Algebra 1 • Lesson 5.1* (Domain and Range) Day 51 Notes Before we begin exploring domain and range, we need a working definition for relation. A relation is a ! ! !
!
!
!
!
of
Q2. The domain of a relation is the set of all !! values (i.e., ! ! ! values).
.
Discovering Domain • Graphically (Q1-Q3): Q1. The domain of the first relation is {1, 3, 4, 9 } . Find the
Q3. The domain of the first relation is { !3,!2,!1 } . Find
domain of the second and third relations.
the domain of the second and third relations.
Page 1 of 4!
Updated: October 31, 2009
Algebra 1 • Lesson 5.1* (Domain and Range) Day 51 Notes Discovering Range • Graphically (Q4-Q6): Q4. The range of the first relation is
{!5, 0, 2, 4 } . Find the
range of the second and third relations.
Q5. The range of the first relation is
{ 2, 4, 6 } . Find the
range of the second and third relations.
Q6. The range of a relation is the set of all ! ! values (i.e., ! ! ! values). Page 2 of 4!
Updated: October 31, 2009
Algebra 1 • Lesson 5.1* (Domain and Range) Day 51 Notes Discovering Domain • Numerically (Q7-Q8): Q7. The domain of the first relation is
{ 2, 3, 4, 5 } . Find the
domain of the second, third, and fourth relations.
Q9. The range of the first relation is { !7, 0, 2, 3 } . Find the range of the second, third, and fourth relations.
x
y
x
y
2
–7
2
–7
3
3
3
3
4
0
4
0
5
2
5
2
x
y
x
y
6
4
6
4
11
3
11
3
16
2
16
2
21
1
21
1
x
y
x
y
1.3
–1.3
1.3
–1.3
1.7
–1.7
1.7
–1.7
2.1
–2.1
2.1
–2.1
2.5
–2.5
2.5
–2.5
x
y
x
y
1 2
2
1 2
2
3 4
4 3
3 4
4 3
7 8
8 7
7 8
8 7
15 16
16 15
15 16
16 15
Q8. The domain of a relation is the set of all !! values (i.e., ! ! ! values).
Page 3 of 4!
Discovering Range • Numerically (Q9-Q10):
Q10. The range of a relation is the set of all !! values (i.e., ! ! ! values).
Updated: October 31, 2009
Algebra 1 • Lesson 5.1* (Domain and Range) Day 51 Notes Domain and Range • Graphically (Q11-Q14): Q11. Find the domain and range of the relation below.
Q13. Find the domain and range of the relation below.
Domain: !
!
!
!
!
Domain: !
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
Range: ! !
!
!
!
!
Range: ! !
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
We can also say…
We can also say…
Domain: !
!
!
!
!
Domain: !
!
!
!
!
Range: ! !
!
!
!
!
Range: ! !
!
!
!
!
Q12. Find the domain and range of the relation below.
Q14. Find the domain and range of the relation below.
Domain: !
!
!
!
!
Domain: !
!
!
!
!
Range: ! !
!
!
!
!
Range: ! !
!
!
!
!
We can also say…
We can also say…
Domain: !
!
!
!
!
Domain: !
!
!
!
!
Range: ! !
!
!
!
!
Range: ! !
!
!
!
!
Page 4 of 4!
Updated: October 31, 2009
Algebra 1 • Lesson 5.2* (Relations and Functions • Introduction) Day 52 Notes A function is a rule or relationship where there is exactly one output value for each input value. In other words, a relation (graph, table, formula, etc.) is a function if each input gives exactly one output.
Q5. Does the relationship ( social security number, person ) represent a function? If not, provide a counterexample (that is, an example of one input that has two or more outputs).
If one input produces more than one output, the relation(ship) is not a function. These ideas take time to sink in, and are best explored using all four representations (verbal, graphical, numerical, and algebraic). Let!s get started! Functions • Verbally (Q1-Q9): Q1. Does the relationship (city, ZIP code ) represent a function? If not, provide a counterexample (that is, an example of one input that has two or more outputs).
Q2. Does the relationship
( person,
birth date ) represent a
Q6. Does the relationship ( person, passport number ) represent a function? If not, provide a counterexample (that is, an example of one input that has two or more outputs).
Q7. Does the relationship ( apartment building, tenant )
function? If not, provide a counterexample (that is, an example of one input that has two or more outputs).
represent a function? If not, provide a counterexample (that is, an example of one input that has two or more outputs).
Q3. Does the relationship ( last name, first name )
Q8. Does the relationship ( wedding ring, finger ) represent
represent a function? If not, provide a counterexample (that is, an example of one input that has two or more outputs).
a function? If not, provide a counterexample (that is, an example of one input that has two or more outputs).
Q4. Does the relationship ( state, capital ) represent a
Q9. Does the relationship ( game, final score ) represent a
function? If not, provide a counterexample (that is, an example of one input that has two or more outputs).
function? If not, provide a counterexample (that is, an example of one input that has two or more outputs).
Page 1 of 2"
Updated: November 1, 2009
Algebra 1 • Lesson 5.2* (Relations and Functions • Introduction) Day 52 Notes Functions • Graphically (Q10-Q11):
Functions • Numerically (Q12-Q14):
Q10. Does the relation shown below represent a function? Explain why or why not.
Q12. Does the relation shown below represent a function? Explain why or why not.
x
y
2
–7
3
3
4
0
5
2
Q13. Does the relation shown below represent a function? Explain why or why not.
Q11. Does the relation shown below represent a function? Explain why or why not.
Page 2 of 2!
x
y
6
4
6
3
11
2
11
1
Q14. Does the relation shown below represent a function? Explain why or why not.
x
y
1.1
–1.3
1.2
–1.3
1.3
–1.3
1.4
–1.3
Updated: November 1, 2009
Algebra 1 • Lesson 5.3* (Identifying Functions Quickly) Day 53 Notes Our goal today is to reinforce the concept of a function, and to develop ways to identify them quickly.
Q3. Does the relation shown below represent a function? Explain why or why not.
Functions • Graphically (Q1-Q5): Q1. Does the relation shown below represent a function? Explain why or why not.
Q4. Does the relation shown below represent a function? Explain why or why not.
Q11. Does the relation shown below represent a function? Explain why or why not.
Q5. How can you tell quickly which relations above are functions? Can you create a visual test?
Page 1 of 2!
Updated: November 1, 2009
Algebra 1 • Lesson 5.3* (Identifying Functions Quickly) Day 53 Notes Functions • Numerically (Q6-Q10): Q6. Does the relation shown below represent a function?
x
y
1
2
2
2
3
3
4
3
Q7. Does the relation shown below represent a function?
x
y
2
6
2
5
6
4
6
3
Q8. Does the relation shown below represent a function?
x
y
1
2
2
4
4
6
4
8
Q9. Does the relation shown below represent a function?
x
y
0
3
0
3
1
4
1
4
Q10. How can you tell quickly which relations above are functions? Can you create a quick “input-focused” test?
Page 2 of 2!
Updated: November 1, 2009
Algebra 1 • Lesson 5.4* (Squares and Square Roots) Day 54 Notes Squares (Q1-Q8):
2
Q1. The square of a number n is n ! n , which can also be
Q5. True or false: ( !a ) = a 2 .
written as n 2 . Find the square of each number in the table without using a calculator. number ( n )
( )
square n 2
Q6. True or false: !4 2 = !16 .
1 2 3 4
Q7. True or false: 62 = 36 .
5 6 7 8
Q8. True or false: !a 2 = a 2 .
9 10 Q2. Use your calculator to find the square of each number in the table. number ( n )
( )
square n 2
–3
Square Roots (Q9-Q16):
12
Q9. The positive square root of a number n is n . The square root (positive or negative) is the number you would have to square (multiply by itself) to obtain n . Find the positive square root of each number in the table without using a calculator. Then verify your results using a calculator.
–8 0 1.5 –2
number (n )
11 –7
positive square root
( n)
1
100
4
–15
9 2
Q3. True or false: ( !4 ) = !16 .
16 25 36 49 64
2
Q4. True or false: ( !6 ) = 36 .
81 100
Page 1 of 2!
Updated: November 1, 2009
Algebra 1 • Lesson 5.4* (Squares and Square Roots) Day 54 Notes Q10. The negative square root of a number n is ! n . The square root (positive or negative) is the number you would have to square (multiply by itself) to obtain n . Find the negative square root of each number in the table without using a calculator. Then verify your results using a calculator. number (n )
Graphs of Squares and Square Roots (Q17-Q19): Q17. Enter the table from Q1 into a list or spreadsheet on your calculator. Make a scatter plot. Describe and sketch what you see.
negative square root
(! n )
1 4 9 16 25 36 49 64
Q18. Enter the table from Q9 into a list or spreadsheet on your calculator. Make a scatter plot. Describe and sketch what you see.
81 100 Q11. Evaluate
49 .
Q12. Evaluate
100 .
Q13. Evaluate ! 1 .
Q14. Evaluate
Q19. Enter the table from Q10 into a list or spreadsheet on your calculator. Make a scatter plot. Describe and sketch what you see.
9.
Q15. Evaluate ! 1, 000, 000 .
Q16. Evaluate
Page 2 of 2!
!25 .
Updated: November 1, 2009
Algebra 1 • Lesson 5.5* (Cubes and Cube Roots) Day 55 Notes Review • Squares and Square Roots (Q1-Q7):
9=
Q1. We say that
because
Cube Roots (Q9-Q14):
2
(
)
= 9.
49 =
because
(
)
Q3. We say that
16 =
because
(
)
Q4. We say that ! 81 =
because
Q5. We say that ! 1 =
because
Q6. We say that ! 36 =
Q7. We say that !
!
!25 has !
!
(
2
because
)
= 49 .
2
= 16 . 2
(
) 2
(
because
Now we will find cube roots
2
Q2. We say that
)
= 81 .
= 1. 2
(
In the previous lesson, we found square roots by focusing on their connection to square numbers.
)
= 36 .
!
!
!
!
3
connection to cube numbers (perfect cubes). Q9. We say that
3
Q10. We say that
Q11. We say that
Q12. We say that
Q13. We say that
1=
because
3
(
)
=1. 3
3
64 =
because
(
)
3
!8 =
because
(
)
3
0=
3
!125 =
3
27 =
because
b/c
3
3
(
) 3
(
)
= !25 has ! Q14. We say that
!
( n ) by focusing on their
because
(
= !125 . 3
)
.
!64 has !
!
(
2
!
!
because
!
!
!
!
)
!
Q15.
4+ 9 =
Q16.
81 ! 49 =
= !64 has !
.
Cube Hunt (Q9): Q9. Type factor(9) on your calculator, and you will see that 9 is a square number (we sometimes say perfect square), since 9 = 32 .
Q17.
3
27 ! 1 =
Find the first five perfect cubes using a similar technique. Summarize your results in the space below. (Hint: The first perfect cube is 1, since 1 = 13 .)
Q18. True or false:
3
64 > 16
3
125 < 100
First perfect cube: 1 Second perfect cube:
Q19. True or false:
Third perfect cube: Fourth perfect cube:
Q20. True or false: 12 + 23 ! 4 ! 3 125 =
1+ 31
Fifth perfect cube:
Page 1 of 1!
= !8 .
=0.
Evaluating Roots/Radicals (Q15-Q20): Q8. We say that
= 64 .
Updated: November 1, 2009
= 27 .
Algebra 1 • Lesson 5.6* (Exponents 1) Day 56 Notes Introduction • Exponents (Q1-Q10):
Multiplying Powers (Q11-Q20):
Use your calculator to rewrite each expression as a power (that is, using a base and an exponent).
Use your calculator to simplify each product of powers. Type in what is written below, and write down the result.
Q1. When I enter x ! x ! x the calculator displays...
Q11. x 2 ! x 3
Q2. When I enter a ! a ! a ! a ! a the calculator displays...
Q12. a 5 ! a 3
Q13. b10 !b 5 Q3. When I enter b !b !b !b the calculator displays...
Q14. y 7 ! y 8 Q4. When I enter n ! n the calculator displays... Q15. c 270 !c 3 Q5. When I enter c !c !c !c !c the calculator displays... Q16. t 5 ! t 5 Q6. To display x 5 enter… Q17. n 30 ! n 20
Q7. To display a 3 enter…
Q18. w 2 ! w 3 ! w 4
Q19. x 3 ! x 7 ! x 11
Q8. To display n 7 enter…
Q20. x a ! x b Q9. To display b1 enter…
We summarize these results by saying that when
Q10. To display c 8 enter...
!
!
!
powers with the same
!
!
, we !
!
the exponents.
We summarize this by saying that x n is equivalent to an expression with ! !
factors of !
. Algebraically, ab ! ac =
Page 1 of 1!
Updated: November 1, 2009
Algebra 1 â&#x20AC;˘ Lesson 5.7* (Exponents 2) Day 57 Notes Simplifying Exponents (Q1-Q6): Use your calculator to rewrite each quotient as a power. Q1.
Q10.
x !x !x !x !x = x !x Q11.
Q2.
a !a !a !a !a !a !a = a !a !a !a !a Q12.
Q3.
Q4.
Q5.
y8 = y
b !b !b !b = b !b !b
y !y !y !y !y = y !y !y !y !y
c !c !c = c
t !t !t !t !t !t Q6. = t !t
Dividing Powers (Q7-Q12):
c10
=
c4
t6 t6
=
Dividing Powers Quickly (Q13-Q16): Based on your observations made while working on Q1 through Q12, simplify these quotients of powers quickly without a calculator. Verify your results at the end using a calculator. Q13.
Q14.
Q15.
Q16.
a 11
=
a7
b19
=
b6
c 77
=
c 67
d 100 d 99
=
Rewrite each numerator and denominator in expanded form (so that the problems look like Q1 through Q6). Then simplify without using a calculator. Q7.
Q8.
x7 x
2
a5 a3
=
We summarize these results by saying that when !
!
!
powers with the same
!
!
, we !
!
= Algebraically,
Q9.
b9 b4
the exponents.
ab ac
=
=
Bonus. What does x 0 equal? (Hint: Study Q4.) Page 1 of 1!
Updated: November 1, 2009
Algebra 1 â&#x20AC;˘ Lesson 5.8* (Exponents 3) Day 58 Notes Powers of Powers (Q1-Q8):
Review â&#x20AC;˘ Exponent Rules (Q9-Q18):
Use your calculator to rewrite each expression. Be sure to write a multiplication symbol in between each variable.
Simplify each expression using the exponent rules learned in the last three lessons.
2
Q9. a m ! a n =
3
Q10.
4
Q11. a m
Q1. ( a !b ) =
Q2. (c ! d ) =
(
2
)
4
)
(
)
(
)
Q6. g 10 ! h 11
Q7. a !b 2 !c 3
(
5
2
Q8. x ! y ! z
=
n
Q13.
=
3
4
x 31 x 28
=
2
( )
Q14. x 3
=
7
=
Q12. x 17 ! x 3 =
=
(
Q5. m 2 ! n 3
an
( )
Q3. (e ! f ) =
Q4. x 2 ! y
am
Q15. y 7 ! y 5 ! y 6 =
=
10
)
=
Q16.
=
We summarize these results by saying that when raising a power to a power, !
!
!
y8 y5
=
4
( )
Q17. y 7
=
each inside
exponent with the outside exponent.
(
Q18. a 2b 4c 3 c
( )
Algebraically, ab
Page 1 of 1!
5
)
=
=
Updated: November 1, 2009