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1.5.1: Transformation of Functions

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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).

屏幕快照 2019-06-14 下午3.07.52.png

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).

屏幕快照 2019-06-14 下午3.08.46.png

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

17. g(x)=2^{x} +1屏幕快照 2019-06-14 下午3.10.53.png

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. 屏幕快照 2019-06-14 下午3.13.43.png

26. 屏幕快照 2019-06-14 下午3.14.17.png

27. 屏幕快照 2019-06-14 下午3.14.58.png

28.屏幕快照 2019-06-14 下午3.15.37.png

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

29. 屏幕快照 2019-06-14 下午3.16.39.png

30. 屏幕快照 2019-06-14 下午3.17.08.png

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

屏幕快照 2019-06-14 下午3.18.02.png

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. 屏幕快照 2019-06-14 下午3.21.50.png 36. 屏幕快照 2019-06-14 下午3.22.23.png

37. 屏幕快照 2019-06-14 下午3.23.15.png 38.屏幕快照 2019-06-14 下午3.23.48.png

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.屏幕快照 2019-06-14 下午3.38.50.png76.屏幕快照 2019-06-14 下午3.39.25.png77.屏幕快照 2019-06-14 下午3.40.01.png

78.屏幕快照 2019-06-14 下午3.40.25.png79. 屏幕快照 2019-06-14 下午3.41.11.png80. 屏幕快照 2019-06-14 下午3.41.43.png

81.屏幕快照 2019-06-14 下午3.42.36.png82.屏幕快照 2019-06-14 下午3.43.25.png 83. 屏幕快照 2019-06-14 下午3.43.59.png

84. 屏幕快照 2019-06-14 下午3.44.41.png85.屏幕快照 2019-06-14 下午3.45.12.png 86.屏幕快照 2019-06-14 下午3.45.47.png

Write an equation for each transformed toolkit function graphed below.

87. 屏幕快照 2019-06-14 下午3.47.01.png88.屏幕快照 2019-06-14 下午3.47.33.png 89. 屏幕快照 2019-06-14 下午3.47.59.png

90. 屏幕快照 2019-06-14 下午3.48.35.png91. 屏幕快照 2019-06-14 下午3.49.16.png92.屏幕快照 2019-06-14 下午3.49.47.png

93. 屏幕快照 2019-06-14 下午3.53.14.png94.屏幕快照 2019-06-14 下午3.53.57.png 95. 屏幕快照 2019-06-14 下午3.54.44.png

96. 屏幕快照 2019-06-14 下午3.55.27.png97. 屏幕快照 2019-06-14 下午3.57.57.png98.屏幕快照 2019-06-14 下午3.59.00.png

Write a formula for the piecewise function graphed below.

99. 屏幕快照 2019-06-14 下午3.59.40.png100. 屏幕快照 2019-06-14 下午4.00.25.png

101. 屏幕快照 2019-06-14 下午4.01.04.png102. 屏幕快照 2019-06-14 下午4.01.26.png

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?

Answer

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. Screen Shot 2019-10-01 at 8.56.34 AM.png19. Screen Shot 2019-10-01 at 8.56.55 AM.png

21. Screen Shot 2019-10-01 at 8.57.26 AM.png23. Screen Shot 2019-10-01 at 8.57.54 AM.png

25. y = |x - 3| - 2

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

29. y = -\sqrt{x}

31. Screen Shot 2019-10-01 at 8.58.26 AM.png

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

Screen Shot 2019-10-01 at 9.00.14 AM.png becomes Screen Shot 2019-10-01 at 9.00.39 AM.png

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

Screen Shot 2019-10-01 at 9.01.08 AM.png becomes Screen Shot 2019-10-01 at 9.01.37 AM.png

61. Vertically compress y values by 1/2

Screen Shot 2019-10-01 at 9.06.17 AM.png becomes Screen Shot 2019-10-01 at 9.06.43 AM.png

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

Screen Shot 2019-10-01 at 9.07.21 AM.png becomes Screen Shot 2019-10-01 at 9.07.43 AM.png

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

Screen Shot 2019-10-01 at 9.08.13 AM.png becomes Screen Shot 2019-10-01 at 9.08.42 AM.png

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. 2f(-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


1.5.1: Transformation of Functions is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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