1. Functions and Their Graphs
Advanced Math
Chapter 2

2. Linear Equations in Two Variables
Advanced Math
Section 2.1

3. Slope-intercept form m is slope b is y-intercept
Advanced Math chapter 2

4. Slope
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5. Vertical and horizontal lines
Vertical line have undefined slope
Example: x = 6
Horizontal lines have zero slope
Example: y = 2
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6. Examples
Find the slope of the line connecting points (1, 3) and (-2, 1)
Draw a line through the point (2, 1) that has a slope of -1
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7. Point-slope form Point (x1, y1) is on the line
Most useful for finding the equation of a line when you know the slope and a point on the line
Advanced Math chapter 2

8. Examples
Write the equation of a line with a slope of 2 that goes through the point (3, 4).
Write the equation of a line that goes through the point (-5, 3) that has a slope of -12.
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9. Parallel lines
Have equal slopes
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10. Perpendicular lines
Have slopes that are negative reciprocals of each other
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11. Example
Find the slope-intercept forms of the equations of the lines that pass through the point (3, 4) and are (a) parallel and (b) perpendicular to the line 4x + 3y = 7
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12. Slope A ratio if both axes have the same unit of measure
Example: slope of a ramp
A rate of change if they have different units of measure
Example: straight-line depreciation
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13. Functions
Advanced Math
Section 2.2

14. Relation When two quantities are related to each other
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15. Function Special type of relation
For every x value there is exactly one y value
It’s ok if two or more x values have the same y value
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16. Ways to represent a function
Verbally
A sentence
Numerically
A table or list of ordered pairs
Graphically
Points on a graph
Algebraically
An equation
Advanced Math chapter 2

17. Testing for functions
Decide whether each input value is matched with exactly one output value
Or each value in the domain is matched with exactly one value in the range
Examples: exercises 5 and 7
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18. Testing for functions algebraically
Solve the equation for y
If each x value corresponds to only one y value, it is a functions
If any x value has more than one y value, it’s not a function
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19. Examples
Is it a function?
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20. Function notation f(x) Read “f of x”
Tells you that x is the independent variable and f(x) is the dependent variable
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21. Finding the value of a function
Find the value of the function at whatever is inside the parentheses
Replace each x is the original equation with whatever is inside the parentheses
Advanced Math chapter 2

22. Examples
Find the following:
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23. Example
Piecewise function
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24. Implied domain
Set of all real numbers for which the expression is defined
Not stated
In general, excludes values that would cause division by zero or that would result in the even root of a negative number
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25. Examples
Find the domain of each function
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26. Domain limitations Domains can be limited by physical context Examples
Length, radius, volume, etc. can’t be zero or negative
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27. Example
An open box is to be made from a square piece of material 24 centimeters on a side by cutting equal squares from the corners and turning up the sides. Write the volume, V as a function of x and determine its domain.
24-2x x
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28. Example
Evaluate the difference quotient
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29. Analyzing graphs of functions
Advanced Math
Section 2.3

30. Graphs of functions f(x) = y Domain is possible x values
Range is possible y values
Examples: Exercises 2 and 4
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31. Vertical line test
y is a function of x if no vertical line intersects the graph at more than one point
Examples: exercises10, 12, 14
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32. Zeros of a function Where f(x) = 0 x-intercepts
To find them, set f(x) = 0 and solve for x
Example: Find the zeros of the function
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33. Increasing functions
A function is increasing over an interval if, for any x1 and x2 in the interval, x1 < x2 implies that f(x1) < f(x2)
The graph is going up
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34. Decreasing functions
A function is increasing over an interval if, for any x1 and x2 in the interval, x1 < x2 implies that f(x1) > f(x2)
The graph is going down
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35. Constant functions
A function is constant over an interval if, for any x1 and x2 in the interval, f(x1) = f(x2)
The graph is horizontal
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36. Examples
Exercises 32, 34
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37. Relative minimum AKA local minimum
Lowest point of the graph on some interval
f(a) is a relative minimum if f(a) ≤ f(x)
Approximate from a graph (for now)
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38. Relative maximum AKA local maximum
Highest point of the graph on some interval
f(a) is a relative maximum if f(a) ≥ f(x)
Approximate from a graph (for now)
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39. Example Estimate any relative maxima or relative minima
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40. Average rate of change
The slope of the line between two points on a curve (a secant line)
Example: Find the average rate of change of the following function from x1 = 1 to x2 = 5
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41. Even functions Symmetric with respect to the y-axis
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42. Odd functions Symmetric with respect to the origin
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43. Examples
Determine whether the following functions are even, odd, or neither.
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44. A Library of parent functions
Advanced Math
Section 2.4

45. Linear functions Slope m y-intercept (0, b) Domain is all real numbers
Range is all real numbers
x-intercept (-b/m, 0)
Graph is increasing if m > 0, decreasing if m < 0
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46. Example Write the linear function for which f(5) = - 4 and f(-2) = 17
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47. Two special linear functions
Constant function
f(x) = c
Horizontal line
Domain is all real numbers
Range is a single value, c
Identity function
f(x) = x
Slope of 1
Passes through origin
Advanced Math chapter 2

48. Squaring function Domain is all real numbers
Range is all nonnegative real numbers
Even function
Intercept at (0, 0)
Decreasing on (-∞, 0), increasing on (0, ∞)
Symmetric with respect to y-axis
Relative minimum and (0, 0)
Advanced Math chapter 2

49. Squaring function
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50. Cubic function Domain is all real numbers Range is all real numbers
Odd function
Intercept at (0, 0)
Increasing on (-∞, ∞)
Symmetric with respect to the origin
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51. Cubic function
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52. Square Root function Domain is all nonnegative real numbers
Range is all nonnegative real numbers
Intercept at (0, 0)
Increasing on the interval (0, ∞)
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53. Square Root function
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54. Reciprocal function Domain is all real numbers, x ≠ 0
Range is all real numbers, y ≠ 0
Odd function
No intercepts
Decreasing on intervals (-∞, 0) and (0, ∞)
Symmetric with respect to origin
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55. Reciprocal function
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56. Step functions Resemble stairsteps
Most famous is greatest integer function
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57. Greatest integer function
Domain is all real numbers
Range is all integers
y-intercept at (0, 0)
x-intercepts in the interval [0, 1)
Constant between each pair of consecutive integers
Jumps vertically one unit at each integer value
Advanced Math chapter 2

58. Greatest integer function
Advanced Math chapter 2

59. Parent functions
It is important that you are familiar with the 8 parent functions on page 219.
This will help you analyze more complicated graphs in section 2.5
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60. Transformations of functions
Advanced Math
Section 2.5

61. Rigid transformations
The graph shifts, but doesn’t change size
Slides around or flips, but doesn’t get distorted
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62. Vertical shift upward
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63. Vertical shift downward
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64. Horizontal shift to the right
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65. Horizontal shift to the left
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66. Reflection in the x-axis
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67. Reflection in the y-axis
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68. Examples Exercises 6 (skip c and g), 12, 38, 44, 48
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69. Nonrigid transformations
Cause distortion
Stretch
Shrink
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70. Vertical stretch
Each y multiplied by c
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71. Vertical shrink
Each y multiplied by c
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72. Horizontal stretch
Each y multiplied by f(c)
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73. Horizontal shrink
Each y multiplied by f(c)
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74. Examples
Exercises 6c and g, 24, 28, 42
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75. Combinations of functions: Composite functions
Advanced Math
Section 2.6

76. Arithmetic combinations
Addition
Subtraction
Multiplication
Division
Domain of the combination consists of all real numbers that are common to the domains of the original functions.
When dividing, the denominator can’t equal 0
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77. Finding the sum
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78. Finding the difference
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79. Finding the product
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80. Finding the quotient
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81. Finding the quotient
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82. Composition of functions
Domain is limited by both functions
The inside function can’t equal something that’s not in the domain of the outside function.
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83. Examples
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84. Inverse functions
Advanced Math
Section 2.7

85. Inverse function Switch domain and range Interchange x and y values
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86. Finding inverse functions informally
Think of how to “undo” the function
f -1(x) always indicates the inverse of f(x), never the reciprocal
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87. Verifying inverse functions
For inverse functions
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88. Inverse functions
If g is the inverse of f, then f is also the inverse of g
They are inverses of each other
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89. Examples
Determine whether the two functions are inverses of each other.
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90. Graphs of inverse functions
Reflect in the line y = x
If (a, b) is on the graph of a function, then (b, a) is on the graph of its inverse.
Advanced Math chapter 2

91. Examples
Sketch the graphs of f and g and show that they are inverse functions.
Advanced Math chapter 2

92. Horizontal Line Test
A function has an inverse if and only if no horizontal line intersects the graph at more than one point
Like the vertical line test, but with a horizontal line
Advanced Math chapter 2

93. One-to-One function Passes the horizontal line test Has an inverse
Each y corresponds to exactly one x
[Also, each x corresponds to exactly one y (passes the vertical line test) or it wouldn’t be a function at all]
Advanced Math chapter 2

94. Examples
Exercises 26, 30, 32
First, is it a function (vertical line test)
Next, does it have an inverse (horizontal line test)
Advanced Math chapter 2

95. Finding inverses algebraically
Use the horizontal line test to decide whether f has an inverse
In the equation for f(x), replace f(x) with y
Switch x and y, and solve for y Replace y with f -1(x).
Verify that they are inverses.
Advanced Math chapter 2

96. Examples Find the inverses of the following functions algebraically
Advanced Math chapter 2