Future Mathematical courses Book
Advanced Mathematical Concepts
By: Quentin Nathan Watson Ms. Simpson AFM 7 February 2012 1
Table of Content 1. General Techniques of Graphing a. The coordinate system; page- 3 – 6 - Relations and functions, domain and range - One-to-one functions - Vertical Line Test - Horizontal Line Test - Inverse Functions - Periodic Functions b. Properties of symmetry about the x-axis, y-axis, and origin; page- 7 - 10 - Test for symmetry on the Graph - Test for symmetry in the equation - Even and odd functions (equations and graphs) 2. Common Functions and Graphs (Include domain, range and important values where appropriate) a. Linear Functions; page- 11 - 13 - Forms of linear equations a. General or standard form b. Slope-intercept form c. Point-slope form d. Equation of vertical and horizontal lines b. Slope of a line; page- 13 - 15 a. Equation b. Meaning of slope c. Positive, negative slope d. Slope of parallel line e. Slope of horizontal and vertical lines c. X and Y intercepts; paged. Linear inequalities; page- 16 3. Quadratic Functions, pages 18-19 a. Equation in Standard Form -How to find the vertex and intercepts -Graphs including: vertex, axis of symmetry, intercepts -Quadratic Formula and implications of the effect of the discriminate on the graph 4. Transformations, pages 20-21 5. Polynomial Functions pages 22-23 a. Effects of n, where n is the largest exponent b. Effects of a, where a is the leading coefficient c. Effects of the multiplicity of a factor on d. Maximum number of turning points
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6. Rational Functions, pages 24-30 a. Vertical asymptotes b. Intercepts c. Horizontal asymptotes d. Slant asymptotes e. Holes f. Translations g. Types of discontinuity 7. Square Root Functions, pages31 a. Equation and graph b. Domain and range
8. Trigonometric Functions pages 32-35 a. General graph and equation - Domain and range - Odd or Even - Fundamental period 9. Exponential Functions (Domain, range, intercepts, asymptotes for each) pages 36-37 1. y a x 2. y a x 3. y e x 4. Translations 10. Logarithmic Functions (Domain, range, intercepts, asymptotes for each), pages 38-40 1. y log 2 x 2. y log 12 x 3. y ln x 4. Translations 11. Library of Functions (Write the equation and give the graph.), pages41-47 12. Optional Graphs (Not Functions), pages 48-49 13. Work cited pages 50- 51
General Techniques of Graphing 3
The Coordinate System Relations and functions Relation- A pair of elements of one set with elements of a second set, as well as, a set of order pair. Function- Is a relation in which each element of the domain is paired with exactly one element in the range. Domain- Is the set of all abscissas of the ordered pairs. Range- Is the set of all ordinates of the ordered pairs. a. The relation is a function, y = 3x+5
b. This relation is not a function, y = │x + 2│- 2
Example 2: State the domain and range of the relations, then state whether the relation is a function. {(2, -3), (9, 0), (8, -3), (-9, 8)} 4
Domain: {-9, 2, 8, 9} Range: {-3, 0, 8} Yes, each x value is paired with exactly one y value
One-to-one functions One-to-one function - is a function in which every element in the range of the function corresponds with one and only one element in the domain. Example: of a one-to-one function {(0, 1), (5, 2), (6, 4)} Domain: 0, 5, 6 Range: 1, 2, 4 Each element in the domain (0, 5, and 6) corresponds with a unique element in the range. Therefore this function is a one-to-one function
Vertical line test Vertical line test- If every vertical line drawn on the graph of a relation passes through no more than one point on the graph, then the relation is a function a. A relation that is a function; y = 2x- 3; x= -2
b. A relation that is not a function; y = sqrt(3x), y = - sqrt(3x); x= 4
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Horizontal line test Horizontal line test- Is used to determine if a function has an inverse that is also a function;
Inverse Function Inverse function- A function obtained by expressing the dependent variable of one function as the independent variable of another; f and g are inverse functions if f(x)=y and g(y)=x. The graph of f-1 is the reflection about the line y = x of the graph of f.
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Example: State whether F(x) = -3x+2 is an inverse Y= -3x+2 Y= -3x+2 X= -3y+2 -X= -3y+2 x-2= -3y -x-2= -3y y= -x/3 + 2/3 y= x/3 + 2/3 No, F(x) ≠Finv.(x)
Periodic Function Periodic function- A function returning to the same value at regular intervals For n= 1, 2 ... For example, the sine function sin x, illustrated above, is periodic with least period (often simply called "the" period) (as well as with period -2π, 4π, 6π, etc.). Sin(-x)
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Properties of symmetry about the x-axis, y-axis, and origin Test for symmetry on the graph Point symmetry- Two distinct points P and P’ are symmetric with respect to point M if and only if M id the midpoint of PP’ (with line over the PP). Point M is symmetric with respect to itself Symmetry with respect to the origin- The graph of a relation S is symmetric with respect to the origin if and only if (a, b) E S implies that (-a, -b) E S. A function has a graph that is symmetric with respect to the origin if and only if f(-x)= -f(x) for all X in the domain of f. Example: Determine whether the graph is symmetric with the origin Y = x^7
Find –f(x) -f(x)= -(x)^7 -f(x)= -x^7
Find (-x) f(-x)= (-x)^7 f(-x)= -x^7 f(-x) = -f(x)
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Line symmetry- Two distinct points P and P’ are symmetric with respect to a line l if and only if l is the perpendicular bisector of PP’ (with line over the PP). A point P is symmetric to itself with respect to the line l if and only if P is on l. X-axis; y = x
Y-axis; y = -x
y=x
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y = -x
Example: Determine whether the graph of xy=-2 is symmetric with respect to the x-axis, the yaxis, y = x, and y = -x x-axis – x(-y)= -4 -xy= -4 xy= -4
y-axis- (-x)y = - 4 -xy = -4 xy = 4
y = x- yx = -4 xy = -4
y = -x – y(-x) = -4 -xy = -4 xy = -4
Therefore, the graph of xy = -4 is symmetric with respect to the line y = x and the line y = -x. A sketch of the graph verifies the algebraic tests.
Even functions- Functions whose graphs are symmetric with respect to the y-axis
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Odd functions- Functions whose graphs are symmetric with respect to the origin
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Common Functions and Graphs Linear Functions Forms of linear equations
Standard form- A linear equation written in the form Ax + By + C = 0, where A, B, and C are real numbers and A and B are not both zero. X- intercept- Is the point where the line crosses the x-axis Y- intercept- Is the point where the line crosses the y-axis
X-intercept 5x-y-4=0 5x-(0)-4=0 5x=4 X= 5/4
Example: graph 5x – y- 4 using the x and y intercepts Y-intercept 5x-y-4=0 5(0)-y-4=0 -y=4 y= -4
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Slope-intercept form- if a line has a slope of m and a y-intercept b, the slope-intercept form of the equation of the line can be written as; y = mx + b Example: a slope of 6, and passed through A(4, 5) y = mx + b (5) = 6(4) + b 5 = 24 +b -19 = b
y = 926(30) + 9916 y = 37, 696 y = 6x - 19
y = 6x - 19
Point-slope form- If the point with coordinates (x1, y1) lies on a line m, the point-slope form of the equation of the line can be written as follows y - y1 = m(x – x1) Example: the points on the graph (0, 9916), (17, 25,660) a). Find the linear equation that can be used as a model to perdict the average person income per year b). Assume the tae of growth is constanct over time so from 1980 to 2010 = (25,660 – 9,916)/ (17-0) ≈926 y – y1 = m(x – x1) y – 9916 = 926(x – 0) y = 926x + 9916 2010 – 1980 = 30
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Equations of vertical and horizontal lines Vertical lines- a vertical line is a line that only passes through the x- axis unless x is equal to zero. Example: x=0
x=5
Horizontal lines- A horizontal line is a line that only passes through the y- axis unless y is equal to zero. Example: y=0
y = -5
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Slope of a line Equations Slope- the slope, m, of the line through (x1, y1), (x2, y2) is given by the following equation, if x1≠x2. M= (y2-y1)/ (x2- x1) Positive slope- A slope of a curve that is greater than zero, representing a positive or direct relationship between two variables y = 7x + 5; positive slope, because there is a positive in front of the coefficient of x
Negative slope- A slope of a curve that is less than zero, representing a negative or inverse relationship between two variables Y = -3x + 3; negative slope, because there is a negative in front of the coefficient of x
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Slopes of parallel lines and perpendicular lines Parallel lines Coincide- The graph of two equations that represent the same line Parallel lines- Two non-vertical lines in a place are parallel if and only if their slopes are equal and they have no points in common. Two vertical lines are always parallel. Example: Parallel lines 3x – 4y = 12 9x – 12y = 72 3x – 4y = 12 -4y = -3x + 12 y = (3/4)x – 3
9x – 12y = 72 -12y = -9x + 72 y = (3/4)x - 6
Example: Coincide 15x + 12y = 36 5x + 4y = 12 15x + 4y = 36 4y = -15x + 36 y = -(5/4)x +3
5x + 4y = 12 4y = -5x +12 y = -(5/4)x +3
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Perpendicular lines Perpendicular lines- Two vertical lines in a plane are perpendicular if and only if their slopes are opposite reciprocals A horizontal and vertical line is always perpendicular The slope is the negative inverse so if you have -2/3 then you would have 3/2 as the new slope Example: write in standard form of the equation of the line that passes through the point (7, -2) and is perpendicular to the graph 6x – y – 3 = 0 6x – y – 3 = 0 -y = -6x + 3 y – y1 = m(x – x1) y = 6x – 3 y – (-2) = -1/6 (x – 7) The slope is -(1/6) y = -(1/6)x + 19/6
Linear inequalities Linear inequality- A relation whose boundary is a straight line *≥ ≤ = dark line; > < = then your line is dotted *To find the shaded region plug in two points to your equation if it’s true shade hat region if false shade the opposite region x>4 y ≥ 1/3x + 5 y ≤ │x - 3│
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Quadratic Functions Equation in Standard Form Standard Form- A linear equation written in the form of Ax + By +C= 0, where A, B, and C are real numbers and A and B are not both zero. Example: y = -2 4y=3x – 8 -3x + 4y = -8 -3x + 4y + 8 = 0 (standard form) Graph:
How to find the vertex and intercepts -To determine the vertex of the graph of a quadratic function, f(x) = ax+ bx + c, you can either: use the method of completing the square to rewrite the function in the form of f(x) = a(x – h) + k. The vertex is (h, k) or uses the method of factoring. Example: Method one -Completing the square
x- 3 = ± 5 x-3 = 5 or x-3 = -5 x=8 x = -2 Method two -Factoring (x + 2)(x - 8) = 0 x + 2 = 0 or x–8=0 x = -2 x=8
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Graphs including: vertex, axis of symmetry, intercepts Vertex: (3,-9) Axis of symmetry: x=3 Intercepts: Y-intercept: Let x=0, then y=0 X-intercepts: Let x=0, then y = 0 or 6
F(x) = x^2-6x
Quadratic Formula and implications of the effect of the discriminant on the graph Discriminate is the expression
â&#x20AC;&#x201C; 4ac. â&#x2C6;&#x161;
The discriminate can be used to determine the number and type of roots of a quadratic equation.
Discriminate Consider + bx + c = 0 Value of discriminate
Type and Number of roots
Example of related graph of related function
2 real, rational roots Is a perfect square.
Is not a perfect
2 real irrational roots
square.
1 real, rational root
2 complex roots
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Transformations The parent graph is y=x
The blue graph is y f ( x)
The orange graph is y f ( x c)
What happens is that x coordinates move right by how much the C value is. My C value is 2 so it moves 2 units right from parent graph. The green graph is y f ( x) c
What happens is that all the x coordinates are a reversal of the parent graph so the graph would flip over the x- axis The red graph is y f ( x c) What happens is that the x coordinates move up by how much the C value is. My C value is three so the graph would move 3 units up from parent graph.
What happens is that all the x coordinates would move left by how much the C value is. My C value is 2 so it moves 2 units left from parent graph. 20
The yellow graph is y f ( x) c
What happens is that the x coordinates move down by how much the C value is. My C value is three so the graph would move 3 units down from parent graph. The blue graph is y a f ( x)
The light blue graph is y f ( x)
What happens is that the x coordinates are shrunk to size because of the A value. Depending on the A value is how much it will shorten or shrink. My A value is 2 so it shortened by 2 units. Also because it is negative the x coordinates are a reversal of the parent graph so the graph would flip over the x- axis. The green graph is y f ( x)
What happens is that the x coordinates are shrunk to size because of the A value. Depending on the A value is how much it will shorten or shrink. My A value is 2 so it shortened by 2 units.
(1) What happens is that none of the y coordinates are negative like the parent graph. In quadrant one whatever x is y is also, however; in quadrant two when x is negative y is the same x number but positive.
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Polynomial Functions Effects of n, where n is the largest exponent - The effect on n when it is the largest is that it depicts every aspect when graphing your polynomial function. The degree of a polynomial in one variable is the greatest exponent in this case it is n. N also shows the leading coefficient, which is the coefficient of the variable with the greatest exponent. Leading coefficient → A. n is even When n is even and positive it goes up left and up right, (-∞, ∞) (∞, ∞) When n is even and negative goes down left and down right, (-∞, -∞) (∞, -∞) B. n is odd When n is even and positive it goes down left and up right, (-∞, ∞) (∞, -∞) When n is even and negative goes up left and down right, (∞, -∞) (-∞, ∞)
Effects of a, where a is the leading coefficient -
a, effects what direction the graph will point.
A. a is positive The graph will point upward y=
B. a is negative The graph will point downward y= -
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Effects of the multiplicity of a factor
ď&#x20AC;¨x ď&#x20AC; rď&#x20AC;Š
n
- A zero has a "multiplicity", which refers to the number of times that its associated factor appears in the polynomial. Depending on the degree of n if it will factor into a binominal or trinomial will represent the real numbers or imaginary numbers in the function. A. n is even When n is even you have turns in your graph And the graph of is symmetric with respect to the y-axis and contains the point (-1, 1).
B. n is odd When n is odd, the graph of (-1, -1).
is symmetric with respect to the origin and contains the point
Maximum number of turning points - The maximum possible number of turning points is one less than the degree of the polynomial For example: The maximum turning points for this graph is three 4 -1 = 3
(1) 23
Rational Functions Vertical asymptotes -
â&#x20AC;&#x153;Are vertical lines which correspond to the zeroes of the denominator of a rational function. (They can also arise in other contexts, such as logarithms, but you'll almost certainly first encounter asymptotes in the context of rationales.)â&#x20AC;? (3) =y
=y (x+2)(x+3) = 0 X= -2 or -3 So x cannot be -2 or -3, because then it would be undefined The vertical asymptotes are the dotted line and the graph is avoiding them Graph:
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Intercepts Finding the x intercept of ration functions Y intercept set x = 0 X intercept set to y = 0 f(x) =
y intercept f(0) = *cannot divide by zero because it’s undefined no y intercept
X intercept: x(x+1) 0=
=
= x-3, x = 3, the coordinate is (3,0) Short cut: set the numerator to 0.
Horizontal asymptotes - “Horizontal asymptotes are just useful suggestions. Whereas you can never touch a vertical asymptote, you can (and often do) touch and even cross horizontal asymptotes. Whereas vertical asymptotes indicate very specific behavior (on the graph), usually close to the origin, horizontal asymptotes indicate general behavior far off to the sides of the graph” (4). y= *This will not factor so you can create a table x
y=
-100 1.9971026… -10 1.7339449… -1 -0.9 0 -1.2222222… 1 -0.9 10 1.7339449… 100 1.9971026… y is close to 2 but does not pass 2 so the horizontal asymptote is at 2 Graph:
(4) 25
Slant asymptotes To find slant asymptotes you must do long division first Original function:
Graph of the original function:
y=
Long Division:
Rearranged function:
Polynomial part: y = â&#x20AC;&#x201C;3x â&#x20AC;&#x201C; 3
Graph of polynomial part:
Note the similarity between the two graphs. Except for where the vertical asymptote causes a break in the middle, the two graphs are practically the same, as you can see from the overlay.
(5) 26
Holes Finding holes in a ration function Example: y = =
=
x≠3
x intercept (3, 0); y intercept (0, -3); vertical asymptote x = -1;
Horizontal asymptote = 1 y = 1; hole x = 3
To find the hole you plug in 3 to get Which is our x intercept
Graph:
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Translations There are three different types of rational function translational they are vertical, horizontal and oblique translations. Original graph: f(x) =
Horizontal Translation: f(x) = The center is at (-b, 0) b > 0, y = is shifted to the left b unites f(x) = Graph:
Vertical Translation: f(x) = The center is at (0, a) a > 0, y = moves upward a unites f(x) = Graph:
+4
Oblique Translation: f(x) = The center is (-b, a), y = moves upward a unites and shifts to the left b unites f(x) = Graph:
(6) 28
Types of Discontinuity Continuity -
A function f(x) is continuous at a point where x = c when the following three conditions are satisfied.
We can also define continuity on an interval. - A function f(x) is continuous on the open interval (a, b) if it is continuous at every point x=c contained in that interval
Discontinuity “An infinite discontinuity exists when one of the one-sided limits of the function is infinite. In other words, limx→c+f(x) = ∞, or one of the other three varieties of infinite limits”.
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“A finite discontinuity exists when the two-sided limit does not exist, but the two one-sided limits are both finite, yet not equal to each other. The graph of a function having this feature will show a vertical gap between the two branches of the function”.
“An oscillating discontinuity exists when the values of the function appear to be approaching two or more values simultaneously. A standard example of this situation is the function f(x) = pictured below”.
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Square Root Functions Equation and graph Parent graph y = √ Graph:
Graph y = √ + 1 Graph:
Domain and Range Determine the domain and range of the given function: y=
√
The domain is all values that x can take on. The main problem is the negative inside square root and we cannot have this. So I'll set the insides greater-than-or-equal-to zero, and solve. The result will be my domain: –2x + 3 > 0 –2x > –3 2x < 3 x < 3/2 = 1.5
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The domain is “all x ≤ Range is y = 0 so the graph goes down from that Graph:
Trigonometric Function General Graph and equation General Graphs Sin Graph:
Cosine Graph:
Csc Graph:
Tangent Graph:
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Sec Graph:
Cot Graph Domain and Range
Function Domain f(x) = sin ( x ) (-infinity , + infinity) [-1 , 1]
Range
f(x) = cos ( x ) (-infinity, + infinity) [-1 , 1] f(x) = tan ( x )
All real numbers except pi/1 + n*Pi
(-infinity , + infinity )
f(x) = sec ( x )
All real numbers except pi/1 + n*Pi
(-infinity, -1] U [1 , + infinity )
f(x) = csc ( x )
All real numbers except n*Pi
(-infinity, -1] U [1 , + infinity)
f(x) = cot ( x )
All real numbers except n*Pi
(-infinity, + infinity) ( )
Odd or Even Function Definition of even: f(x) is even if f(-x) = f(x). If you graph this, you'll see it has the usual interpretation of "symmetrical about y-axis" Definition of odd: f(x) is odd if f(-x) = -f(x). If you are careful about graphing this, you see the same agreement with "symmetrical about the origin." Function Sin(-x)= -sin(x) Cos(-x)= cos(x) Tan(-x)=- tan(x) Sec(-x)= sec(x) Csc(-x)=-csc(x) Cot(-x)= -cot(x)
Even
Odd Yes
Yes Yes Yes Yes Yes
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Fundamental period
Fundamental Trigonometric Identity sin2t + cos2t = 1
()
http://people.hofstra.edu/Stefan_Waner/trig/trig2.html http://www.analyzemath.com/DomainRange/domain_range_functions.html
Transformations for y = sin(x) Original graph y = sin(x):
Y =2sin(x):
Y= sin(x +Ď&#x20AC;):
Y = sin(2x):
Y=sin(x)-2:
Y = -sin(x):
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Y= 2sin(2x+Ď&#x20AC;)-2:
Y=cos(3x):
Y= cos(-x):
Transformation for y = cos(x) Original graph y = cos(x):
Y=3cos(x): Y= cos(x- Ď&#x20AC;):
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Y= cos(x)+2:
Y= 3cos(-3x-π) +2:
Exponential Functions y ax
Domain: All real numbers Range: All real numbers > 0 Intercepts: (0, 1) Horizontal asymptote: negative x-axis Vertical asymptote: none Graph:
y a x Domain: All real numbers Range: All real numbers < 0 Intercepts: (0, 1) Horizontal asymptote: positive x- axis Vertical asymptote: none 36
Graph:
y ď&#x20AC;˝ ex Domain: all real numbers Range: all real numbers > 0 Intercepts: (0, 1) Horizontal asymptotes: negative x-axis Vertical asymptotes: none Graph:
Translations y= Domain: all real numbers Range: all real numbers > 0 Intercepts: (0, 2) Horizontal asymptotes: y = 1 Vertical asymptotes: none Graph:
Domain: all real numbers Range: all real numbers < 0 Intercepts: (-1, 0); (0,-1) 37
Horizontal asymptotes: y = -2 Vertical asymptotes: none Graph:
Domain: all real numbers Range: all real numbers > 0 Intercepts: (0, 2) Horizontal asymptotes: y = 1 Vertical asymptotes: none Graph:
Logarithmic Functions y ď&#x20AC;˝ log 2 x
Domain: x > 0 Range: (-inf , +inf) Intercepts: Asymptotes: x = 0 Graph:
("Table of Contents") 38
y log 12 x Domain: x > 0 Range: (-inf , +inf) Intercepts: Asymptotes: x = 0 Graph:
y ln x
(“SOLUTION”)
Domain: x > 0 Range: (-inf , +inf). Intercepts: (1, 0) Asymptotes: x = 0 Graph:
Translations log2 (x + 2) 39
Domain: x > -2 Range: (-inf , +inf) Intercepts: x intercept is at (-1 , 0), y intercept is (0 , 1) Asymptotes: x = -2 Graph:
("Graphs of Logarithmic Functions")
Domain: x > 0 Range: (-inf , +inf) Intercepts: (0,-1) Asymptotes: x = 0 -3ln (x - 4) Domain: x > 4 Range: (-inf , +inf) Intercepts: x intercept is at (5 , 0), y intercept is undefined since x = 0 Asymptotes: x = 4 Graph:
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("Graphs of Logarithmic Functions”)
Library of Functions (Write the equation and give the graph.) A.
y c; c = 2
41
B.
yx
C.
y x
( "Algebra/Other Types of Graphs")
D.
y x 42
("Greatest Integer Function and Graph" ) E.
y x2
F.
x y2
( Barron, T., and S. Kastberg) G.
y x
43
H.
y x3
I.
y x1/ 3
J.
y x 2/ 3
44
K.
y
1 x
L.
y
1 x2
45
M. y sin x
N.
y cos x
O.
y tan x
46
P.
y csc x
Q. y sec x
R.
y cot x
47
S.
y 2x
T.
y log 2 x
U.
ye
V.
y ln x
(Logarithmic Graphs) x
48
Optional Graphs (Not Functions) Conic Sections (include standard forms of the equations and graphs of equations) 1. circle 2. ellipse 3. hyperbola 4. parabola Include terms such as center, axes, vertices, foci, etc. when appropriate for each of the above 1. Circle
2. Ellipse
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3. Hyperbola
4. Parabola
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Works Cited 1. "About Polynomials." Math Homework Help – Math Tutor Software – Algebra Help. Web. 24 Mar. 2012. <http://www.teacherschoice.com.au/maths_library/functions/about_polynomials.htm>. 2. "Algebra II: Graphs and Functions." ThinkQuest. Oracle Foundation. Web. 24 Mar. 2012. <http://library.thinkquest.org/20991/alg2/graphs.html>. 3. "Algebra/Other Types of Graphs." - Wikibooks, Open Books for an Open World. Web. 09 May 2012. <http://en.wikibooks.org/wiki/Algebra/Other_Types_of_Graphs>. 4. "Continuity and Discontinuity." Milefoot. Web. 25 Mar. 2012. <http://www.milefoot.com/math/calculus/limits/Continuity06.htm>. 5. "Graphs of Logarithmic Functions." Graphs of Logarithmic Functions. Web. 09 May 2012. <http://www.analyzemath.com/Graphing/GraphLogarithmicFunction.html>. 6. "Greatest Integer Function and Graph." Greatest Integer Function and Graph. Web. 08 May 2012. <http://www.mathwarehouse.com/step-function/greatest-integer-functionand-graph.php>. 7. "Rational Function." Diccionario De Matemáticas. Web. 25 Mar. 2012. <http://www.ditutor.com/functions/rational_function.html>. 8. "SOLUTION: Please Graph This Logarithmic Function.. or Give Me the Points.:) Y=log[1/2]x [] Number in It Means Base. Thanks for Helpin Me:)." SOLUTION: Please Graph This Logarithmic Function.. or Give Me the Points.:)y=log[1/2]x[] Number in It Means Base. Thanks for Helpin Me:). Web. 09 May 2012. <http://www.algebra.com/algebra/homework/Exponential-and-logarithmicfunctions/Exponential-and-logarithmic-functions.faq.question.198276.html>.
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9. Stapel,, Elizabeth. "Horizontal Asymptotes." Http://www.purplemath.com/modules/asymtote2.htm. Purplemath, 2003. Web. 25 Mar. 2012. 10. Stapel, Elizabeth. "Slant, or Oblique, Asymptotes." Http://www.purplemath.com/modules/asymtote3.htm. Purplemath, 2003. Web. 25 Mar. 2012. 11.
Stapel, Elizabeth. "Vertical Asymptotes." Http://www.purplemath.com/modules/asymtote.htm. Purplemath, 2003. Web. 25 Mar. 2012.
12. "Table of Contents." BHS-Methods34. Web. 09 May 2012. <http://bhsmethods34.wikispaces.com/04Bloggraphs>.
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