2. Packet #2 Graphs of Functions
Math 160
Packet #2
Graphs of Functions

3. Ex 1. Graph 𝑓 𝑥 = 𝑥 .

4. Ex 1. Graph 𝑓 𝑥 = 𝑥 .

5. Ex 1. Graph 𝑓 𝑥 = 𝑥 .

6. Ex 1. Graph 𝑓 𝑥 = 𝑥 .

7. Ex 1. Graph 𝑓 𝑥 = 𝑥 .

8. Ex 1. Graph 𝑓 𝑥 = 𝑥 .

9. Ex 1. Graph 𝑓 𝑥 = 𝑥 .

10. Ex 1. Graph 𝑓 𝑥 = 𝑥 .

11. Ex 1. Graph 𝑓 𝑥 = 𝑥 .

12. Ex 1. Graph 𝑓 𝑥 = 𝑥 .

13. Ex 1. Graph 𝑓 𝑥 = 𝑥 .

14. Ex 1. Graph 𝑓 𝑥 = 𝑥 .

15. Ex 1. Graph 𝑓 𝑥 = 𝑥 .

16. Ex 1. Graph 𝑓 𝑥 = 𝑥 .

17. Graphs of Basic Functions

18. Graphs of Basic Functions

19. Graphs of Basic Functions

20. Graphs of Basic Functions

21. Graphs of Basic Functions

22. Graphs of Basic Functions

23. Graphs of Basic Functions

24. Graphs of Basic Functions

25. Graphs of Basic Functions

26. Graphs of Basic Functions

27. Graphs of Basic Functions

28. Graphs of Basic Functions

29. What happens to the graph of 𝑓 𝑥 = 𝑥 if we add 2 to all the outputs/𝑦-values? In other words, what does 𝑔 𝑥 = 𝑥 +2 look like?

30. What happens to the graph of 𝑓 𝑥 = 𝑥 if we add 2 to all the outputs/𝑦-values? In other words, what does 𝑔 𝑥 = 𝑥 +2 look like?

32. 𝒙
𝒙 +𝟐
0+2=2 1
1+2=3 4 2
2+2=4

33. 𝒙
𝒙 +𝟐
0+2=2 1
1+2=3 4 2
2+2=4

34. 𝒙
𝒙 +𝟐
0+2=2 1
1+2=3 4 2
2+2=4

35. 𝒙
𝒙 +𝟐
0+2=2 1
1+2=3 4 2
2+2=4

36. 𝒙
𝒙 +𝟐
0+2=2 1
1+2=3 4 2
2+2=4

37. What if we multiplied the outputs of 𝑓 𝑥 = 𝑥 by negative 1
What if we multiplied the outputs of 𝑓 𝑥 = 𝑥 by negative 1? In other words, what does 𝑔 𝑥 =− 𝑥 look like?

38. What if we multiplied the outputs of 𝑓 𝑥 = 𝑥 by negative 1
What if we multiplied the outputs of 𝑓 𝑥 = 𝑥 by negative 1? In other words, what does 𝑔 𝑥 =− 𝑥 look like?

40. 𝒙
− 𝒙
−0=0 1 −1 4 2 −2

41. 𝒙
− 𝒙
−0=0 1 −1 4 2 −2

42. 𝒙
− 𝒙
−0=0 1 −1 4 2 −2

43. 𝒙
− 𝒙
−0=0 1 −1 4 2 −2

44. 𝒙
− 𝒙
−0=0 1 −1 4 2 −2

45. Now, what happens if we replace 𝑥 with 𝑥−1. So, 𝑔 𝑥 = 𝑥−1
Now, what happens if we replace 𝑥 with 𝑥−1? So, 𝑔 𝑥 = 𝑥−1 . Notice that plugging in 𝑥=1 into 𝑥−1 is like plugging in 𝑥=0 into 𝑥 . And plugging in 𝑥=5 into 𝑥−1 is like plugging in 𝑥=4 into 𝑥 . So, the effect of 𝑥−1 is to “pull” the 𝑦-values of 𝑥 to the right by 1.

46. Now, what happens if we replace 𝑥 with 𝑥−1. So, 𝑔 𝑥 = 𝑥−1
Now, what happens if we replace 𝑥 with 𝑥−1? So, 𝑔 𝑥 = 𝑥−1 . Notice that plugging in 𝑥=1 into 𝑥−1 is like plugging in 𝑥=0 into 𝑥 . And plugging in 𝑥=5 into 𝑥−1 is like plugging in 𝑥=4 into 𝑥 . So, the effect of 𝑥−1 is to “pull” the 𝑦-values of 𝑥 to the right by 1.

47. Now, what happens if we replace 𝑥 with 𝑥−1. So, 𝑔 𝑥 = 𝑥−1
Now, what happens if we replace 𝑥 with 𝑥−1? So, 𝑔 𝑥 = 𝑥−1 . Notice that plugging in 𝑥=1 into 𝑥−1 is like plugging in 𝑥=0 into 𝑥 . And plugging in 𝑥=5 into 𝑥−1 is like plugging in 𝑥=4 into 𝑥 . So, the effect of 𝑥−1 is to “pull” the 𝑦-values of 𝑥 to the right by 1.

48. Now, what happens if we replace 𝑥 with 𝑥−1. So, 𝑔 𝑥 = 𝑥−1
Now, what happens if we replace 𝑥 with 𝑥−1? So, 𝑔 𝑥 = 𝑥−1 . Notice that plugging in 𝑥=1 into 𝑥−1 is like plugging in 𝑥=0 into 𝑥 . And plugging in 𝑥=5 into 𝑥−1 is like plugging in 𝑥=4 into 𝑥 . So, the effect of 𝑥−1 is to “pull” the 𝑦-values of 𝑥 to the right by 1.

51. 𝒙
𝒙−𝟏 1
1−1 = 0 =0 2
2−1 = 1 =1 5
5−1 = 4 =2

52. 𝒙
𝒙−𝟏 1
1−1 = 0 =0 2
2−1 = 1 =1 5
5−1 = 4 =2

53. 𝒙
𝒙−𝟏 1
1−1 = 0 =0 2
2−1 = 1 =1 5
5−1 = 4 =2

54. 𝒙
𝒙−𝟏 1
1−1 = 0 =0 2
2−1 = 1 =1 5
5−1 = 4 =2

55. 𝒙
𝒙−𝟏 1
1−1 = 0 =0 2
2−1 = 1 =1 5
5−1 = 4 =2

56. 𝒙
𝒙−𝟏 1
1−1 = 0 =0 2
2−1 = 1 =1 5
5−1 = 4 =2

57. 𝒙
𝒙−𝟏 1
1−1 = 0 =0 2
2−1 = 1 =1 5
5−1 = 4 =2

58. Transformations of Functions For a function 𝑓(𝑥), and 𝑐>0, 𝑓 𝑥 +𝑐 shifts up 𝑓 𝑥 −𝑐 shifts down 𝑐𝑓(𝑥) stretches vertically (if 𝑐>1), or shrinks vertically (if 0<𝑐<1) −𝑓(𝑥) reflects about 𝑥-axis 𝑓 𝑥−𝑐 shifts right 𝑓 𝑥+𝑐 shifts left 𝑓(𝑐𝑥) stretches horizontally (if 0<𝑐<1), or shrink horizontally (if 𝑐>1) 𝑓(−𝑥) reflects about 𝑦-axis

59. Transformations of Functions For a function 𝑓(𝑥), and 𝑐>0, 𝑓 𝑥 +𝑐 shifts up 𝑓 𝑥 −𝑐 shifts down 𝑐𝑓(𝑥) stretches vertically (if 𝑐>1), or shrinks vertically (if 0<𝑐<1) −𝑓(𝑥) reflects about 𝑥-axis 𝑓 𝑥−𝑐 shifts right 𝑓 𝑥+𝑐 shifts left 𝑓(𝑐𝑥) stretches horizontally (if 0<𝑐<1), or shrink horizontally (if 𝑐>1) 𝑓(−𝑥) reflects about 𝑦-axis

60. Transformations of Functions For a function 𝑓(𝑥), and 𝑐>0, 𝑓 𝑥 +𝑐 shifts up 𝑓 𝑥 −𝑐 shifts down 𝑐𝑓(𝑥) stretches vertically (if 𝑐>1), or shrinks vertically (if 0<𝑐<1) −𝑓(𝑥) reflects about 𝑥-axis 𝑓 𝑥−𝑐 shifts right 𝑓 𝑥+𝑐 shifts left 𝑓(𝑐𝑥) stretches horizontally (if 0<𝑐<1), or shrink horizontally (if 𝑐>1) 𝑓(−𝑥) reflects about 𝑦-axis

61. Note: To graph a transformed function, plot key points of basic function and then move points based on transformations.

62. Ex 2. Graph 𝑓 𝑥 =− 𝑥+4 2 . Be sure to describe the transformations to the basic function.

63. Ex 2. Graph 𝑓 𝑥 =− 𝑥+4 2 . Be sure to describe the transformations to the basic function.

64. Ex 2. Graph 𝑓 𝑥 =− 𝑥+4 2 . Be sure to describe the transformations to the basic function.

65. Ex 2. Graph 𝑓 𝑥 =− 𝑥+4 2 . Be sure to describe the transformations to the basic function.

66. Ex 2. Graph 𝑓 𝑥 =− 𝑥+4 2 . Be sure to describe the transformations to the basic function.

67. Ex 2. Graph 𝑓 𝑥 =− 𝑥+4 2 . Be sure to describe the transformations to the basic function.

68. Ex 2. Graph 𝑓 𝑥 =− 𝑥+4 2 . Be sure to describe the transformations to the basic function.

69. Ex 2. Graph 𝑓 𝑥 =− 𝑥+4 2 . Be sure to describe the transformations to the basic function.

70. Ex 3. Graph 𝑔 𝑥 = − 1 2 𝑥 +3. Be sure to describe the transformations to the basic function.

71. Ex 3. Graph 𝑔 𝑥 = − 1 2 𝑥 +3. Be sure to describe the transformations to the basic function.

72. Ex 3. Graph 𝑔 𝑥 = − 1 2 𝑥 +3. Be sure to describe the transformations to the basic function.

73. Ex 3. Graph 𝑔 𝑥 = − 1 2 𝑥 +3. Be sure to describe the transformations to the basic function.

74. Ex 3. Graph 𝑔 𝑥 = − 1 2 𝑥 +3. Be sure to describe the transformations to the basic function.

75. Ex 3. Graph 𝑔 𝑥 = − 1 2 𝑥 +3. Be sure to describe the transformations to the basic function.

76. Ex 4. Graph ℎ 𝑥 = 1 2 𝑥−1 −2. Be sure to describe the transformations to the basic function.

77. Ex 4. Graph ℎ 𝑥 = 1 2 𝑥−1 −2. Be sure to describe the transformations to the basic function.

78. Ex 4. Graph ℎ 𝑥 = 1 2 𝑥−1 −2. Be sure to describe the transformations to the basic function.

79. Ex 4. Graph ℎ 𝑥 = 1 2 𝑥−1 −2. Be sure to describe the transformations to the basic function.

80. Ex 4. Graph ℎ 𝑥 = 1 2 𝑥−1 −2. Be sure to describe the transformations to the basic function.

81. Ex 4. Graph ℎ 𝑥 = 1 2 𝑥−1 −2. Be sure to describe the transformations to the basic function.

82. Piecewise-defined function
Ex 5. Graph 𝑓 𝑥 = 2𝑥+7 4 𝑥 2 − if 𝑥≤−3 if−3<𝑥≤0 if 𝑥>0

83. Piecewise-defined function
Ex 5. Graph 𝑓 𝑥 = 2𝑥+7 4 𝑥 2 − if 𝑥≤−3 if−3<𝑥≤0 if 𝑥>0

84. Piecewise-defined function
Ex 5. Graph 𝑓 𝑥 = 2𝑥+7 4 𝑥 2 − if 𝑥≤−3 if−3<𝑥≤0 if 𝑥>0

85. Piecewise-defined function
Ex 5. Graph 𝑓 𝑥 = 2𝑥+7 4 𝑥 2 − if 𝑥≤−3 if−3<𝑥≤0 if 𝑥>0

86. Piecewise-defined function
Ex 5. Graph 𝑓 𝑥 = 2𝑥+7 4 𝑥 2 − if 𝑥≤−3 if−3<𝑥≤0 if 𝑥>0

87. Piecewise-defined function
Ex 5. Graph 𝑓 𝑥 = 2𝑥+7 4 𝑥 2 − if 𝑥≤−3 if−3<𝑥≤0 if 𝑥>0

88. Piecewise-defined function
Ex 5. Graph 𝑓 𝑥 = 2𝑥+7 4 𝑥 2 − if 𝑥≤−3 if−3<𝑥≤0 if 𝑥>0

89. Piecewise-defined function
Ex 5. Graph 𝑓 𝑥 = 2𝑥+7 4 𝑥 2 − if 𝑥≤−3 if−3<𝑥≤0 if 𝑥>0

90. Piecewise-defined function
Ex 5. Graph 𝑓 𝑥 = 2𝑥+7 4 𝑥 2 − if 𝑥≤−3 if−3<𝑥≤0 if 𝑥>0

91. Piecewise-defined function
Ex 5. Graph 𝑓 𝑥 = 2𝑥+7 4 𝑥 2 − if 𝑥≤−3 if−3<𝑥≤0 if 𝑥>0

92. Piecewise-defined function
Ex 5. Graph 𝑓 𝑥 = 2𝑥+7 4 𝑥 2 − if 𝑥≤−3 if−3<𝑥≤0 if 𝑥>0

93. To test if a graph is a function we can use the ________________
To test if a graph is a function we can use the ________________. If any vertical line intersects a graph in two or more points, then the graph does not represent a function.

94. To test if a graph is a function we can use the ________________
To test if a graph is a function we can use the ________________. If any vertical line intersects a graph in two or more points, then the graph does not represent a function.
vertical line test

95. To test if a graph is a function we can use the ________________
To test if a graph is a function we can use the ________________. If any vertical line intersects a graph in two or more points, then the graph does not represent a function.
vertical line test

96. Ex 6. Which of the following graphs are functions?

97. Ex 6. Which of the following graphs are functions?

98. Ex 6. Which of the following graphs are functions?

99. Ex 6. Which of the following graphs are functions?

100. Ex 6. Which of the following graphs are functions?

101. Even and Odd Functions
𝑓 is an ___________________ if 𝑓 −𝑥 =𝑓(𝑥) for all 𝑥 in the domain of 𝑓. if 𝑓 −𝑥 =−𝑓(𝑥) for all 𝑥 in the domain of 𝑓.

102. even function Even and Odd Functions
𝑓 is an ___________________ if 𝑓 −𝑥 =𝑓(𝑥) for all 𝑥 in the domain of 𝑓. if 𝑓 −𝑥 =−𝑓(𝑥) for all 𝑥 in the domain of 𝑓.
even function

103. even function odd function Even and Odd Functions
𝑓 is an ___________________ if 𝑓 −𝑥 =𝑓(𝑥) for all 𝑥 in the domain of 𝑓. if 𝑓 −𝑥 =−𝑓(𝑥) for all 𝑥 in the domain of 𝑓.
even function
odd function

104. Even and Odd Functions
Ex 7. Determine whether 𝑓 𝑥 = 𝑥 3 is even, odd, or neither. Ex 8. Determine whether 𝑓 𝑥 = 𝑥 4 −2 𝑥 2 is even, odd, or neither.

105. Note: Even functions are symmetric about the _________
Note: Even functions are symmetric about the _________. Odd functions are symmetric about the _________.

106. 𝒚-axis
Note: Even functions are symmetric about the _________. Odd functions are symmetric about the _________.

107. 𝒚-axis
Note: Even functions are symmetric about the _________. Odd functions are symmetric about the _________.
origin