answer: A Type: BI Var: 50+ Objective: Write a Logarithmic Expression in Expanded Form

3) log2

p5 q9

A) 5log2 p - 9log2 q

B) -4(log2 p - log2 q)

5 C) (log2 p - log2 q) 9

5 D) (log2 p + log2 q) 9

Answer: A Type: BI Var: 50+ Objective: Write a Logarithmic Expression in Expanded Form

4) log 5

a8 b 7c

A) 8log 5a + 7log 5b - log 5c

B) 8log 5a + 7log 5b + log 5c

C) 8log 5a - 7log 5b + log 5c

D) 8log 5a - 7log 5b - log 5c

Answer: D Type: BI Var: 50+ Objective: Write a Logarithmic Expression in Expanded Form

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5) log3 4 A)

x 3

log x - 1 4 3 4

C) log3

x - log3

1 4

log x - 1 3

D) log3 x - log3 4 - 1

Answer: A Type: BI Var: 16 Objective: Write a Logarithmic Expression in Expanded Form

6) log4(6ab) 6

A) 24log4 a + 24log4 b

B) (log4 a + log4 b)

C) log4 6 + log4 a + log4 b

D) log4 6 + log4 (a + b)

Answer: C Type: BI Var: 44 Objective: Write a Logarithmic Expression in Expanded Form

x2 7) log6 yz A) log6 2x - log6 y - log6 z

B) 2log6 x - log6 y - log6 z

C) log6 2x + log6 y + log6 z

D) 2log6 x + log6 y + log6 z

Answer: B Type: BI Var: 50+ Objective: Write a Logarithmic Expression in Expanded Form

125a5 8 - b 8) log5 c(b + 7)2 1 A) 3 + 5log a + 1 log 8 - log5 b - log5 c - 2log5b - 2log57 5 2 5 2 1 B) 3 + 5log a + log (8 - b) - log c - 2log (b + 7) 5 5 5 2 5 1 1 C) 15log5 a + log5 8 + log5 b + log5 c + 2log5b + 2log57 2 2 1 D) 15log5 a + log5(8 - b) - log5 c - 2log5(b + 7) 2 Answer: B Type: BI Var: 50+ Objective: Write a Logarithmic Expression in Expanded Form

Page 397

Write as the sum or difference of logarithms and fully simplify, if possible. Assume the variable represents a positive real number. 3 ab 9) log c5 1

log a + log b - 5log c

B) 3log a + log b - 5log c

1 1 C) log a + log b - 5log c 3 3

D) 3 log a + log b + 5log c

Answer: C Type: BI Var: 45 Objective: Write a Logarithmic Expression in Expanded Form

10) ln

ab cd 1 1 A) ln a + ln b - 4 ln c - 4 ln d 5 5 4

C) 5 ln a + 5 ln b - 4 ln c - ln d

B) D)

1 5 1

ln a + ln a +

1 5 1

ln b - 4 ln c - ln d ln b - 4 ln c + ln d

A) 3 - log(x + 5)

B) 3 +

1 log(x2 + 25) 2

1 2 C) 3 - log(x + 25) 2

D) 3 - log( x2 + 25)

Answer: B Type: BI Var: 49 Objective: Write a Logarithmic Expression in Expanded Form

11) log

1,000 x2 + 25

Answer: C Type: BI Var: 25 Objective: Write a Logarithmic Expression in Expanded Form

Page 398

12) ln

e7 x4 + y

A) C)

7 8 7 8

1 ln e +

1 - ln(x4 + y) 8 8 7 4 1 D) - ln x + ln y 8 8 8

ln(x4 + y)

Answer: B Type: BI Var: 50+ Objective: Write a Logarithmic Expression in Expanded Form

Write the logarithm as a sum or difference of logarithms. Simplify each term as much as possible. 5x(x9 + 4)9 13) ln 9 3 + 5x A) ln 5 + ln x + 729ln x · ln 4 -

ln 3 · ln 5 -

81 B) ln 5x + 9 ln (x9 + 4 -

ln 3 · ln x

ln 3+ 5x)

9 C) ln 5 + ln x + 9ln (x9 + 4) -

ln (3+ 5x)

9 1 D) ln 5 + ln x + 81ln x + 9 ln 4 -

ln 3 -

1 9

ln 5 -

1 9

ln x

Answer: C Type: BI Var: 50+ Objective: Write a Logarithmic Expression in Expanded Form 3

14) log8 y 8 1 A) log y - 1 3 8 6

log y - 1 3 8 2

log y + 1 3 8 6

log y + 1 3 8 2

Answer: C Type: BI Var: 50+ Objective: Write a Logarithmic Expression in Expanded Form 3 Write a Logarithmic Expression as a Single Logarithm

Write the logarithmic expression as a single logarithm with coefficient 1, and simplify as much as possible. 1) log4 112 - log4 7 A) log4 119

B) log4 105

Answer: C Type: BI Var: 50+ Objective: Write a Logarithmic Expression as a Single Logarithm

Page 399

C) 2

D) log4 784

2) ln p + ln 2 2 A) ln p

B) ln p2

C) ln 2p

D) ln 2 + p

C) 3

D) log9 571,536

C) log35 420

D) 420

C) 15 logb (yz)

D) logb (y5 + z3)

Answer: C Type: BI Var: 28 Objective: Write a Logarithmic Expression as a Single Logarithm

3) log9 20,412 - log9 7 - log9 4 A) 20,401

B) log9 20,423

Answer: C Type: BI Var: 50+ Objective: Write a Logarithmic Expression as a Single Logarithm

4) log35 245 + log35 175 A) 3

B) log35 70

Answer: A Type: BI Var: 50+ Objective: Write a Logarithmic Expression as a Single Logarithm

5) 5logb y + 3logb z A) 8 logb (y + z)

B) logb (y5z3)

Answer: B Type: BI Var: 50+ Objective: Write a Logarithmic Expression as a Single Logarithm

6) 3log5 m - 8log5 n m3 A) log5 n8 Answer: A

3 B) log5 (3m - 8n)

C) log5 8 n

D) log5 (m - n )

C) log5 37

D) 2

Type: BI Var: 50+ Objective: Write a Logarithmic Expression as a Single Logarithm

7) log5 5 + log5 40 - log5 8 A) log5 1600

B) log5

45 8

Answer: D Type: BI Var: 8 Objective: Write a Logarithmic Expression as a Single Logarithm

Page 400

8) 6 log7t - 4 log7u - 5 log7v t6v5 u4

A) log7

B) log7

15tv 2u

C) log7

t6 u4v5

D) log7

3t 10uv

Answer: C Type: BI Var: 50+ Objective: Write a Logarithmic Expression as a Single Logarithm

9) log5 x - 8log5 y - 9log5 z A) log (x - 8y - 9z)

B) log

C) log

5 y8z9

5 72yz

Answer: B Type: BI Var: 50+ Objective: Write a Logarithmic Expression as a Single Logarithm

1 10)

log2 p + 4log2 3 A) log2 (81 p)

B) log2 ( p + 81)

C) log2 6p

D) log2

p + 24 2

Answer: A Type: BI Var: 50+ Objective: Write a Logarithmic Expression as a Single Logarithm

1 11) [6 ln (x - 3) + ln x - 2ln x] 2 x (x - 3)3 2x

A) ln

(x - 3)3 B) ln

(x - 3)6 C) ln

Answer: B Type: BI Var: 50+ Objective: Write a Logarithmic Expression as a Single Logarithm

1 2 12) [log (a + 8a) - log (a + 8)] 3 (a2 + a - 8)

A) log 1/3

C) log

3 2

a +a-8

Answer: B Type: BI Var: 40 Objective: Write a Logarithmic Expression as a Single Logarithm

Page 401

B) log 3 a D) log(a3 + 8a2 + 64a)1/3

D) log (x - y - z )

(x - 3)3 D) ln

13) 3[ ln x - ln(x + 9) - ln (x - 9)] 3 x 3 x A) ln 2 B) ln 2+x x - 81

x 3 C) ln 2+x

x D) ln 2 x - 81

Answer: D Type: BI Var: 42 Objective: Write a Logarithmic Expression as a Single Logarithm

14) 7 log x -

log y -

7 A) log

log z

7 x7

B) log (x y

7 4

z )

yz4

x7 7

D) log(x7

yz4

Answer: A Type: BI Var: 50+ Objective: Write a Logarithmic Expression as a Single Logarithm

15) log7 a + log7 (b 3 A) log7

- 25) - log7 (b + 5)

a (b3 + 5b2 - 25b - 125)

B) log7

a (b - 5)

3 C)

a (b - 5)

D) log7

a (b2 - 25) (b + 5)

Answer: B Type: BI Var: 50+ Objective: Write a Logarithmic Expression as a Single Logarithm

16) log (7y5 - 6y3) + log y-3 A) log (7y2- 6) 5 3 C) log 7y - 6y +

B) log 1 3

Answer: A Type: BI Var: 50+ Objective: Write a Logarithmic Expression as a Single Logarithm

Page 402

7 6y

D) log (7y5- 6y3 -3y)

yz4)

Solve the problem. 17) Use logb 2 ≈ 0.333 and logb 6 ≈ 0.862 to approximate the value of the given logarithm. logb12 A) 0.529 B) 1.195 C) 2.589 D) 0.287 Answer: B Type: BI Var: 50+ Objective: Write a Logarithmic Expression as a Single Logarithm

18) Given that log 5 ≈ 0.6990 and log 9 ≈ 0.9542, use the properties of logarithms to approximate the following. Do not use a calculator. 5 log 81 A) -1.2094

B) 2.6074

C) -0.2115

D) 0.7677

Answer: A Type: BI Var: 36 Objective: Write a Logarithmic Expression as a Single Logarithm

19) Use logb 3 ≈ 0.319, use the properties of logarithms to approximate the following. Do not use a calculator. logb 81 A) 1.276 B) -3.681 C) 4.319 D) 0.08 Answer: A Type: BI Var: 50+ Objective: Write a Logarithmic Expression as a Single Logarithm 4 Apply the Change-of-Base Formula

Use the change-of-base and a calculator to approximate the logarithm to 4 decimal places. 1) log59 A) 0.7325

B) 1.3652

C) 0.4394

D) 0.1788

C) -4.3013

D) -0.2325

C) -1.7659

D) -6.1355

Answer: B Type: BI Var: 50+ Objective: Apply the Change-of-Base Formula

2) log9 0.6 A) -0.0568

B) -1.1224

Answer: D Type: BI Var: 50+ Objective: Apply the Change-of-Base Formula

3) log8(7.32 · 10-7) A) -6.7939

B) -0.1472

Answer: A Type: BI Var: 50+ Objective: Apply the Change-of-Base Formula

Page 403

5 Mixed exercises

Determine if the statement is true or false. 1 1) log e = ln 10 A) True Answer: A

B) False

Type: MC Var: 1 Objective: Mixed exercises

1 2) ln 10 = log e A) True Answer: A

B) False

Type: MC Var: 1 Objective: Mixed exercises

1 3) log5

1 log5 x

A) True Answer: B

B) False

Type: MC Var: 1 Objective: Mixed exercises

1 4) log4

= - log4 p

A) True Answer: A

B) False

Type: MC Var: 1 Objective: Mixed exercises

5) log (xy) - (log x)(log y) A) True Answer: B

B) False

Type: MC Var: 1 Objective: Mixed exercises

6) log

x y

log x log y

A) True Answer: B Type: MC Var: 1 Objective: Mixed exercises

Page 404

B) False

7) log2(7y) + log2 2 = log2(7y) A) True

B) False

Answer: A Type: MC Var: 1 Objective: Mixed exercises

4.5 Exponential and Logarithmic Equations 0 Concept Connections

Provide the missing information. 1) An equation such as 4x = 9 is called an in the exponent.

equation because the equation contains a variable

Answer: exponential Type: SA Var: 1 Objective: Concept Connections

2) The equivalence property of exponential expressions indicates that if bx = by then

3) The equivalence property of logarithmic expressions indicates that if logb x = logb y, then .

Answer: x ; y Type: SA Var: 1 Objective: Concept Connections

4) An equation containing a variable within a logarithmic expression is called a

equation.

Answer: logarithmic Type: SA Var: 1 Objective: Concept Connections 1 Solve Exponential Equations

Solve the equation. 1) 52y = 625 A) {25}

B) {4}

C) {3}

D) {2}

Answer: D Type: BI Var: 8 Objective: Solve Exponential Equations

2) 22x = 4x2 A) {0}

B) {4}

Answer: D Type: BI Var: 9 Objective: Solve Exponential Equations

Page 405

1 ,2 3

D) {0, 1}

3) 49x+8 = 74x 56 A) 3

10 3

C) {8}

D) No solution

Answer: C Type: BI Var: 50+ Objective: Solve Exponential Equations

4) 52z + 3 = 625 A) 311

B) 4

1 5

D) 1 2

Answer: D Type: BI Var: 50+ Objective: Solve Exponential Equations

49 = 7t A) {5}

2 5

1 C) 5

D) {10}

C) {3}

D) { }

C) {3}

Answer: B Type: BI Var: 50+ Objective: Solve Exponential Equations

6) 32x + 5 = 43x - 1 A) {27}

B) {12}

Answer: A Type: BI Var: 50+ Objective: Solve Exponential Equations

7) 9x =

1 729

A) {-3}

1 3

Answer: A Type: BI Var: 30 Objective: Solve Exponential Equations

8) 4(3x + 3) = A) - 2

1 (x - 9) 64 B) 2

Answer: C Type: BI Var: 50+ Objective: Solve Exponential Equations

Page 406

C) 4

D) -4

3 4y +6 9 5y = 4 16 A) {4}

B) No real solution

C) {1}

D) {0}

Answer: C Type: BI Var: 50+ Objective: Solve Exponential Equations

10) 10,0004x + 5 = 104 - x 16 A) 17

B) -

1 17

C) -

17 16

D) -

1 5

Answer: A Type: MC Var: 50+ Objective: Solve Exponential Equations

Solve the equation. Write the solution set with the exact values given in terms of natural or common logarithms. Also give approximate solutions to 4 decimal places, if necessary. 11) 4t = 77 ; use natural logarithms ln 4 ln 4 A) ; t = 0.3191 B) ; t = -2.9575 ln 77 ln 77 C)

ln 77 ; t = 3.1334 ln 4

ln 77 ; t = 2.9575 ln 4

ln 1,469 ; x ≈ 0.4051 18

Answer: C Type: BI Var: 50+ Objective: Solve Exponential Equations

12) 1,462 = 18x + 7 ; use natural logarithms ln 1,455 A) ; x ≈ 2.5197 ln 18 C)

ln 1,469 ; x ≈ 2.5230 ln 18

D) { }

Answer: A Type: BI Var: 50+ Objective: Solve Exponential Equations

13) e2x - 3ex - 10 = 0 ; use natural logarithms A) {ln 5, ln (-2)}; x ≈ 1.6094 or -0.6931 C) {ln 5, -ln 2}; x ≈ 1.6094 or -0.6931 Answer: B Type: BI Var: 42 Objective: Solve Exponential Equations

Page 407

B) {ln 5}; x ≈ 1.6094 D) { }

14) 109x - 1 - 8,000 = 110,000 ; use common logarithms log 102,000 + 1 log 118,000 + 1 A) ; x ≈ 0.6676 B) ; x ≈ 0.6747 9 9 C)

log 118,000 + 1 ; x ≈ 6.3630 log 9

D) { }

Answer: B Type: BI Var: 50+ Objective: Solve Exponential Equations

15) 4e 5m - 5 - 2 = 14 ln 3 + 5 A) ; m = 1.2197 5 C)

ln 4 + 5 ; m = 1.2773 5

ln 3 + 5 ln 5

; m = 3.7893

ln 4 + 5 ln 5

; m = 3.9680

ln 3 ; x ≈ 1.5694 0.7

Answer: C Type: BI Var: 50+ Objective: Solve Exponential Equations

16) 8,000 = 24,000e-0.7t ln 3 A) ; x ≈ -1.5694 -0.7 C)

ln 3 ; x ≈ -3.0801 ln 0.7

D) { }

Answer: B Type: BI Var: 50+ Objective: Solve Exponential Equations

17) 33x - 3 = 25x - 4 -4 ln 2 - 3 ln 3 ; x ≈ -0.5521 A) 3 ln 3 + 5 ln 2 C)

-4 ln 2 + 3 ln 3 ; x ≈ -3.0798 3 ln 3 - 5 ln 2

Answer: C Type: BI Var: 50+ Objective: Solve Exponential Equations

Page 408

-4 ln 2 + 3 ln 2 ; x ≈ 0.3155 3 ln 3 - 5 ln 3

D) { }

18) e7x = -9e ln 9 ; x ≈ 2.2756 A) 5 C)

ln -9 ; x ≈ 2.2756 5

ln 9 ; x ≈ -2.2756 -5

D) { }

Answer: D Type: BI Var: 50+ Objective: Solve Exponential Equations

19) 55x + 6 = 48x ; use natural logarithms ln 6 A) ; x ≈ 3.8122 ln 8 - ln 5 6 ln 5

; x ≈ 3.1732

6 ln 5 ; x ≈ -3.1732 5 ln 5 - 8 ln 4

D) { }

8 ln 4 - 5 ln 5 Answer: C Type: BI Var: 50+ Objective: Solve Exponential Equations 2 Solve Logarithmic Equations

Determine if the given value of x is a solution to the logarithmic equation. 1) log2(x - 63) = 6 - log2 x; x = 64 A) Yes

B) No

Answer: A Type: BI Var: 30 Objective: Solve Logarithmic Equations

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. 2) 4 log2(9p - 95) = 8 A)

97 ; p ≈ 10.7778 9

B) {11}

4 ; p ≈ 0.0676 9 log2 95

D) { }

Answer: B Type: BI Var: 50+ Objective: Solve Logarithmic Equations

Page 409

3) 2 log 6(6p - 8) - 4 = -2 A)

7 3

B) -

1 3

C) -

7 6

3 D) 2

Answer: A Type: BI Var: 50+ Objective: Solve Logarithmic Equations

4) log(p + 8) = 4.8 A) {log 4.8 - 8}; p ≈ -7.3188 C) {10-3.2}; p ≈ 0.0006

B) {104.8 - 8}; p ≈ 63,087.7344 D) { }

Answer: B Type: BI Var: 50+ Objective: Solve Logarithmic Equations

5) log (x2 - 2x) = log 15 A) {5}

B) {5, -3}

C) {-5, 3}

D) { }

C) 4

D) -24

C) 5

D) -5, 7

Answer: B Type: BI Var: 24 Objective: Solve Logarithmic Equations

6) log 5(6x - 14) = 1 + log 5(x - 2) 4 A) 11

17 5

Answer: C Type: BI Var: 50+ Objective: Solve Logarithmic Equations

7) ln x + ln(x + 4) = ln(2x + 35) A) 5, -7 B) -7 Answer: C Type: BI Var: 50+ Objective: Solve Logarithmic Equations

Solve the logarithmic equation. 8) 19 = 17 - log2(2x + 4) 15 A) 8

B) -

Answer: B Type: BI Var: 50+ Objective: Solve Logarithmic Equations

Page 410

15 8

C) -

8 15

D) { }

9) log9 t = log9 t2 - 2 A) {81}

1 81

C) { }

D) { 2}

C) {2}

D) {2, -5}

C) {1}

D) {8.5}

C) ∅

D) {12}

C) {-36, -1}

D) {1, 36}

Answer: B Type: BI Var: 50+ Objective: Solve Logarithmic Equations

10) log x = 1 - log(x + 3) 3 A) , 5 2

B) {1, -3}

Answer: C Type: BI Var: 1 Objective: Solve Logarithmic Equations

Solve the equation. 11) log3 r + log3 (r - 8) = 2 A) {9}

B) ∅

Answer: A Type: BI Var: 11 Objective: Solve Logarithmic Equations

12) log2 (72x) - log2 (x + 96) = 3 A)

96 7

104 71

Answer: D Type: BI Var: 50+ Objective: Solve Logarithmic Equations

13) log6 (37 - z) + log6 z = 2 A) {-36, 1}

B) ∅

Answer: D Type: BI Var: 25 Objective: Solve Logarithmic Equations

Solve the equation. Write the solution set with the exact solutions. 14) ln x + ln(x - 4) = ln(5x - 14) A) {2, 7} B) {2} C) {7} Answer: C Type: BI Var: 50+ Objective: Solve Logarithmic Equations

Page 411

D) { }

Solve the equation. 15) log3 t + log3 (t + 2) = log3 80 A) {39}

B) {10}

C) {-10, 8}

D) {8}

C) {-7, -5}

D) {5, 7}

C) {4, 6}

D) {-6}

C) { }

Answer: D Type: BI Var: 50+ Objective: Solve Logarithmic Equations

16) log5 y + log5 (12 - y) = log5 35 A) {-23, 35}

B) { }

Answer: D Type: BI Var: 50+ Objective: Solve Logarithmic Equations

17) log5 (-x) + log5 (x - 10) = log5 24 A) {-6, -4}

B) { }

Answer: B Type: BI Var: 50+ Objective: Solve Logarithmic Equations

18) log5 (5p + 3) + log5 p = log5 14 A) {2}

7 5

11 6

Answer: B Type: BI Var: 50+ Objective: Solve Logarithmic Equations

19) log x + log(x - 11) = log(x - 35) A) {7} B) {7, 5}

C) {-7, -5}

D) { }

C) {-5, 3}

D) {3}

Answer: D Type: BI Var: 25 Objective: Solve Logarithmic Equations

20) log3(n - 3) + log3(n + 5) = 2 A) {4}

B) {-6, 4}

Answer: A Type: BI Var: 10 Objective: Solve Logarithmic Equations

Page 412

3 Use Exponential and Logarithmic Equations in Applications

Solve the problem. 1) Sunlight is absorbed in water, and as a result the light intensity in oceans, lakes, and ponds decreases exponentially with depth. The percentage of visible light P (in decimal form) at a depth of x meters is given by P = e-kx, where k is a constant related to the clarity and other physical properties of the water. The model for a particular lake is P = e-0.1284x. Determine the depth at which the light intensity is 80% of the value from the surface for this lake. Round to the nearest tenth of a meter. A) 0.8 m

B) 1.7 m

C) 16.2 m

D) 34.1 m

Answer: B Type: BI Var: 50+ Objective: Use Exponential and Logarithmic Equations in Applications

2) The formula L = 10 log

I I0

gives the loudness of sound L (in dB) based on the intensity of the

sound (in W/m2). The value I0 = 10-12 W/m2 is the minimal threshold for hearing for mid-frequency sounds. Hearing impairment is often measured according to the minimal sound level (in dB) detected by an individual for sounds of various frequencies. If the minimum loudness of sound detected by an individual is 60 dB, determine the corresponding intensity of sound. A) 100.5 W/m2

B) 10-0.5 W/m2

C) 10-6 W/m2

D) 106 W/m2

Answer: C Type: BI Var: 7 Objective: Use Exponential and Logarithmic Equations in Applications

3) In the formula pH = -log[H+], the variable pH represents the level of acidity or alkalinity of a liquid on the pH scale, and H+ is the concentration of hydronium ions in the solution. Determine the value of H+ (in mol/L) for a liquid, given its pH value. Round to 2 significant digits. pH = 10.1 A) -7.9 · 10-10 mol/L -10 C) 7.9 · 10 mol/L

B) 7.9 · 10-11 mol/L -11 D) -7.9 · 10 mol/L

Answer: B Type: BI Var: 50+ Objective: Use Exponential and Logarithmic Equations in Applications

Page 413

4) If $22,000 is invested in an account earning 4.5% interest compounded continuously, determine how long it will take the money to quadruple. Round to the nearest year. Use the model A = Pert where A represents the future value of P dollars invested at an interest rate r compounded continuously for t years. A) 3 years

B) 31 years

C) 36 years

D) 308 years

Answer: B Type: BI Var: 50+ Objective: Use Exponential and Logarithmic Equations in Applications

5) If $18,000 is put aside in a money market account with interest reinvested monthly at 2.3%, find the time required for the account to earn $2,000. Round to the nearest month. Use the model A = r nt P1+ where A represents the future value of P dollars invested at an interest rate r n compounded n times per year for t years. A) 97 months

B) 96 months

C) 55 months

D) 5 months

Answer: C Type: BI Var: 50+ Objective: Use Exponential and Logarithmic Equations in Applications

6) The enrollment at one Midwestern college is approximated by the formula P = 3000(1.08)t where t is the number of years after 2000. What is the first year in which you would expect enrollment to surpass 3,700? A) 2,002

B) 2,003

C) 2,004

D) 2,001

Answer: B Type: BI Var: 14 Objective: Use Exponential and Logarithmic Equations in Applications

7) Carbon dating determines the approximate age of an object made from materials that were once alive by measuring the remaining percentage of a radioactive isotope, carbon-14. The age is calculated using the formula ln P A=0.000121 where P is the percentage of remaining carbon-14 (in decimal form). A specimen is determined to be 2,800 years old. What is the percentage of remaining carbon-14? Round to the nearest tenth of a percent. A) 31.3%

B) 51.3%

C) 84.3%

Answer: D Type: BI Var: 21 Objective: Use Exponential and Logarithmic Equations in Applications

Page 414

D) 71.3%

8) A pie comes out of the oven at 275°F and is placed to cool in a 70°F kitchen. The temperature of the pie T (in °F) after t minutes is given by T = 70 + 205e-0.018t. The pie is cool enough to cut when the temperature reaches 130°F. How long will this take? Round to the nearest minute. A) 56 minutes

B) 25 minutes

C) 68 minutes

D) 85 minutes

Answer: C Type: BI Var: 50+ Objective: Use Exponential and Logarithmic Equations in Applications

9) A new teaching method to teach vocabulary to sixth-graders involves having students work in groups on an assignment to learn new words. After the lesson was completed, the students were tested at 1-month intervals. The average score for the class S(t) can be modeled by S(t) = 96 - 10 ln(t+1) where t is the time in months after completing the assignment. If the average score is 73, how many months had passed since the students completed the assignment? A) 9 months

B) 12 months

C) 7 months

D) 11 months

Answer: A Type: BI Var: 50+ Objective: Use Exponential and Logarithmic Equations in Applications

10) The limiting magnitude L of an optical telescope with lens diameter D (in inches) is given by L(D) = 8.8 + 5.1 log D. a. Find the limiting magnitude for a telescope with a lens 11 in. in diameter. Round to 1 decima place. b. Find the lens diameter of a telescope whose limiting magnitude is 10.7. Round to the nearest inch. A) a. 21 b. 1 in.

B) a. 2.4 b. 14 in.

C) a. 5.3 b. 2 in.

D) a. 14.1 b. 2 in.

Answer: D Type: BI Var: 50+ Objective: Use Exponential and Logarithmic Equations in Applications

11) A $3,000 bond grows to $4,475.47 in 8 yr under continuous compounding. Find the interest rate. Round our answer to the nearest whole percent. A) 5% B) 6% C) 7% D) 8% Answer: A Type: BI Var: 50+ Objective: Use Exponential and Logarithmic Equations in Applications

Page 415

12) $14,000 is invested at 6% interest compounded quarterly and grows to $16,738.65. For how long was the money invested? Round to the nearest year. A) 12 yr B) 3 yr C) 9 yr D) 2 yr Answer: B Type: BI Var: 50+ Objective: Use Exponential and Logarithmic Equations in Applications

13) $20,000 is invested at 5.5% interest compounded monthly. How long will it take for the investment to double? Round to the nearest tenth of a year A) 13.2 yr B) 10.7 yr C) 12.6 yr D) 13.5 yr Answer: C Type: BI Var: 12 Objective: Use Exponential and Logarithmic Equations in Applications

14) A $45,000 inheritance is invested for 15 yr compounded quarterly and grows to $109,940. Find the interest rate. Round to the nearest percent A) 6% B) 4% C) 3% D) 5% Answer: A Type: BI Var: 50+ Objective: Use Exponential and Logarithmic Equations in Applications

15) If $4,000 is put aside in a money market account with interest compounded continuously at 3.1%, find the time required for the account to earn $1000. Round to the nearest month. A) 7 yr, 2 months B) 7 yr, 10 months C) 8 yr, 2 months D) 5 yr, 11 months Answer: A Type: BI Var: 46 Objective: Use Exponential and Logarithmic Equations in Applications 4 Mixed Exercises

Find an equation for the inverse function. 1) f (x) = 8x - 3 A) f -1(x) = log (x + 3) 8

B) f -1(x) = log (x) + 3 8

C) f -1(x) = log (x - 8)

D) f -1(x) = log (x - 3)

Answer: A Type: BI Var: 50+ Objective: Mixed Exercises

2) f (x) = ln(x + 3) A) f -1(x) = ex + 3 Answer: B Type: BI Var: 18 Objective: Mixed Exercises

Page 416

B) f -1(x) = ex - 3

C) f -1(x) = ex - 3

D) f -1(x) = ex + 3

3) f (x) = 10x + 1 + 5 A) f -1(x) = log(x - 1) - 5 C) f -1(x) = log(x - 5) - 1

B) f -1(x) = log(x + 1) + 5 D) f -1(x) = log(x + 5) + 1

Answer: C Type: BI Var: 50+ Objective: Mixed Exercises

4) f (x) = log(x + 5) + 4 A) f -1(x) = 10x + 4 + 5 C) f -1(x) = 10x + 5 + 4

B) f -1(x) = 10x - 5 - 4 D) f -1(x) = 10x - 4 - 5

Answer: D Type: BI Var: 50+ Objective: Mixed Exercises

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. 5) 3|x| - 6 = 3 A) {2} B) {-2} C) {2, -2} D) { } Answer: C Type: BI Var: 50+ Objective: Mixed Exercises

6) 3x2 - 7 = 9 A) {3}

B) {3, -3}

C) {-3}

D) { }

B) {768}

C) {64,000}

D) {192}

C) {4}

D) { }

Answer: B Type: BI Var: 50+ Objective: Mixed Exercises

7) log x - 2 log 8 = 3 A) {8,000} Answer: C Type: BI Var: 30 Objective: Mixed Exercises

1 8) log5 x - log5(2x + 6) = log5 4 2 A) {-4, 4} B) {-4} Answer: D Type: BI Var: 48 Objective: Mixed Exercises

Page 417

1 9) log4 x - log4(4x - 7) = log4 4 2 A) {2, -2} B) {2}

C) {-2}

D) { }

B) {1, 9}

C) {1}

D) {-1}

B) {8}

C) {8, -8}

D) { }

B) {0}

C) {1}

D) {3}

Answer: B Type: BI Var: 50+ Objective: Mixed Exercises

10) log7 |x + 5| = log7 4 A) {-1, -9} Answer: A Type: BI Var: 50+ Objective: Mixed Exercises

11) x2ex = 64ex A) {-8} Answer: C Type: BI Var: 8 Objective: Mixed Exercises

12) log3(log3 x) = 0 A) {3, -3} Answer: D Type: BI Var: 8 Objective: Mixed Exercises

13) 3ex(ex - 8) = 5ex - 40 5 A) ln , ln(8) ; x ≈ 0.5108, x ≈ 2.0794 3 C)

5 3

B) -

5 3

,-8

D) { }

Answer: A Type: BI Var: 50+ Objective: Mixed Exercises 5 Expanding Your Skills

Solve the equation. 10x - 31 · 10-x 1) =6 5 A) {31} C) {31, -1} Answer: D Type: BI Var: 22 Objective: Expanding Your Skills

Page 418

B) { } D) {log 31}; x ≈ 1.4914

ex - 19 · e-x 3

=6 B) {ln 19}; x ≈ 2.9444 D) { }

A) {19} C) {19, -1} Answer: B Type: BI Var: 22 Objective: Expanding Your Skills

3) (ln x)2 - ln x5 = -6 A) {2, 3} C) {e-2, e-3}; x ≈ 0.1353, x ≈ 0.0498

B) { } D) {e2, e3}; x ≈ 7.3891, x ≈ 20.0855

Answer: D Type: BI Var: 12 Objective: Expanding Your Skills

4) (log x)2 = log x2 A) { }

B) {1}

C) {1, 100}

D) {100}

B) {-9, 2}

C) { }

D) {10,000}

Answer: C Type: BI Var: 4 Objective: Expanding Your Skills

5) log x + 7 log x - 18 = 0 A) {2} Answer: D Type: BI Var: 13 Objective: Expanding Your Skills

6) e2x - 14ex + 9 = 0 A) {ln(7 + 2 10)}; x ≈ 2.5896 C) {ln(7 ± 2 10)}; x ≈ 2.5896, x ≈ -0.3924

B) { } D) {7 ± 2 10}; x ≈ 13.3246, x ≈ 0.6754

Answer: C Type: BI Var: 50+ Objective: Expanding Your Skills

7) log2 x + 3 + log2 x = 1 A) {1} Answer: A Type: MC Var: 8 Objective: Expanding Your Skills

Page 419

B) { }

C) {-4, 1}

D) {-4}

4.6 Modeling with Exponential and Logarithmic Functions 0 Concept Connections

Provide the missing information. 1) If k > 0, the equation y = y0ekt is a model for exponential (growth/decay), whereas if k < 0, the equation is a model for exponential (growth/decay). Answer: growth; decay Type: SA Var: 1 Objective: Concept Connections

2) A function defined by y = abx can be written in terms of an exponential function base e as . Answer: y = ae(ln b)x Type: SA Var: 1 Objective: Concept Connections

3) A function defined by y =

c is called a 1 + ae-bt

growth model and imposes a limiting

value on y. Answer: logistic Type: SA Var: 1 Objective: Concept Connections

4) Given a logistic growth function y =

c -bt

, the limiting value of y is

1 + ae Answer: c Type: SA Var: 1 Objective: Concept Connections 1 Solve Literal Equations for a Specific Variable

Solve for the indicated variable. 1) N = N0 e-0.0361t for t N ln N0 A) t = 0.0361

ln B) t =

ln N + ln N0 0.0361

C) t =

ln N0 + ln N 0.0361

Answer: A Type: BI Var: 50+ Objective: Solve Literal Equations for a Specific Variable

2) M = 1.5 + 9.8 log D for D A) D = 10M - 9.8 - 10-8.3 C) D = 10M - 9.8 - e-8.3 Answer: B Type: BI Var: 50+ Objective: Solve Literal Equations for a Specific Variable

Page 420

B) D = 10((M - 1.5)/9.8) D) D = e((M - 1.5)/9.8)

D) t =

N N0

0.0361

1 3) ln P = A for P 16.6 k A) P = 16.6kekA

B) P = ekA + 16.6k

C) P = ekA + 16.6

D) P = 16.6ekA

Answer: D Type: BI Var: 50+ Objective: Solve Literal Equations for a Specific Variable

4) Q = Q0e-kt for k Q ln Q0 A) k = t

ln B) k =

ln Q - Q0

C) k = -

ln Q - Q0

D) k = -

Q Q0 t

Answer: D Type: BI Var: 1 Objective: Solve Literal Equations for a Specific Variable

5) log E - 12.8 = 1.41M for E A) E = 101.41M - 12.8 C) E = 101.41M + 12.8

B) E = 101.41M + 12.8 D) E = 109.08M

Answer: C Type: BI Var: 50+ Objective: Solve Literal Equations for a Specific Variable

6) pH = -log(H+) for H+ A) H+ = 10pH

B) H+ = 10-pH

C) H+ = -10pH

D) H+ = log(pH)

C) I = I0 - 10(L/10)

D) I = I0 10L

Answer: B Type: BI Var: 1 Objective: Solve Literal Equations for a Specific Variable

7) L = 10 log

I I0

for I

A) I = I0 10(L/10)

B) I = I0 10(L-10)

Answer: A Type: BI Var: 1 Objective: Solve Literal Equations for a Specific Variable

8) A = P(1 + r)t for t ln A) t =

ln(P - A) ln(1 + r)

B) t =

ln(A - P) ln(1 + r)

Answer: C Type: BI Var: 1 Objective: Solve Literal Equations for a Specific Variable

Page 421

C) t =

A P

ln(1 + r)

ln D) t =

A P

1+r

9) A = Pert for r P ln A A) r = t

ln B) r =

A P

ln C) r =

ln(t)

A P D) r =

ln(A - P) t

Answer: C Type: BI Var: 1 Objective: Solve Literal Equations for a Specific Variable

10) ln

-E for k RT

A) k = -Ae

E/(RT)

B) k = Ae

-E/(RT)

C) k = A ln -E RT

e-D) k =

E/(RT)

Answer: B Type: BI Var: 1 Objective: Solve Literal Equations for a Specific Variable

11) -

1 k

= A for P

14.7

e-kA A) P = 14.7

-A/k

B) P = 14.7e

-kA

C) P = 14.7e

D) P = 14.7 ln(-kA)

Answer: C Type: BI Var: 1 Objective: Solve Literal Equations for a Specific Variable 2 Create Models for Exponential Growth and Decay

Solve the problem. 1) Suppose that $16,000 is invested in a bond fund and the account grows to $18,552.21 in 4 yr. Use the model A = Pert to determine the average rate of return under continuous compounding. Round t the nearest tenth of a percent. A) 1.8%

B) 1.2%

C) 3.7%

D) 1.5%

Answer: C Type: BI Var: 50+ Objective: Create Models for Exponential Growth and Decay

2) Suppose that P dollars in principal is invested in an account earning 5.3% interest compounded continuously. At the end of 3 yr, the amount in the account has earned $1,378.70 in interest. Find the original principal. Round to the nearest dollar. (Hint: Use the model A = Pert and substitute P + 1,378.70 for A.) A) $8,000

B) $1,176

Answer: A Type: BI Var: 50+ Objective: Create Models for Exponential Growth and Decay

Page 422

C) $9,379

D) $2,555

3) The population of a country on January 1, 2000, is 20.7 million and on January 1, 2010, it has risen to 22.6 million. Write a function of the form P(t) = P0ert to model the population P(t) (in millions) t years after January 1, 2000. Then use the model to predict the population of the country on January 1, 2,017. round to the nearest hundred thousand. A) P = 20.7e0.00878t; 119 million C) P = 22.6e0.00878t; 130 million

B) P = 20.7e0.00878t; 24 million D) P = 22.6e0.00878t; 26.2 million

Answer: B Type: BI Var: 50+ Objective: Create Models for Exponential Growth and Decay

4) The population of a country is modeled by the function P(t) = 15.9e0.01462t where P(t) is the population (in millions) t years after January 1, 2000. Use the model to predict the year during which year the population will reach 30 million if this trend continues. A) 2,043

B) 2,004

C) 2,084

D) 2,014

Answer: A Type: BI Var: 50+ Objective: Create Models for Exponential Growth and Decay

5) The function P(t) = 8(1.0134)t represents the population (in millions) of a country t years after January 1, 2000. Write an equivalent function using base e; that is, write a function of the P(t) = P0ert. Round r to 5 decimal places. Also, determine the population for the year 2000. A) P(t) = 2.1e0.10649t; population in 2000: 2.1 million B) P(t) = 8e0.01331t; population in 2000: 8 million C) P(t) = 8e0.01331t; population in 2000: 8.1 million D) P(t) = 2.1e0.10649t; population in 2000: 2.3 million Answer: B Type: BI Var: 50+ Objective: Create Models for Exponential Growth and Decay

6) If the half-life of an element is 67 yr and the initial quantity is 3 kg, write a function of the form Q(t) = Q0e-kt to model the quantity of the element left after t years. Round k to 4 decimal places. A) Q(t) = 67e-0.0034t C) Q(t) = 67e-0.0103t Answer: D Type: BI Var: 50+ Objective: Create Models for Exponential Growth and Decay

Page 423

B) Q(t) = 3e-0.0034t D) Q(t) = 3e-0.0103t

7) If the function Q(t) = 4e-0.00938t models the quantity (in kg) of an element in a storage unit after t years, how long will it be before the quantity is less than 1.5 kg? Round to the nearest year. A) 4 yr

B) 31 yr

C) 45 yr

D) 105 yr

Answer: D Type: BI Var: 50+ Objective: Create Models for Exponential Growth and Decay

8) Eight million E. coli bacteria are present in a laboratory culture. An antibacterial agent is introduced and the population of bacteria P(t) decreases by half every 5 hr. The population can be 1 t/5 represented by P(t) = 8,000,000 . Convert this to an exponential function using base e. 2 A) P(t) = 4,000,000e-0.13863t -0.1t C) P(t) = 4,000,000e

B) P(t) = 8,000,000e-0.1t -0.13863t D) P(t) = 8,000,000e

Answer: D Type: BI Var: 35 Objective: Create Models for Exponential Growth and Decay 3 Apply Logistic Growth Models

Solve the problem. 1) After a new product is launched the cumulative sales S(t) (in $1000) t weeks after launch is given by: 74 S(t) = 1 + 14e-0.4t Determine the cumulative amount in sales 4 weeks after launch. Round to the nearest thousand. A) $55,000

B) $10,000

C) $19,000

D) $1,000

Answer: C Type: BI Var: 50+ Objective: Apply Logistic Growth Models

2) After a new product is launched the cumulative sales S(t) (in $1000) t weeks after launch is given by: 54 S(t) = 1 + 11e-0.32t Determine the amount of time for the cumulative sales to reach $48,000. Round to the nearest week A) 27 weeks

B) 54 weeks

Answer: C Type: BI Var: 50+ Objective: Apply Logistic Growth Models

Page 424

C) 14 weeks

D) 1 week

3) After a new product is launched the cumulative sales S(t) (in $1000) t weeks after launch is given by: 64 S(t) = 1 + 8e-0.47t What is the limiting value in sales? A) $8,000

B) $64,000

C) $47,000

D) $6,000

Answer: B Type: BI Var: 50+ Objective: Apply Logistic Growth Models 4 Create Exponential and Logarithmic Models Using Regression

Graph the points and from visual inspection, select the model that would best fit the data. Choose from y = mx + b (linear) y = abx (exponential) c y = a + b ln x (logarithmic) y= -bx (logistic) 1 + ae Then use a graphing utility to find a function that fits the data. (Hint: For a logistic model, go to STAT, CALC, Logistic.) 1) x y 4 4.4 5 5.2 9 10.5 14 25.1 20 71.3 24 143.1 425 A) y = B) y = 6.2x - 35.27 1 + 214e-0.11x C) y = 2.2(1.19)x

D) y = -101.2 + 61.88 ln x

Answer: C Type: BI Var: 8 Objective: Create Exponential and Logarithmic Models Using Regression

Page 425

2) x 10 20 30 40 50 60

y 41.2 47.5 51.2 53.8 55.9 57.5

A) y = 0.31x + 40.3

B) y = 20.2 + 9.11 ln x

C) y = 40.8(1.01)x

D) y =

138 -0.09x

1 + 0.23e Answer: B Type: BI Var: 8 Objective: Create Exponential and Logarithmic Models Using Regression

Solve the problem. 3) The monthly costs for a small company to do business has been increasing over time in part due to inflation. The table gives the monthly cost y (in $) for the month of January for selected years. The variable t represents the number of years since 2008. Year Monthly (t = 0 is 2008) Costs ($) y 0 14,000 1 14,200 2 14,400 3 14,600 Monthly Cost (Jan.) for Selected Years y 15000

Monthly Costs ($) 12000 9000 6000 3000

Year (t = 0 represents 2008) a. Use a graphing utility to find a model of the form y = abt. b. Write the function from part (a) as an exponential function with base e. c. Use the model to predict the monthly cost for January in the year 2,017 if this trend continues. Page 426

Round to the nearest hundred dollars. A) a. y = 14,001(1.014)t b. y = 10e1.014t c. $91,900 C) a. y = 14,001(1.014)t b. y = 14,001e0.0140t c. $15,900

B) a. y = 14,001(0.0140)t b. y = 10e0.0140t c. $91,900 D) a. y = 14,001(1.014)t b. y = 10e0.0140t c. $15,900

Answer: C Type: BI Var: 50+ Objective: Create Exponential and Logarithmic Models Using Regression

4) The sales of a book tend to increase over the short-term as word-of-mouth makes the book "catch on." The number of books sold N(t) for a new novel t weeks after release at a certain book store is given in the table for the first 6 weeks. Weeks t Number Sold N(t) 1 20 2 28 3 33 4 36 5 38 6 41 Book Sales vs. Weeks After Release 50

N(t)

Number Sold 40 30 20 10

Week a. Find a model of the form y = a + b ln t. b. Use the model to predict the sales in week 11. Round to the nearest whole unit. c. Is it reasonable to assume that the logarithmic trend will continue? Why or why not? A) a. y = 20.1 + 11.5 ln t b. 48 books c. No. The trend cannot continue indefinitely. At some point, book sales will begin to decrease as most reading enthusiasts have read the book. Page 427

B) a. y = 3.9 + 19 ln t b. 49 books c. Yes. The number of books sold will continue to increase although the sales will approach a limit. C) a. y = 3.9 + 19 ln t b. 49 books c. No. The trend cannot continue indefinitely. At some point, book sales will begin to decrease as most reading enthusiasts have read the book. D) a. y = 20.1 + 11.5 ln t b. 48 books c. Yes. The number of books sold will continue to increase although the sales will approach a limit. Answer: A Type: BI Var: 50+ Objective: Create Exponential and Logarithmic Models Using Regression

Page 428

Chapter 6 Matrices and Determinants and Applications 6.1 Solving Systems of Linear Equations Using Matrices 0 Concept Connections

Provide the missing information. 1) A rectangular array of elements is called a

Answer: matrix Type: SA Var: 1 Objective: Concept Connections

2) An matrix is used to represent a system of linear equations written in standard form. Answer: augmented Type: SA Var: 1 Objective: Concept Connections

row operations performed on an augmented matrix results in a new augmented matrix that represents an equivalent system of equations. Answer: Elementary Type: SA Var: 1 Objective: Concept Connections

4) Identify the elements on the main diagonal. -4 -1 2 0 0 1

1 8 5 11 -7 -6

Answer: -4; 0; -7 Type: SA Var: 1 Objective: Concept Connections

5) Explain the meaning of the notation R2⇔ R3. Answer: Interchange rows 2 and 3. Type: SA Var: 1 Objective: Concept Connections

6) Explain the meaning of the notation -R2→ R2. Answer: Multiply row 2 by -1 and replace the original row 2 with the result. Type: SA Var: 1 Objective: Concept Connections

7) Explain the meaning of the notation 3R1 → R1. Answer: Multiply row 1 by 3 and replace the original row 1 with the result. Type: SA Var: 1 Objective: Concept Connections

Page 1

8) Explain the meaning of R1⇔ R3. Answer: Interchange rows 1 and 3. Type: SA Var: 1 Objective: Concept Connections

9) Explain the meaning of the notation 3R1 + R2 → R2. Answer: Add 3 times row 1 to row 2 and replace the original row 2 with the result. Type: SA Var: 1 Objective: Concept Connections

10) Explain the meaning of the notation 4R2 + R3 → R3. Answer: Add 4 times row 2 to row 3 and replace the original row 3 with the result. Type: SA Var: 1 Objective: Concept Connections 1 Write an Augmented Matrix

Write the augmented matrix for the given system. 1) x + 3y + 5z = -3 -5y + 3z = -8 -8z = -6 1 A) 0 1 1 C) 0 0

3 5 -3 -5 3 -8 0 -8 -6 3 5 -3 -5 3 -8 0 -8 -6

Answer: C Type: BI Var: 50+ Objective: Write an Augmented Matrix

Page 2

1 B) 0 1 1 D) 1 1

3 -5 1 3 -5 1

5 3 -8 5 3 -8

-3 -8 -6 -3 -8 -6

2) -3x - 6y + 9z = 12 -x + 2y = -10 x - 8z = -2 A) -3 -6 9 12 -1 2 0 -10 1 0 -8 -2

B) -3 -6 9 -1 2 0 1 0 -8 12 -10 -2

D) -3 -6 9 -1 2 0 1 -8 0

12 -10 -2

-3 -6 9 12 0 -1 2 -10 0 1 -8 -2

Answer: A Type: BI Var: 50+ Objective: Write an Augmented Matrix

3) 4(x - 3y) = 9y + 6 8x = 9y + 6 4 -21 6 A) 8 9 6

4 8

-21 6 -9 6

4 8

-3 6 9 6

Answer: B Type: BI Var: 50+ Objective: Write an Augmented Matrix

4) 7x + 9 = -7z -4x + 4z =-8 - 6y -3x - 7y + 9z = 2 A) 7 0 7 -9 -4 6 4 -8 -3 -7 9 2 C) 7 0 7 0 -4 -6 4 0 -3 -7 9 0 Answer: A Type: BI Var: 50+ Objective: Write an Augmented Matrix

Page 3

B) 7 0 -7 9 -4 -6 4 -8 -3 -7 9 2 D) 7 0 9 -4 -6 4 -3 -7 9

-7 -8 2

4 8

-3 6 -9 6

Write a system of linear equations represented by the augmented matrix. -7 9 -1 5) 5 -6 7 A) -7x + 9y = -1 5x - 6y = 7

B) -7x + 9y = -1 5x - 6y = -7

C) -7x - 9y = 1 5x + 6y = 7

D) -7x - 9y = 1 5x + 6y = -7

Answer: A Type: BI Var: 50+ Objective: Write an Augmented Matrix

5 8 6 0 -5 11 A) 5x + 8y = 6 -5x = 11

B) 5x = 6 8x - 5y = 11

5x + 8y = 6 x - 5y = 11

Answer: D Type: BI Var: 50+ Objective: Write an Augmented Matrix

7) 4 9 4

9 -4 -12 2 7 6 -2 2 -1

A) 4x = -12 2y = 6 2z = -1 C) 4x + 9y - 4z = -12 9x + 2y + 7z= 6 4x - 2y + 2z= -1 Answer: C Type: BI Var: 50+ Objective: Write an Augmented Matrix

Page 4

B) 4x + 9y - 4z = 12 9x + 2y + 7z= -6 4x - 2y + 2z= 1 D) 4x + 9y + 4z = -12 9x + 2y - 2z= 6 -4x + 7y + 2z= -1

5x + 8y = 6 -5y = 11

1 0 0 1

0 0

7 -9 1 9

A) x = 7 y=9 1 z= 9

B) x = -7 y = -9 1 z= 9

C) x = 7 y = -9 1 z= 9

D) x = -7 y=9 1 z=9

Answer: C Type: BI Var: 50+ Objective: Write an Augmented Matrix 2 Use Elementary Row Operations

Perform the elementary row operation on the given matrix. 1 1) - R2 → R2 2 9 11 16 32

1 20

9 32

11 64

1 40

9 8

11 16

1 10

9 11 -8 -16

5 8

1 2

2 -2 0 -2 -7 -1

1 -10

18 16

22 2 32 20

-10 -26 -4 -2 -7 -1

Answer: C Type: BI Var: 50+ Objective: Use Elementary Row Operations

2) 3R2 + R1 → R1 4 5 1 -2 -7 -1 A)

-2 -16 -2 -2 -7 -1

4 10

Answer: A Type: BI Var: 50+ Objective: Use Elementary Row Operations

Page 5

3) R2 ⇔ R3 -8 2 -9 -4 -3 7 -6 6 5 0 3 6 -8 A) 2 5 -8 C) -3 2

2 7 0 2 7 7

-9 -4 -3 9 3 6 -9 -4 -6 6 -3 9

-3 B) -8 5 -8 D) 5 -3

7 2 0 2 0 7

-6 6 -9 -4 3 6 -9 -4 3 6 -6 6

1 B) 7 0 1 D) 2 5

16 5 3 18 5 8 -68 -16 16 16 5 3 -62 -20 -7 12 9 13

Answer: D Type: BI Var: 50+ Objective: Use Elementary Row Operations

4) -5R1 + R3 → R3 1 7 5

16 18 12

1 A) 7 6 1 C) 7 0

5 5 9

3 8 13

16 5 3 18 5 8 28 14 16 16 5 3 18 5 8 -68 -16 -2

Answer: C Type: BI Var: 50+ Objective: Use Elementary Row Operations

Page 6

Perform the indicated row operations, then write the new matrix. 1 1 1 -1 2R1 + R2 → R2, 5) -2 3 5 3 -3R1 + R3 → R3 3 2 4 1 A)

B) 1 0 0

1 1 5 7 -1 1

-1 1 4

1 0 3

1 5 2

1 7 4

-1 1 1

1 0 0

1 1 5 7 -1 1

-1 3 1

D) 1 1 -2 3 0 -1

1 5 1

-1 3 1

Answer: A Type: BI Var: 50+ Objective: Use Elementary Row Operations 3 Use Gaussian Elimination and Gauss-Jordan Elimination

Determine if the matrix is in row-echelon form. 1 -6 -8 -7 1) 0 1 0 8 0 0 2 3 A) Yes

B) No

Answer: B Type: BI Var: 50+ Objective: Use Gaussian Elimination and Gauss-Jordan Elimination

1 0 0 0 1 0 2) 0 0 -1 0 0 0 A) Yes

0 0 0 1

2 4 9 -6 B) No

Answer: B Type: BI Var: 50+ Objective: Use Gaussian Elimination and Gauss-Jordan Elimination

Solve the system using Gaussian elimination or Gauss-Jordan elimination. 3) 5x - 7y = -3 -6x + 5y = 7 A) {(-1, 2)}

B) {(-1, -2)}

C) {(-2, -1)}

Answer: C Type: BI Var: 50+ Objective: Use Gaussian Elimination and Gauss-Jordan Elimination

Page 7

D) {(2, 1)}

x + 4y = -7 4x + 3y = 11 65 17 A) - , 19 19

B) {(5, -3)}

C) {(-3, 5)}

17 65 , 19 19

Answer: B Type: BI Var: 50+ Objective: Use Gaussian Elimination and Gauss-Jordan Elimination

5) -5y = 24 - 3x 2(x - 4y) = 49 - y A) {(-9, -7)}

B) {(9, 7)}

C) {(7, 9)}

D) {(-7, -9)}

Answer: D Type: BI Var: 50+ Objective: Use Gaussian Elimination and Gauss-Jordan Elimination

6) -5x + 9y - 9z = -106 2x - 2y - 5z = -14 -5x - 8y + 2z = 11 A) {(5, -3, 6)}

B) {(-3, 5, 6)}

C) {(-3, -5, -6)}

D) {(-5, 3, 6)}

Answer: A Type: BI Var: 50+ Objective: Use Gaussian Elimination and Gauss-Jordan Elimination

7) 5x + 9z = -16 + 6y 8x = 7y + 7z - 141 8z = -2y - 2x + 42 A) {(-8, 5, 6)}

B) {(5, -8, 6)}

C) {(5, 8, -6)}

D) {(8, -5, 6)}

Answer: A Type: BI Var: 50+ Objective: Use Gaussian Elimination and Gauss-Jordan Elimination

8) w - 2x + 3y = 8 -5x - 5y = -25 -2x + 3y - 4z = 17 -3w - 5x + 2y - 3z = -4 A) {(-3, -2, -3, 3)}

B) {(-3, -2, 3, -3)}

C) {(3, 2, -3, 3)}

Answer: D Type: BI Var: 50+ Objective: Use Gaussian Elimination and Gauss-Jordan Elimination

Page 8

D) {(3, 2, 3, -3)}

9) 2(x - 2z) = 3y + x + 30 x = 2y - 2z - 3 -5x + 2y + 6z = -41 A) {(-1, 3, 5)}

B) {(1, -3, -5)}

C) {(-3, 1, -5)}

D) {(3, -1, 5)}

Answer: B Type: BI Var: 50+ Objective: Use Gaussian Elimination and Gauss-Jordan Elimination

10) x1 - 5x2 + 2x3 = 6 5x2 - 2x3 = -1 -3x2 - 4x3 - 5x4 = -30 5x1 + 4x2 + 4x3 + 5x4 = 56 A) {(5, 1, -3, -3)}

B) {(-5, -1, -3, -3)}

C) {(5, 1, 3, 3)}

D) {(-5, -1, 3, 3)}

Answer: C Type: BI Var: 50+ Objective: Use Gaussian Elimination and Gauss-Jordan Elimination 4 Mixed Exercises

Solve the problem. 1) Andre borrowed $40,000 to buy a truck for his business. He borrowed from his parents who charge him 3% simple interest. He borrowed from a credit union that charges 4% simple interest, and he borrowed from a bank that charges 6% simple interest. He borrowed four times as much from his parents as from the bank, and the amount of interest he paid at the end of 1 yr was $1,560. How much did he borrow from each source? A) He borrowed 6,000 from his parents, 32,500 from the credit union, and 1,500 from the bank. B) He borrowed $8,000 from his parents, $30,000 from the credit union, and $2,000 from the bank. C) He borrowed 1,500 from his parents, 32,500 from the credit union, and 6,000 from the bank. D) He borrowed $4,000 from his parents, $35,000 from the credit union, and $1,000 from the bank. Answer: B Type: BI Var: 50+ Objective: Mixed Exercises

Page 9

2) Danielle stayed in three different cities (Washington, D.C., Atlanta, Georgia, and Dallas, Texas) for a total of 22 nights. She spent twice as many nights in Dallas as she did in Washington. The total cost for 22 nights (excluding tax) was $3,100. Determine the number of nights that she spent in eac city. City Cost per Night Washington $100 Atlanta $175 Dallas $150 A) 4 nights in Washington, 10 nights in Atlanta, and 8 nights in Dallas B) 5 nights in Washington, 7 nights in Atlanta, and 10 nights in Dallas C) 1 night in Washington, 19 nights in Atlanta, and 2 nights in Dallas D) 6 nights in Washington, 4 nights in Atlanta, and 12 nights in Dallas Answer: D Type: BI Var: 50+ Objective: Mixed Exercises

Find the partial fraction decomposition for the given rational expression. Use the technique of Gaussian elimination to find A, B, and C. -3x2 + 7x - 10 A B C 3) + 2= (x + 2)(x - 1) x + 2 x - 1 (x - 1)2 A)

-4 1 2 + + x + 2 x - 1 (x - 1)2

-4 1 2 + x + 2 x - 1 (x - 1)2

4 1 2 + x + 2 x - 1 (x - 1)2

-4 2 2 + x + 2 x - 1 (x - 1)2

Answer: B Type: BI Var: 50+ Objective: Mixed Exercises 5 Technology Connections

Use a calculator to approximate the reduced row-echelon form of the augmented matrix representing the given system. Give the solution set where x, y, and z are rounded to 2 decimal places. 0.52x - 3.79y - 4.67z = 9.15 1) 0.03x + 0.06y + 0.13z = 0.53 0.974x + 0.813y + 0.419z = 0.189 A) {(-4.49, -6.63, 2.49)} C) {(7.91, -1.03, 3.18)} Answer: D Type: BI Var: 50+ Objective: Technology Connections

Page 10

B) {(-15.61, -1.39, -2.84)} D) {(6.18, -11.2, 7.82)}

6.2 Inconsistent Systems and Dependent Equations 0 Concept Connections

Provide the missing information. 1) True or false? A system of linear equations in three variables may have no solution. Answer: True Type: SA Var: 1 Objective: Concept Connections

2) True or false? A system of linear equations in three variables may have exactly one solution. Answer: True Type: SA Var: 1 Objective: Concept Connections

3) True or false? A system of linear equations in three variables may have exactly two solutions. Answer: False Type: SA Var: 1 Objective: Concept Connections

4) True or false? A system of linear equations in three variables may have infinitely many solutions. Answer: True Type: SA Var: 1 Objective: Concept Connections

5) If a system of linear equations has no solution, then the system is said to be

Answer: inconsistent Type: SA Var: 1 Objective: Concept Connections

6) If a system of linear equations has infinitely many solutions, then the equations are said to be . Answer: dependent Type: SA Var: 1 Objective: Concept Connections 1 Identify Inconsistent Systems and Systems with Dependent Equations

For the given augmented matrix, determine the number of solutions to the corresponding system of equations. 1 0 7 -2 1) 0 1 -8 1 0 0 0 0 A) Infinitely many solutions B) One solution C) No solution Answer: A Type: BI Var: 50+ Objective: Identify Inconsistent Systems and Systems with Dependent Equations

Page 11

1 0 0 -3 2) 0 1 0 -9 0 0 1 0 A) Infinitely many solutions B) One solution C) No solution Answer: B Type: BI Var: 50+ Objective: Identify Inconsistent Systems and Systems with Dependent Equations

Determine the solution set for the system represented by the augmented matrix. 1 0 -7 4 3) 0 1 6 1 0 0 0 1 A) { } B) {(11, -5, 1)} C) {(-7, 6, 0)}

D) {(4, 1, 0)}

Answer: A Type: BI Var: 50+ Objective: Identify Inconsistent Systems and Systems with Dependent Equations

1 0 -4 7 4) 0 1 8 -7 0 0 0 0 A) {(7 - 4z, -7 + 8z, z) | z is any real number} C) { }

B) {(7 + 4z, -7 - 8z, z) | z is any real number} D) {(7, -7, 0)}

Answer: B Type: BI Var: 50+ Objective: Identify Inconsistent Systems and Systems with Dependent Equations

Solve the system using Gaussian elimination or Gauss-Jordan elimination. 5) 8x + 10y = -17 4x + 5y = -9 A) { }

B) {(-17, -9)}

5 9 C) {(- x - , y) | y is any real number} 4 4

D) {(-7, 7)}

Answer: A Type: BI Var: 50+ Objective: Identify Inconsistent Systems and Systems with Dependent Equations

Page 12

6) -6x + 15y = 15 2x - 5y = -5 A) { }

5 B) {( x - 5 , y) | y is any real number} 2 2

C) {(15, -5)}

D) {(-9, -5)}

Answer: B Type: BI Var: 50+ Objective: Identify Inconsistent Systems and Systems with Dependent Equations

7) -3x - 7y + 7z = 71 -2x + 7y - 3z = -22 -3x - 9y + 9z = 87 A) { }

B) {(5, 2, 6)}

C) {(-5, 2, 6)}

D) {(-5, -2, 6)}

Answer: D Type: BI Var: 50+ Objective: Identify Inconsistent Systems and Systems with Dependent Equations

8) -4x - 6y + 5z = 13 8x + 4y - 7z = -7 4x - 2y - 2z = -15 A) {(8, 4, 5)}

B) { }

C) {(8, 5, 4)}

Answer: B Type: BI Var: 5 Objective: Identify Inconsistent Systems and Systems with Dependent Equations

9) -2x - y + 3z = 13 x - 3y - 3z = -4 A) { } B) {(1, 2, -3)} C) {(-1, -2, 3)} 12 43 5 D) {( z - , - 3 z - , z) | z is any real number} 7 7 7 7 Answer: D Type: BI Var: 50+ Objective: Identify Inconsistent Systems and Systems with Dependent Equations

10) -3x - 3y - 3z = 30 -9x - 9y - 9z = 90 -1.5x - 1.5y - 1.5z = 15 A) {(2, 2, 6)} C) {(x, y, z) | -3x - 3y - 3z = 30}

B) {(-2, -2, -6}) D) { }

Answer: C Type: BI Var: 50+ Objective: Identify Inconsistent Systems and Systems with Dependent Equations

Page 13

D) {(5, 8, 4)}

11) 3x + 2y + 7z = 38 9x + 6y + 21z = 114 0.6x + 0.4y + 1.4z = 7.6 A) {(-1, -4, 7}) C) { }

B) {(1, 4, -7)} D) {(x, y, z) | 3x + 2y + 7z = 38}

Answer: D Type: BI Var: 50+ Objective: Identify Inconsistent Systems and Systems with Dependent Equations

The solution set to a system of dependent equations is given. Write three ordered triples that are solutions to the system. Answers may vary. 6 - 4y - 15z 12) , y, z y and z are any real numbers 3 A) (2, 0, 0), (2, 3 ,0), (1, 0, 1) C) (5, 0, 0), (2, 3 ,0), (-3, 0, 1)

B) (2, 0, 0), (-2, 3 ,0), (-3, 0, 1) D) (5, 0, 0), (-2, 3 ,0), (1, 0, 1)

Answer: B Type: BI Var: 50+ Objective: Identify Inconsistent Systems and Systems with Dependent Equations

The solution set to a system of dependent equations is given. Write the specific solution corresponding to the given value of z. 13) {(4z - 7, z - 4, z) | z is any real number} z = -5 A) (-27, -9, -5)

B) (-13, -9, -5)

C) (-13, 1, -5)

Answer: A Type: BI Var: 50+ Objective: Identify Inconsistent Systems and Systems with Dependent Equations

Page 14

D) (-27, 1, -5)

2 Solve Applications of Systems of Equations

Solve the problem. 1) Assume that traffic flows freely through intersections A, B, and C. The values x1, x2, x3, and all other numbers in the figure represent flow rates in vehicles per hour. 130

110 x1

140

150 A

B x3

C 160

170

a. Write an equation representing equal flow into and out of intersection A. b. Write an equation representing equal flow into and out of intersection B. c. Write an equation representing equal flow into and out of intersection C. d. Write the system of equations from parts (a)-(c) in standard form. e. Write the reduced row-echelon form of the augmented matrix representing the system of equation from part (d). f. If the flow rate between intersections A and C is 120 vehicles per hour, determine the flow rates x1 and x2. g. If the flow rate between intersections A and C is between 150 and 240 vehicles per hour, inclusive, determine the flow rates x1 and x2. Answer: a. 130 + 140 = x1 + x3 b. x1 + x2 = 110 + 150 c. x3 + 160 = x2 + 170 d. x1 + x3 = 270 x1 + x2 = 260 x3 - x2 = 10 1 0 1 270 e. 0 1 -1 -10 0 0 0 0 f. x1 = 150; x2 = 110 g. 30 ≤ x1 ≤ 120; 140 ≤ x2 ≤ 230 Type: SA Var: 50+ Objective: Solve Applications of Systems of Equations

Page 15

2) An accountant checks the reported earnings for a concert venue for three different performers again the number of tickets sold. Performer Children Student General Total Tickets Tickets Admission Revenue 1 1,000 250 1,000 $24,750 2 400 300 300 $10,700 3 400 300 300 $8,700 Let x, y, and z represent the cost for children tickets, student tickets, and general admission tickets, respectively. Set up an augmented matrix for the system and solve for x, y, and z. Explain what the accountant knows about the reported earnings. A) {(6, 6, 15)}; There were $6 children's tickets, $6 student tickets, and $15 general admission tickets sold. B) { }; The system of equations reduces to a contradiction. There are no values for x, y, and z that can simultaneously meet the conditions of this problem. C) {(7, 7, 16)};There were $7 children's tickets, $7 student tickets, and $16 general admission tickets sold. D) {(8, 8, 17)}; There were $8 children's tickets, $8 student tickets, and $17 general admission tickets sold. Answer: B Type: BI Var: 50+ Objective: Solve Applications of Systems of Equations

Page 16

3) 402

425 x1

242

275 x4

281

357 x3 331

397

a. Assume that traffic flows at a rate of 215 vehicles per hour on the stretch of road between intersections D and A. Find the flow rates x1, x2, and x3. b. If traffic flows at a rate of between 195 and 270 vehicles per hour inclusive between intersections D and A, find the flow ratesx1, x2, and x3. A) a. x1 = 55, x2 = 205, and x3 = 165; b. 40 ≤ x1 ≤ 115, 180 ≤ x2≤ 255, and 155 ≤ x3≤ 230 B) a. x1 = 55, x2 = 205, and x3 = 165; b. 35 ≤ x1 ≤ 110, 185 ≤ x2≤ 260, and 145 ≤ x3≤ 220 C) a. x1 = 65, x2 = 195, and x3 = 170; b. 35 ≤ x1 ≤ 110, 185 ≤ x2≤ 260, and 145 ≤ x3≤ 220 D) a. x1 = 65, x2 = 195, and x3 = 170; b. 40 ≤ x1 ≤ 115, 180 ≤ x2≤ 255, and 155 ≤ x3≤ 230 Answer: B Type: BI Var: 50+ Objective: Solve Applications of Systems of Equations

6.3 Operations on Matrices 0 Concept Connections

Provide the missing information. 1) If the of a matrix is p × q, then p represents the number of represents the number of .

and q

Answer: order; rows; columns Type: SA Var: 1 Objective: Concept Connections

2) A matrix with the same number of rows and columns is called a Answer: square Type: SA Var: 1 Objective: Concept Connections

Page 17

matrix.

3) What are the requirements for two matrices to be equal? Answer: The order of the matrices must be the same, and the corresponding elements must be equal. Type: SA Var: 1 Objective: Concept Connections

4) An n × m matrix whose elements are all zero is called a

matrix.

Answer: zero Type: SA Var: 1 Objective: Concept Connections

5) To multiply two matrices A and B, the number of of B.

of A must equal the number of

Answer: columns; rows Type: SA Var: 1 Objective: Concept Connections

6) If A is a 5 × 3 matrix and B is a 3 × 7 matrix, then the product AB will be a matrix of order . The product BA (is/is not) defined. Answer: 5 × 7; is not Type: SA Var: 1 Objective: Concept Connections

7) True or false: Matrix multiplication is a commutative operation. Answer: False Type: SA Var: 1 Objective: Concept Connections

8) True or false: If a row matrix A and a column matrix B have the same number of elements, then the product AB is well defined. Answer: True Type: SA Var: 1 Objective: Concept Connections 1 Determine the Order of a Matrix

Give the order of the matrix. 8 7 7 1) -4 8 -6 A) 2×3

B) 6

Answer: A Type: BI Var: 50+ Objective: Determine the Order of a Matrix

Page 18

C) 5

D) 3×2

Classify the matrix as a square matrix, row matrix, column matrix, or none of these. 1 5 2) 5 -3 A) Row matrix Answer: B

B) Square matrix

C) Column matrix

D) None of these

Type: BI Var: 50+ Objective: Determine the Order of a Matrix

Give the order of the matrix. Classify the matrix as a square matrix, row matrix, column matrix, or none of these. -3 -8 -2 3) 5 2 5 A) 3×2, none of these C) 2×3, row matrix Answer: B

B) 2×3, none of these D) 3×2, column matrix

Type: BI Var: 50+ Objective: Determine the Order of a Matrix

4) 3

-5 -5

A) 4×1, column matrix C) 1×4, row matrix Answer: C

B) 1×4, column matrix D) 4×1, row matrix

Type: BI Var: 50+ Objective: Determine the Order of a Matrix

4 -1 2 5) 8 4 -5 -7 6 -4 A) 7, none of these C) 3×3, square matrix Answer: C Type: BI Var: 50+ Objective: Determine the Order of a Matrix

Page 19

B) 3×3, none of these D) 9, square matrix

6) -2

8 2

7 π

1 0.6 11 A) 3×4, none of these C) 4×3, column matrix Answer: A

B) 3×4, column matrix D) 3×4, row matrix

Type: BI Var: 50+ Objective: Determine the Order of a Matrix

Determine the value of the given element of the matrix. 7) a 34

3 -8 9 -2 A = 15 -3 10 8 -7 2 -5 13 A) 13

B) -5

C) -7

D) 24

Answer: A Type: BI Var: 50+ Objective: Determine the Order of a Matrix 2 Add and Subtract Matrices

For what values of x, y, and z will A = B? y 3 -2 x B= 1) A = 8 9 z 9 A) x = 3, y = -2, z = 8 C) x = 8, y = -2, z = 9

B) x = 9, y = -2, z = 8 D) x = 3, y = 9, z = 3

Answer: A Type: BI Var: 50+ Objective: Add and Subtract Matrices

Find the additive inverse of A. 9 2 2) A = -4 5 A)

-9 -2 4 -5

Answer: A Type: BI Var: 50+ Objective: Add and Subtract Matrices

Page 20

9 -2 4 5

-9 2 -4 -5

-9 2 4 5

Find A + B. -6 -2 -2 7 3) A = 9 4 , B = -8 -1 0 2 8 4 -8 5 A) 1 3 8 6

-8 -9 B) 1 5 8 -2

-4 C) 17 -8

5 3 6

-4 -9 D) 17 5 -8 -2

-7 12 C) -5 0 7 -1

-5 12 D) -1 0 9 -1

-12 -7 C) -12 -6 1 -7

Answer: A Type: BI Var: 50+ Objective: Add and Subtract Matrices

Find B - A. 2 4) A = 4 2

6 -7 6 4 , B = -5 -4 -7 7 6

9 0 A) 9 8 -5 -13

-9 0 B) -9 -8 5 13

Answer: B Type: BI Var: 50+ Objective: Add and Subtract Matrices

Find A + B. 5) A =

-9 -6 , B = -5 -3 -1 -2 -5 6

-6 -9 -5 2

A) Undefined

-15 -15 -5 B) -3 -4 -2

Answer: A Type: BI Var: 50+ Objective: Add and Subtract Matrices

Page 21

-15 -15 -3 -4

Find C - A + B -4 -3 6.5 6 ,B= 1 3 14

5 6) A = 6 6

3 1

2 1 ,C= 1- 3 11

-3 10 3.5 -6 A) 8 2 + 2 11 3

2+ 5

4.2 10 3

14 9.6 -2

B) 2 + 11 27 -6

17 C)

1.5

D) 10 3

8 16 3

2 - 14 + 11

Answer: D Type: BI Var: 50+ Objective: Add and Subtract Matrices 3 Multiply a Matrix by a Scalar

Perform the indicated operations. 1) Find 2A -2

A = -1

3 3 2

-4 2 -2 2 3

6 3

-4 0 C) -2 2 3

6 3

Answer: C Type: BI Var: 50+ Objective: Multiply a Matrix by a Scalar

Page 22

-1

3 2

1 2

3 2

3 4

D) 1

2+ 3

5 7 2

2) Find -5A - 4B. A=

3 -8 -4 -4 5 and B = -4 6 9 -4 5

-1 -6

1 20 24 36 -50 -21 -31 60 16 C) 4 -10 -69

-1 -20 -24 -36 50 21 8 7 21 D) 36 -49 -6

Answer: A Type: BI Var: 50+ Objective: Multiply a Matrix by a Scalar

Find the indicated matrix. 3) Find -3(A- B). A=

1 9 -9 4

3 3

-7 -2 8 and B = 3 8 4

-51 15 -24 18

2 -9 3 1 -17 0

-9 12

-9 -3 27 51 -0 -36

Answer: C Type: BI Var: 50+ Objective: Multiply a Matrix by a Scalar

4) Find -4A - 5(A - B). A=

4 8 0 8 2 -1 and B = 9 8 -4 -6 5 8

A) -76 -82 5 -51 -97 -4 -36 -2 5 C) 39 -17 -44 Answer: D Type: BI Var: 50+ Objective: Multiply a Matrix by a Scalar

Page 23

-4 62 5 111 47 -76 4 -62 -5 D) -111 -47 76 B)

9 3 -51 0

-27 36

Given the matrices A and B, solve for X. 5) 2X = 2A - B. A=

-6 -4 -8 -18 and B = 6 4 6 16 -

1 2

1 5

1 3

1 4

-2 -5 -3 -4

-2 3

5 -4

-4 10 6 -8

-6 9 -6 1

0 0

-54 -11 -120 -19

D) Not possible

Answer: C Type: BI Var: 50+ Objective: Multiply a Matrix by a Scalar

6) 5X - B = A. A=

-3 5 and B = -3 4 -3 -4 -3 5 1 5

9 5

6 5

9 5

6 5

1 5

1 -9

Answer: B Type: BI Var: 50+ Objective: Multiply a Matrix by a Scalar 4 Multiply Matrices

Find AB, if possible. 1) A =

-6 7 and B = 3 8 8 1 7 1

-11 67 -19 145

Answer: A Type: BI Var: 50+ Objective: Multiply Matrices

Page 24

-37 -11 -65 -47

2 -1 and B = 6 2 65 16 11 -2 A) 41 42

2) A =

12 -2 6 30

C) Not possible

-2 12 30 6

C) Not possible

-23 7 2 35

26 B) 84 -33

7 C) 27 12

D) 26

B) Not possible

65 C) 36 82

Answer: A Type: BI Var: 50+ Objective: Multiply Matrices

3) A =

4 3 -3 8

7 -6 7 and B = 0 7 1

-2 10 4 9

5 -6

-2 -6 -25 49 34 -77

Answer: C Type: BI Var: 50+ Objective: Multiply Matrices

-7 7 -5 5 4) A = 3 9 3 and B = 8 8 -9 1 -1

A) Not possible

84 -33

Answer: B Type: BI Var: 50+ Objective: Multiply Matrices

Find A2 if possible. 9 -1 8 5) A = 8 -5 -2 -1 5 1 65 A) 9 30

36 82 7 72 -19 -17

Answer: A Type: BI Var: 50+ Objective: Multiply Matrices

Page 25

34 7 72

30 -19 -17

81 1 D) 64 25 1 25

64 4 1

Find AB and BA. 2 -2 1 6) A = and B = -3 9 0

0 1 2

-2

2 -2 -3 9 -3 9 2 0 2 0 ; BA = D) AB = 0 9 0 9

1 0 1 0 ; BA = 0 1 0 1 2 -2 ; BA = 2 -3 C) AB = -2 9 -3 9

A) AB =

B) AB =

; BA =

Answer: B Type: BI Var: 50+ Objective: Multiply Matrices

1 7) A = 0 0

0 1 0

1 1 -5 0 0 and B = 8 -9 -5 1 -1 4 5

1 1 -5 1 8 -1 A) AB = 8 -9 -5 ; BA = 1 -9 4 -1 4 5 -5 -5 5

1 1 -5 1 1 -5 B) AB = 8 -9 -5 ; BA = 8 -9 -5 -1 4 5 -1 4 5

1 C) AB = 0 0

1 D) AB = 0 0

0 0 1 ; BA = -9 0 0 0 5 0

Answer: B Type: BI Var: 50+ Objective: Multiply Matrices

Page 26

0 0 -9 0 0 5

0 1 0

0 1 ; BA = 0 0 1 0

0 0 1 0 0 1

5 Apply Operations on Matricies

Solve the problem. 1) In matrix C, a coffee shop records the cost to produce a cup of coffee and the cost to produce a cup of hot chocolate. Matrix P contains the selling prices to the customer. coffee chocolate coffee chocolate $0.93 $0.86 Small $3.00 $2.40 Small C = $1.21 $1.11 Medium; P = $3.75 $3.15 Medium $1.51 $1.41 Large $4.40 $3.90 Large a. Compute P - C and interpret its meaning. b. If the tax rate in a certain city is 6%, use scalar multiplication to find a matrix F that gives the final price to the customer (including sales tax) for both beverages for each size. Round each entry to the nearest cent. $2.07 $1.54 A) a. P - C = $2.54 $2.04 ; this represents the profit that the store makes for both beverages fo $2.89 $2.49 each size. $2.19 $1.63 b. F = $2.69 $2.16 $3.06 $2.64 $2.07 $1.54 B) a. P - C = $2.54 $2.04 ; this represents the profit that the store makes for both beverages fo $2.89 $2.49 each size. $3.18 $2.54 b. F = $3.98 $3.34 $4.66 $4.13 $3.93 $3.26 C) a. P - C = $4.96 $4.26 ; this represents the profit that the store makes for both beverages fo $5.91 $5.31 each size. $3.18 $2.54 b. F = $3.98 $3.34 $4.66 $4.13 $2.07 $1.54 D) a. P - C = $2.54 $2.04 ; this represents the profit that the store makes for both beverages fo $2.89 $2.49 each size. $0.18 $0.14 b. F = $0.23 $0.19 $0.26 $0.23 Answer: B Page 27

Type: BI Var: 50+ Objective: Apply Operations on Matricies

2) A street vendor at a parade sells fresh lemonade, soda, bottled water, and iced-tea, and the unit price for each item is given in matrix P. The number of units sold of each item is given in matrix N. Compute NP and interpret the result. Lemonade Soda N = 120 280

Water 370

Tea 70

$3.00 Lemonade $2.00 Soda P= $1.00 Water $1.50 Tea A) [$900]; The value $900 represents the total revenue from the sale of these four items B) [$6,300]; The value $6,300 represents the total revenue from the sale of these four items C) [$1,410]; The value $1,410 represents the total revenue from the sale of these four items D) [$1,395]; The value $1,395 represents the total revenue from the sale of these four items Answer: D Type: BI Var: 50+ Objective: Apply Operations on Matricies

3) A math course has 4 exams weighted 15%, 10%, 20%, and 55%, respectively, toward the final grade. Suppose that a student earns grades of 75, 84, 92, and 86, respectively, on the four exams. The weights are given in matrix W and the test grades are given in matrix G. Compute WG and interpret the result. 75 Test 1 , G = 84 Test 2 0.55] 92 Test 3 86 Test 4

Test 1 Test 2 Test 3 Test 4 W=

[0.15

0.10

0.20

A) [84.25]; The value 84.25 is the student's overall course grade. B) [80.15]; The value 80.15 is the student's overall course grade. C) [85.35]; The value 85.35 is the student's overall course grade. D) [50.55]; The value 50.55 is the student's overall course grade. Answer: C Type: BI Var: 12 Objective: Apply Operations on Matricies

Page 28

4) A gas station manager records the number of gallons of Regular, Plus, and Premium gasoline sold during the week (Monday–Friday) and on the weekends (Saturday–Sunday) in matrix A. The selling price and profit for 1 gal of each type of gasoline is given in matrix B. Regular 4,270 A= 2,320

Plus Premium 1,760 810 Weekdays 610 410 Weekend

Selling Price $3.49 B = $3.69 $4.09

Profit $0.26 Regular $0.28 Plus $0.20 Premium

a. Compute AB. b. Determine the profit for the weekend. c. Determine the revenue for the entire week. $24,709.60 $12,024.60 $1,765.00 $856.00 b. $12,880.60 c. $1,765.00 $24,709.60 $1,765.00 C) a. AB = $12,024.60 $856.00 b. $2,621.00 c. $12,024.60

A) a. AB =

Answer: B Type: BI Var: 50+ Objective: Apply Operations on Matricies

Page 29

$24,709.60 $1,765.00 $12,024.60 $856.00 b. $856.00 c. $36,734.20 $24,709.60 $12,024.60 D) a. AB = $1,765.00 $856.00 b. $856.00 c. $26,474.60 B) a. AB =

5) The labor costs per hour for an electrician, plumber, and air-conditioning/heating expert are given in matrix L. The time required from each specialist for three new model homes is given in matrix T. Cost/hr $40 Electrician L = $38 Plumber $34 AC/heating Time(hr) Electrician Plumber AC/heating 165 14 16 Model 1 T = 25 22 19 Model 2 17 13 8 Model 3 Find a matrix that gives the total cost for these three services for each model. $8,128 A) [$11,604] B) [$11,600] C) $1,838 $1,634

$7,676 D) $2,482 $1,446

Answer: D Type: BI Var: 50+ Objective: Apply Operations on Matricies

6) A student researches the cost for three cell phone plans. Matrix C contains the cost per text message and the cost per minute over the maximum number of minutes allowed in each plan. Matrix N represents the number of text messages and number of minutes over the maximum for 3 months. Cost/text $0.20 C = $0 $0.15

Cost/min $0.35 Plan A $0.35 Plan B $0 Plan C

Month 1 Month 2 Month 3 27 60 25 Number of texts N= 105 27 0 Minutes over Find the product CN and interpret its meaning. $42.15 $21.45 $5.00 A) $36.75 $9.45 $0.00 $4.05 $9.00 $3.75 The matrix CN represents the additional cost per month for each plan. For example column 1 represents the cost for Plan A for months 1, 2, and 3, respectively.

Page 30

$42.15 $36.75 $4.05 B) $21.45 $9.45 $9.00 $5.00 $0.00 $3.75 The matrix CN represents the additional cost per month for each plan. For example row 1 represents the cost for Plan A for months 1, 2, and 3, respectively. $42.15 $21.45 $5.00 C) $36.75 $9.45 $0.00 $4.05 $9.00 $3.75 The matrix CN represents the additional cost per month for each plan. For example row 1 represents the cost for Plan A for months 1, 2, and 3, respectively. $42.15 $36.75 $4.05 D) $21.45 $9.45 $9.00 $5.00 $0.00 $3.75 The matrix CN represents the additional cost per month for each plan. For example column 1 represents the cost for Plan A for months 1, 2, and 3, respectively. Answer: C Type: BI Var: 50+ Objective: Apply Operations on Matricies

7) Matrix G represents the gas mileage (in gal/mi) for city and highway driving for four models of car. Matrix N1 represents the number of city and highway miles driven for one trip. Matrix N3 represents the number of city and highway miles driven for three trips. City 1 23

Highway 1 Car 1 automatic 30

1 24

1 Car 1 manual 37

1 24

1 Car 2 automatic 30

1 25

1 Car 2 manual 31

110 Number of city miles 260 Number of highway miles Trip 1 Trip 2 Trip 3 110 120 60 Number of city miles N3 = 260 350 290 Number of highway miles a. Find the product GN1 and interpret its meaning. Round each entry to 1 decimal place. b. Find the product GN3 and interpret its meaning. Round each entry to 1 decimal place. N1 =

Page 31

13.4 11.6 . A) a. GN1 = 13.3 12.8 This matrix represents the number of gallons of gasoline used for each model of car for 110 m driven in the city and 260 mi driven on the highway. 13.4 16.9 12.3 11.6 14.5 10.3 b. GN3 = . 13.3 16.7 12.2 12.8 16.1 11.8 This matrix represents the number of gallons of gasoline used for each model of car for each o the three trips. 13.4 11.6 . B) a. GN1 = 13.3 12.8 This matrix represents the number of gallons of gasoline used for each model of car for 110 m driven in the city and 260 mi driven on the highway. 12.3 16.9 13.4 10.3 14.5 11.6 b. GN3 = . 12.2 16.7 13.3 11.8 16.1 12.8 This matrix represents the number of gallons of gasoline used for each model of car for each o the three trips. 15.0 13.8 . C) a. GN1 = 14.5 13.9 This matrix represents the number of gallons of gasoline used for each model of car for 260 m driven in the city and 110 mi driven on the highway. 13.4 16.9 12.3 11.6 14.5 10.3 b. GN3 = . 13.3 16.7 12.2 12.8 16.1 11.8 This matrix represents the number of gallons of gasoline used for each model of car for each o the three trips.

Page 32

13.4 11.6 . D) a. GN1 = 13.3 12.8 This matrix represents the number of gallons of gasoline used for each model of car for 110 m driven in the city and 260 mi driven on the highway. 16.9 13.4 12.3 14.5 11.6 10.3 b. GN3 = . 16.7 13.3 12.2 16.1 12.8 11.8 This matrix represents the number of gallons of gasoline used for each model of car for each o the three trips. Answer: A Type: BI Var: 50+ Objective: Apply Operations on Matricies

Page 33

8) 5 y

(1, 5)

4 3 (-1, 1)

2 1 (3, 1)

-5 -4 -3 -2 -1 -1

-2 -3 -4 -5

a. Write a matrix A that represents the coordinates of the triangle. Place the x-coordinate of each point in the first row of A and the corresponding y-coordinate in the second row of A. b. Use addition of matrices to shift the triangle 5 units to the right and 4 units down. -1 1 3 1 5 1 -6 -4 -2 b. 5 9 5

A) a. A =

-1 1 1 5 4 6 8 b. 5 9 5

C) a. A =

3 1

Answer: B Type: BI Var: 50+ Objective: Apply Operations on Matricies

Page 34

-1 1 3 1 5 1 4 6 8 b. -3 1 -3

B) a. A =

1 5 1 -1 1 3 -3 1 -3 b. 4 6 8

D) a. A =

9) 5 y 4

(1, 4)

3 (4, 3)

2 (-1, 1)

-5 -4 -3 -2 -1 -1

-2 -3 -4 -5

1 -1 1 4 represents the coordinates of the triangle. Find the product 0 1 4 3 and explain the effect on the graph of the triangle. The matrix A =

-1 1 -1 -4

4 ; This matrix represents the reflection of the triangle across the y-axis. -3

1 1

-1 -4 ; This matrix represents the reflection of the triangle across the y-axis. 4 3

1 1

-1 -4 ; This matrix represents the reflection of the triangle across the x-axis. 4 3

-1 1 -1 -4

4 ; This matrix represents the reflection of the triangle across the x-axis. -3

Answer: A Type: BI Var: 50+ Objective: Apply Operations on Matricies

Page 35

0 ·A -1

10) 5 y

(1, 5)

4 3 (-3, 2)

(4, 3)

2 1

-5 -4 -3 -2 -1 -1

-2 (-2, -2)

-3 -4 -5

a. Write a matrix A that represents the coordinates of the quadrilateral. Place the x-coordinate of each point in the first row of A and the corresponding y-coordinate in the second row of A. b. What operation on A will shift the graph of the quadrilateral 2 units upward? -3 2 2 b. A + 0

1 4 -2 5 3 -2 2 2 2 0 0 0

-3 2 0 b. A + 2

1 4 -2 5 3 -2 0 0 0 2 2 2

A) a. A =

C) a. A =

2 -3 2 b. A + 0

B) a. A =

5 3 -2 1 4 -2 2 2 2 0 0 0

-3 1 4 -2 2 5 3 -2 0 0 0 0 b. A + -2 -2 -2 -2

D) a. A =

Answer: C Type: BI Var: 50+ Objective: Apply Operations on Matricies

6.4 Inverse Matrices and Matrix Equations 0 Concept Connections

Provide the missing information. 1) The symbol In represents the

matrix of order n.

Answer: identity Type: SA Var: 1 Objective: Concept Connections

2) In is an n × n matrix with Answer: 1; 0 Type: SA Var: 1 Objective: Concept Connections

Page 36

’s along the main diagonal and

’s elsewhere.

3) Given an n × n matrix A, if there exists a matrix A-1 such that A · A-1 = In and A-1· A = In, then A-1 is called the of A. Answer: inverse Type: SA Var: 1 Objective: Concept Connections

4) A matrix that does not have an inverse is called a inverse is said to be invertible or .

matrix. A matrix that does have an

Answer: singular; nonsingular Type: SA Var: 1 Objective: Concept Connections

5) Let A =

be an invertible matrix. Then a formula for the inverse A-1 is given by .

Answer:

d -b ad - bc -c b

Type: SA Var: 1 Objective: Concept Connections

6) Suppose that the matrix equation AX = B represents a system of n linear equations in n variables with a unique solution. Then X = . Answer: A-1B Type: SA Var: 1 Objective: Concept Connections

Page 37

1 Identify Identity and Inverse Matrices

Verify AIn = A. -9 4 6 1) A = -6 3 -1 -9 -3 5 1 0 A) AIn = 0 1 0 0

-9 4 6 0 -9 4 6 0 -6 3 -1 = -6 3 -1 1 -9 -3 5 -9 -3 5

-9 4 6 1 B) AIn = -6 3 - 1 0 -9 -3 5 0

-9 4 6 0 0 1 0 = -6 3 -1 0 1 -9 -3 5

0 -9 0 -6 1 -9

4 6 9 = 3 -1 6 -3 5 9

-4 -6 -3 1 3 -5

-9 4 6 1 D) AIn = -6 3 - 1 0 -9 -3 5 0

9 0 0 1 0 = 6 0 1 9

-4 -6 -3 1 3 -5

1 0 C) AIn = 0 1 0 0

Answer: B Type: BI Var: 50+ Objective: Identify Identity and Inverse Matrices

Determine whether A and B are inverses. 2) 1 2 11 11 5 1 A= and B = 1 5 -1 2 11 11 A) Yes Answer: A

B) No

Type: BI Var: 50+ Objective: Identify Identity and Inverse Matrices

3 -2 4 1 and B = 2 2 -11 8 A) No Answer: A

3) A =

Type: BI Var: 50+ Objective: Identify Identity and Inverse Matrices

Page 38

B) Yes

4) 5 16 7 -2 -1 -5 A = 2 -1 3 and B = 32 1 4 -2 9 32

11 16

1 4

9 32 7

1 8

A) No Answer: B

-1 8 B) Yes

Type: BI Var: 50+ Objective: Identify Identity and Inverse Matrices

0 3 1 -4 -7 2 5) A = 1 -2 0 and B = -2 -4 1 4 -1 2 7 12 -3 A) No Answer: B

B) Yes

Type: BI Var: 50+ Objective: Identify Identity and Inverse Matrices 2 Determine the Inverse of a Matrix

Determine the inverse of the given matrix, if possible. Otherwise, state the matrix is singular. -3 4 1) A = -5 3 4 3 3 4 11 11 11 11 A) A-1 =

5 11

3 11

C) Singular matrix

Answer: A Type: BI Var: 50+ Objective: Determine the Inverse of a Matrix

Page 39

B) A-1 =

D) A-1 =

5 11

3 11

4 11

5 11

3 11

2) A =

-5 -4 -25 -20 1 8

B) A-1 =

A) Singular matrix

1 8

C) A-1 = 1 8

D) A-1 =

1 8

1 10

1 8

1 10

1 8

5 16

1 8

5 16

13 16

1 8

3 16

5 16

1 10 -

1 8

Answer: A Type: BI Var: 50+ Objective: Determine the Inverse of a Matrix

4 -1 5 3) A = 1 1 -2 -1 1 0 1 5 8 16 1 A) A = 8 -

3 16

3 16 13 16

5 16

-1

1 8

-1

B) A =

5 16

C) Singular matrix

Type: BI Var: 50+ Objective: Determine the Inverse of a Matrix

Page 40

3 16

1 8

5 16

1 D) A = 8

5 16

13 16

3 16

5 16

-1

1 8 Answer: D

3 16

-5 0 4) A = 3 0

5 1 2 1

3 -4 -2 0 1 3 1 0 10 1 9 0

A) A-1 =

C) A-1 =

4 3

59 9

1 3

2 3

1 3

-1 -

11 9

5 3

64 9

-1 -

10 9

4 3

59 9

-1

10 9

1 3

2 3

1 3

11 9

5 3

64 9

Answer: D Type: BI Var: 50+ Objective: Determine the Inverse of a Matrix

Page 41

B) Singular matrix

D) A-1 =

0 1

4 3

59 9

1 3

2 3

1 3

11 9

5 3

64 9

5) A =

1 4

1 2

3 8

1 4

1 8

7 8

5 16

17 16

1 4

A) Singular matrix

3 1 2 B) A = -4 -3 -2 1 5 2

3 C) A = 4 1

3 D) A = 1 2

-1

-1 2 -3 2 -5 2

-1

-4 1 -3 5 -2 2

Answer: B Type: BI Var: 50+ Objective: Determine the Inverse of a Matrix 3 Solve Systems of Linear Equations Using the Inverse of a Matrix

Write the system of equations as a matrix equation of the form AX = B, where A is the coefficient matrix, X is the column matrix of variables, and B is the column matrix of constants. 1) -5x - 2y = -7 3x + 3y = 6 -5 -2 x -5 -2 x 7 -7 = = A) B) 3 3 y -6 3 3 y 6 C)

-5 2 x -7 = 3 -3 y 6

5 2 x -7 = -3 -3 y 6

Answer: B Type: BI Var: 50+ Objective: Solve Systems of Linear Equations Using the Inverse of a Matrix

Page 42

2) -5x - 8y + 9z = 16 2x - z = 5 -7y + z = 8 5 8 9 x 16 A) 2 0 1 y = 5 0 7 1 z 8

-16 -5 -8 9 x B) 2 0 -1 y = -5 0 -7 1 z -8

16 -5 -8 9 x C) 2 0 -1 y = 5 0 -7 1 z 8

16 -5 -8 9 x D) 2 -1 0 y = 5 -7 1 0 z 8

Answer: C Type: BI Var: 50+ Objective: Solve Systems of Linear Equations Using the Inverse of a Matrix

Solve the system by using the inverse of the coefficient matrix. 3) 4x - 5y = -11 -2x + 3y = 5 A) {(-4, -1)}

B) {(4, 1)}

C) {(-1, -4)}

D) {(1, 4)}

Answer: A Type: BI Var: 50+ Objective: Solve Systems of Linear Equations Using the Inverse of a Matrix

4) -2x + 5y + 2z = -24 x + y - 4z = 14 5x + 3y - z = 17 A) {(4, -2, -3)}

B) {(4, -2, 3)}

C) {(-4, -2, 3)}

D) {(-4, -2, -3)}

Answer: A Type: BI Var: 50+ Objective: Solve Systems of Linear Equations Using the Inverse of a Matrix

5) w - 2x - 3y = -5 -4x - 2y = -16 -2x + 4y - 5z = -28 5w - 2x + 2y - 5z = -13 A) {(-3, -4, 0, 4)}

B) {(3, 4, 0, 4)}

C) {(3, 4, 0, -4)}

Answer: B Type: BI Var: 50+ Objective: Solve Systems of Linear Equations Using the Inverse of a Matrix

Page 43

D) {(-3, -4, 0, -4)}

6) r - 3s + t = 8 -r + 6s + t = 3 3r -3s - t = 6 1 A) 4, , 5 3

1 B) -5,

, -4

1 C) -4,

Answer: A Type: BI Var: 50+ Objective: Solve Systems of Linear Equations Using the Inverse of a Matrix 4 Mixed Exercises

Determine if the statement is true or false. 1) A 3 × 2 matrix has a multiplicative inverse. Answer: FALSE Type: TF Var: 1 Objective: Mixed Exercises

2) A 2 × 3 matrix has a multiplicative inverse. Answer: FALSE Type: TF Var: 1 Objective: Mixed Exercises

3) Every square matrix has an inverse. Answer: FALSE Type: TF Var: 1 Objective: Mixed Exercises

4) Every singular matrix has an inverse. Answer: FALSE Type: TF Var: 1 Objective: Mixed Exercises

5) Every nonsingular matrix has an inverse. Answer: TRUE Type: TF Var: 1 Objective: Mixed Exercises

6) The matrix

-a -b is invertible. a b

Answer: FALSE Type: TF Var: 1 Objective: Mixed Exercises

7) The matrix

x y is invertible. 2x 2y

Answer: FALSE Type: TF Var: 1 Objective: Mixed Exercises

Page 44

, -5

D) 5,

1 3

8) The inverse of the matrix I4 is itself. Answer: TRUE Type: TF Var: 1 Objective: Mixed Exercises

Solve the problem. 6 6 9) A = -6 -8 a. Find A-1 b. Find (A-1)-1 1 2

2 3 A) a. A-1 =

-1 2

b. (A-1)-1 =

C) a. A-1 =

b. (A-1)-1 =

1 2 1 2

1 2

2 3

6 6 -6 -8

b. (A-1)-1 =

1 2

2 3 6 6 -6 -8

Answer: A Type: BI Var: 50+ Objective: Mixed Exercises

Page 45

B) a. A-1 =

D) a. A-1 =

b. (A-1)-1 =

6 6 -6 -8 1 2 1 2

1 2 2 3

-8 6 -6 6

5 Expanding Your Skills

Physicists know that if each edge of a thin conducting plate is kept at a constant temperature, then the temperature at the interior points is the mean (average) of the four surrounding points equidistant from the interior point. Use this principle in to find the temperature at points x1, x2, x3, and x4. 1) 44°F

40°F

48°F

60°F A) x1 = 45°F; x2 = 46°F; x3 = 50°F; x4 = 51°F; B) x1 = 45°F; x2 = 47°F; x3 = 49°F; x4 = 51°F; C) x1 = 42°F; x2 = 47°F; x3 = 49°F; x4 = 54°F; D) x1 = 42°F; x2 = 46°F; x3 = 50°F; x4 = 54°F; Answer: B Type: BI Var: 50+ Objective: Expanding Your Skills

6.5 Determinants and Cramer’s Rule 0 Concept Connections

Provide the missing information. 1) Associated with every square matrix A is a real number denoted by A called the

of A.

Answer: determinant Type: SA Var: 1 Objective: Concept Connections

2) For a 2 × 2 matrix A =

a c

, A = a c d

b = d

Answer: ad - bc Type: SA Var: 1 Objective: Concept Connections

3) Given A = aij , the and j th column. Answer: minor Type: SA Var: 1 Objective: Concept Connections

Page 46

of the element aij is the determinant obtained by deleting the i th row

4) Given A = aij , then the value (-1)i + j Mij is called the

of the element aij

Answer: cofactor Type: SA Var: 1 Objective: Concept Connections

5) The determinant of a 3 × 3 matrix A = a2 a3

c2 is given by

A =a □□ -a □ □ +a □□ 1 2 3 □□ □□ □□ Answer:

;

Type: SA Var: 1 Objective: Concept Connections

6) Suppose that the given system has one solution. a1x + b1y = c1 a2x + b2y = c2 □ □ Cramer’s rule gives the solution as x = and y = □ □ □□ □□ □□ ,D = , and D = where D = x y □□ □□ □□ Answer:

a Dx Dy ; ;D= 1 a2 D D

b1 b2

, Dx =

, and D = y

Type: SA Var: 1 Objective: Concept Connections

7) When applying Cramer’s rule, if D = 0, then the system (does / does not) have a unique solution. Answer: does not Type: SA Var: 1 Objective: Concept Connections

8) When applying Cramer’s rule, if D = 0, and all other determinants Di in the numerator are zero, then the system as (no solution / infinitely many) solutions. Answer: infinitely many Type: SA Var: 1 Objective: Concept Connections

Page 47

1 Evaluate the Determinant of a 2 x 2 Matrix

Evaluate the determinant of the given matrix. -1 6 1) A = -7 -9 A) -51

B) 51

C) -32

D) -50

C) 5

D) -5

C) u2 + 45

D) u2 - 45

C) 11log m

D) -18log m

C) -7e9x

D) -e8x - 6e9x

Answer: B Type: BI Var: 50+ Objective: Evaluate the Determinant of a 2 x 2 Matrix

2) A =

5 2

3 4

A) 25

B) -25

Answer: D Type: BI Var: 50+ Objective: Evaluate the Determinant of a 2 x 2 Matrix

3) A =

-5

A) -u2 + 45

B) -u2 - 45

Answer: C Type: BI Var: 50+ Objective: Evaluate the Determinant of a 2 x 2 Matrix

4) A =

log m 2

log m -9

A) -7log m

B) -11log m

Answer: B Type: BI Var: 50+ Objective: Evaluate the Determinant of a 2 x 2 Matrix

ex A= 5) 6

e9x 8x -e

A) 7e9x

B) -5e9x

Answer: C Type: BI Var: 36 Objective: Evaluate the Determinant of a 2 x 2 Matrix

Page 48

2 Evaluate the Determinant of an n x n Matrix

Find the minor of the given element. 1) a11 -2 4 A = [aij] = 8 1 -2 9

-2 -9 7

A) 83

B) -88

C) 93

D) 88

C) -13

D) 13

Answer: D Type: BI Var: 50+ Objective: Evaluate the Determinant of an n x n Matrix

Find the cofactor of the given element. 2) a13 7 -2 5 A = [aij] = 2 5 4 -1 -9 3 A) -8

B) -18

Answer: C Type: BI Var: 50+ Objective: Evaluate the Determinant of an n x n Matrix

Evaluate the determinant of the given matrix and state whether the matrix is invertible. -5 -3 1 3) A = -4 3 -9 -5 6 4 A) -156, Yes

B) -522, Yes

C) -522, No

D) -156, No

C) -153, No

D) -363, No

Answer: B Type: BI Var: 50+ Objective: Evaluate the Determinant of an n x n Matrix

74 5 4) A = 5 -5 8 15 1 A) -153, Yes

B) -363, Yes

Answer: A Type: BI Var: 50+ Objective: Evaluate the Determinant of an n x n Matrix

Page 49

7 0 2 -3 -4 9 5) A = 0 -1 6 9 5 6

-7 5 5 2

A) 648, Yes

B) 648, No

C) 1,026, No

D) 1,026, Yes

C) -4; Yes

D) -228; Yes

Answer: A Type: BI Var: 50+ Objective: Evaluate the Determinant of an n x n Matrix

1 2 0 4 0 3 -1 3 6) A = 2 1 0 4 -4 0 4 0 A) 0; No

B) -68; No

Answer: C Type: BI Var: 50+ Objective: Evaluate the Determinant of an n x n Matrix 3 Apply Cramer's Rule

Solve the system, if possible, by using Cramer's rule. If Cramer's rule does not apply, solve the system by using another method. 1) -6x - 3y = 2 -12x + 15y = -6 A) -

2 10 ,21 21

C) { }

B) {(D)

1 - y, y) | y is any real number} 3 2

2 10 , 21 21

Answer: A Type: BI Var: 50+ Objective: Apply Cramer's Rule

2) -12(x + y) = -4x - 28 -2x = 3y - 7 A) {(-

3 2

) | y is any real y + 7) number} 2

C) {5, -1} Answer: A Type: BI Var: 50+ Objective: Apply Cramer's Rule

Page 50

B) {-5, 1} D) { }

3) -9x + 27y + 9z = 7 18x + 45y = 14 27x + 9y - 18z = -7 A) { }

, 0,

9 7 14 C) - , 0, 9 9

D) {(-

14 9

7 9

+ z, 0, z | z is any real number}

Answer: B Type: BI Var: 50+ Objective: Apply Cramer's Rule

4) 6x - 9y + 9z = 3 -4x + 6y - 6z = -2 2x - 3y + 3z = 2 3 A) {(1 +

y - 3 z) | y and z are any real numbers} 2 2

B) { } C) {(1, -1, -2)} D) {(-1, 1, 2)} Answer: B Type: BI Var: 50+ Objective: Apply Cramer's Rule

5) x1 + 5x2 - 3x3 - 5x4 = -6 -3x2 + x4 = 4 x1 + 5x3 = -2 2x3 - 4x4 = -8 Solve for x3. 28 A) 29 Answer: A Type: BI Var: 50+ Objective: Apply Cramer's Rule

Page 51

44 29

36 29

82 29

4 Mixed Exercises

x1 Using the determinant x2

y1 y2

1 1 , determine if the points are collinear.

1) (8, 9), (2, 2), (-4, -5) A) No

B) Yes

Answer: B Type: BI Var: 50+ Objective: Mixed Exercises

x1 1 x Use the formula Area = ± 2 2 x3

1 to find the area of the triangle with the given vertices. Choose the

+ or - sign so that the value of the area is positive. 2) (5, 4), (-5, 1), (-2, -2) A) -39

B) 39

C) -

39 2

Answer: D Type: BI Var: 50+ Objective: Mixed Exercises

x The equation x1 x2

y y1

1 1 = 0 represents an equation of the line passing through distinct points (x , y )

and (x2, y2). a. Use the determinant equation to write an equation of the line passing through the given points. b. Write the equation of the line in slope-intercept form. 3) (-2, 6) and (-3, 2) x y 1 x y 1 A) a. -2 -3 1 = 0; b. y = -8x - 10 B) a. -2 -3 1 = 0; b. y = 4x + 14 6 2 1 6 2 1 x y 1 x y 1 C) a. -2 6 1 = 0; b. y = 4x + 14 D) a. -2 6 1 = 0; b. y = -8x - 10 -3 2 1 -3 2 1 Answer: C Type: BI Var: 50+ Objective: Mixed Exercises

Page 52

5 Technology Connections

Evaluate the determinant of the given matrix and state whether the matrix is invertible. 0.9 1.7 8 12.7 -3.5 0.4 -1.4 5 1) A = 6 8.8 -4.4 2 -2 3.5 11.4 -8.2 A) 10,144

B) -1,482

Answer: A Type: BI Var: 50+ Objective: Technology Connections

Page 53

C) 10,784

D) -2,122

Chapter 7 Analytic Geometry 7.1 The Ellipse 0 Concept Connections

Provide the missing information. 1) The circle, the ellipse, the hyperbola, and the parabola are categories of

sections.

Answer: conic Type: SA Var: 1 Objective: Concept Connections

2) An is a set of points (x, y) in a plane such that the sum of the distances between (x, y) and two fixed points called is a constant. Answer: ellipse; foci Type: SA Var: 1 Objective: Concept Connections

3) The line through the foci intersects an ellipse at two points called

Answer: vertices Type: SA Var: 1 Objective: Concept Connections

4) The line segment with endpoints at the vertices of an ellipse is called the

axis.

Answer: major Type: SA Var: 1 Objective: Concept Connections

5) The center of an ellipse is the midpoint of the

axis.

Answer: major Type: SA Var: 1 Objective: Concept Connections

6) The line segment perpendicular to the major axis, with endpoints on the ellipse, and passing through the center of the ellipse is called the axis. Answer: minor Type: SA Var: 1 Objective: Concept Connections

7) The standard form of an equation of an ellipse centered at the origin with a horizontal major axis is , where a > b > 0. If the major axis is vertical, then the equation is . Answer:

x2 a2

y2 b2

= 1;

x2 b2

Type: SA Var: 1 Objective: Concept Connections

Page 1

y2 a2

8) The standard form of an equation of an ellipse centered at (h, k) with a horizontal major axis is , where a > b > 0. If the major axis is vertical, then the equation is . (x - h)2 Answer:

(y - k)2 +

(x - h)2 = 1;

(y - k)2 +

Type: SA Var: 1 Objective: Concept Connections

x2 y2 9) Given 2 + 2 = 1 where a > b > 0, the ordered pairs representing the vertices are a b . The ordered pairs representing the endpoints of the minor axis are .

and and

Answer: (a, 0); (-a, 0); (0, b); (0, -b) Type: SA Var: 1 Objective: Concept Connections

(x - h)2 10) Given

(y - k)2 +

vertices are axis are

= 1 where a > b > 0, the ordered pairs representing the endpoints of the

and and

. The ordered pairs representing the endpoints of the minor .

Answer: (h + a, k); (h - a, k); (h, k + b); (h, k - b) Type: SA Var: 1 Objective: Concept Connections

(x - h)2 11) Given

vertices are axis are

(y - k)2 +

= 1 where a > b > 0, the ordered pairs representing the endpoints of the and

and

. The ordered pairs representing the endpoints of the minor .

Answer: (h, k + a); (h, k - a); (h + b, k); (h - b, k) Type: SA Var: 1 Objective: Concept Connections

12) When referring to the standard form of an equation of an ellipse, the □ as e = . □ c Answer: eccentricity; a Type: SA Var: 1 Objective: Concept Connections

Page 2

, e, is defined

1 Graph an Ellipse Centered at the Origin

From the equation of the ellipse, determine if the major axis is horizontal or vertical. x2 y2 1) + =1 11 8 A) Horizontal

B) Vertical

Answer: A Type: BI Var: 50+ Objective: Graph an Ellipse Centered at the Origin

Graph the ellipse. Identify the center and vertices. x2 y2 2) + =1 36 49 A) center: (0, 0); vertices (0, -7), (0, 7)

B) center: (6, 7); vertices (-7, 0), (7, 0)

10 8 6

2 2 4

-10 -8 -6 -4 -2 -2 -4

6 8 10 x

-6

-8

-10

C) center: (0, 0); vertices (0, -7), (0, 7)

8 10 x

D) center: (6, 7); vertices (0, -7), (0, 7)

10 8

4 2

4 2 2 4

6 8 10 x

-10 -8 -6 -4 -2 -2 -4

-6 -8

-10

Answer: A Type: BI Var: 36 Objective: Graph an Ellipse Centered at the Origin

Page 3

-6

-8

-10 -8 -6 -4 -2 -2 -4

3) 4x 2 + 25y 2 = 100 A) center: (0, 0); vertices (0, -2), (0, 2)

B) center: (0, 0); vertices (-5, 0), (5, 0)

10 8

4 2

4 2 2 4

-10 -8 -6 -4 -2 -2 -4 -6 -8

6 8 10 x

-10

8 10 x

D) center: (0, 0); vertices (-5, 0), (5, 0)

10 8 6

2 2 4

6 8 10 x

-10 -8 -6 -4 -2 -2 -4

-6

-8

-10

Answer: D Type: BI Var: 21 Objective: Graph an Ellipse Centered at the Origin

Page 4

-10

C) center: (0, 0); vertices (0, -2), (0, 2)

-10 -8 -6 -4 -2 -2 -4

-10 -8 -6 -4 -2 -2 -4 -6 -8

Graph the ellipse. Identify the foci and vertices. 4) 25x2 + 9y2 = 225 A) foci: (-4, 0), (4, 0); vertices: (0, -5), (0, 5)

B) foci: (0, -4), (0, 4); vertices: (-5, 0), (5, 0)

10 8

4 2

2 4

-10 -8 -6 -4 -2 -2 -4 -6

6 8 10 x

-10 -8 -6 -4 -2 -2 -4 -6

-8

-10

C) foci: (0, -4), (0, 4); vertices: (0, -5), (0, 5)

8 10 x

D) foci: (-4, 0), (4, 0); vertices: (-5, 0), (5, 0)

10 8

-10 -8 -6 -4 -2 -2 -4

2 4

6 8 10 x

-10 -8 -6 -4 -2 -2 -4

-6

-8

-8 -10

-10

Answer: C Type: BI Var: 8 Objective: Graph an Ellipse Centered at the Origin

Determine the length of the major and minor axis. 5) 4x2 + 81y2 = 324 A) length of major axis: 81; length of minor axis: 4 C) length of major axis: 2; length of minor axis: 9 Answer: D Type: BI Var: 50+ Objective: Graph an Ellipse Centered at the Origin

Page 5

B) length of major axis: 9; length of minor axis: 2 D) length of major axis: 18; length of minor axis: 4

Identify the vertices and the foci. x2 y2 6) + =1 14 2 A) vertices: ( 14, 0) and (- 14, 0); foci: (2 3, 0) and (- 2 3, 0) C) vertices: (14, 0) and (-14, 0); foci: (12, 0) and (-12, 0)

B) vertices: (0, 14) and (0, -14); foci: (0, 12) and (0, -12) D) vertices: (0, 14) and (0, - 14) foci: (0, 2 3) and (0, - 2 3)

Answer: A Type: BI Var: 50+ Objective: Graph an Ellipse Centered at the Origin

Graph the ellipse. x2 y2 7) + =1 18 7 A)

B) 25 y

5 y

-25 -20 -15 -10 -5 -5

5 10 15 20 25 x

-1

-10

-2

-15

-3

-20

-4

-25

-5

D) 25 y

5 y

-25 -20 -15 -10 -5 -5

5 10 15 20 25 x

-5 -4 -3 -2 -1 -1

-10

-2

-15

-3

-20

-4

-25

-5

Answer: B Type: BI Var: 50+ Objective: Graph an Ellipse Centered at the Origin

Page 6

-5 -4 -3 -2 -1

2 Graph a Ellipse Centered at (h, k)

Graph the ellipse. 25(x - 5)2 25(y + 1)2 1) + =1 81 196 A)

B) y

-12

-8

4 4

-4

-12

-8

-4

-8

-12

D) y

-12

-8

4 4

-4

-12

-8

-4

-8

-12

Answer: C Type: BI Var: 50+ Objective: Graph a Ellipse Centered at (h, k)

Page 7

Graph the ellipse. Identify the center and the endpoints of the minor axis. (x + 1)2 (y + 4)2 2) + =1 36 25 A) center: (-1, -4); endpts of minor axis: (-1, -9), (-1, 1)

B) center: (-1, -4); endpts of minor axis: (4, -4), (-6, -4)

10 8

2 2 4

-10 -8 -6 -4 -2 -2 -4

6 8 10 x

-10 -8 -6 -4 -2 -2 -4

-6 -8

-10

C) center: (1, 4); endpts of minor axis: (6, 4), (-4, 4)

D) center: (1, 4); endpts of minor axis: (1, 9), (1, -1)

10 8 6

-10 -8 -6 -4 -2 -2 -4

2 4 6

8 10 x

-6

-10 -8 -6 -4 -2 -2 -4

-6

-8

-10

Answer: A Type: BI Var: 50+ Objective: Graph a Ellipse Centered at (h, k)

Identify the vertices and the foci. y2 3) (x - 6)2 + =1 25 A) vertices: (0, 5) and (0, -5); foci: (0, 2) and (0, - 2) C) vertices: (0, 5) and (0, -5); foci: (0, 2 6) and (0, - 2 6) Answer: B Type: BI Var: 50+ Objective: Graph a Ellipse Centered at (h, k)

Page 8

8 10 x

B) vertices: (6, 5) and (6, -5); foci: (6, 2 6) and (6, - 2 6) D) vertices: (6, 5) and (6, -5); foci: (6, 2) and (6, - 2)

8 10 x

Identify the center of the ellipse and the foci. 25(x - 3)2 49( y + 6)2 4) + =1 169 225 A) center: (3, -6); 4 166 4 166 , -6 foci: 3 + , -6 and 3 35 35

B) center: (6, -3); foci: 6 + 4 166, -3 and 6 - 4 166, -3

C) center: (-6, 3); foci: -6 + 4 166, 3 and -6 - 4 166, 3

D) center: (-3, 6); 4 166 4 166 , 6 and -3 ,6 foci: -3 + 35 35

Answer: A Type: BI Var: 50+ Objective: Graph a Ellipse Centered at (h, k)

Page 9

Graph the ellipse. Identify the foci and vertices. 5) 25(x + 2)2 + 9(x + 4)2 = 225 A) foci: (-2, 0), (-2, -8); vertices: (-2, -9), (-2, 1)

B) foci: (2, 8), (2, 0); vertices: (2, 9), (2, -1)

-12

-8

-4

12 x

-12

-8

-4

-8

-12

C) foci: (6, 4), (-2, 4); vertices: (7, 4), (-3, 4)

-8

12 x

-4

12 x

D) foci: (2, -4), (-6, -4); vertices: (3, -4), (-7, -4)

-12

12 x

-12

-8

-4

-8

-12

Answer: A Type: BI Var: 50+ Objective: Graph a Ellipse Centered at (h, k)

Write the equation of the ellipse in standard form. Identify the center and vertices. 6) 4x2 + 16y2 - 40x + 96y + 180 = 0 (x - 5)2 (y + 3)2 (x - 5)2 (y + 3)2 A) + =1 B) + =1 16 4 4 16 center: (-5, 3); vertices:(-9, 3), (-1, 3) (x - 5)2 (y + 3)2 C) + =1 4 16

center: (5, -3); vertices:(5, -5), (5, -1) (x - 5)2 (y + 3)2 D) + =1 16 4

center: (-5, 3); vertices:(-5, 1), (-5, 5)

center: (5, -3); vertices:(1, -3), (9, -3)

Answer: D Type: BI Var: 50+ Objective: Graph a Ellipse Centered at (h, k)

Page 10

Write the equation of the ellipse in standard form. Identify the vertices and foci. 7) 25x2 + y2 + 14y + 24 = 0 (y + 7)2 (y - 7)2 2 2 =1 =1 A) x + B) x + 25 25 vertices: (0, -12), (0, -2) foci (0, -7 - 2 6), (0, -7 + 2 6) (y - 7)2 C) x2 + =1 25 vertices: (0, 2), (0, 12) foci (0, 7 - 2 6), (0, 7 + 2 6)

vertices: (2, 0), (12, 0) foci (7 - 2 6, 0), (7 + 2 6, 0) (y + 7)2 D) x2 + =1 25 vertices: (-12, 0), (-2, 0) foci (-7 - 2 6, 0), (-7 + 2 6, 0)

Answer: A Type: BI Var: 50+ Objective: Graph a Ellipse Centered at (h, k)

8) x2 + 81y2 - 14x - 32 = 0 (x - 7)2 A) + y2 = 1 81

B) x +

(y - 7)2 81

vertices: (-2, 0), (16, 0) foci (7 - 4 5, 0), (7 + 4 5, 0) (y + 7)2 2 =1 C) x + 81

vertices: (0, -16), (0, 2) foci (0, 7 - 4 5), (0, 7 + 4 5) (x + 7)2 D) + y2 = 1 81

vertices: (0, -16), (0, 2) foci (0, -7 - 4 5), (0, -7 + 4 5)

vertices: (-16, 0), (2, 0) foci (-7 - 4 5, 0), (-7 + 4 5, 0)

Answer: A Type: BI Var: 50+ Objective: Graph a Ellipse Centered at (h, k)

Page 11

9) 36x2 + 25y2 + 100y + 99 = 0 2

A) 36x + 25(y + 2) = 1 vertices: (36, -2), (-36, -2) 11 11 foci: 0, 2 + , 0, 2 30 30

(y + 2) x2 B) + =1 1 1 36 25 9

, 0, - 11 5 5

vertices: 0, -

11 foci: 0, 2 +

11 30

, 0, 2 -

C) 36x + 25(y + 2) = 1 vertices: 0, -

, 0, -

(y + 2) x2 + =1 1 1 36 25

11 foci: 0, 2 + , 0, 2 30 30

9 11 vertices: - , 0 , - , 0 5 5 11 foci: 2 +

Answer: B Type: BI Var: 40 Objective: Graph a Ellipse Centered at (h, k)

Page 12

,0, 2-

11 30

Write the equation of the ellipse in standard form. Identify the center and foci. 10) 4x2 + 9y2 + 12x - 108y + 297 = 0 32 32 x+ x+ 2 2 (y - 6)2 (y - 6)2 A) + B) + =1 =1 9 4 9 4 3 3 center: - , 6 center: , -6 2 2 3 3 3 9 + 5, -6 foci: - 5, -6 , foci: - , 6 , , 6 2 2 2 2 2 32 x+ x- 3 2 2 (y + 6)2 (y - 6)2 C) + D) + =1 =1 9 4 9 4 3 3 center: , -6 center: - , 6 2 2 3 3 3 3 + 5, -6 foci: - 5, -6 , foci: - - 5, 6 , - + 5, 6 2 2 2 2 Answer: D Type: BI Var: 50+ Objective: Graph a Ellipse Centered at (h, k)

Write the standard form of an equation of the ellipse subject to the given conditions. 11) Vertices: (-10, 0), (10, 0) Foci: (-6, 0), (6, 0) x2 y2 x2 y2 x2 y2 x2 y2 A) + + = 1 C) + + =1 =1 B) =1 D) 64 100 100 36 36 100 100 64 Answer: D Type: BI Var: 12 Objective: Graph a Ellipse Centered at (h, k)

12) Endpoints of minor axis: ( 31,0) and (- 31,0) Foci: (0, 10) and (0, -10) x2 y2 x2 y2 A) + =1 B) + =1 131 31 31 41 Answer: C Type: BI Var: 50+ Objective: Graph a Ellipse Centered at (h, k)

Page 13

x2 31

y2 131

x2 41

y2 31

13) Foci: (-5, 5) and (3, 5) Length of minor axis: 4 (x - 5)2 ( y + 1)2 A) + =1 20 4 (x - 5)2 ( y + 1)2 C) + =1 4 20

B) D)

(x + 1)2 4 (x + 1)2 20

+ +

( y - 5)2 20 ( y - 5)2 4

=1 =1

Answer: D Type: BI Var: 50+ Objective: Graph a Ellipse Centered at (h, k)

14) Vertices: (0, 10) and (0,-10) 32 Passes through - , 6 5 A)

x2 y2 + =1 36 100

x2 y2 + B) =1 100 36

y2 x2 + =1 C) 100 64

x2 y2 + =1 D) 64 100

Answer: D Type: BI Var: 24 Objective: Graph a Ellipse Centered at (h, k)

15) Vertices: (6, 1) and (-12, 1) Foci: (-3 - 65, 1) and (-3 + 65, 1) (x + 3)2 ( y - 1)2 A) + =1 9 4 (x - 3)2 ( y + 1)2 C) + =1 9 4 Answer: D Type: BI Var: 50+ Objective: Graph a Ellipse Centered at (h, k)

Page 14

B) D)

(x - 3)2 81 (x + 3)2 81

+ +

( y + 1)2 16 ( y - 1)2 16

=1 =1

3 Use Ellipses in Applications

Solve the problem. 1) The reflective property of an ellipse is used in lithotripsy. Lithotripsy is a technique for treating kidney stones without surgery. Instead, high-energy shock waves are emitted from one focus of an elliptical shell and reflected painlessly to a patient's kidney stone located at the other focus. The vibration from the shock waves shatters the stone into pieces small enough to pass through the patient's urine.

21.3 cm.

26 cm. A vertical cross section of a lithotripter is in the shape of a semiellipse with the dimensions shown. Approximate the distance from the center along the major axis where the patient's kidney stone should be located so the shock waves will target the stone. Round to two decimal places. A) 40.57 cm. below the center B) 24.95 cm. below the center C) 14.91 cm. below the center D) 16.87 cm. below the center Answer: D Type: BI Var: 49 Objective: Use Ellipses in Applications

2) The reflective property of an ellipse is the principle behind "whispering galleries". These are rooms with elliptically shaped ceilings such that a person standing at one focus can hear even the slightest whisper spoken by another person standing at the other focus. Suppose that a dome has a semielliptical ceiling, 94 ft long and 22 ft high. Approximately how far from the center along the major axis should each person be standing to hear the "whispering" effect? Round to one decimal place. A) 25.0 feet B) 91.4 feet C) 51.9 feet D) 41.5 feet Answer: D Type: BI Var: 50+ Objective: Use Ellipses in Applications

Page 15

3) A homeowner wants to make an elliptical rug from a 30-foot by 10-foot rectangular piece of carpeting. a. What lengths of the major and minor axes would maximize the area of the new rug? b. Write an equation of the ellipse with maximum area. Use a coordinate system with the origin at the center of the rug and horizontal major axis. A) a. Major axis: 32 feet. Minor axis: 16 feet B) a. Major axis: 30 feet. Minor axis: 10 feet x2 y2 x2 y2 b. + =1 b. + =1 256 64 225 25 C) a. Major axis: 30 feet. Minor axis: 10 feet x2 y2 b. + =1 900 100

D) a. Major axis: 15 feet. Minor axis: 5 feet x2 y2 b. + =1 225 25

Answer: B Type: BI Var: 17 Objective: Use Ellipses in Applications

4) A window above a door is to be made in the shape of a semiellipse. If the window is 12 feet at the base and 4 feet high at the center, determine the distance from the center at which the foci are located. Round to one decimal place. A) 20.0 feet B) 11.3 feet C) 4.5 feet D) 8.9 feet Answer: C Type: BI Var: 27 Objective: Use Ellipses in Applications 4 Determine and Apply the Eccentricity of an Ellipse

Determine the eccentricity of the ellipse. (x + 4)2 (y + 5)2 1) + =1 144 169 4 12 A) e = B) e = 5 5

C) e =

D) e =

5 13

Answer: D Type: BI Var: 50+ Objective: Determine and Apply the Eccentricity of an Ellipse

x- 1 7

y- 4 3 +

25 A) e =

=1 9

3 5

B) e =

3 28

Answer: C Type: BI Var: 50+ Objective: Determine and Apply the Eccentricity of an Ellipse

Page 16

C) e =

4 5

D) e =

3 4

Solve the problem. 3) A planet's moon has an orbit that is elliptical with eccentricity 0.054 and with the planet at one focus. If the distance between the moon and the planet at perihelion (the closest point) is 364,100 km, determine the distance at aphelion (the farthest point). Round to the nearest 100 km. A) 749,000 km B) 405,700 km C) 769,800 km D) 384,900 km Answer: B Type: BI Var: 50+ Objective: Determine and Apply the Eccentricity of an Ellipse

Write the standard form of an equation of the ellipse subject to the given conditions. 24 4) Center (0, 0); Eccentricity: ; Major axis vertical of length 50 units 25 A)

x2 y2 + =1 49 625

x2 y2 + =1 49 576

y2 x2 + =1 C) 625 49

x2 y2 + =1 D) 576 625

Answer: A Type: BI Var: 9 Objective: Determine and Apply the Eccentricity of an Ellipse

5) Foci: (-13, 4) and (19, 4); Eccentricity:

4 5

A) C)

(x + 3)

400 (x + 3)2 144

+ +

( y + 4)

144 ( y + 4)2 400

Answer: D Type: BI Var: 50+ Objective: Determine and Apply the Eccentricity of an Ellipse

Page 17

(x - 3)2 144 (x - 3)2 400

+ +

( y - 4)2 400 ( y - 4)2 144

=1 =1

Solve the problem. 6) A park has an elliptical shape with a major axis of 970 feet and a minor axis of 917 feet. Find the equation of the elliptical boundary. a. Take the horizontal axis to be the major axis and locate the origin of the coordinate system at the center of the ellipse. b. Approximate the eccentricity of the ellipse. Round to two decimal places. x2 y2 x2 y2 a. B) + =1 A) a. + =1 4852 458.52 9702 9172 b. e ≈ 0.33 x2 y2 C) a. + =1 4852 458.52

b. e ≈ 0.33 x2 y2 D) a. + =1 4852 458.52 b. e ≈ 0.34

b. e ≈ 0.11 Answer: B Type: BI Var: 50+ Objective: Determine and Apply the Eccentricity of an Ellipse 5 Mixed Exercises

Solve the system of equations. x2 y2 1) + =1 81 49 -7x + 9y = -63 A) {(0,-7), (9, 0)}

B) {(7, 0), (0, -9)}

C) {(-7, 0), (0, 9)}

D) {(0,7), (-9, 0)}

Answer: A Type: BI Var: 50+ Objective: Mixed Exercises

x2 y2 + =1 4 16 y = -x2 + 4 A) {(0, 4), (2, 0)} C) {(0, 4), (-2, 0), (2, 0)} Answer: C Type: BI Var: 8 Objective: Mixed Exercises

Page 18

B) {(0, -4), (0, 4), (-2, 0), (2, 0)} D) {(0, -4), (-2, 0), (2, 0)}

Given an ellipse with major axis of length 2a and minor axis of length 2b, the area is given by A = πab. The perimeter is approximated by P ≈ π 2(a2 + b2). a. Determine the area of the ellipse. b. Approximate the perimeter. 2 2 (y 6) x 3) + =1 4 12 A) a. A = 4π 3 square units b. P ≈ 12π units

B) a. A = 48π square units b. P ≈ 4π 2 units

C) a. A = 48π square units b. P ≈ 12π units

D) a. A = 4π 3 square units b. P ≈ 4π 2 units

Answer: D Type: BI Var: 50+ Objective: Mixed Exercises

Solve the problem. 4) a. A circular vent pipe with diameter 4 inches is placed on a flat roof. Write an equation of the circular cross section that the pipe makes with the roof. Assume the origin is placed at the center of the circle. b. Suppose the pipe is instead placed on a roof with a slope of

3 4

. The cross-section of the pipe

where it intersects the roof is an ellipse. Determine the lengths of the major and minor axes of this ellipse. A) a. x2 + y2 = 4 B) a. x2 + y2 = 4 b. Major axis ≈ 5.0 in; minor axis = 4 in b. Major axis = 4 in; minor axis ≈ 5.0 in 2 2 C) a. x + y = 16 D) a. x2 + y2 = 16 b. Major axis ≈ 6.7 in; minor axis = 4 in b. Major axis ≈ 2.5 in; minor axis = 2 in Answer: A Type: BI Var: 50+ Objective: Mixed Exercises

7.2 The Hyperbola 0 Concept Connections

Provide the missing information. 1) A is the set of point (x, y) in a plane such that the difference in distances between (x, y) and two fixed points (called ) is a positive constant. Answer: hyperbola; foci Type: SA Var: 1 Objective: Concept Connections

2) The points where a hyperbola intersects the line through the foci are called the Answer: vertices Type: SA Var: 1 Objective: Concept Connections

Page 19

3) The line segment between the vertices of a hyperbola is called the

axis.

Answer: transverse Type: SA Var: 1 Objective: Concept Connections

4) The midpoint of the transverse axis is the

of the hyperbola.

Answer: center Type: SA Var: 1 Objective: Concept Connections

5) The equation

x2 a2

y2 b2

= 1 represents a hyperbola with a (horizontal / vertical) transverse axis. The

vertices are given by the ordered pairs equations and . Answer: horizontal; (a, 0); (-a, 0); y =

and

. The asymptotes are given by the

b b x and y = - x a a

Type: SA Var: 1 Objective: Concept Connections

6) The equation

y2 a2

x2 b2

= 1 represents a hyperbola with a (horizontal / vertical) transverse axis. The

vertices are given by the ordered pairs and equations and . a Answer: vertical; (0, a); (0, -a); y = x and y = - a x b b

. The asymptotes are given by the

Type: SA Var: 1 Objective: Concept Connections

7) The line segment perpendicular to the transverse axis passing through the center of a hyperbola, and with endpoints on the reference rectangle is called the axis. Answer: conjugate Type: SA Var: 1 Objective: Concept Connections

8) When referring to the standard form of an equation of a hyperbola, the □ = . □ c Answer: eccentricity; a Type: SA Var: 1 Objective: Concept Connections

Page 20

, e, is defined as e

(y - k)2 9) Given

(x - h)2 -

= 1, the ordered pairs representing the vertices are

and

. Answer: (h, k + a), (h, k - a) Type: SA Var: 1 Objective: Concept Connections

(x - h)2 10) Given

(y - k)2 -

= 1, the ordered pairs representing the vertices are

and

. Answer: (h + a, k); (h - a, k) Type: SA Var: 1 Objective: Concept Connections 1 Graph a Hyperbola Centered at the Origin

Determine whether the transverse axis and foci of the hyperbola are on the x-axis or the y-axis. x2 y2 1) =1 9 100 A) y-axis Answer: B Type: BI Var: 50+ Objective: Graph a Hyperbola Centered at the Origin

Page 21

B) x-axis

Graph the hyperbola. Identify the center and vertices. x2 y2 2) =1 49 64 A) center: (7, 8); vertices: (-7, 0), (7, 0)

B) center: (0, 0); vertices: (0, -7), (0, 7) y

y 10 8

2 4

-10 -8 -6 -4 -2 -2 -4

6 8 10 x

-6 -8

-10

C) center: (7, 8); vertices: (0, -7), (0, 7)

8 10 x

y 8 6

10 8 6

2 2 4

6 8 10 x

-10 -8 -6 -4 -2 -2 -4

-6

-8

-10

Answer: D Type: BI Var: 50+ Objective: Graph a Hyperbola Centered at the Origin

Page 22

D) center: (0, 0); vertices: (-7, 0), (7, 0)

-10 -8 -6 -4 -2 -2 -4

Graph the hyperbola. Identify the foci and write the equations for the asymptotes. y2 x2 3) =1 144 25 A) foci: (0, -13), (0,13); 12 12 asymptotes: y = - x, y = x 5 5

-16 -12

-8

B) foci: (0, -13), (0,13); 12 12 asymptotes: y = - x, y = x 5 5

16 y

4 4

-4

-8

-4 -4

-8

-12

-16

16 y

4 4

-4

D) foci: (0, -12), (0,12); 5 5 asymptotes: y = - x, y = x 12 12

-16 -12

-8

-4

-8

-12

-16

Answer: A Type: BI Var: 8 Objective: Graph a Hyperbola Centered at the Origin

Page 23

-8

-4

C) foci: (0, -12), (0,12); 5 5 asymptotes: y = - x, y = x 12 12

-16 -12

Identify the vertices and the foci. 4) -10x2 + 8y2 = -80 A) vertices: ( 8, 0), (- 8, 0) foci: ( 2, 0), (- 2, 0) C) vertices: ( 10, 0), (- 10, 0) foci: (3 2, 0), (-3 2, 0)

B) vertices: ( 8, 0), (- 8, 0) foci: (3 2, 0), (-3 2, 0) D) vertices: (0, 10), (0, - 10) foci: (0, 3 2), (0, -3 2)

Answer: B Type: BI Var: 15 Objective: Graph a Hyperbola Centered at the Origin

Identify the vertices and the foci, and write equations for the asymptotes. 49x2 49y2 5) =1 36 64 7 7 7 7 A) vertices: , 0 , - , 0 B) vertices: , 0 , - , 0 8 8 6 6 10 10 ,0, - ,0 7 7 4 4 asymptotes: y = x and y = - x 3 3 7 7 C) vertices: , 0 , - , 0 6 6 foci:

10 10 ,0, - ,0 7 7 3 3 asymptotes: y = x and y = - x 4 4 foci:

Answer: B Type: BI Var: 38 Objective: Graph a Hyperbola Centered at the Origin

Page 24

10 10 ,0, - ,0 7 7 4 4 asymptotes: y = x and y = - x 3 3 7 7 D) vertices: , 0 , - , 0 6 6 foci:

70 70 ,0 ,0, 2,401 2,401 4 4 asymptotes: y = x and y = - x 3 3 foci:

2 Graph a Hyperbola Centered at (h, k)

Graph the hyperbola. Identify the center and vertices. (y - 1)2 (x + 1)2 1) =1 81 49 A) center: (-1, 1); vertices: (-1, -8), (-1, 10)

-16 -12

-8

B) center: (1, -1); vertices: (-8, -1), (10, -1)

16 y

4 4

-4

-8

-4

-8

-12

-16

16 y

4 4

-4

D) center: (-1, 1); vertices: (-10, 1), (8, 1)

-16 -12

-8

-4

-8

-12

-16

Answer: A Type: BI Var: 50+ Objective: Graph a Hyperbola Centered at (h, k)

Page 25

-8

-4

C) center: (1, -1); vertices: (1, -10), (1, 8)

-16 -12

-4

Graph the hyperbola. Identify the foci and write the equations for the asymptotes. (x + 5)2 (y + 1)2 2) =1 64 36 A) foci: (-5, -9), (-5, 7); asymptotes: 11 , y = 3 x - 19 y = - 3x + 4 4 4 4

B) foci: (-15, -1), (5, -1); asymptotes: 3 19 3 11 y=- x- ,y= x+ 4 4 4 4

4 4

-16 -12 -8 -4

12 16

-16 -12 -8 -4

12 16

-4

-8

-12

-16

C) foci: (-13, -1), (3, -1); asymptotes: 3 11 , y = 3 x - 19 y=- x+ 4 4 4 4

D) foci: (-5, -11), (-5, 9); asymptotes: 3 19 3 11 y=- x- ,y= x+ 4 4 4 4

4 4

-16 -12 -8 -4

12 16

-16 -12 -8 -4

-4

-8

-12

-16

Answer: B Type: BI Var: 50+ Objective: Graph a Hyperbola Centered at (h, k)

Page 26

-4

12 16

Identify the foci and write equations for the asymptotes. 3) 9(y - 4)2 - 100(x + 4)2 = -900 A) foci: (-4 + 109, 4), (-4 - 109, 4) 10 52 10 28 asymptotes: y = x + and y = x 3 3 3 3 B) foci: (0, 4 + 109), (0, 4 - 109) 3 26 3 14 asymptotes: y = x + and y = - x + 10 5 10 5 C) foci: (0, -4 + 109), (0, -4 - 109) 3 14 3 26 asymptotes: y = x and y = - x 10 5 10 5 D) foci: (4 + 109, 0), (4 - 109, 0) 10 28 10 52 asymptotes: y = x and y = x + 3 3 3 3 Answer: A Type: BI Var: 50+ Objective: Graph a Hyperbola Centered at (h, k)

Identify the vertices and the foci. (y + 5)2 2 =1 4) x 3 A) vertices: (0, -5 + 3), (0, -5 - 3) foci: (0, -7), (0, -3) C) vertices: (1, -5), (-1, -5) foci: (2, -5), (-2, -5)

B) vertices: (0, 5 + 3), (0, 5 - 3) foci: (0, 7), (0, 3) D) vertices: (1, 5), (-1, 5) foci: (2, 5), (-2, 5)

Answer: C Type: BI Var: 26 Objective: Graph a Hyperbola Centered at (h, k)

Write the equation of the hyperbola in standard form. Identify the center and vertices. 5) 4y2 - 16x2 + 40y - 32x + 20 = 0 (y + 5)2 (x + 1)2 (y - 5)2 (x - 1)2 A) =1 B) =1 16 4 16 4 center: (-1, -5); vertices: (-1, -9), (-1, -1) (y + 5)2 (x + 1)2 C) =1 16 4 center: (1, 5); vertices: (1, 1), (1, 9) Answer: A Type: BI Var: 50+ Objective: Graph a Hyperbola Centered at (h, k)

Page 27

center: (1, 5); vertices: (1, 1), (1, 9) (y - 5)2 (x - 1)2 D) =1 16 4 center: (-1, -5); vertices: (-1, -9), (-1, -1)

Write the equation of the hyperbola in standard form. Identify the center and foci. 6) 6x2 - 5y2 + 60x - 20y + 100 = 0 (x + 5)2 (y + 2)2 (x - 5)2 (y - 2)2 A) =0 B) =0 5 6 5 6 center: (-5, -2) foci: (-5 + 11, -2), (-5 - 11, -2) (x - 5)2 (y - 2)2 C) =1 5 6

center: (5, 2) foci: (5 + 11, 2), (5 - 11, 2) (x + 5)2 (y + 2)2 D) =1 5 6

center: (5, 2) foci: (5 + 11, 2), (5 - 11, 2)

center: (-5, -2) foci: (-5 + 11, -2), (-5 - 11, -2)

Answer: D Type: BI Var: 50+ Objective: Graph a Hyperbola Centered at (h, k)

Write the equation of the hyperbola in standard form. Identify the center and vertices. 7) -144x2 + 9y2 + 384x + 36y - 1,516 = 0 42 42 x+ x+ 3 3 (y - 2)2 (y - 2)2 A) =1 B) =0 144 9 144 9 4 4 center: - , 2 center: - , 2 3 3 4 4 4 4 vertices: - , 14 and - , -10 vertices: - , 14 and - , -10 3 3 3 3

(y + 2)2

144 center:

42 3

9 4 3

, 10 and

center: 4 , -14 3

Answer: C Type: BI Var: 50+ Objective: Graph a Hyperbola Centered at (h, k)

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(y + 2)2

42 3

144

, -2 4

vertices:

9 4 3

, -2 4

vertices:

, 10 and

4 , -14 3

Write the standard form of the equation of the hyperbola subject to the given conditions. 8) Vertices: (0, -6), (0, 6); 3 3 Asymptotes: y = - x, y = x 4 4 A)

y2 x2 =1 36 64

x2 y2 =1 64 36

x2 y2 =1 36 64

y2 x2 =1 D) 64 36

Answer: A Type: BI Var: 8 Objective: Graph a Hyperbola Centered at (h, k)

9) Vertices: (-10, -5), (2, -5) Slope of the asymptotes: ±

1 2

A) C)

(y + 5)2 9 (x + 4)2 36

(x + 4)2 36 (y + 5)2 9

(x + 4)2 9 (y + 5)2 36

(y + 5)2 36 (x + 4)2 9

=1 =1

Answer: C Type: BI Var: 50+ Objective: Graph a Hyperbola Centered at (h, k)

10) Vertices: (3, 0), (-3, 0); Foci: (5, 0), (-5, 0) y2 x2 x2 y2 A) =1 B) =1 9 16 9 25

y2 x2 =1 16 9

Answer: D Type: BI Var: 9 Objective: Graph a Hyperbola Centered at (h, k)

11) Vertices: (-5, -1), (-5, -5); Foci (-5, -3 + 15), (-5, -3 - 15) 2 (y - 3)2 (x - 5)2 (x + 3)2 (y + 5) A) =1 =1 B) 4 4 11 11 2 (x + 5)2 (y + 3)2 (y + 3)2 (x + 5) C) =1 =1 D) 4 11 4 11 Answer: D Type: BI Var: 50+ Objective: Graph a Hyperbola Centered at (h, k)

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x2 y2 =1 9 16

12) Corners of the reference rectangle: (2, 10), (2, 0), (-10, 10), (-10, 0); Horizontal transverse axis 2 (x + 4)2 (y - 5)2 (x - 4)2 (y + 5) A) =1 =1 B) 36 36 25 25 2 (y + 4)2 (x - 5)2 (y - 5)2 (x + 4) C) =1 =1 D) 25 36 25 36 Answer: A Type: BI Var: 50+ Objective: Graph a Hyperbola Centered at (h, k) 3 Determine the Eccentricity of a Hyperbola

Determine the eccentricity of the hyperbola. (x - 4)2 (y + 2)2 1) =1 144 25 A) e = 2

B) e =

13 12

C) e =

D) e =

12 5

Answer: B Type: BI Var: 50+ Objective: Determine the Eccentricity of a Hyperbola

y+ 7 9

52 9

196 A) e =

2,304 25 7

B) e =

7 25

C) e =

D) e =

Answer: A Type: BI Var: 50+ Objective: Determine the Eccentricity of a Hyperbola

Write the standard form of the equation of the hyperbola subject to the given conditions. 5 3) Vertices: (-2, 5), (10, 5); eccentricity 3 2 2 2 (x - 4) (y - 5) (y - 5)2 (x - 4) A) =1 =1 B) 64 36 64 36 2 (x + 4)2 (y + 5)2 (y - 4)2 (x - 5) C) =1 =1 D) 36 36 64 64 Answer: A Type: BI Var: 50+ Objective: Determine the Eccentricity of a Hyperbola

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7 24

4 Use Hyperbolas in Applications

Solve the problem. 1) The cross section of a cooling tower of a nuclear power plant is in the shape of a hyperbola, and can be modeled by the equation 2 x2 (y - 83) =1 576 2,304 where x and y are measured in meters. The base of the tower is located at y = 0, and the top of the tower is 110 m above the base. Determine the diameter of the tower at the top. Round to the nearest meter. A) 28 m B) 96 m C) 48 m D) 55 m Answer: D Type: BI Var: 50+ Objective: Use Hyperbolas in Applications

2) Suppose that two microphones 2,800 m apart at points A = (1,400, 0) and B = (-1,400, 0) detect the sound of a rifle shot. The time difference between the sound detected at A and the sound detected at B is 2 sec. If sound travels at approximately 330 m/sec, find an equation of the hyperbola with foci at A and B defining the points where the shooter may be located. x2 y2 x2 y2 A) = 1 B) =1 3302 1,3612 2,8002 2,7212 C)

x2 y2 =1 2 1,400 3302

Answer: A Type: BI Var: 40 Objective: Use Hyperbolas in Applications

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y2 x2 =1 21,361 3302

3) In 1911, Ernest Rutherford discovered the nucleus of the atom. Experiments leading to this discovery involved the scattering of alpha particles by the heavy nuclei in gold foil. When alpha particles are thrust towards the gold nuclei, the particles are deflected and follow a hyperbolic path. Suppose the minimum distance that the alpha particles get to the gold nucleus is 6 microns, and that 1 the hyperbolic path has asymptotes of y = ± x. Determine an equation of the path of the particle 2 shown. y 10 8 6 4

Alpha

2 Nucleus -2

8 10 12 14 16 18 x

-4 -6 -8

y2 36

x2 9

x2 36

y2 9

x2 36

Answer: C Type: BI Var: 26 Objective: Use Hyperbolas in Applications

4) Atomic particles with like charges tend to repel one another. Suppose that two beams of like-charged particles are hurled towards each other from two parallel atomic accelerators. The path defined by the particles is x2 - 4y2 = 36, where x and y are measured in microns. What is the minimum distance between the particles? A) 6 microns B) 36 microns C) 12 microns D) 4 microns Answer: C Type: BI Var: 16 Objective: Use Hyperbolas in Applications

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5) The path of a comet can be modeled by the equation x2 y2 =1 (1,169.7)2 (26)2 where x and y are measured in AU (astronomical units). Determine the distance (in AU) at perihelion. Round to one decimal place. A) 0.6 AU B) 1,170.0 AU C) 1,169.7 AU D) 0.3 AU Answer: D Type: BI Var: 50+ Objective: Use Hyperbolas in Applications 5 Mixed Exercises

Solve the system of equations. 1) x2 + 4y2 = 103 4x2 - y2 = 4 A) {(2 6, 7), (2 6, - 7), (- 2 6, 7), (- 2 6, - 7)} B) {(7, 24), (-7, 24), (7, -24), (-7, -24)} C) {(24, 7), (-24, 7), (24, -7), (-24, -7)} D) {( 7, 2 6), (- 7, 2 6), ( 7, - 2 6), (- 7, - 2 6)} Answer: D Type: BI Var: 50+ Objective: Mixed Exercises

Find the standard form of the equation of the ellipse or hyperbola shown. 2) 10 y 8 6 4 2 -10 -8 -6 -4 -2 -2

8 10 x

-4 -6 -8 -10

A) C)

(x + 2)

16 (x - 2)2 4

+ +

(y + 1)2 25 (y - 1)2 5

Answer: B Type: BI Var: 50+ Objective: Mixed Exercises

Page 33

(x - 2)2 16 (x - 2)2 16

+ -

(y - 1)2 25 (y - 1)2 25

=1 =1

3) 10 y 8 6 4 2 -10 -8 -6 -4 -2 -2

8 10 x

-4 -6 -8 -10

A) C)

(x + 1)2 4 (x + 1)2 16

(y - 1)2 2 (y - 1)2 4

Answer: B Type: BI Var: 50+ Objective: Mixed Exercises

Page 34

=1 =1

B) D)

(x + 1)2 16 (x - 1)2 16

(y - 1)2 4 (y + 1)2 4

=1 =1

6 Expanding your Skills

Solve the problem. 1) Radio signals emitted from points (10, 0) and (-10, 0) indicate that a plane is 8 miles closer to (10, 0) than to (-10, 0). Find an equation of the hyperbola that passes through the plane's location with foci (10, 0) and (-10, 0). All units are in miles. Radio signals emitted from points (0, 8) and (0, -8) indicate that a plane is 4 miles farther from (0, 8) than (0, -8). Find an equation of the hyperbola that passes through the plane's location with foci (0, 8) and (0, -8). From the figure, the plane is located in the fourth quadrant of the coordinate system. Solve the system of equations defining the two hyperbolas for the point of intersection in the fourth quadrant. This is the location of the plane. Round the coordinates to the nearest tenth of a mile. 10 y 8 6 4 2 -10 -8 -6 -4 -2 -2 -4

8 10 x

6 Plane

-6 -8 -10

A) a.

B) a.

x2 y2 =1 16 84

C) a.

x2 y2 =1 16 84

D) a.

y2 x2 =1 16 84

x2 y2 =1 4 60

y2 x2 =1 4 60

x2 y2 =1 4 60

y2 84

c. (4.1, -2.3)

c. (4.1, 2.3)

c. (4.1, -2.3)

c. (-2.3, 4.1)

Answer: C Type: BI Var: 49 Objective: Expanding your Skills

7.3 The Parabola 0 Concept Connections

Provide the missing information. 1) A is the set of all points in a plane that are equidistant from a fixed line (called the ) and a fixed point (called the ). Answer: parabola; directrix; focus Type: SA Var: 1 Objective: Concept Connections

Page 35

2) The line perpendicular to the directrix and passing through the focus is called the axis of . Answer: symmetry Type: SA Var: 1 Objective: Concept Connections

3) The

of a parabola is the point of intersection of the parabola and the axis of symmetry.

Answer: vertex Type: SA Var: 1 Objective: Concept Connections

4) The distance between the vertex and the focus of a parabola is called the often represented by p .

length and is

Answer: focal Type: SA Var: 1 Objective: Concept Connections

5) The standard form of an equation of a parabola with vertex (0, 0) and vertical axis of symmetry is . Answer: x2 = 4py Type: SA Var: 1 Objective: Concept Connections

6) Given a parabola defined by y2 = 4px, the directrix is given by x = -p and the focus is given by the ordered pair . Answer: (p, 0) Type: SA Var: 1 Objective: Concept Connections

7) Given y2 = 4px with p < 0, the parabola opens (upward/downward/ left/right). Answer: left Type: SA Var: 1 Objective: Concept Connections

8) Given x2 = 4py with p > 0, the parabola opens (upward/downward/ left/right). Answer: upward Type: SA Var: 1 Objective: Concept Connections

9) The line segment perpendicular to the axis of symmetry, passing through the focus and with endpoints on the parabola is called the . Answer: latus rectum Type: SA Var: 1 Objective: Concept Connections

Page 36

10) The length of the latus rectum is called the

diameter.

Answer: focal Type: SA Var: 1 Objective: Concept Connections

11) Given (x - h)2 = 4p(y - k), the ordered pairs representing the vertex and focus are , respectively. The directrix is the line defined by the equation .

and

Answer: (h, k); (h, k + p); y = k - p Type: SA Var: 1 Objective: Concept Connections

12) Given (y - k)2 = 4p(x - h), the ordered pairs representing the vertex and focus are , respectively. The directrix is the line defined by the equation

and .

Answer: (h, k); (h + p, k); x = h - p Type: SA Var: 1 Objective: Concept Connections

13) If the directrix is horizontal, then the parabola opens (horizontally/vertically). Answer: vertically Type: SA Var: 1 Objective: Concept Connections

14) If the line of symmetry is horizontal, then the parabola opens (horizontally/vertically). Answer: horizontally Type: SA Var: 1 Objective: Concept Connections 1 Identify the Focus and Directrix of a Parabola

A model of the form x2 = 4py is given. Identify the focus, and write an equation for the directrix. 1) x2 = 32y A) focus: (0, -8); directrix: y = 8 B) focus: (0, 8); directrix: y = -8 C) focus: (8, 0); directrix: y = -8 D) focus: (-8, 0); directrix: y = 8 Answer: B Type: BI Var: 18 Objective: Identify the Focus and Directrix of a Parabola

A model of the form y2 = 4px is given. Identify the focus, and write an equation for the directrix. 2) y2 = -3x 3 3 3 A) focus: 0, ; directrix: y = B) focus: - 3 , 0 ; directrix: x = 4 4 4 4 3 3 3 C) focus: 0, - ; directrix: y = D) focus: - 3 , 0 ; directrix: x = 4 4 4 4 Answer: D Type: BI Var: 22 Objective: Identify the Focus and Directrix of a Parabola

Page 37

Solve the problem. 3) A 20-in. satellite dish for a television has parabolic cross sections. A coordinate system is chosen so that the vertex of a cross section through the center of the dish is located at (0, 0). The equation of the parabola is modeled by x2 = 26.4y where x and y are measured in inches. Where should the receiver be placed to maximize signal strength? That is, where is the focus? A) Focus: (0, 6.4); Place the receiver 6.4 in. above the center of the dish. B) Focus: (0, 12.8); Place the receiver 12.8 in. above the center of the dish. C) Focus: (0, 13.2); Place the receiver 13.2 in. above the center of the dish. D) Focus: (0, 6.6); Place the receiver 6.6 in. above the center of the dish. Answer: D Type: BI Var: 8 Objective: Identify the Focus and Directrix of a Parabola

4) If a cross-section of the parabolic mirror in a flashlight has the equation y2 = 7x, where should the bulb be placed? 7 7 A) units above the vertex B) units to the right of the vertex 4 4 7 7 C) units to the left of the vertex D) units to the right of the vertex 4 2 Answer: B Type: BI Var: 13 Objective: Identify the Focus and Directrix of a Parabola 2 Graph a Parabola with Vertex at the Origin

An equation of the form x2 = 4py is given. Identify the focus and the endpoints of the latus rectum. 1) x2 = -12y A) focus: (0, -3); endpoints of latus rectum: (-6, -3); (6, -3) B) focus: (0, 3); endpoints of latus rectum: (-6, 3); (6, 3) C) focus: (0, 3); endpoints of latus rectum: (-12, 3); (12, 3) D) focus: (0, -3); endpoints of latus rectum: (-12, -3); (12, -3) Answer: A Type: BI Var: 18 Objective: Graph a Parabola with Vertex at the Origin

An equation of the form y2 = 4px is given. Identify the vertex, value of p, focus, focal diameter, and endpoints of the latus rectum. 2) 4y2 = 64x A) vertex: (0, 0); p = 4; focus: (4, 0); focal diameter: 16, endpoints of latus rectum: (4, -8); (4, 8) B) vertex: (0, 0); p = 4; focus: (0, 4); focal diameter: 8, endpoints of latus rectum: (4, -4); (4, 4) C) vertex: (0, 0); p = 4; focus: (4, 0); focal diameter: 8, endpoints of latus rectum: (4, -4); (4, 4) D) vertex: (0, 0); p = 4; focus: (0, 4); focal diameter: 16, endpoints of latus rectum: (8, -4); (8, 4) Answer: A Type: BI Var: 40 Objective: Graph a Parabola with Vertex at the Origin

Page 38

An equation of the form y2 = 4px is given. Write equations for the directrix and axis of symmetry. 3) y2 = -7x 7 7 ; axis of symmetry: x = 0 B) directrix: x = 4 A) directrix: x = ; axis of symmetry: y = 0 4 7 7 C) directrix: x = - ; axis of symmetry: x = 0 D) directrix: x = - ; axis of symmetry: y = 0 4 4 Answer: A Type: BI Var: 22 Objective: Graph a Parabola with Vertex at the Origin

Graph the parabola. Identify the focus. 4) 2y2 = -4x A)

5 y

-5 -4 -3 -2 -1 -1

-5 -4 -3 -2 -1

-1

-2

-3

-4

-5

1 focus: (0, - ) 2

1 focus: (0, ) 2

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

focus: (- , 0) 2 Answer: C Type: BI

Var: 21

Objective: Graph a Parabola with Vertex at the Origin

Page 39

-5

focus: ( , 0) 2

5) 2x2 = 6y A)

B) 5 y

5 y

-5 -4 -3 -2 -1 -1

-5 -4 -3 -2 -1

-1

-2

-3

-4

-5

3 focus: ( , 0) 4

3 focus: (0, ) 4

D) 5 y

5 y

-5 -4 -3 -2 -1 -1

-2

-3

-4

-5

focus: (- , 0) 4 Answer: B Type: BI Var: 21 Objective: Graph a Parabola with Vertex at the Origin

Page 40

-5

focus: (0, - ) 4

3 Graph a Parabola with Vertex at (h, k)

An equation of a parabola (x - h)2 = 4p(y - k) is given. Graph the parabola. Identify the vertex and the focal diameter. 1) (x + 3)2 = -4(y + 4) A) vertex: (3, 4); focal diameter: 1 B) vertex: (-3, -4); focal diameter: 4 y

10 8

6 4

2 2 4

-10 -8 -6 -4 -2 -2 -4

6 8 10 x

-6 -8

-8 -10

C) vertex: (3, 4); focal diameter: 4

8 10 x

D) vertex: (-3, -4); focal diameter: 1

10 8

6 4

2 2 4

6 8 10 x

-6

-10 -8 -6 -4 -2 -2 -4 -6

-8

-10

Answer: B Type: BI Var: 50+ Objective: Graph a Parabola with Vertex at (h, k)

Page 41

-6

-10

-10 -8 -6 -4 -2 -2 -4

8 10 x

Graph the parabola. Identify the directrix and focus. 2) (x - 1)2 = -4(y - 2) A) directrix: y = 3 focus: (1, 1) 10 y

10 y

-10 -8 -6 -4 -2 -2

8 10 x

-10 -8 -6 -4 -2 -2

-4

-6

-8

-10

C) directrix: y = 1 focus: (1, 3)

8 10 x

D) directrix: y = 6 focus: (1, -2) 10 y

10 y

-10 -8 -6 -4 -2 -2

8 10 x

-10 -8 -6 -4 -2 -2

-4

-6

-8

-10

Answer: A Type: BI Var: 50+ Objective: Graph a Parabola with Vertex at (h, k)

Page 42

B) directrix: y = -1 focus: (1, -3)

An equation of a parabola (y - k)2 = 4p(x - h) is given. Graph the parabola. Write the equation for the directrix and axis of symmetry. 3) (y - 1)2 = 12(x - 2) A) directrix: y = -2; B) directrix: x = -1; axis of symmetry: x = 2 axis of symmetry: y = 1 y

10 8

4 2

4 2 2 4

-10 -8 -6 -4 -2 -2 -4 -6

6 8 10 x

-8

-10

C) directrix: x = -5; axis of symmetry: y = -1 y

8 10 x

y 10 8 6

2 2 4

6 8 10 x

-10 -8 -6 -4 -2 -2 -4

-6

-8

-10

Answer: B Type: BI Var: 50+ Objective: Graph a Parabola with Vertex at (h, k)

Page 43

D) directrix: y = -4; axis of symmetry: x = -2

10 8 6

-10 -8 -6 -4 -2 -2 -4

-10 -8 -6 -4 -2 -2 -4 -6

Graph the parabola. Find the focus and the endpoints of the latus rectum. 4) (y - 1)2 = -12(x - 1) A) focus: (-4, 1) B) focus: (1, 1) latus rectum: (-4, 5), (-4, -7) latus rectum: (1, 7), (1, -5) 10 y

10 y

-10 -8 -6 -4 -2 -2

8 10 x

-10 -8 -6 -4 -2 -2

-4

-6

-8

-10

C) focus: (4, 1) latus rectum: (4, 7), (4, -5)

10 y

2 2

8 10 x

D) focus: (-2, 1) latus rectum: (-2, 7), (-2, -5)

10 y

-10 -8 -6 -4 -2 -2

8 10 x

-10 -8 -6 -4 -2 -2

-4

-6

-8

-10

8 10 x

Answer: D Type: BI Var: 50+ Objective: Graph a Parabola with Vertex at (h, k)

An equation of a parabola is given. Write the equation of the parabola in standard form, and identify the vertex and the equation of the directrix. 5) y2 + 16x - 4y + 52 = 0 A) (y + 2)2 = -16(x - 3) B) (y - 2)2 = -16(x + 3) vertex: (3, -2); focus: (7, -2) vertex: (3, -2); focus: (-7, -2) 2 C) (y - 2) = -16(x + 3) D) (y + 2)2 = -16(x - 3) vertex: (-3, 2); focus: (-7, 2) vertex: (-3, 2); focus: (7, 2) Answer: C Type: BI Var: 50+ Objective: Graph a Parabola with Vertex at (h, k)

Page 44

An equation of a parabola is given. Write the equation of the parabola in standard form, and identify the focus and the equation of the directrix. 6) 4x2 + 28x + 40y + 209 = 0 72 13 2 = -10(y + 4) A) (x + 14) = -40 y + B) x 2 40 387 413 3 7 ; directrix: y = focus: -14, focus: , - 13 ; directrix: y = 40 40 2 2 2 72 = -10(y - 4) C) x + 2 7 3 13 focus: - , ; directrix: y = 2 2 2

72 = -10(y + 4) D) x + 2 7 13 3 ; directrix: y = focus: - , 2 2 2

Answer: D Type: BI Var: 50+ Objective: Graph a Parabola with Vertex at (h, k)

An equation of a parabola is given. Write the equation of the parabola in standard form, and identify the vertex and the focus. 7) 16y2 - 104y + 288x + 1,033 = 0 13 2 13 2 A) y = -18(x + 3) B) x = -18(y + 3) 4 4 13 15 13 , ; focus 2 4 4

vertex: -3, C) x +

13 2 = -18(y + 3) 4

vertex: -

13 4

, -3 ; focus -

vertex: D) y -

, - 15 2 4

13 4

, -3 ; focus

13 4

13 2 = -18(x - 3) 4

vertex: 3,

13 15 13 , ; focus 4 2 4

Answer: A Type: BI Var: 50+ Objective: Graph a Parabola with Vertex at (h, k)

8) x2 - 8x - 16y - 16 = 0 A) (x - 4)2 = 16(y + 2) vertex: (-4, 2); focus (-4, 6) C) (x + 4)2 = 16(y - 2) vertex: (-4, 2); focus (-4, 6) Answer: B Type: BI Var: 50+ Objective: Graph a Parabola with Vertex at (h, k)

Page 45

15 2

B) (x - 4)2 = 16(y + 2) vertex: (4, -2); focus (4, 2) D) (x - 4)2 = 16(y + 2) vertex: (-2, 4); focus (2, 4)

An equation of a parabola is given. Write the equation of the parabola in standard form, and identify the vertex and the focal diameter. 9) y2 - 8y - 12x - 8 = 0 A) (y + 4)2 = 12(x - 2) B) (y - 4)2 = 12(x + 2) vertex: (-4, 2); focal diameter: 6 vertex: (-2, 4); focal diameter: 6 2 C) (y - 4) = 12(x + 2) D) (y + 4)2 = 12(x - 2) vertex: (-2, 4); focal diameter: 12 vertex: (-4, 2); focal diameter: 12 Answer: C Type: BI Var: 50+ Objective: Graph a Parabola with Vertex at (h, k)

Determine the standard form of an equation of the parabola subject to the given conditions. 10) Vertex: (0, 0); Directrix: y = -8 A) x2 = -8y B) y2 = 32x C) x2 = 32y D) y2 = -8x Answer: C Type: BI Var: 36 Objective: Graph a Parabola with Vertex at (h, k)

11) Vertex: (-1, -3); Directrix: x = -5 A) (x + 1)2 = -5(y + 3) C) (y + 3)2 = 16(x + 1)

B) (y + 3)2 = -5(x + 1) D) (x + 1)2 = 16(y + 3)

Answer: C Type: BI Var: 50+ Objective: Graph a Parabola with Vertex at (h, k)

12) Focus: (-4, -3); Vertex: (-4, -5) A) (x + 4)2 = 8(y + 5) C) (x + 4)2 = 2(y + 5) Answer: A Type: BI Var: 50+ Objective: Graph a Parabola with Vertex at (h, k)

13) Vertex: (-3, -1); passes through (9, 11) A) (x + 3)2 = 12(y + 1) or (y + 1)2 = 12(x + 3) B) (x + 3)2 = 3(y + 1) or (y + 1)2 = 3(x + 3) C) (x - 3)2 = 3(y - 1) or (y - 1)2 = 3(x - 3) D) (x - 3)2 = 12(y - 1) or (y - 1)2 = 12(x - 3) Answer: A Type: BI Var: 50+ Objective: Graph a Parabola with Vertex at (h, k)

Page 46

B) (x - 4)2 = 2(y - 5) D) (x - 4)2 = 8(y - 5)

Fill in the blanks. Let |p| represent the focal length (the distance between the vertex and the focus). 14) If the directrix of a parabola is given by y = -1 and the focus is (-3, 5), then the vertex is given by the ordered pair and the value of p is . A) (-3, 6); -2 B) (-3, 2); 3 C) (-2, 2); -1 D) (3, 2), -3 Answer: B Type: BI Var: 50+ Objective: Graph a Parabola with Vertex at (h, k)

15) If the vertex of a parabola is (-2, -4) and the focus is (-4, -4), then the directrix is given by the equation and the value of p is . A) x = -4; -2 B) x = 0; -4 C) y = 0; -2 D) x = 0; -2 Answer: D Type: BI Var: 50+ Objective: Graph a Parabola with Vertex at (h, k)

16) If the focal diameter of a parabola is 8 units, then the focal length is A) 4 B) -2 C) 32 Answer: D Type: BI Var: 9 Objective: Graph a Parabola with Vertex at (h, k)

Page 47

units. D) 2

4 Use Parabolas in Applications

Solve the problem. 1) Suppose a solar cooker has a parabolic mirror (see figure).

7 in.

14 in. a. Use a coordinate system with the origin at the vertex of the mirror and write an equation of a parabolic cross section of the mirror. b. Where should a pot be placed so it receives maximum heat? A) a. x2 = 28y for -14 ≤ x ≤ 14 b. The pot should be placed 28 in. above the vertex B) a. x2 = 28y for -14 ≤ x ≤ 14 b. The pot should be placed 7 in. above the vertex C) a. x2 = 14y for -14 ≤ x ≤ 14 b. The pot should be placed 3.5 in. above the vertex D) a. x2 = 14y for -14 ≤ x ≤ 14 b. The pot should be placed 14 in. above the vertex Answer: B Type: BI Var: 16 Objective: Use Parabolas in Applications

2) The quality of the images from an orbital telescope result from a large parabolic mirror, 3.8 meters in diameter, that collects light from space. Suppose that a coordinate system is chosen so that the vertex of a cross section through the center o the mirror is located at (0,0). Furthermore, the focal length of the mirror is 56.8 m. Assume that x and y are in meters. Write an equation of the parabolic cross section of the mirror for -1.9 ≤ x ≤ 1.9. A) x2 = 56.8y for -1.9 ≤ x ≤ 1.9 B) y2 = 227.2x for -1.9 ≤ y ≤ 1.9 C) x2 = 113.6y for -1.9 ≤ x ≤ 1.9 D) x2 = 227.2y for -1.9 ≤ x ≤ 1.9 Answer: D Type: BI Var: 50+ Objective: Use Parabolas in Applications

Page 48

3) A parabolic mirror on a telescope 30 cm in diameter has a focal length of 15 cm. For the coordinate system shown, write an equation of the parabolic cross section of the mirror. y

(15, 0)

A) y2 = 15x for -15 ≤ y ≤ 15 C) x2 = 60y for -15 ≤ x ≤ 15

B) y2 = 60x for -15 ≤ y ≤ 15 D) y2 = 60x for -30 ≤ y ≤ 30

Answer: B Type: BI Var: 50+ Objective: Use Parabolas in Applications

4) A parabolic arch forms a structure that allows equal vertical loading along its length.

(45, -35)

Take the origin at the vertex of the parabolic arch, and write an equation of the parabola. 7 405 9 405 A) x2 = y B) x2 = y D) x2 = y C) x2 = - y 405 7 7 7 Answer: D Type: BI Var: 17 Objective: Use Parabolas in Applications

Page 49

5 Mixed Exercises

Solve the system of equations. 1) (y - 3)2 = -8(x + 3) -4x - 6y = -22 A) {(-11, 11), (-7, 5)} C) {(11, -11), (7, -5)} Answer: D Type: BI Var: 50+ Objective: Mixed Exercises

Page 50

B) {(11, -11), (5, -7)} D) {(-11, 11), (-5, 7)}

Chapter 8 Sequences, Series, Induction, and Probability 8.1 Sequences and Series 0 Concept Connections

Provide the missing information. 1) An infinite is a function whose domain is the set of positive integers. A sequence is a function whose domain is the set of the first n positive integers. Answer: sequence; finite Type: SA Var: 1 Objective: Concept Connections

2) The expression an is called the

term or general term of a sequence.

Answer: nth Type: SA Var: 1 Objective: Concept Connections

3) An

sequence is a sequence in which consecutive terms alternate in sign.

Answer: alternating Type: SA Var: 1 Objective: Concept Connections

4) A formula defines the nth term of a sequence as a function of one or more terms preceding it. Answer: recursion Type: SA Var: 1 Objective: Concept Connections

5) For n ≥ 1, the expression represents the product of the first n positive integers n(n - 1)(n - 2)···(2)(1). For n = 0, we have 0! = . Answer: n!; 1 Type: SA Var: 1 Objective: Concept Connections

6) Given an infinite sequence {an}= a1, a2, a3, … the sum of the terms of the sequence is called an infinite . The notation Sn is called the nth of the sequence and is called a finite series. Answer: series; partial sum Type: SA Var: 1 Objective: Concept Connections

7) Given

∑ ai , the variable i is called the

. The value 1 is called the

i=1 limit of summation. The value n is called the upper Answer: index; summation; lower; limit Type: SA Var: 1 Objective: Concept Connections

Page 1

of summation.

8) One property of summation indicates that

∑c =

i=1 Answer: cn Type: SA Var: 1 Objective: Concept Connections 1 Write Terms of a Sequence from the nth Term

The nth term of a sequence is given. Write the first four terms of the sequence. 1) an = 9n - 7 A) -2, -4, -6, -8 B) 2, 4, 6, 8 C) -2, -11, -20, -29

D) 2, 11, 20, 29

Answer: D Type: BI Var: 50+ Objective: Write Terms of a Sequence from the nth Term

nn + 3

an = (-1)

n+4

7 A) - , 11 , - 15 , 19 4 4 4 4

4 5 B) , - , 6 , - 7 5 6 7 8 4 D) - , 5 , - 6 , 7 5 6 7 8

7 11 15 19 ,C) , - , 4 4 4 4 Answer: D Type: BI Var: 50+ Objective: Write Terms of a Sequence from the nth Term

an = n(n3 - 8) A) -7, 0, 57, 224

B) -6, 0, 4, 8

C) -7, -4, 3, 16

D) -6, 2, 22, 60

C) 3, -9, 27, -81

D) 3, -6, 9, -12

Answer: A Type: BI Var: 50+ Objective: Write Terms of a Sequence from the nth Term

an = (-1)n+1 3n A) -3, 6, -9, 12

B) -3, 9, -27, 81

Answer: C Type: BI Var: 5 Objective: Write Terms of a Sequence from the nth Term

Page 2

(-1)n 5) an =

n2 + 4

1 1 1 1 A) , - , , 5 8 13 20 1 C) - , 1 , - 1 , 1 4 5 8 13

1 1 1 1 B) - , , - , 5 8 13 20 1 1 D) , - , 1 , - 1 4 5 8 13

Answer: B Type: BI Var: 16 Objective: Write Terms of a Sequence from the nth Term

6) an = e2ln n A) e, e2, e3, e4

B) 1, 4, 9, 16

C) 2, 4, 8, 16, 32

D) 2, 4, 6, 8

Answer: B Type: BI Var: 4 Objective: Write Terms of a Sequence from the nth Term

The nth term of a sequence is given. Find the indicated term. 7) an = 2n + 9; a6 A) 108 Answer: C

B) 21

C) 73

D) 576

Type: BI Var: 50+ Objective: Write Terms of a Sequence from the nth Term

- 1; a10 n 7 A) 10

8) an =

1 B) 5

Answer: D Type: BI Var: 50+ Objective: Write Terms of a Sequence from the nth Term 2 Write Terms of a Sequence Defined Recursively

Write the first five terms of the sequence defined recursively. 1) b1 = 5; bn = 5bn-1 - 3 A) b1 = 5, b2 = 22, b3 = 107, b4 = 532, b5 = 2,657 B) b1 = 5, b2 = 10, b3 = 15, b4 = 20, b5 = 25 C) b1 = 5, b2 = 25, b3 = 110, b4 = 535, b5 = 2,660 D) b1 = 5, b2 = 2, b3 = -1, b4 = -4, b5 = -7 Answer: A Type: BI Var: 50+ Objective: Write Terms of a Sequence Defined Recursively

Page 3

13 10

D) -

7 10

1 2) a = 20; a = a + 3 1 n 4 n-1 A) a = 20, a = 8, a = 5, a = 17 , a = 65 1 2 3 4 4 5 16 5 5 5 B) a1 = 20, a2 = 5, a3 = , a4 = , a5 = 4 16 64 C) a = 20, a = 23 , a = 13 , a = 29 , a = 1 1 2 4 3 8 4 64 5 8 D) a1 = 20, a2 = 23, a3 = 26, a4 = 29, a5 = 32 Answer: A Type: BI Var: 50+ Objective: Write Terms of a Sequence Defined Recursively

3) a1 = 6, an = -

1 an-1

1 1 1 1 A) a1 = 6, a2 = - , a3 = , a4 = , a5 = 6 36 216 1,296 1 1 B) a1 = 6, a2 = - , a3 = 6, a4 = - , a5 = 6 6 6 1 1 C) a1 = 6, a2 = , a3 = 6, a4 = , a5 = 6 6 6 1 1 1 1 D) a1 = 6, a2 = , a3 = , a4 = , a5 = 6 36 216 1,296 Answer: B Type: BI Var: 8 Objective: Write Terms of a Sequence Defined Recursively 3 Use Factorial Notation

Evaluate the expression. 9! 1) 5! · 4! A) 126 Answer: A Type: BI Var: 22 Objective: Use Factorial Notation

Page 4

B) 24

C) 1

D) 362,736

(n + 2)! (n + 4)! A)

1 (n + 4)

1 (n + 3)!

B) (n + 4)(n + 3)

n! B) (n + 1)!

1 C) 6n

1 (n + 4)(n + 3)

Answer: D Type: BI Var: 13 Objective: Use Factorial Notation

(6n)! (6n + 1)! 1 A) 2

1 D) 6n + 1

Answer: D Type: BI Var: 8 Objective: Use Factorial Notation

The nth term of a sequence is given. Find the indicated term. 2n 4) an = ; a (n + 1)! 4 16 1 B) 2 C) A) 15 5 Answer: D

2 15

Type: BI Var: 24 Objective: Use Factorial Notation

5) cn =

(2n)! 5n

; c5

A) 8064

B) 3,628,800

2 5

D) 145,152

Answer: D Type: BI Var: 18 Objective: Use Factorial Notation

Find the nth term an of a sequence whose first four terms are given. 6) 4, 16, 64, 256, ... a = 2n A) n Answer: A Type: BI Var: 10 Objective: Use Factorial Notation

Page 5

B) an = n + 2

C) an = 2n

D) an = 2n-1

7) -

6 12

7 24

A) an =

36 n-7

9 48

, ...

12n

7-n B) an =

12n

n+5 C) an = -

12n

D) an =

n+5 12n

D) an =

(n + 4)! 2n

Answer: C Type: BI Var: 50+ Objective: Use Factorial Notation

8) 1, -16, 81, -256 A) an = -n4 C) a = (-1)n +1(n4)

B) an = n4 D) an = (-1)n(n4)

Answer: C Type: BI Var: 6 Objective: Use Factorial Notation

1· 2· 3· 4 1· 2· 3· 4· 5 1· 2· 3· 4· 5· 6 1· 2· 3· 4· 5· 6· 7 , , , , ... 8 2 4 6 (n + 3)! (n + 4)! (n + 3)! A) an = B) an = C) a = n 2n 2n 2n Answer: A Type: BI Var: 12 Objective: Use Factorial Notation

10)

10 15 20 , , , ... 16 25 36 49 5n A) an = (n + 3)2 ,

5n B) an =

(n + 2)2

5n C) an =

(n + 2)2

D) an =

Answer: A Type: BI Var: 12 Objective: Use Factorial Notation 4 Use Summation Notation

Find the sum. 4 1)

∑ (5i + 1) i=1 A) 21

B) 27

Answer: C Type: BI Var: 50+ Objective: Use Summation Notation

Page 6

C) 54

D) 33

(n + 3)2

45 2)

∑7

i=1 A) 308

B) 1,035

C) 315

D) 7

C) -12

D) 12

C) 540

D) -540

Answer: C Type: BI Var: 50+ Objective: Use Summation Notation

3) ∑(-1) j(4j) j=1 A) 84

B) -84

Answer: D Type: BI Var: 8 Objective: Use Summation Notation

4) ∑ (-4k3) k=1 A) -1,764

B) 1,764

Answer: A Type: BI Var: 32 Objective: Use Summation Notation

1k ∑ 3 k=2 5 A) 128

40 243

121 243

121 1024

Answer: B Type: BI Var: 8 Objective: Use Summation Notation

6) ∑ j( j + 8) j=1 A) 48

B) 36

Answer: D Type: BI Var: 6 Objective: Use Summation Notation

Page 7

C) 120

D) 110

∑ (n2 + 5) n=2 A) 35

B) 74

C) 50

D) 59

C) 45

C) 1,200

D) 130

Answer: B Type: BI Var: 50+ Objective: Use Summation Notation

j+4 j

∑

j=1 54 A) 5

79 5

15 2

Answer: B Type: BI Var: 5 Objective: Use Summation Notation

9) ∑ (k + 3)(k + 5) k=1 A) 70

B) 250

Answer: B Type: BI Var: 50+ Objective: Use Summation Notation

Write the sum using summation notation. 3 1 10) + 4 + 27 + ... + n n+4 7 5 3 n n n3 i3 A) ∑ B) ∑ i+4 n+4 i=1 i=1

n C)

n+1

∑ i+4

i=0

i=1

∑4

D) ∑ 4ii

i=1

Answer: A Type: BI Var: 10 Objective: Use Summation Notation

11) 4 + 4 + 4 + 4 + 4 + 4 n A) ∑ 4 i=1

4 B)

Answer: C Type: BI Var: 25 Objective: Use Summation Notation

Page 8

∑6 i=1

12) -3 + 9 - 27 + 81 4 A) ∑ (-1)n+1n n=1

B) ∑ (-1)nn n=1

C) ∑ (-1)n3n n=1

4 D)

∑ (-1)n+13n n=1

Answer: C Type: BI Var: 10 Objective: Use Summation Notation

13) -

1 1 1 1 1 + + 3 9 27 81 243 5 i+1 1 A) ∑ (-1) 3i i=1 5 1 (-1)i C) ∑ 3i i=1

5 B)

i+1 1 3i

∑ (-1) i=1 5

∑ (-1) 3i

i=1

Answer: C Type: BI Var: 3 Objective: Use Summation Notation

14)

x2 x4 x6 x8 x10 + + + + 1 2 6 24 120 5 4 x2i x2i A) ∑ i - 1 B) ∑ 2 i! i=1 i=0

∑ i=1

x2i i!

∑ i=0

x2i (i+1)!

Answer: C Type: BI Var: 4 Objective: Use Summation Notation

15) a + ar + ar2 + ar3 + ... + ar19 20 19 i-1 i A) ∑ ar B) ∑ ar i=1 i=1 Answer: D Type: BI Var: 11 Objective: Use Summation Notation

Page 9

19 C)

∑ ari i=1

20 D)

∑ ari - 1 i=1

16)

1 2 6 24 120 + + + + x + 3 x + 6 x + 9 x + 12 x + 15 5

2i - 1 A) ∑ x + 3i i=1

i! B) ∑ x + 3i i=1

i! C) ∑ x+i i=1

∑ x+3

i=1

Answer: B Type: BI Var: 7 Objective: Use Summation Notation

Rewrite the series as an equivalent series with the new index of summation. 7 ? 17) ∑(9i) = ∑ ( ? ) k=3 i=1 9 9 9 A)

∑ [9(k + 3)]

k=3

∑ [9(k - 3)] k=3

∑ [9(k - 2)] k=3

9 D)

∑ [9(k + 2)] k=3

Answer: C Type: BI Var: 50+ Objective: Use Summation Notation 5 Mixed Exercises

Use the provided sums to evaluate the given expression. 40 40 40 2 1) ∑ i = 820 and ∑ i2 = 22,140; evaluate ∑ (5i + 9i) i=1 i=1 i=1 A) 203,360 B) 118,080 C) 22,960

D) 111,520

Answer: B Type: BI Var: 50+ Objective: Mixed Exercises

Solve the problem. 2) Expenses for a company for year 1 are $20,000. Every year thereafter, expenses increase by $1,500 plus 2% of the cost of the prior year. Let a1 represent the original cost for year 1; that is a1 = 20,000. Use a recursive formula to find the cost an in terms of an-1 for each subsequent year, n ≥ 2. A) an = 0.02an-1 + 21,500, n ≥ 2 C) an = 0.02an-1 + 1,500, n ≥ 2 Answer: D Type: BI Var: 50+ Objective: Mixed Exercises

Page 10

B) an = 1.02an-1 + 21,500, n ≥ 2 D) an = 1.02an-1 + 1,500, n ≥ 2

Determine whether the statement is true or false. n n 3) ∑ (3i + 7) = 3 ∑(i) + 7n i=1 i=1 Answer: TRUE Type: TF Var: 1 Objective: Mixed Exercises

∑(i - 4i + 5) = ∑ i - 4 ∑(i) + 5n 2

i=1

Answer: TRUE Type: TF Var: 1 Objective: Mixed Exercises

5) ∑(aibi) = ∑(ai) ∑(bi) i=1 i=1 i=1 Answer: FALSE Type: TF Var: 1 Objective: Mixed Exercises

∑(ai)

n a

i=1

6) ∑ = b n i=1 i

∑(bi)

i=1 Answer: FALSE Type: TF Var: 1 Objective: Mixed Exercises

Solve the problem. 7) In a business meeting, every person at the meeting shakes every other person’s hand exactly one 1 time. The total number of handshakes for n people at the meeting is given by a = n(n - 1). n 2 Evaluate a12 and interpret its meaning in the context of this problem. A) 78; If 12 people are present at the meeting there will be 78 handshakes. B) 132; If 12 people are present at the meeting there will be 132 handshakes. C) 66; If 12 people are present at the meeting there will be 66 handshakes. D) 11; The 12th person shakes performs 11 handshakes. Answer: C Type: BI Var: 1 Objective: Mixed Exercises

Page 11

8.2 Arithmetic Sequences and Series 0 Concept Connections

Provide the missing information. 1) An sequence is a sequence in which each term after the first differs from its predecessor by a fixed constant. Answer: arithmetic Type: SA Var: 1 Objective: Concept Connections

2) The difference between consecutive terms in an arithmetic sequence is called the , and is denoted by d. Answer: common difference Type: SA Var: 1 Objective: Concept Connections

3) Given an arithmetic sequence with first term a1, and common difference d, the nth term is represented by the formula an = . Answer: a1 + (n - 1)d Type: SA Var: 1 Objective: Concept Connections

4) An arithmetic sequence is a linear function whose domain is the set of

integers.

Answer: positive Type: SA Var: 1 Objective: Concept Connections

5) The sum of the first n terms of a sequence is called the nth

sum and is denoted by Sn.

Answer: partial Type: SA Var: 1 Objective: Concept Connections

6) Given an arithmetic sequence with first term a1 and nth term an, the nth partial sum is given by the formula Sn = n Answer: (a + a ) n 2 1 Type: SA Var: 1 Objective: Concept Connections 1 Identify Specific and General Terms of an Arithmetic Sequence

Determine whether the sequence is arithmetic. If so, find the common difference. 1) 4, -16, 64, -256, . . . A) arithmetic; d = -4 B) arithmetic; d = 4 C) not arithmetic D) arithmetic; d = -2 Answer: C Type: BI Var: 50+ Objective: Identify Specific and General Terms of an Arithmetic Sequence

Page 12

2) 5,

25 125 625 , , 6 36 216 A) arithmetic; d = C) arithmetic; d =

5 B) not arithmetic

6 6

D) arithmetic; d = 5

5 Answer: B

Type: BI Var: 50+ Objective: Identify Specific and General Terms of an Arithmetic Sequence

Write the first four terms of an arithmetic sequence, {an}, based on the given information about the sequence. 3) a1 = 3, d = 6 A) 3, 9, 15, 21 C) 18, 108, 648, 3,888

B) 9, 15, 21, 27 D) 3, 18, 108, 648

Answer: A Type: BI Var: 50+ Objective: Identify Specific and General Terms of an Arithmetic Sequence

4) Write the first five terms of the arithmetic sequence. a1 = 3, d = -4 A) 3, -1, -5, -9, -13 B) -4, -1, 2, 5, 8 C) -4, -1, 5, 23, 77 D) 3, -1, 19, -61, 259 Answer: A Type: BI Var: 42 Objective: Identify Specific and General Terms of an Arithmetic Sequence

Write the first four terms of the arithmetic sequence with the given first term and common difference. 5) a1 = 9, d = -7 A) -7, 2, 11, 20 B) -7, -16, -25, -34 C) 9, 16, 23, 30 D) 9, 2, -5, -12 Answer: D Type: BI Var: 50+ Objective: Identify Specific and General Terms of an Arithmetic Sequence

Find the indicated term of the arithmetic sequence based on the given information. 6) a1 = 38, d = 3; Find a25. A) 107

B) 104

C) 113

Answer: D Type: BI Var: 50+ Objective: Identify Specific and General Terms of an Arithmetic Sequence

Page 13

D) 110

7) a1 = A)

4 5

1 ,d=

; Find a 24

142 15

44 5

127 15

137 15

Answer: C Type: BI Var: 50+ Objective: Identify Specific and General Terms of an Arithmetic Sequence

Write a recursive formula to define the sequence. 8) a1 = 5, d = 4 A) a1 = 5 and an = an+1 + 4 for n ≥ 2 C) a1 = 5 and an = an-1 - 4 for n ≥ 2

B) a1 = 5 and an = an-1 + 4 for n ≥ 2 D) a1 = 5 and an = 4an-1 for n ≥ 2

Answer: B Type: BI Var: 50+ Objective: Identify Specific and General Terms of an Arithmetic Sequence

9) a1 = 2, d = -9 A) a1 = 2 and an = an-1 - 9 for n ≥ 2 C) a1 = 2 and an = an+1 - 9 for n ≥ 2

B) a1 = 2 and an = an-1 + 9 for n ≥ 2 D) a1 = 2 and an = -9an-1 for n ≥ 2

Answer: A Type: BI Var: 50+ Objective: Identify Specific and General Terms of an Arithmetic Sequence

Write a nonrecursive formula for the nth term of the arithmetic sequence {an} based on the given information. 10) a1 = 6, d = 4 A) an = 4n - 2

B) an = 2n - 4

C) an = 4n + 2

D) an = 2n + 4

Answer: C Type: BI Var: 50+ Objective: Identify Specific and General Terms of an Arithmetic Sequence

1 11) a1 =

2 ,d= 2

3 1

A) an = n 3 6

B) an = - n + 6 3

C) an = n + 3 6

Answer: A Type: BI Var: 50+ Objective: Identify Specific and General Terms of an Arithmetic Sequence

Page 14

D) an = - n 6 3

Find the indicated term. 12) a1 = -11, d = 9; Find a18 A) 142

B) -351

C) 182

D) 133

Answer: A Type: BI Var: 50+ Objective: Identify Specific and General Terms of an Arithmetic Sequence

13) a1 = A)

1 2

,d=

1 5

; Find a

19 10

29 10

5 2

27 10

Answer: C Type: BI Var: 50+ Objective: Identify Specific and General Terms of an Arithmetic Sequence

Solve the problem. 14) A physical activity class requires students to jog around an indoor track. For the first week of class the students jog 300 m around the track each day. Each week thereafter, the students increase the distance jogged by 125 m. Write the nth term of a sequence defining the number of meters jogged each day by the students in the nth week of class. A) an = 125n + 300 B) an = 300n + 125 C) an = 175n + 125 D) an = 125n + 175 Answer: D Type: BI Var: 20 Objective: Identify Specific and General Terms of an Arithmetic Sequence

15) A student studying to be a veterinarian's assistant keeps track of a kitten's weight each week for a 5-week period after birth. Week number 1 2 3 4 5 Weight (lb) 0.7 0.97 1.24 1.51 1.78 a. Write an expression for the nth term of the sequence representing the kitten's weight, n weeks aft birth. b. If the weight of the kitten continues to increase linearly for 3 months, predict the kitten's weight 1 weeks after birth. A) a. an = 0.27n + 0.7 b. 2.7 lb C) a. an = 0.27n + 0.43 b. 2.7 lb

B) a. an = 0.27n + 0.43 b. 3.13 lb D) a. an = 0.27n + 0.7 b. 3.4 lb

Answer: B Type: BI Var: 50+ Objective: Identify Specific and General Terms of an Arithmetic Sequence

Page 15

Find the indicated term of the arithmetic sequence based on the given information. 16) a1 = 36 and a6 = 11; Find a33 . A) -124

B) -114

C) -119

D) -129

Answer: A Type: BI Var: 50+ Objective: Identify Specific and General Terms of an Arithmetic Sequence

17) a1 = -11 and a16 = 19; Find a13 A) 25

B) 19

C) 13

D) 15

Answer: C Type: BI Var: 50+ Objective: Identify Specific and General Terms of an Arithmetic Sequence

18) a1 = 13 and a18 = -55; Find a9 A) -23

B) -55

C) -91

D) -19

Answer: D Type: BI Var: 50+ Objective: Identify Specific and General Terms of an Arithmetic Sequence

19) a15 = 67 and a41 = 223; Find a108 A) 671

B) 625

C) 637

D) 631

Answer: B Type: BI Var: 50+ Objective: Identify Specific and General Terms of an Arithmetic Sequence

20) a17 = 9.76 and a41 = 10.72; Find a108 A) 13.44

B) -4.76

C) 13.48

D) 13.4

Answer: D Type: BI Var: 50+ Objective: Identify Specific and General Terms of an Arithmetic Sequence

Find the number of terms of the finite arithmetic sequence. 21) 12, 21, 30, 39,..., 525 A) 57 B) 58 C) 60

D) 59

Answer: B Type: BI Var: 50+ Objective: Identify Specific and General Terms of an Arithmetic Sequence

22) 12, 11.7, 11.4, 11.1,..., -2.4 A) 50 B) 51

C) 48

Answer: D Type: BI Var: 50+ Objective: Identify Specific and General Terms of an Arithmetic Sequence

Page 16

D) 49

From the given terms of the arithmetic sequence, find a1 and d. 23) a12 = 50 and a25 = 89 A) a1 = 20, d = 3

B) a1 = 14, d = 3

C) a1 = 17, d = 3

D) a1 = 2, d = 4

Answer: C Type: BI Var: 50+ Objective: Identify Specific and General Terms of an Arithmetic Sequence 2 Evaluate a Finite Arithmetic Series

Find the sum. 1) 3 + 4.6 + 6.2 + 7.8 + ... + 55.8 A) 1,002.8 B) 999.6

C) 387

D) 77.4

C) 4,572

D) 9,108

Answer: B Type: BI Var: 50+ Objective: Evaluate a Finite Arithmetic Series

2) Find the sum of the first 36 terms of the sequence. {1, 8, 15, 22, 29,... } A) 9,144 B) 4,446 Answer: B Type: BI Var: 50+ Objective: Evaluate a Finite Arithmetic Series

Find the indicated terms. 3) If the third and fourth terms of an arithmetic sequence are -7 and -10, what are the first and second terms? A) a1 = -4, a2 = -1 B) a1 = 7, a2 = 10 C) a1 = -1, a2 = -4 D) a1 = 10, a2 = 7 Answer: C Type: BI Var: 32 Objective: Evaluate a Finite Arithmetic Series

Find the sum. 25 4) ∑ (5n - 3) n=1 A) 1,550

B) 3,100

Answer: A Type: BI Var: 50+ Objective: Evaluate a Finite Arithmetic Series

Page 17

C) 3,247

D) 1,650

Evaluate the sum. 14 5)

∑ (24 - 4n) n=1 A) -88

B) -92

C) -80

D) -84

C) -1,994.75

D) -2,034.25

Answer: D Type: BI Var: 50+ Objective: Evaluate a Finite Arithmetic Series

158 6)

∑ 7 - 4k

k=1 A) 4,246.25

B) 3,140.25

Answer: D Type: BI Var: 50+ Objective: Evaluate a Finite Arithmetic Series 3 Apply Arithmetic Sequences and Series

Solve the problem. 1) A lecture hall has 11 rows of seats. The first row has 15 seats, and each row after that has 4 more seats than the previous row. How many seats are in the last row? How many seats are in the lecture hall? A) last row: 59; total seats: 209 B) last row: 59; total seats: 407 C) last row: 55; total seats: 385 D) last row: 55; total seats: 209 Answer: C Type: BI Var: 50+ Objective: Apply Arithmetic Sequences and Series

2) Elizabeth must choose between two job offers. The first job offers $42,000 for the first year and a $3,300 raise each year thereafter. The second job offers $48,000 for the first year and a $2,100 raise each year thereafter. a. If she anticipates working for the company for 4 years, find the total amount she would earn from each job. b. If she anticipates working for the company for 12 years, find the total amount she would earn from each job. A) a. Job 1: $204,600; Job 2: $187,800 B) a. Job 1: $189,000; Job 2: $225,000 b. Job 1: $714,600; Job 2: $721,800 b. Job 1: $667,800; Job 2: $833,400 C) a. Job 1: $187,800; Job 2: $204,600 D) a. Job 1: $225,000; Job 2: $189,000 b. Job 1: $721,800; Job 2: $714,600 b. Job 1: $833,400; Job 2: $667,800 Answer: C Type: BI Var: 50+ Objective: Apply Arithmetic Sequences and Series

Page 18

3) An object in free fall is dropped from a tall cliff. It falls 16 ft in the first second, 48 ft in the second second, 80 ft in the third second, and so on. Write a formula for the nth term of an arithmetic sequence that represents the distance dn (in ft) that the object will fall in the nth second. A) dn = 32n - 16 Answer: A

B) dn = 32n + 16

C) dn = 16n

D) dn = 16n2

C) 650

D) 8,450

Type: BI Var: 1 Objective: Apply Arithmetic Sequences and Series 4 Mixed Exercises

Solve the problem. 1) Find the sum of the integers from -60 to 70. A) 655 B) 8,515 Answer: A Type: BI Var: 50+ Objective: Mixed Exercises

2) Find the sum of the first 50 integers that are exactly divisible by 3. A) 3,825 B) 3,978 C) 1,275

D) 3,675

Answer: A Type: BI Var: 36 Objective: Mixed Exercises

3) The arithmetic mean (average) of two numbers c and d is given by x =

c+d

. The value x is

2 equidistant between c and d, so the sequence c, x, d is an arithmetic sequence. Inserting k equally spaced values between c and d, yields the arithmetic sequence c, x1, x2, x3, x4, ... , xn, d. Use this information to insert five means between 22 and 70 A) The common difference is 8. B) The common difference is 24. C) The five arithmetic means between 22 and 70 are 30, 38, 46, 54, and 62. D) The arithmetic mean of 22 and 70 is 46. Answer: C Type: BI Var: 50+ Objective: Mixed Exercises

8.3 Geometric Sequences and Series 0 Concept Connections

Provide the missing information. 1) A sequence is a sequence in which each term after the first is the product of the preceding term and a fixed nonzero real number, called the common , r. Answer: geometric; ratio Type: SA Var: 1 Objective: Concept Connections

Page 19

2) The nth term of a geometric sequence with first term a1 and common ratio r is given by an = . Answer: a1r n-1 Type: SA Var: 1 Objective: Concept Connections

3) The nth partial sum Sn of the first n terms of a geometric sequence is a (finite/infinite) geometric series. Answer: finite Type: SA Var: 1 Objective: Concept Connections

4) The sum Sn of the first n terms of a geometric sequence with first term a1 and common ratio r is given by the formula Sn = . Answer:

a1(1 - rn) 1-r

Type: SA Var: 1 Objective: Concept Connections

5) Given a geometric sequence with |r| < 1, the value of rn →

as n → ∞.

Answer: 0 Type: SA Var: 1 Objective: Concept Connections

6) Given an infinite geometric series with first term a1 and common ratio r, if |r| < 1, then the sum S is given by the formula S = . If |r| > 1, then the sum (does/does not) exist. a1 Answer: ; does not 1-r Type: SA Var: 1 Objective: Concept Connections

7) A sequence of payments made at equal intervals over a fixed period of time is called an Answer: annuity Type: SA Var: 1 Objective: Concept Connections

Page 20

8) If P dollars is invested at the end of each compounding period n times per year at interest rate r, then the value A of the annuity (in $) after t years is given by the formula A= . P 1+

r nt -1 n

Answer: r n Type: SA Var: 1 Objective: Concept Connections

9) Suppose that an infinite series a1 + a2 + a3 + ... + an approaches a value L as n → ∞. Then the series . Otherwise, the series . Answer: converges; diverges Type: SA Var: 1 Objective: Concept Connections 1 Identify Specific and General Terms of a Geometric Sequence

Determine whether the sequence is geometric. If so, find the value of r. 1) 2, -6, 18, -54, . . . A) not geometric B) geometric; r = -8 C) geometric; r = -3 D) geometric; r = 2 Answer: C Type: BI Var: 50+ Objective: Identify Specific and General Terms of a Geometric Sequence

40, 5,

5 5 , , ... 8 64

A) geometric; r = C) geometric; r =

1 8 5 8

B) not geometric D) geometric; r = 8

Answer: A Type: BI Var: 15 Objective: Identify Specific and General Terms of a Geometric Sequence

Page 21

3) -2,

2 2 ,- , 3 9 27

A) geometric, r =

1 3

C) geometric, r = 3

B) not geometric D) geometric; r = -

1 3

Answer: D Type: BI Var: 50+ Objective: Identify Specific and General Terms of a Geometric Sequence

4) 5, 5, 5 5, 25 A) geometric, r = 5 C) geometric, r = 5 + 5

B) not geometric D) geometric, r = 5

Answer: A Type: BI Var: 10 Objective: Identify Specific and General Terms of a Geometric Sequence

15 75 375 1,875 , , , t16 t4 t8 t12 15 A) geometric, r =

B) geometric, r =

t4 5

C) geometric, r = 4 t

1 t4

D) not geometric

Answer: C Type: BI Var: 48 Objective: Identify Specific and General Terms of a Geometric Sequence

6 9 12 15 6) 2 , 4 , 6 , 8 t t t t 6

A) geometric, r = 2 t

B) geometric, r = 2 t 1 D) geometric, r = t2

C) not geometric Answer: C

Type: BI Var: 48 Objective: Identify Specific and General Terms of a Geometric Sequence

Page 22

Write the first five terms of the geometric sequence. 7) Write the first five terms of the geometric sequence. a1 = -6, r = -2 A) -6, 12, -24, 48, -96 B) -2, 12, -72, 432, -2,592 C) -6, -8, -10, 9, -14 D) -2, -8, -14, -20, 10 Answer: A Type: BI Var: 46 Objective: Identify Specific and General Terms of a Geometric Sequence

Write the first four terms of a geometric sequence, {an}, based on the given information about the sequence. 8) a1 = -5, r = -4 A) -9, -13, -17, -21, ... B) -5, -9, -13, -17, ... C) 20, -80, 320, -1,280, ... D) -5, 20, -80, 320, ... Answer: D Type: BI Var: 50+ Objective: Identify Specific and General Terms of a Geometric Sequence

Write the first four terms of the geometric sequence. 1 9) a = - 1 and r = . 1 2 3 1 3 9 -27 A) - , - , - , ,... 2 2 2 2 1 3 9 C) - , , - , -27 ,... 2 2 2 2

1 B) -

,...

2 6 18 54 1 1 1 1 D) - , , - , ,... 2 6 18 54

Answer: D Type: BI Var: 10 Objective: Identify Specific and General Terms of a Geometric Sequence

10) a = 40, a = 1 a 1 n 5 n-1 8 8 A) 40, 8, , 5 25 C) 40,

201 202 203 , , 5 5 5

B) 40, 200, 1,000, 5,000 D) 8, 8 , 8 , 8 5 25 125

Answer: A Type: BI Var: 22 Objective: Identify Specific and General Terms of a Geometric Sequence

Page 23

Write a formula for the nth term of the geometric sequence. 11) 1,000, 200, 40, 8, ... 1 n-1 B) an = 1,000(-800)n A) an = 1,000 5 1n D) an = 1,000(-800)n-1 C) an = 1,000 5 Answer: A Type: BI Var: 32 Objective: Identify Specific and General Terms of a Geometric Sequence

12) 8, 4, 2, 1, ... A) an = 8

1n 2

B) an = 8

1 n-1 2

C) an = 8

1 n+1 2

D) an = 8 -

1 n-1 2

Answer: B Type: BI Var: 22 Objective: Identify Specific and General Terms of a Geometric Sequence

13)

64 5

, 8, 5,

25 8

1 n-1 A) an = 64 8

64 5 n B) an = 5 8

64 5 C) an = 5 8

n-1

D) an = 8

Answer: C Type: BI Var: 20 Objective: Identify Specific and General Terms of a Geometric Sequence

Solve the problem. 14) Doctors in a certain city report 30 confirmed cases of the flu to the health department. If the number of reported cases increases by roughly 35% each week thereafter, find the number of new cases reported 7 weeks after the initial report. Round to the nearest whole unit. A) 613 cases B) 833 cases C) 182 cases D) 245 cases Answer: D Type: BI Var: 50+ Objective: Identify Specific and General Terms of a Geometric Sequence

Page 24

15) A farmer depreciates a $100,000 tractor. He estimates that the resale value of the tractor n years after purchase is 83% of its value from the previous year. a. Write a formula for the nth term of a sequence that represents the resale value of the tractor n years after purchase. b. What will the resale value be after 7 years? Round to the nearest $1000. A) a. an = 100,000(83)n B) a. an = 100,000(0.83)n b. $33,000 b. $27,000 n-1 C) a. an = 100,000(0.83) D) a. an = 100,000(0.83)n-1 b. $33,000 b. $27,000 Answer: B Type: BI Var: 50+ Objective: Identify Specific and General Terms of a Geometric Sequence

Find the indicated term of a geometric sequence from the given information. 16) a1 = 3 and a4 = 81. Find a10. A) 177,147

B) 783

C) 705

D) 59,049

Answer: D Type: BI Var: 50+ Objective: Identify Specific and General Terms of a Geometric Sequence

17) an = -4(2)n-1, find a4. A) 2

B) 4

C) -32

D) -64

Answer: C Type: BI Var: 50+ Objective: Identify Specific and General Terms of a Geometric Sequence

18) a1 = 24 and a2 = -18. Find the sixth term. 729 B) -186 A) 128

C) -

729 128

243 32

Answer: C Type: BI Var: 28 Objective: Identify Specific and General Terms of a Geometric Sequence

19) a = 81

and r = - . Find a1. 16 4 64 A) B) 16 3

C) -12

Answer: B Type: BI Var: 28 Objective: Identify Specific and General Terms of a Geometric Sequence

Page 25

D) -16

Find a1 and and r for a geometric sequence {an} from the given information. 20) a2 = 18 and a7 = 4,374 A) a1 = 18 and r = 3 C) a1 = 18 and r = 12

B) a1 = 6 and r = 12 D) a1 = 6 and r = 3

Answer: D Type: BI Var: 38 Objective: Identify Specific and General Terms of a Geometric Sequence

21) a2 = -6 and a 6 = A) a = 1

243 128

and r = -

B) a = 8 and r = - 3 1 4 3 D) a1 = 8 and r = 4

4 4

C) a1 = 8 and r = -

Answer: B Type: BI Var: 28 Objective: Identify Specific and General Terms of a Geometric Sequence 2 Evaluate Finite and Infinite Geometric Series

Find the sum of the geometric series, if possible. 7 2 n-1 1) ∑ 6 3 n=1 A) 35

B) 28

C) 18

4118 243

Answer: D Type: BI Var: 50+ Objective: Evaluate Finite and Infinite Geometric Series

4 i-1 2) ∑ 7 5 i=1 432187 A) 15625

11836867 390625

Answer: C Type: BI Var: 50+ Objective: Evaluate Finite and Infinite Geometric Series

Page 26

2275623 78125

325089 78125

3) 24 - 12 + 6 - 3 + ... A) 48

B) 15

C) 16

D) Sum does not exist

Answer: C Type: BI Var: 7 Objective: Evaluate Finite and Infinite Geometric Series

4) 10 + 2 + A)

2 5

2 25

2 125

18 125

1562 125

18 155

16 155

Answer: B Type: BI Var: 9 Objective: Evaluate Finite and Infinite Geometric Series

5) 9 + 45 + 225 + 1,125 + 5,625 + . . . 9 A) 4 C) Sum does not exist

9 4

D) -

5 8

Answer: C Type: BI Var: 5 Objective: Evaluate Finite and Infinite Geometric Series

∞ 6)

∑ -2

n=1

A) -

B) 2

C) - 1

3 Type: BI Var: 36 Objective: Evaluate Finite and Infinite Geometric Series

∞

3 n-1 7) ∑ 4 n=1 B)

1 4

C) 3 Answer: D Type: BI Var: 7 Objective: Evaluate Finite and Infinite Geometric Series

Page 27

2 3

Answer: A

A) Sum does not exist

D) 4

∞

3 n-1 8) ∑ 2 n=1 1 A) 2

B) Sum does not exist D) -

C) -2

1 2

Answer: B Type: BI Var: 7 Objective: Evaluate Finite and Infinite Geometric Series

9) ∑ 5(2)n n=1 A) 2,550

B) 2,555

C) 1,022

D) 5,110

Answer: D Type: BI Var: 50 Objective: Evaluate Finite and Infinite Geometric Series

Write the repeating decimal as a fraction. 10) 0.6 6 66 A) B) 10 999

100

2 3

Answer: D Type: BI Var: 8 Objective: Evaluate Finite and Infinite Geometric Series

Write the given rational number as the quotient of two integers in simplest form. 11) 0.77 7 77 77 A) 9 B) C) 1000 100 Answer: A

77 999

2209 999

Type: BI Var: 50+ Objective: Evaluate Finite and Infinite Geometric Series

12) 2.231 A)

2231 1000

2209 1000

Answer: C Type: BI Var: 50+ Objective: Evaluate Finite and Infinite Geometric Series

Page 28

2209 990

Solve the problem. 13) An event in Daytona Beach brings an estimated 550,000 people to the town. Suppose that each person spends an average of $330. a. How much money is infused into the local economy during the event? b. If the money is respent in the community over and over again at a rate of 64%, determine the tota amount spent. Assume that the money is respent an infinite number of times. A) a. $181,500,000 B) a. $181,500,000 b. $2,880,952.38 b. $504,166,667 C) a. $18,150,000 D) a. $181,500,000 b. $50,416,667 b. $283,593,750 Answer: B Type: BI Var: 50+ Objective: Evaluate Finite and Infinite Geometric Series

14) Rafael received an inheritance of $27,000. He saves $13,700 and then spends $13,300 of the money on college tuition, books, and living expenses for school. If the money is respent over and over again in the community an infinite number of times, at a rate of 62%, determine the total amount spent. A) $43,548 B) $71,053 C) $21,452 D) $35,000 Answer: D Type: BI Var: 50+ Objective: Evaluate Finite and Infinite Geometric Series 3 Find the Value of an Annuity

Find the value of an ordinary annuity in which regular payments of P dollars are made at the end of each compounding period, n times per year, at an interest rate r for t years. 1) P = $250, n = 4, r = 4%, t = 35 yr A) $756.77 B) $10,415.07 C) $75,677.48 D) $100,677.48 Answer: C Type: BI Var: 50+ Objective: Find the Value of an Annuity

Page 29

Solve the problem. 2) Suppose that P dollars is invested at the end of each compounding period n times per year at interest rate r. Then the value A (in $) of the annuity after t years is given by r nt P 1+ -1 n A= . r n An employee invests $150 per month in an ordinary annuity. If the interest rate is 4%, find the value of the annuity after 25 yr. A) $97,695.55 B) $97,858.15 C) $76,121.46 D) $77,119.43 Answer: D Type: BI Var: 50+ Objective: Find the Value of an Annuity 4 Mixed Exercises

Solve the problem. 1) A ball drops from a height of 6 ft. With each bounce, the ball rebounds to

2 5

of its height. The total

vertical distanced traveled is given by 2 22 23 6 + (2)(6) + 2(6) + 2(6) ... 5 5 5 After the first term, the series is an infinite geometric series. Compute the total vertical distance traveled. A) The total vertical distance traveled is 14 ft. B) The total vertical distance traveled is 4 ft. C) The total vertical distance traveled is 10 ft. D) The total vertical distance traveled is 8 ft. Answer: A Type: BI Var: 45 Objective: Mixed Exercises

Page 30

2) The yearly salary for job A is $57,000 initially with an annual raise of $3,200 every year thereafter. The yearly salary for job B is $53,000 initially with an annual raise of 5%. a. Consider a sequence representing the salary for job A for year n. Is this an arithmetic or geometri sequence? Find the total earnings for job A over 20 years. Round to the nearest dollar. b. Consider a sequence representing the salary for job B for year n. Is this an arithmetic or geometri sequence? Find the total earnings for job B over 20 years. Round to the nearest dollar. A) a. Arithmetic; $1,748,000 b. Arithmetic; $1,643,500 C) a. Arithmetic; $1,748,000 b. Geometric; $1,752,496

B) a. Geometric; $2,011,827 b. Geometric; $1,752,496 D) a. Geometric; $2,011,827 b. Arithmetic; $1,643,500

Answer: C Type: BI Var: 50+ Objective: Mixed Exercises

3) The initial swing (one way) of a pendulum makes an arc of 20 in. Each swing (one way) thereafter makes an arc of 98% of the length of the previous swing. What is the total arc length that the pendulum travels? A) 1,400 in. B) 1,000 in. C) 1,100 in. D) 500 in. Answer: B Type: BI Var: 18 Objective: Mixed Exercises

4) Suppose that an individual is paid $0.02 on day 1 and every day thereafter, the payment is doubled. a. Write a formula for the nth term of a sequence that gives the payment (in $) on day n. b. How much will the individual earn on day 10? day 20? A) a. an = 0.02(2)n-1 b. $20.46 on day 10, $20,971.5 on day 20 B) a. an = 0.02(2)n b. b. $20.48 on day 10, $20,971.52 on day 20 C) a. an = 0.02(2)n-1 b. $10.24 on day 10, $10,485.76 on day 20 D) a. an = 0.02(2)n b. b. $40.94 on day 10, $41,943.02 on day 20 Answer: C Type: BI Var: 5 Objective: Mixed Exercises

Page 31

8.4 Mathematical Induction 0 Concept Connections

Provide the missing information. 1) Let Pn be a statement involving the positive integer n, and let k be an arbitrary positive integer. Proof by mathematical indicates that Pn is true for all positive integers n if 1. is true, and 2.The truth of Pk implies the truth of

Answer: induction; P1; Pk+1 Type: SA Var: 1 Objective: Concept Connections

2) The statement that Pk is true is called the

hypothesis.

Answer: inductive Type: SA Var: 1 Objective: Concept Connections 1 Prove a Statement Using Mathematical Induction

Use mathematical induction to prove the given statement for all positive integers n. 1) 3 + 7 + 11 + ... + (4n - 1) = n(2n + 1) Answer: Proofs will vary. Type: SA Var: 1 Objective: Prove a Statement Using Mathematical Induction

2) 3 + 5 + 7 + ... + (2n + 1) = n(n + 2) Answer: Proofs will vary. Type: SA Var: 1 Objective: Prove a Statement Using Mathematical Induction

3) 8 + 11 + 14 + ... + (3n + 5) =

n (3n + 13) 2

Answer: Proofs will vary. Type: SA Var: 1 Objective: Prove a Statement Using Mathematical Induction

4) 16 + 12 + 8 + ... + (-4k + 20) = -2n(n - 9) Answer: Proofs will vary. Type: SA Var: 1 Objective: Prove a Statement Using Mathematical Induction

5) 1 + 4 + 42 + 43 + ... + 4n-1 =

4n - 1 3

Answer: Proofs will vary. Type: SA Var: 1 Objective: Prove a Statement Using Mathematical Induction

Page 32

2 6)

2 27

+ ... +

=1-

Answer: Proofs will vary. Type: SA Var: 1 Objective: Prove a Statement Using Mathematical Induction

7) 1 · 4 + 2 · 7 + 3 · 10 + ... + n(3n + 1) = n(n + 1)2 Answer: Proofs will vary. Type: SA Var: 1 Objective: Prove a Statement Using Mathematical Induction

8) ∑ 2 = 2n i=1 Answer: Proofs will vary. Type: SA Var: 6 Objective: Prove a Statement Using Mathematical Induction

9) 1 -

1 3

1 1 1 2 1= ... 1 4 5 n+2 n+2

Answer: Proofs will vary. Type: SA Var: 1 Objective: Prove a Statement Using Mathematical Induction

10) ∑ 2i = n(n + 1) i=1 Answer: Proofs will vary. Type: SA Var: 1 Objective: Prove a Statement Using Mathematical Induction

11) ∑ 2i = i=1

n(n + 1)(2n + 1) 3

Answer: Proofs will vary. Type: SA Var: 1 Objective: Prove a Statement Using Mathematical Induction

Page 33

Solve the problem. 12) Suppose you wish to prove the statement that follows using mathematical induction. n 4 + 9 + 14 + ... + (5n - 1) = (5n + 3), for all positive integers n. 2 n Let Sn be the statement 4 + 9 + 14 + ... + (5n - 1) = (5n + 3). Show that S1 is true. 2 2 A) Since 4 + 9 = 13 = (5(2) + 3), S1 is true. 21 1 B) Since (5(1) + 3) = (8) = 4, S1 is true. 2 2 C) Since (5(1) - 1) = 4, S1 is true. 2 D) Since 4 +(5(1) - 1) = (4 + 5(1) - 1), S1 is true. 2 Answer: B Type: BI Var: 7 Objective: Prove a Statement Using Mathematical Induction

13) Suppose you wish to prove the statement that follows using the extended principle of mathematical induction. 2 is a factor of 5n - 3, for all natural numbers n. Assume that Sk is true for k ≥ 1 where Sk is the statement that 2 is a factor of 5k - 3, and write the statement Sk+1. A) Sk+1: 2 is a factor of 6k - 3. C) Sk+1: 2 is a factor of 5k+1 - 3.

B) Sk+1: 2 is a factor of 6k+1 - 3. D) Sk+1: 2 is a factor of 5k - 2.

Answer: C Type: BI Var: 30 Objective: Prove a Statement Using Mathematical Induction 2 Prove a Statement Using the Extended Principle of Mathematical Induction

Use trial-and-error to determine the smallest integer n for which the given statement is true. 1) (n + 1)! > 2n A) n = 2 B) n = 4 C) n = 3 D) n = 1 Answer: A Type: BI Var: 8 Objective: Prove a Statement Using the Extended Principle of Mathematical Induction

2) 2n < 2n A) 5

B) 2

C) 3

Answer: C Type: BI Var: 15 Objective: Prove a Statement Using the Extended Principle of Mathematical Induction

Page 34

D) 4

Solve the problem. 3) Suppose you wish to prove the statement that follows using the extended principle of mathematical induction. (n + 1)! > 3n, for positive integers n ≥ 4. Assume that Sk is true for k ≥ 4 where Sk is the statement (n + 1)! > 3n, and write the statement Sk+1. A) Sk+1: (k + 1)! > 4k C) Sk+1: (k + 2)! > 3k+1

B) Sk+1: (k + 1)! > 3k+1 D) Sk+1: (k + 2)! > 3k

Answer: C Type: BI Var: 8 Objective: Prove a Statement Using the Extended Principle of Mathematical Induction

Use mathematical induction to prove the given statement. 4) n! > 2n for postive integers n ≥ 4. Answer: Proofs will vary. Type: SA Var: 1 Objective: Prove a Statement Using the Extended Principle of Mathematical Induction

5) (n + 2)! > 7n for positive integers n ≥ 10. Answer: Proofs will vary. Type: SA Var: 1 Objective: Prove a Statement Using the Extended Principle of Mathematical Induction

6) 10n < 4n for positive integers n ≥ 3. Answer: Proofs will vary. Type: SA Var: 1 Objective: Prove a Statement Using the Extended Principle of Mathematical Induction 3 Mixed Exercises

Use mathematical induction to prove the given statement for all positive integers n and real numbers x and y. 1) (xy)n = xnyn Answer: Proofs will vary. Type: SA Var: 1 Objective: Mixed Exercises

n xn x = n provided that y ≠ 0. y y

Answer: Proofs will vary. Type: SA Var: 1 Objective: Mixed Exercises

Page 35

3) If x < 1, then xn > xn-1. Answer: Proofs will vary. Type: SA Var: 1 Objective: Mixed Exercises

4) If 0 < x < 1, then xn < xn-1. Answer: Proofs will vary. Type: SA Var: 1 Objective: Mixed Exercises

8.5 The Binomial Theorem 0 Concept Connections

Provide the missing information. 1) The expression a3 + 3a2b + 3ab2 + b3 is called the

expansion of (a + b)3.

Answer: binomial Type: SA Var: 1 Objective: Concept Connections

2) Consider (a + b)n where n is a whole number. How many terms are in the binomial expansion? Answer: n + 1 Type: SA Var: 1 Objective: Concept Connections

3) Consider (a + b)n where n is a whole number. What is the degree of each term in the expansion? Answer: n Type: SA Var: 1 Objective: Concept Connections

4) Consider (a + b)n where n is a whole number. The coefficients of the terms in the expansion can be n found by using triangle or by using . r Answer: Pascal’s Type: SA Var: 1 Objective: Concept Connections

5) For positive integers n and k (k ≤ n + 1), the kth term of (a + b)n is given by Answer:

n an-(k-1) bk-1 k-1

Type: SA Var: 1 Objective: Concept Connections

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a□b□.

6) Given (a + b)17 the 12th term is given by Answer:

a□b□.

a6b11

11 Type: SA Var: 1 Objective: Concept Connections 1 Determine Binomial Coefficients

Use Pascal's triangle to write the expansion of the the given expression. 1) (x - y)4. A) x4 + 4x3y - 6x2y2 + 4xy3 - y4 B) x4 - y4 C) x4 - 4x3y + 6x2y2 - 4xy3 + y4 D) x4 - xy + y4 Answer: C Type: BI Var: 6 Objective: Determine Binomial Coefficients

Evaluate the expression. 8 2) 3 A) 24

B) 11

C) 56

D) 336

Answer: C Type: BI Var: 28 Objective: Determine Binomial Coefficients 2 Apply the Binomial Theorem

Expand the binomial by using the binomial theorem. 1) (5 + y)4 A) 625 + 125y + 25y2 + 5y3 + y4 C) 625 + 500y + 150y2 + 20y3 + y4

B) 625 + 4y + 6y2 + 4y3 + y4 D) 625 + y4

Answer: C Type: BI Var: 4 Objective: Apply the Binomial Theorem

2) (5y + 1)4 A) 625y4 + 20y3 + 150y2 + 500y + 1 C) 625y4 + 500y3 + 150y2 + 20y + 1 Answer: C Type: BI Var: 14 Objective: Apply the Binomial Theorem

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B) 625y4 + 1 D) 625y4 + 500y3 + 200y2 + 20y + 1

3) (4p - 5q)5 A) 1,024p5 - 6,400p4q + 16,000p3q2 - 20,000p2q3 + 12,500pq4 - 3,125q5 B) 1,024p5 - 12,500p4q + 20,000p3q2 - 16,000p2q3 + 6,400pq4 - 3,125q5 C) 1,024p5 - 6,400p4q + 8,000p3q2 - 10,000p2q3 + 12,500pq4 - 3,125q5 D) 1,024p5 - 3,125q5 Answer: A Type: BI Var: 10 Objective: Apply the Binomial Theorem

4) x - 2y3 4 A) x4 - 2x3y + 4x2y2 - 8xy3 + 16y4 C) x4 - 8x3y3 + 24x2y6 - 32xy9 + 16y12

B) x4 + 2x3y3 + 4x2y6 + 8xy9 + 16y12 D) x4 - 8x3y + 24x2y2 - 32xy3 + 16y4

Answer: C Type: BI Var: 20 Objective: Apply the Binomial Theorem

5) (x5 + y)6 A) x6 + 6x5y + 15x4y2 + 20x3y3 + 15x2y4 + 6xy5 + y6 B) x11 + 6x10y + 15x9y2 + 20x8y3 + 15x7y4 + 6x5y5 + y6 C) x30 + 6x25y + 15x20y2 + 20x15y3 + 15x10y4 + 6x5y5 + y6 D) x30 + y6 Answer: C Type: BI Var: 4 Objective: Apply the Binomial Theorem

6) (3 - y3)5 A) [3 + (-y3)]5; 243 - 5y3 + 10y6 - 10y9 + 5y12 - y15 B) [3 + (-y3)]5; 243 - 405y + 270y2 - 90y3 + 15y5 - y6 C) [3 + (-y3)]5; 243 - 5y + 10y2 - 10y3 + 5y4 - y6 D) [3 + (-y3)]5; 243 - 405y3 + 270y6 - 90y9 + 15y12 - y15 Answer: D Type: BI Var: 12 Objective: Apply the Binomial Theorem

7) (0.2a + 0.5b)4 A) 0.0016a4 + 0.016a3b + 0.06a2b2 + 0.1ab3 + 0.0625b4 B) 0.16a4 + 0.4a3b + 1a2b2 + 2.5ab3 + 6.25b4 C) 0.0016a4 + 0.004a3b + 0.01a2b2 + 0.025ab3 + 0.0625b4 D) 0.16a4 + 1.6a3b + 6a2b2 + 10ab3 + 6.25b4 Answer: A Type: BI Var: 12 Objective: Apply the Binomial Theorem

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4 1 8) v - 4w 3

1 A) v4 - 4w4 3 1 256 3 32 2 2 16 3 B) v4 v w+ v w - vw + 256w2 81 3 3 27 1 4 v + 256w2 81 1 16 32 2 2 256 3 D) v4 - v3w + vw vw + 256w2 81 27 3 3 C)

Answer: D Type: BI Var: 10 Objective: Apply the Binomial Theorem

5 a -b 3

1 5 5 4 10 3 2 20 3 3 10 2 3 5 4 a - a b+ a b a b + a b - ab + b5 3 3 3 3 3 3 1 5 5 4 10 3 2 10 2 3 5 4 B) a - a b + a b a b - ab - b5 3 3 3 3 3 1 5 5 4 10 3 2 20 3 3 10 2 3 5 4 C) a a b+ ab a b + a b - ab + b5 243 81 27 27 9 3

1 5 5 4 10 3 2 10 2 3 5 ab4 - b5 a a b+ ab a b + 243 81 27 9 3

Answer: D Type: BI Var: 6 Objective: Apply the Binomial Theorem 3 Find a Specific Term in a Binomial Expansion

Find the indicated term of the binomial expansion. 1) (x - y)11; sixth term A) -462x6y5 B) 462x6y5

C) 462x5y6

D) -462x5y6

C) 90r2s8

D) 960r3s14

Answer: A Type: BI Var: 42 Objective: Find a Specific Term in a Binomial Expansion

2) (2r + s2)10; eighth term A) 240r3s7

B) 180r2s16

Answer: D Type: BI Var: 20 Objective: Find a Specific Term in a Binomial Expansion

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3) x3 - 2y2 13; eighth term A) -219,648x18y14

B) -329,472x15y16

C) -3,432x18y14

D) -2,574x15y16

C) -2,912x33y6

D) -728x33y6

C) 13,608x13y14

D) 2,268x13y14

Answer: A Type: BI Var: 50+ Objective: Find a Specific Term in a Binomial Expansion

4) x3 - 2y2 14; Find the term containing x33. A) -2,002x30y8

B) -16,016x30y8

Answer: C Type: BI Var: 50+ Objective: Find a Specific Term in a Binomial Expansion 3

5 9

5) ( 3x + y ) ; fourth term A) 13,608x12y15 B) 2,268x12y15 Answer: B Type: BI Var: 16 Objective: Find a Specific Term in a Binomial Expansion 4 Mixed Exercises

Expand the binomial by using the binomial theorem. 1) (ex - e-x)4 4 1 A) e4x + 4e2x + 6 + + e2x e4x C) e4x + 4e3x + 6e2x + 4ex + 1

B) e4x - 4e2x + 6 -

4 e2x

1 e4x

D) e4x - 4e3x + 6e2x - 4ex + 1

Answer: B Type: BI Var: 1 Objective: Mixed Exercises

2) (x + y - 9)3 A) x3 + y3 - 729 B) x2 + 2xy - 18x + y2 - 18y + 81 C) x3 + 3x2y + 27x2 + 3xy2 + 54xy - 243x + y3 + 27y2 - 243y + 729 D) x3 + 3x2y - 27x2 + 3xy2 - 54xy + 243x + y3 - 27y2 + 243y - 729 Answer: D Type: BI Var: 18 Objective: Mixed Exercises

Use the Binomial Theorem to find the value of the number raised to the given power. 3) (1.1)3 Hint: Write the expression as (1 + 0.1)3. A) 3.3 B) 1.331 C) 1.21 D) 1.001 Answer: B Type: BI Var: 4 Objective: Mixed Exercises

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Simplify the difference quotient:

f (x + h) - f (x) h

4) 6x3 + 6 A) x3 + 18x2 + 18xh + 6h2 C) 18x2 + 18xh + 6h2

B) 18x2 + 18xh + 6h2 + 6 D) 18hx2 + 18xh2 + 6h3

Answer: C Type: BI Var: 45 Objective: Mixed Exercises

5) x4 - 3x2 - 1 A) 4x3h + 6x2h2 + 4xh3 + h4 - 6x2h + h2 C) 4x3 + 6x2h + 4xh2 + h3 - 6x2 + h - 1

B) 4x3 + 6x2h + 4xh2 + h3 - 6x - 3h D) x4 + 4x3 + 6x2h + 4xh2 + h3 - 6x2 + h

Answer: B Type: BI Var: 50+ Objective: Mixed Exercises

Use the Binomial Theorem to find the value of the complex number raised to the given power. Recall that i = -1. 6) (-5 + 3i)4 A) -644 - 960i B) 644 + 960i C) -644 + 960i D) 644 - 960i Answer: A Type: BI Var: 48 Objective: Mixed Exercises 5 Expanding Your Skills

Stirling’s formula (named after Scottish mathematician, James Stirling: 1692–1770) is used to nn approximate large values of n!. Stirling’s formula is n! ≈ 2πn . e a. Use Stirling’s formula to approximate the given expression. Round to the nearest whole unit. b. Compute the actual value of the expression. 1) 8! A) a. 40,111 B) a. 40,111 C) a. 39,902 D) a. 39,902 b. 40,320 b. 40,312 b. 40,320 b. 40,312 Answer: C Type: BI Var: 5 Objective: Expanding Your Skills

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Using calculus, we can show that (1 + x)n = 1 + nx +

n(n - 1) 2 n(n - 1)(n - 2) 3 x + x + ..., for |x| < 1. This 2! 3!

formula can be used to evaluate binomial expressions raised to noninteger exponents. Use the first four terms of this infinite series to approximate the given expression. Round to 3 decimal places if necessary. 2) (1.3)4/3 A) 1.423 B) 1.422 C) 1.419 D) 1.416 Answer: C Type: BI Var: 50+ Objective: Expanding Your Skills

8.6 Principles of Counting 0 Concept Connections

Provide the missing information. 1) The Fundamental of indicates that if one event can occur in m different ways, and a second event can occur in n different ways, then the two events can occur in sequence in different ways. Answer: Principle; Counting, m · n Type: SA Var: 1 Objective: Concept Connections

2) A

diagram can be used to illustrate the possible outcomes in a sequence of events.

Answer: tree Type: SA Var: 1 Objective: Concept Connections

3) If n items are arranged in order, then each arrangement is called a

of n items.

Answer: permutation Type: SA Var: 1 Objective: Concept Connections

4) The number of ways that n distinguishable items can be arranged in various order is

Answer: n Type: SA Var: 1 Objective: Concept Connections

5) Consider a set of n elements of which one element is repeated r1 times. Then the number of □ permutations of the elements of the set is given by . □ n! Answer: r1 ! Type: SA Var: 1 Objective: Concept Connections

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6) Suppose that n represents the number of distinct elements in a group from which r elements will be chosen in a particular order. Then each arrangement is called a of n items taken at a time. Answer: permutation; r Type: SA Var: 1 Objective: Concept Connections

7) The number of permutations of n elements taken r at a time is denoted by nPr and is computed by □ nPr = □ or nPr = n(n - 1)(n - 2)...(n - r + 1) Answer:

n! (n - r)!

Type: SA Var: 1 Objective: Concept Connections

8) Suppose that n represents the number of elements in a group from which r elements will be selected in no particular order. Then each group selected is called a of n elements taken at a time. Answer: combination; r Type: SA Var: 1 Objective: Concept Connections

9) The number of combinations of n elements taken r at a time is denoted by nCr and is computed by . nPr n! or Answer: r! · (n - r)! r! Type: SA Var: 1 Objective: Concept Connections 1 Apply the Fundamental Principle of Counting

Solve the problem. 1) Consider the set of integers from 1 to 23, inclusive. If one number is selected, in how many ways can we obtain a number that is a multiple of 10? A) 0 B) 3 C) 2 D) 13 Answer: C Type: BI Var: 50+ Objective: Apply the Fundamental Principle of Counting

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2) Consider the set of integers from 1 to 22, inclusive. If one number is selected, in how many ways can we obtain a number that is divisible by 7? A) 3 B) 0 C) 1 D) 15 Answer: A Type: BI Var: 50+ Objective: Apply the Fundamental Principle of Counting

3) A school cafeteria has 5 choices for the main dish, 3 choices of vegetable, 4 choices of beverage, and 3 choices of dessert. How many different meals can be formed if a student chooses one item from each category? A) 156 B) 15 C) 59 D) 180 Answer: D Type: BI Var: 50+ Objective: Apply the Fundamental Principle of Counting

4) Alex has 6 shirts, 7 pairs of pants, and 3 pairs of shoes. How many outfits can he wear? A) 16 outfits B) 5,766 outfits C) 126 outfits D) 3 outfits Answer: C Type: BI Var: 50+ Objective: Apply the Fundamental Principle of Counting

5) A teacher is going to assign each student a 3-digit code using the digits 0 through 4. How many codes are possible if no repetitions are allowed? A) 24 B) 60 C) 64 D) 125 Answer: B Type: BI Var: 6 Objective: Apply the Fundamental Principle of Counting

6) A quiz consists of 5 multiple-choice questions, each with five possible responses, and 5 true/false questions. In how many ways can a student answer the questions on the test? A) 3,157 B) 100,000 C) 3,150 D) 78,125 Answer: B Type: BI Var: 50+ Objective: Apply the Fundamental Principle of Counting

7) In how many ways can 5 students stack their homework assignments? A) 500 B) 32 C) 120

D) 15

Answer: C Type: BI Var: 7 Objective: Apply the Fundamental Principle of Counting

8) A license plate has 2 letters followed by 4 digits. How many license plates can be made if there are no restrictions on the letters or digits. A) 3,407,040 B) 6,500,000 C) 6,760,000 D) 3,276,000 Answer: C Type: BI Var: 4 Objective: Apply the Fundamental Principle of Counting

Page 44

9) A license plate has 2 letters followed by 4 digits. How many license plates can be made if there no digit or letter may be replaced. A) 6,500,000 B) 3,407,040 C) 3,276,000 D) 6,760,000 Answer: C Type: BI Var: 4 Objective: Apply the Fundamental Principle of Counting 2 Count Permutations and Combinations

Evaluate. 1) 10P2 A) 1,814,400

B) 45

C) 90

D) 20

C) 990

D) 165

Answer: C Type: BI Var: 43 Objective: Count Permutations and Combinations

2) 11C8 A) 6,652,800

B) 88

Answer: D Type: BI Var: 44 Objective: Count Permutations and Combinations

3) a. In how many ways can the letters in the word TEXAS be arranged? b. In how many ways can the letters in the word MISSISSIPPI be arranged? A) a. 120 B) a. 60 C) a. 60 b. 34,650 b. 34,650 b. 39,916,800

D) a. 120 b. 39,916,800

Answer: A Type: BI Var: 35 Objective: Count Permutations and Combinations

Solve the problem. 4) Suppose a contest has 11 participants. In how many different ways can first through fifth place be awarded? A) 161,051 B) 55,440 C) 462 D) 39,916,680 Answer: B Type: BI Var: 18 Objective: Count Permutations and Combinations

5) A club must elect a president, vice-president, and treasurer from its 13 members. In how many ways can the positions be filled? A) 286 B) 1,716 C) 2,197 D) 39 Answer: B Type: BI Var: 8 Objective: Count Permutations and Combinations

Page 45

6) Suppose a tournament has 15 participants. In how many different ways can the 15 players be paired to play in the first round of the tournament? Assume that each player can play any other player without regard to seeding. A) 105 B) 210 C) 225 D) 30 Answer: A Type: BI Var: 20 Objective: Count Permutations and Combinations

7) In a state lottery, a player wins the grand prize by choosing the same group of 4 numbers from 1 through 45 as is chosen by the computer. How many 4-number groups are possible? A) 1,080 B) 4,100,625 C) 3,575,880 D) 148,995 Answer: D Type: BI Var: 31 Objective: Count Permutations and Combinations

8) A committee of 4 men and 3 women is to be made from a group of 12 men and 8 women. In how many ways can such a committee be formed? A) 12,216 B) 3,991,680 C) 551 D) 27,720 Answer: D Type: BI Var: 50+ Objective: Count Permutations and Combinations

9) Given {A, B, C, D}, list all of the permutations of three elements from the set. A) ABC, ACB, BAC, BCA, CAB, CBA, ABD, ADB, BAD, BDA, DAB, DBA, ACD, ADC, CAD, CDA, DAC, DCA, BCD, BDC, CBD, CDB, DBC, DCB B) ABC, ABD, ACD, BCD, CBA, DBA, DCA, DCB C) ABC, BAC, CAB, ABD, BAD, DAB, ACD, CAD, DAC, BCD, CBD, DBC D) ABC, ABD, ACD, BCD Answer: A Type: BI Var: 3 Objective: Count Permutations and Combinations

10) Given {A, B, C}, list all of the combinations of two elements from the set. A) AB, AC, BC, BA, CA B) AB, AC, BC C) AB, BC, CA, BA D) AB, BA, AC, CA, BC, CB Answer: B Type: BI Var: 3 Objective: Count Permutations and Combinations

11) In a drama class, 5 students are to be selected from 19 students to perform a synchronized dance. In how many ways can 19 students be selected from the 5? A) 120 B) 1,395,360 C) 2,476,099 D) 11,628 Answer: D Type: BI Var: 14 Objective: Count Permutations and Combinations

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12) Determine the number of ways that 5 students can be selected from the class of 24 to play 5 different roles in a short play. A) 120 B) 15,504 C) 3,200,000 D) 1,860,480 Answer: D Type: BI Var: 14 Objective: Count Permutations and Combinations 3 Mixed Exercises

Solve the problem. 1) Twenty batteries are in a drawer. There are 5 dead batteries among the 20. If five batteries are selected at random, determine the number of ways in which 4 good batteries and 1 dead battery can be selected. A) 6,825 B) 32,765 C) 163,800 D) 1,370 Answer: A Type: BI Var: 50+ Objective: Mixed Exercises

2) In a “Pick-5” game, a player wins a prize for matching a 5-digit number from 0000 to 99,999 with the number randomly selected during the drawing. How many 5-digit numbers can a player choose? Assume that a number can start with a zero or zeros such as 00001. A) 30,240 B) 100,000 C) 10,000 D) 252 Answer: B Type: BI Var: 4 Objective: Mixed Exercises

3) Liza is a basketball coach and must select 5 players out of 17 players to start a game. In how many ways can she select the 5 players if each player is equally qualified to play each position? A) 720 B) 1,419,857 C) 742,560 D) 6,188 Answer: D Type: BI Var: 9 Objective: Mixed Exercises

4) In how many ways can a manager assign 7 employees at a coffee shop to 7 different tasks? Assume that each employee is assigned to exactly one task. A) 5,040 B) 49 C) 128 D) 823,543 Answer: A Type: BI Var: 5 Objective: Mixed Exercises

Page 47

4 Expand Your Skills

Solve the problem. 1) Three biology books, 4 math books, and 2 physics books are to be placed on a book shelf where the books in each discipline are grouped together. In how many ways can the books be arranged on the book shelf? A) 24 B) 1,728 C) 72 D) 288 Answer: B Type: BI Var: 1 Objective: Expand Your Skills

8.7 Introduction to Probability 0 Concept Connections

Provide the missing information. 1) An is a test with an uncertain outcome. Answer: experiment Type: SA Var: 1 Objective: Concept Connections

2) The

of an experiment is the set of all possible outcomes.

Answer: sample space Type: SA Var: 1 Objective: Concept Connections

3) An

is a subset of the sample space of an experiment.

Answer: event Type: SA Var: 1 Objective: Concept Connections

4) The number of elements in event E is often denoted by

Answer: n(E) Type: SA Var: 1 Objective: Concept Connections

5) If S is a sample space with equally likely outcomes and E is an event within the sample space, then □ P(E) is computed by the formula P(E) = . □ n(E) Answer: n(S) Type: SA Var: 1 Objective: Concept Connections

6) If P(E) = 0, then E is called an Answer: impossible; certain Type: SA Var: 1 Objective: Concept Connections

Page 48

event. If P(E) = 1, then E is called a

event.

7) The notation E represents the

of event E. Furthermore, P(E) + P(E) =

Answer: complement; 1 Type: SA Var: 1 Objective: Concept Connections

8) Two events in a sample space are elements. That is, the two events do not overlap.

if they do not share any common

Answer: mutually exclusive Type: SA Var: 1 Objective: Concept Connections

9) If two events A and B are not mutually exclusive, then P(A ∪ B) can be computed by the formula P(A ∪ B) = . Answer: P(A) + P(B) - P(A ∩ B) Type: SA Var: 1 Objective: Concept Connections

10) If two events A and B are mutually exclusive, then P(A ∩ B) = computed from the formula P(A ∪ B) =

. As a result, P(A ∪ B) can be

Answer: 0; P(A) + P(B) Type: SA Var: 1 Objective: Concept Connections

11) Two events A and B are

if neither event affects the probability of the other.

Answer: independent Type: SA Var: 1 Objective: Concept Connections

12) For two independent events A and B, P(A and B) =

Answer: P(A)· P(B) Type: SA Var: 1 Objective: Concept Connections 1 Determine Theoretical Probabilities

Determine which of the following can represent the probability of an event? 4 1) 104%, 0.04, 0.46, 5 4 A) 0.04, 0.46 B) 104%, 0.46 C) 104%, 5 Answer: A Type: BI Var: 50+ Objective: Determine Theoretical Probabilities

Page 49

D) 0.04, -

4 5

Solve the problem. 2) Consider an experiment where a die is rolled that has 16 sides. The outcomes are the numbers 1 to 16. Determine the probability of the given event. A number between 3 and 8, inclusive, is rolled. 5 A) 16

3 B) 8

3 C) 16

1 D) 4

Answer: B Type: BI Var: 50+ Objective: Determine Theoretical Probabilities

3) Consider an experiment where a die is rolled that has 15 sides. The outcomes are the numbers 1 to 15. Determine the probability of the given event. A number less than 4 is rolled. A)

4 B) 5

15 4

1 C) 5

4 D) 15

Answer: C Type: BI Var: 18 Objective: Determine Theoretical Probabilities

4) A course in early civilization has 9 freshmen, 8 sophomores, and 12 juniors. If one student is selected at random, find the probability of the following events. a. A junior is selected. b. A freshman is selected. c. A senior is selected. 12 A) a. 29 b.

9 29

c. 0

B) a.

C) a.

D) a.

8 29

9 29

12 29

8 29

9 29

12 29

c. 0

Answer: A Type: BI Var: 50+ Objective: Determine Theoretical Probabilities

5) If P(E) = 0.552, what is the value of P(E)? A) 0.224 B) 0.552 Answer: D Type: BI Var: 50+ Objective: Determine Theoretical Probabilities

Page 50

C) 1.812

D) 0.448

6) A baseball player with a batting average of 0.304 has a probability of 0.304 of getting a hit for a given time at bat. What is the probability that the player will not get a hit for a given time at bat? A) 0.696 B) 6.96% C) 3.04% D) 0.304 Answer: A Type: BI Var: 50+ Objective: Determine Theoretical Probabilities

7) Consider the sample space when two fair dice are rolled. Determine the probability for the followin event. The sum of the numbers on the dice is not 2. A) 1 17 1 35 B) C) D) 36 18 18 36 Answer: D Type: BI Var: 22 Objective: Determine Theoretical Probabilities

8) Consider the sample space when two fair dice are rolled. Determine the probability for the followin event. The sum of the numbers on the dice is less than 8. 7 5 5 13 B) 12 C) 18 D) 36 A) 18 Answer: B Type: BI Var: 18 Objective: Determine Theoretical Probabilities

9) Suppose that a student group consists of 10 females and 8 males. What is the probability that a committee of 4 people chosen at random will consist of only females? A) 0.05 B) 0.222 C) 0.4 D) 0.069 Answer: D Type: BI Var: 50+ Objective: Determine Theoretical Probabilities

10) An American roulette wheel has 38 slots, numbered 1 through 36, 0, and 00. Eighteen slots are red, 18 are black, and 2 are green. The dealer spins the wheel in one direction and rolls a small ball in the opposite direction until both come to rest. The ball is equally likely to fall in any one of the 38 slots. For a given spin of the wheel, find the probability of the following events. a. The ball lands on a number that is a multiple of 5 (do not include 0 and 00) b. The ball does not land on the number 4. 7 37 7 37 7 35 7 35 A) a. ; b. B) a. ; b. C) a. ; b. D) a. ; b. 36 38 38 38 38 36 36 36 Answer: B Type: BI Var: 50+ Objective: Determine Theoretical Probabilities

Page 51

11) Suppose that a jury pool consists of 19 women and 20 men. a. What is the probability that a jury of 7 people taken at random from the pool will consist only of women? b. What is the probability that the jury will consist only of men? A) a. ≈ 0.51282; b. ≈ 0.01842 B) a. ≈ 0.51282; b. ≈ 0.00504 C) a. ≈ 0.01842; b. ≈ 0.01842 D) a. ≈ 0.00328; b. ≈ 0.00504 Answer: D Type: BI Var: 50+ Objective: Determine Theoretical Probabilities 2 Determine Empirical Probabilities

Solve the problem. 1) A certain city has approximately 2.75 million people. A census indicated that 450,000 people in the city were over the age of 60. If a person is selected at random from the city, what is the probability that the person is over 60 years old? Round to 3 decimal places. A) 8.36 B) 0.836 C) 0.164 D) 1.64 Answer: C Type: BI Var: 50+ Objective: Determine Empirical Probabilities

Page 52

2) The final exam grades for a sample of students in a physical science class resulted in the following grade distribution. F (6)

A (9)

D (7)

B (15) C (18) If one student taking physical science is selected at random, find the probability of the following events. a. The student earned an "A". b. The student earned an "F". 3 9 46 9 A) a. B) a. C) a. D) a. 11 100 55 55 b.

6 55

3 50

49 55

6 55

Answer: D Type: BI Var: 50+ Objective: Determine Empirical Probabilities

3) At a hospital specializing in treating heart disease, it was found that 224 out of 4,612 patients undergoing open heart mitral valve surgery died during surgery or within 30 days after surgery. Determine the probability that a patient will not survive the surgery or 30 days after the surgery. This is called the mortality rate. Round to 3 decimal places. A) 0.055 B) 0.134 C) 0.049 D) 0.042 Answer: C Type: BI Var: 50+ Objective: Determine Empirical Probabilities 3 Find the Probability of the Union of Two Events

Solve the problem. 1) Consider the sample space for a single card drawn from a standard deck. Find the probability that the card drawn is a card numbered between 2 and 10, inclusive or a heart. 1 10 11 17 A) B) C) D) 4 13 13 52 Answer: B Type: BI Var: 50+ Objective: Find the Probability of the Union of Two Events

Page 53

2) Consider the sample space for a single card drawn from a standard deck. Find the probability that the card drawn is a 6 or a queen. 7 1 2 1 A) 52 B) 26 C) 13 D) 52 Answer: C Type: BI Var: 32 Objective: Find the Probability of the Union of Two Events

3) Consider the sample space for a single card drawn from a standard deck. Find the probability that the card drawn is a face card or a black card. 1 11 23 8 A) B) C) D) 4 26 52 13 Answer: D Type: BI Var: 6 Objective: Find the Probability of the Union of Two Events

Use the data in the table categorizing cholesterol levels by the ages of the individuals in a study. If one person from the study is chosen at random, find the probability of the given event. 4) The person has elevated cholesterol.

30 and under 31-60 61 or older Total A)

87 113

Normal Cholesterol

Elevated Cholesterol

Total

11 52 24 87

5 28 80 113

16 80 104 200

87 200

Answer: C Type: BI Var: 50+ Objective: Find the Probability of the Union of Two Events

Page 54

113 200

26 113

5) The person is 60 or under.

30 and under 31-60 61 or older Total A)

Normal Cholesterol

Elevated Cholesterol

Total

17 49 22 88

4 32 76 112

21 81 98 200

21 200

51 100

81 200

49 100

41 50

9 50

Answer: B Type: BI Var: 50+ Objective: Find the Probability of the Union of Two Events

6) The person has normal cholesterol or is 61 or older.

30 and under 31-60 61 or older Total A)

37 100

Normal Cholesterol

Elevated Cholesterol

Total

11 45 18 74

5 31 90 126

16 76 108 200

91 100

Answer: C Type: BI Var: 50+ Objective: Find the Probability of the Union of Two Events

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7) The person is between 31 and 60, inclusive, or elevated cholesterol.

30 and under 31-60 61 or older Total

163 200

Normal Cholesterol

Elevated Cholesterol

Total

15 51 19 85

3 30 82 115

18 81 101 200

21 50

17 100

83 100

Answer: D Type: BI Var: 50+ Objective: Find the Probability of the Union of Two Events 4 Find the Probability of Sequential Independent Events

Solve the problem. 1) A basketball player makes approximately 69% of free throws. If she plays in a game in which she shoots 7 free throws, what is the probability the she will make all 7? A) 0.0745 B) 0.69 C) 0.0443 D) 0.0986 Answer: A Type: BI Var: 50+ Objective: Find the Probability of Sequential Independent Events

2) Suppose a die is rolled followed by the flip of a coin. Find the probability that the outcome is a 3 on the die followed by the coin turning up heads. 1 1 2 1 A) 2 B) 4 C) 3 D) 12 Answer: D Type: BI Var: 12 Objective: Find the Probability of Sequential Independent Events

3) A test has 11 questions. Seven questions are true/false and four are multiple-choice. Each multiple-choice question has 3 possible responses of which exactly one is correct. Find the probability that a student guesses on each question and gets a perfect score. 1 1 A) ≈ 0.000319 B) ≈ 0.000096 3,136 10,368 209 1 C) ≈ 0.020158 D) ≈ 0.005952 10368 168 Answer: B Type: BI Var: 45 Objective: Find the Probability of Sequential Independent Events

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4) The 5-yr survival rate for a type of cancer is 84%. If two people with this type of cancer are selected at random, what is the probability that they both survive 5 yr? A) 0.7056 B) 0.6028 C) 0.5 D) 0.84 Answer: A Type: BI Var: 39 Objective: Find the Probability of Sequential Independent Events

5) A traffic light at an intersection has a 120-sec cycle. The light is green for 80 sec, yellow for 5 sec, and red for 35 sec. a. When a motorist approaches the intersection, find the probability that the light will be red. (Assume that the color of the light is defined as the color when the car is 100 ft from the intersectio This is the approximate distance at which the driver makes a decision to stop or go.) b. If a motorist approaches the intersection twice during the day, find the probability that the light will be red both times. 7 7 1 1 7 49 1 1 A) a. ; b. B) a. ; b. C) a. ; b. D) a. ; b. 24 48 3 9 24 576 3 6 Answer: C Type: BI Var: 1 Objective: Find the Probability of Sequential Independent Events

6) A slot machine in a casino has four wheels that all spin independently. Each wheel has 11 stops, denoted by 0 through 9, and bar. What is the probability that a given outcome is bar-bar-bar? 1 1 1 1 A) 10,000 B) 14,641 C) 44 D) 6,561 Answer: B Type: BI Var: 10 Objective: Find the Probability of Sequential Independent Events

7) Airlines often overbook flights because a small percentage of passengers do not show up (perhaps due to missed connections). Past history indicates that for a certain route, the probability that an individual passenger will not show up is 0.03. Suppose that 66 people bought tickets for a flight that has 65 seats. Determine the probability that there will not be enough seats. Round to 4 decimal places. A) 0.1795 B) 0.1741 C) 0.1381 D) 0.1339 Answer: D Type: BI Var: 30 Objective: Find the Probability of Sequential Independent Events

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