1. Operations on Functions and Analyzing Graphs
College Algebra
Chapter 3
Operations on Functions and Analyzing Graphs

2. Sums and Differences of Functions
College Algebra Chapter 3.1 The algebra and composition of functions
Sums and Differences of Functions
For functions f and g with domains of P and Q respectively, the sum and difference of f and g are defined by:

3. Sums and Differences of Functions
College Algebra Chapter 3.1 The algebra and composition of functions
Sums and Differences of Functions

4. Sums and Differences of Functions
College Algebra Chapter 3.1 The algebra and composition of functions
Sums and Differences of Functions

5. Products and Quotients of Functions
College Algebra Chapter 3.1 The algebra and composition of functions
Products and Quotients of Functions

6. Products and Quotients of Functions
College Algebra Chapter 3.1 The algebra and composition of functions
Products and Quotients of Functions

7. Products and Quotients of Functions
College Algebra Chapter 3.1 The algebra and composition of functions
Products and Quotients of Functions

8. Products and Quotients of Functions
College Algebra Chapter 3.1 The algebra and composition of functions
Products and Quotients of Functions

9. Composition of Functions
College Algebra Chapter 3.1 The algebra and composition of functions
Composition of Functions

10. Composition of Functions
College Algebra Chapter 3.1 The algebra and composition of functions
Composition of Functions

11. Composition of Functions
College Algebra Chapter 3.1 The algebra and composition of functions
Composition of Functions

12. Function Decomposition
College Algebra Chapter 3.1 The algebra and composition of functions
Function Decomposition

13. Function Decomposition
College Algebra Chapter 3.1 The algebra and composition of functions
Function Decomposition

14. College Algebra Chapter 3.1 The algebra and composition of functions
Homework pg

15. Relations Functions One to One Function
College Algebra Chapter 3.2 one to one and inverse functions
Relations
Functions
One to One Function
If a horizontal line intersects a graph at only one point, the function is one to one

16. Relations Functions 1 to 1 functions
College Algebra Chapter 3.2 one to one and inverse functions
Relations
Functions
1 to 1 functions

17. College Algebra Chapter 3.2 one to one and inverse functions

18. Inverse functions An inverse function is denoted by This does not mean
College Algebra Chapter 3.2 one to one and inverse functions
Inverse functions
An inverse function is denoted by
This does not mean
If given coordinates (x,y) the inverse would have coordinates (y,x)
(3,4) (-2,8) (-7,10)

19. College Algebra Chapter 3.2 one to one and inverse functions
An inverse must undo operations taking place in the original equation

20. Inverse functions How to find an inverse Algebraically
College Algebra Chapter 3.2 one to one and inverse functions
Inverse functions
How to find an inverse Algebraically
Use y instead of f(x)
Interchange x and y
Solve for y
The result is the inverse

21. College Algebra Chapter 3.2 one to one and inverse functions

22. College Algebra Chapter 3.2 one to one and inverse functions

23. College Algebra Chapter 3.2 one to one and inverse functions

24. College Algebra Chapter 3.2 one to one and inverse functions
Homework pg

25. Vertical and Horizontal Shifts
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical and Horizontal Shifts
Vertical shift or vertical translation
Given any function whose graph is determined by
and k>0,
The graph of is the graph of
shifted upward k units.
The graph of is the graph of
shifted downward k units.
The amount of shift is equal to the constant added to the function

26. Vertical and Horizontal Shifts
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical and Horizontal Shifts
Vertical shift or vertical translation

27. Vertical and Horizontal Shifts
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical and Horizontal Shifts
Vertical shift or vertical translation

28. Vertical and Horizontal Shifts
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical and Horizontal Shifts
Vertical shift or vertical translation

29. Vertical and Horizontal Shifts
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical and Horizontal Shifts
Vertical shift or vertical translation

30. Vertical and Horizontal Shifts
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical and Horizontal Shifts
Horizontal shift or horizontal translation
Given any function whose graph is determined by
and h>0,
The graph of is the graph of
shifted to the left h units.
The graph of is the graph of
shifted to the right h units.
-Happens when the input values are affected
-Direction of shift is opposite the sign

31. Vertical and Horizontal Shifts
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical and Horizontal Shifts
Horizontal shift or horizontal translation

32. Vertical and Horizontal Shifts
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical and Horizontal Shifts
Horizontal shift or horizontal translation
Graph

33. Vertical and Horizontal Shifts
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical and Horizontal Shifts
Horizontal shift or horizontal translation
Graph

34. Vertical and Horizontal Shifts
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical and Horizontal Shifts
Horizontal shift or horizontal translation
Graph

35. Vertical Reflection over x-axis
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical Reflection over x-axis

36. Vertical Reflection over x-axis
College Algebra Chapter 3.3 Toolbox functions and Transformation
Vertical Reflection over x-axis

37. Horizontal Reflections over y-axis
College Algebra Chapter 3.3 Toolbox functions and Transformation
Horizontal Reflections over y-axis

38. Horizontal Reflections over y-axis
College Algebra Chapter 3.3 Toolbox functions and Transformation
Horizontal Reflections over y-axis

39. College Algebra Chapter 3.3 Toolbox functions and Transformation

40. Ways to graph transformations
College Algebra Chapter 3.3 Toolbox functions and Transformation
Ways to graph transformations
Using a table of values
Applying transformations to a parent graph
Apply stretch or compression
Reflect result
Apply horizontal or vertical shifts usually applied to a few characteristic points

41. College Algebra Chapter 3.3 Toolbox functions and Transformation

42. College Algebra Chapter 3.3 Toolbox functions and Transformation
Homework pg

43. Shifted Form/Vertex Form
College Algebra Chapter 3.4 Graphing General Quadratic Functions
Shifted Form/Vertex Form
Horizontal shift is h units, vertical shift is k units
To put a quadratic equation in shifted form can be done by completing the square

44. Shifted Form/Vertex Form
College Algebra Chapter 3.4 Graphing General Quadratic Functions
Shifted Form/Vertex Form
Completing the square
Group variable terms
Factor our “a”
Add and subtract then regroup
Factor trinomial
Distribute and simplify

45. Shifted Form/Vertex Form
College Algebra Chapter 3.4 Graphing General Quadratic Functions
Shifted Form/Vertex Form
Completing the square

46. Shifted Form/Vertex Form
College Algebra Chapter 3.4 Graphing General Quadratic Functions
Shifted Form/Vertex Form
Completing the square

47. Shifted Form/Vertex Form
College Algebra Chapter 3.4 Graphing General Quadratic Functions
Shifted Form/Vertex Form
Completing the square
Go back 3 pages to find zero’s of each function
Set equation equal to zero and then solve for x

48. Shifted Form/Vertex Form
College Algebra Chapter 3.4 Graphing General Quadratic Functions
Shifted Form/Vertex Form
Completing the square
Group variable terms
Factor our “a”
Add and subtract then regroup
Factor trinomial
Distribute and simplify

49. Shifted Form/Vertex Form
College Algebra Chapter 3.4 Graphing General Quadratic Functions
Shifted Form/Vertex Form
Completing the square

50. Shifted Form/Vertex Form
College Algebra Chapter 3.4 Graphing General Quadratic Functions
Shifted Form/Vertex Form
Completing the square

51. Shifted Form/Vertex Form
College Algebra Chapter 3.4 Graphing General Quadratic Functions
Shifted Form/Vertex Form
Completing the square

52. Standard form for a quadratic function has a vertex at
College Algebra Chapter 3.4 Graphing General Quadratic Functions
Standard form for a quadratic function has a vertex at

53. College Algebra Chapter 3.4 Graphing General Quadratic Functions
Homework pg

54. Reciprocal Quadratic Functions
College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions
Reciprocal Functions
Reciprocal Quadratic Functions
Asymptotes are not part of the graph, but can act as guides when graphing
Asymptotes appear as dashed lines guiding the branches of the graph

55. Direction/Approach Notation
College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions
Direction/Approach Notation
As x becomes an infinitely large negative number, y becomes a very small negative number

56. Horizontal and Vertical asymptotes
College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions
Horizontal and Vertical asymptotes
The line y=k is a horizontal asymptote if, as x increases or decreases without bound, y approaches k
The line x=h is a vertical asymptote if, as x approaches h, |y| increases or decreases without bound

57. Horizontal and vertical shifts of rational functions
College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions
Horizontal and vertical shifts of rational functions
First apply them to the asymptotes, then calculate the x- and y-intercepts as usual

58. To find x intercept; solve
College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions
To find x intercept; solve

59. College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions

60. To find y-intercept; solve
College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions
To find y-intercept; solve
To find x-intercept; solve

61. College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions

62. College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions

63. College Algebra Chapter 3.5 Asymptotes and Simple Rational Functions
Homework pg

64. College Algebra Chapter 3.6 Direct and inverse Variation

65. College Algebra Chapter 3.6 Direct and inverse Variation

66. College Algebra Chapter 3.6 Direct and inverse Variation

67. College Algebra Chapter 3.6 Direct and inverse Variation
Homework pg

68. Piecewise-Defined Functions
College Algebra Chapter 3.7 Piecewise – Defined Functions
Piecewise-Defined Functions
The effective domain is the part of the domain that each piece is graphed over.

69. Piecewise-Defined Functions
College Algebra Chapter 3.7 Piecewise – Defined Functions
Piecewise-Defined Functions
What is the piece-wise function?

70. Piecewise-Defined Functions
College Algebra Chapter 3.7 Piecewise – Defined Functions
Piecewise-Defined Functions
Show what happens when the ends of each line do not meet at the same point

71. Piecewise-Defined Functions
College Algebra Chapter 3.7 Piecewise – Defined Functions
Piecewise-Defined Functions
How to handle which function the end points go with

72. College Algebra Chapter 3.7 Piecewise – Defined Functions
Homework pg

73. Composition of functions Inverse function One-to-one function
College Algebra Chapter 3 Review
Composition of functions
Inverse function
One-to-one function
Transformation
Translation
Reflection
Quadratic
Absolute value
Linear
Reciprocal
Reciprocal quadratic function
Piecewise-defined functions
Effective domain

74. Composition of functions
College Algebra Chapter 3 Review
Composition of functions
Domain and Range?

75. Know them and their graphs
College Algebra Chapter 3 Review
Toolbox Functions
Know them and their graphs

76. College Algebra Chapter 3 Review

77. College Algebra Chapter 3 Review
Variation
The weight of an object on the moon varies directly with the weight of the object on Earth. A 96-kg object on Earth would weigh only 16 kg on the moon. How much would a 250-kg astronaut weigh on the moon?

78. Piece-Wise Defined Functions
College Algebra Chapter 3 Review
Piece-Wise Defined Functions

79. College Algebra Chapter 3 Review