## Section Exercises

1. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal shift from a vertical shift?

2. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal stretch from a vertical stretch?

3. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal compression from a vertical compression?

4. When examining the formula of a function that is the result of multiple transformations, how can you tell a reflection with respect to the x-axis from a reflection with respect to the y-axis?

5. How can you determine whether a function is odd or even from the formula of the function?

6. Write a formula for the function obtained when the graph of $f\left(x\right)=\sqrt{x}$ is shifted up 1 unit and to the left 2 units.

7. Write a formula for the function obtained when the graph of $f\left(x\right)=|x|$
is shifted down 3 units and to the right 1 unit.

8. Write a formula for the function obtained when the graph of $f\left(x\right)=\frac{1}{x}$ is shifted down 4 units and to the right 3 units.

9. Write a formula for the function obtained when the graph of $f\left(x\right)=\frac{1}{{x}^{2}}$ is shifted up 2 units and to the left 4 units.

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function $f$.

10. $y=f\left(x - 49\right)$

11. $y=f\left(x+43\right)$

12. $y=f\left(x+3\right)$

13. $y=f\left(x - 4\right)$

14. $y=f\left(x\right)+5$

15. $y=f\left(x\right)+8$

16. $y=f\left(x\right)-2$

17. $y=f\left(x\right)-7$

18. $y=f\left(x - 2\right)+3$

19. $y=f\left(x+4\right)-1$

For the following exercises, determine the interval(s) on which the function is increasing and decreasing.

20. $f\left(x\right)=4{\left(x+1\right)}^{2}-5$

21. $g\left(x\right)=5{\left(x+3\right)}^{2}-2$

22. $a\left(x\right)=\sqrt{-x+4}$

23. $k\left(x\right)=-3\sqrt{x}-1$

For the following exercises, use the graph of $f\left(x\right)={2}^{x}$ to sketch a graph of each transformation of $f\left(x\right)$.

24. $g\left(x\right)={2}^{x}+1$

25. $h\left(x\right)={2}^{x}-3$

26. $w\left(x\right)={2}^{x - 1}$

For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.

27. $f\left(t\right)={\left(t+1\right)}^{2}-3$

28. $h\left(x\right)=|x - 1|+4$

29. $k\left(x\right)={\left(x - 2\right)}^{3}-1$

30. $m\left(t\right)=3+\sqrt{t+2}$

31. Tabular representations for the functions $f,g$, and $h$ are given below. Write $g\left(x\right)$ and $h\left(x\right)$ as transformations of $f\left(x\right)$.

 $x$ −2 −1 0 1 2 $f\left(x\right)$ −2 −1 −3 1 2
 $x$ −1 0 1 2 3 $g\left(x\right)$ −2 −1 −3 1 2
 $x$ −2 −1 0 1 2 $h\left(x\right)$ −1 0 −2 2 3

32. Tabular representations for the functions $f,g$, and $h$ are given below. Write $g\left(x\right)$ and $h\left(x\right)$ as transformations of $f\left(x\right)$.

 $x$ −2 −1 0 1 2 $f\left(x\right)$ −1 −3 4 2 1
 $x$ −3 −2 −1 0 1 $g\left(x\right)$ −1 −3 4 2 1
 $x$ −2 −1 0 1 2 $h\left(x\right)$ −2 −4 3 1 0

For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.

33.

34.

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40.

For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.

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42.

For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.

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46.

For the following exercises, determine whether the function is odd, even, or neither.

47. $f\left(x\right)=3{x}^{4}$

48. $g\left(x\right)=\sqrt{x}$

49. $h\left(x\right)=\frac{1}{x}+3x$

50. $f\left(x\right)={\left(x - 2\right)}^{2}$

51. $g\left(x\right)=2{x}^{4}$

52. $h\left(x\right)=2x-{x}^{3}$

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function $f$.

53. $g\left(x\right)=-f\left(x\right)$

54. $g\left(x\right)=f\left(-x\right)$

55. $g\left(x\right)=4f\left(x\right)$

56. $g\left(x\right)=6f\left(x\right)$

57. $g\left(x\right)=f\left(5x\right)$

58. $g\left(x\right)=f\left(2x\right)$

59. $g\left(x\right)=f\left(\frac{1}{3}x\right)$

60. $g\left(x\right)=f\left(\frac{1}{5}x\right)$

61. $g\left(x\right)=3f\left(-x\right)$

62. $g\left(x\right)=-f\left(3x\right)$

For the following exercises, write a formula for the function $g$ that results when the graph of a given toolkit function is transformed as described.

63. The graph of $f\left(x\right)=|x|$ is reflected over the $y$ axis and horizontally compressed by a factor of $\frac{1}{4}$ .

64. The graph of $f\left(x\right)=\sqrt{x}$ is reflected over the $x$ -axis and horizontally stretched by a factor of 2.

65. The graph of $f\left(x\right)=\frac{1}{{x}^{2}}$ is vertically compressed by a factor of $\frac{1}{3}$, then shifted to the left 2 units and down 3 units.

66. The graph of $f\left(x\right)=\frac{1}{x}$ is vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.

67. The graph of $f\left(x\right)={x}^{2}$ is vertically compressed by a factor of $\frac{1}{2}$, then shifted to the right 5 units and up 1 unit.

68. The graph of $f\left(x\right)={x}^{2}$ is horizontally stretched by a factor of 3, then shifted to the left 4 units and down 3 units.

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.

69. $g\left(x\right)=4{\left(x+1\right)}^{2}-5$

70. $g\left(x\right)=5{\left(x+3\right)}^{2}-2$

71. $h\left(x\right)=-2|x - 4|+3$

72. $k\left(x\right)=-3\sqrt{x}-1$

73. $m\left(x\right)=\frac{1}{2}{x}^{3}$

74. $n\left(x\right)=\frac{1}{3}|x - 2|$

75. $p\left(x\right)={\left(\frac{1}{3}x\right)}^{3}-3$

76. $q\left(x\right)={\left(\frac{1}{4}x\right)}^{3}+1$

77. $a\left(x\right)=\sqrt{-x+4}$

For the following exercises, use the graph below to sketch the given transformations.

78. $g\left(x\right)=f\left(x\right)-2$

79. $g\left(x\right)=-f\left(x\right)$

80. $g\left(x\right)=f\left(x+1\right)$

81. $g\left(x\right)=f\left(x - 2\right)$