
# 1.5.5E: Transformation of Functions


Section 1.5 Exercises

Describe how each function is a transformation of the original function $$f(x)$$

1. $$f(x - 49)$$

2. $$f(x + 43)$$

3. $$f(x+3)$$

4. $$f(x-4)$$

5. $$f(x)+5$$

6. $$f(x)+8$$

7. $$f(x)-2$$

8. $$f(x)-7$$

9. $$f(x-2)+3$$

10. $$f(x+4)-1$$

11. Write a formula for $$f(x)=\sqrt{x}$$ shifted up 1 unit and left 2 units.

12. Write a formula for $$f(x)=|x|$$ shifted down 3 units and right 1 unit.

13. Write a formula for $$f(x)=\dfrac{1}{x}$$ shifted down 4 units and right 3 units.

14. Write a formula for $$\; f(x)=\dfrac{1}{x^{2} }$$ shifted up 2 units and left 4 units.

15. Tables of values for $$f(x)$$, $$g(x)$$, and $$h(x)$$ are given below. Write $$g(x)$$ and $$h(x)$$ as transformations of $$f(x)$$.

16. Tables of values for $$f(x)$$, $$g(x)$$, and $$h(x)$$ are given below. Write $$g(x)$$ and $$h(x)$$ as transformations of $$f(x)$$.

The graph of $$f(x)=2^{x}$$ is shown. Sketch a graph of each transformation of $$f(x)$$.

17. $$g(x)=2^{x} +1$$

18. $$h(x)=2^{x} -3$$

19. $$w(x)=2^{x-1}$$

20. $$q(x)=2^{x+3}$$

Sketch a graph of each function as a transformation of a toolkit function.

21. $$f(t)=(t+1)^{2} -3$$

22. $$h(x)=|x-1|+4$$

23. $$k(x=(x-2)^{3} -1$$

24. $$m(t)=3+\sqrt{t+2}$$

Write an equation for each function graphed below.

25.

26.

27.

28.

Find a formula for each of the transformations of the square root whose graphs are given below.

29.

30.

The graph of $$f(x)=2^{x}$$ is shown. Sketch a graph of each transformation of $$f(x)$$

31. $$g(x)=-2^{x} +1$$

32. $$h(x)=2^{-x}$$

33. Starting with the graph of $$f(x)= 6^{x}$$ write the equation of the graph that results from

a. reflecting $$f(x)$$ about the $$x$$-axis and the $$y$$-axis

b. reflecting $$f(x)$$ about the $$x$$-axis, shifting left 2 units, and down 3 units

34. Starting with the graph of $$f(x)= 4^{x}$$ write the equation of the graph that results from

a. reflecting $$f(x)$$ about the $$x$$-axis

b. reflecting $$f(x)$$ about the $$y$$-axis, shifting right 4 units, and up 2 units

Write an equation for each function graphed below.

35. 36.

37. 38.

39. For each equation below, determine if the function is Odd, Even, or Neither.

a. $$f(x)=3 x^{4}$$

b. $$g(x)=\sqrt{x}$$

c. $$h(x)=\dfrac{1}{x} +3 x$$

40. For each equation below, determine if the function is Odd, Even, or Neither.

a. $$f(x)=(x-2)^{2}$$

b. $$g(x)=2 x^{4}$$

c. $$h(x)=2 x-x^{3}$$

Describe how each function is a transformation of the original function $$f(x)$$.

41. $$-f(x)$$

42. $$f(-x)$$

43. $$4f(x)$$

44. $$6f(x)$$

45. $$f(5x)$$

46. $$f(2x)$$

47. $$f(\dfrac{1}{3} x)$$

48. $$f(\dfrac{1}{5} x)$$

49. $$3f(-x)$$

50. $$-f(3x)$$

Write a formula for the function that results when the given toolkit function is transformed as described.

51. $$f(x)=|x|$$ reflected over the y axis and horizontally compressed by a factor of $$\dfrac{1}{4}$$.

52. $$f(x)=\sqrt{x}$$ reflected over the x axis and horizontally stretched by a factor of 2.

53. $$f(x)=\dfrac{1}{x^{2} }$$ vertically compressed by a factor of $$\dfrac{1}{3}$$, then shifted to the left 2 units and down 3 units.

54. $$f(x)=\dfrac{1}{x}$$ vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.

55. $$f(x)=x^{2}$$ horizontally compressed by a factor of $$\dfrac{1}{2}$$, then shifted to the right 5 units and up 1 unit.

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

Describe how each formula is a transformation of a toolkit function. Then sketch a graph of the transformation.

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

58. $$g(x)=5(x+3)^{2} -2$$

59. $$h(x)=-2|x-4|+3$$

60. $$k(x)=-3\sqrt{x} -1$$

61. $$m(x)=\dfrac{1}{2} x^{3}$$

62. $$n(x)=\dfrac{1}{3} |x-2|$$

63. $$p(x)=(\dfrac{1}{3} x)^{2} -3$$

64. $$q(x)=(\dfrac{1}{4} x)^{3} +1$$

65. $$a(x)=\sqrt{-x+4}$$

66. $$b(x)=\sqrt[{3}]{-x-6}$$

Determine the interval(s) on which the function is increasing and decreasing.

67. $$f(x)=4(x+1)^{2} -5$$

68. $$g(x)=5(x+3)^{2} -2$$

69. $$a(x)=\sqrt{-x+4}$$

70. $$k(x)=-3\sqrt{x} -1$$

Determine the interval(s) on which the function is concave up and concave down.

71. $$m(x)=-2(x+3)^{3} +1$$

72. $$b(x)=\sqrt[{3}]{-x-6}$$

73. $$p(x)=(\dfrac{1}{3} x)^{2} -3$$

74. $$k(x)=-3\sqrt{x} -1$$

The function $$f(x)$$ is graphed here. Write an equation for each graph below as a transformation of $$f(x)$$.

75.76.77.

78.79. 80.

81.82. 83.

84. 85. 86.

Write an equation for each transformed toolkit function graphed below.

87. 88. 89.

90. 91. 92.

93. 94. 95.

96. 97. 98.

Write a formula for the piecewise function graphed below.

99. 100.

101. 102.

103. Suppose you have a function $$y = f(x)$$ such that the domain of $$f(x)$$ is $$1 \le x \le 6$$ and the range of $$f(x)$$ is (-3 \le y \le 5\). [UW]

a. What is the domain of $$\; f(2(x-3))\;$$?

b. What is the range of $$f(2(x-3))$$ ?

c. What is the domain of $$2f(x)-3$$ ?

d. What is the range of $$2f(x)-3$$ ?

e. Can you find constants $$B$$ and $$C$$ so that the domain of $$f(B(x-C))$$ is $$8 \le x \le 9$$?

f. Can you find constants $$A$$ and $$D$$ so that the range of $$Af(x) + D$$ is 0 $$0 \le y \le 1$$?

1. Horizontal shift right 49 units

3. Horizontal shift left 3 units

5. Vertical shift up 5 units

7. Vertical shift down 2 units

9. Horizontal shift right 2 units, Vertical shift up 3 units

11. $$f(x + 2) + 1 = \sqrt{x + 2} + 1$$

13. $$f(x - 3) - 4 = \dfrac{1}{x - 3} - 4$$

15. $$g(x) = f(x - 1)$$, $$h(x) = f(x) + 1$$

17. 19.

21. 23.

25. $$y = |x - 3| - 2$$

27. $$y = \sqrt{x + 3} - 1$$

29. $$y = -\sqrt{x}$$

31.

33. a. $$-f(-x) = -6^{-x}$$
b. $$-f(x + 2) - 3 = -6^{x + 2} - 3$$

35. $$y = -(x + 1)^2 + 2$$

37. $$y = \sqrt{-x} + 1$$

39. a. Even
b. Neither
c. Odd

41. Reflect $$f(x)$$ about the $$x$$-axis

43. Vertically stretch $$y$$ values by 4

45. Horizontally compress $$x$$ values by 1/5

47. Horizontally stretch $$x$$ values by 3

49. Reflect $$f(x)$$ about the $$y$$-axis and vertically stretch $$y$$ values by 3

51. $$f(-4x) = |-4x|$$

53. $$\dfrac{1}{3} f(x + 2) - 3 = \dfrac{1}{3(x + 2)^2} - 3$$

55. $$f(2(x - 5)) + 1 = (2 (x - 5))^2 + 1$$

57. Horizontal shift left 1 unit, vertical stretch $$y$$ values by 4, vertical shift down 5 units

becomes

59. Horizontal shift right 4 units, vertical stretch $$y$$ values by 2, reflect over $$x$$ axis, vertically shift up 3 units.

becomes

61. Vertically compress $$y$$ values by 1/2

becomes

63. Horizontally stretch $$x$$ values by 3, vertical shift down 3 units

becomes

65. Reflected over the $$y$$ axis, horizontally shift right 4 units $$a(x) = \sqrt{-(x - 4)}$$

becomes

67. This function is increasing on $$(-1, \infty)$$ and decreasing on $$(-\infty, -1)$$

69. This function is decreasing on $$(-\infty, 4)$$

71. This function is concave down on $$(-3, \infty)$$ and concave up on $$(-\infty, -3)$$

73. This function is concave up everywhere

75. $$f(-x)$$

77. $$3f(x)$$

79. 2$$f(-x)$$

81. $$2f(\dfrac{1}{2}x)$$

83. $$2f(x) - 2$$

85. $$-f(x + 1) + 3$$

87. $$y = -2(x + 2)^2 + 3$$

89. $$y = (\dfrac{1}{2} (x - 1))^3 + 2$$

91. $$y = \sqrt{2(x + 2)} + 1$$

93. $$y = \dfrac{-1}{(x - 2)^2} + 3$$

95. $$y = -2|x + 1| + 3$$

97. $$y = \sqrt[3]{-\dfrac{1}{2}(x - 2)} + 1$$

99. $$f(x) = \begin{cases} (x+3)^2 + 1 & if & x \le -2 \\ \dfrac{1}{2}|x - 2| + 3 & if & x > -2 \end{cases}$$

101. $$f(x) = \begin{cases} 1 & if & x < -2 \\ -2(x + 1)^2 + 4 & if & -2 \le x \le 1 \\ \sqrt[3]{x - 2} + 1 & if & x > 1 \end{cases}$$

103a. Domain: $$3.5 \le x \le 6$$
d. Range: $$-9 \le y \le 7$$