Section Exercises
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[latex]x[/latex] | −2 | −1 | 0 | 1 | 2 |
[latex]f\left(x\right)[/latex] | −2 | −1 | −3 | 1 | 2 |
[latex]x[/latex] | −1 | 0 | 1 | 2 | 3 |
[latex]g\left(x\right)[/latex] | −2 | −1 | −3 | 1 | 2 |
[latex]x[/latex] | −2 | −1 | 0 | 1 | 2 |
[latex]h\left(x\right)[/latex] | −1 | 0 | −2 | 2 | 3 |
32. Tabular representations for the functions [latex]f,g[/latex], and [latex]h[/latex] are given below. Write [latex]g\left(x\right)[/latex] and [latex]h\left(x\right)[/latex] as transformations of [latex]f\left(x\right)[/latex].
[latex]x[/latex] | −2 | −1 | 0 | 1 | 2 |
[latex]f\left(x\right)[/latex] | −1 | −3 | 4 | 2 | 1 |
[latex]x[/latex] | −3 | −2 | −1 | 0 | 1 |
[latex]g\left(x\right)[/latex] | −1 | −3 | 4 | 2 | 1 |
[latex]x[/latex] | −2 | −1 | 0 | 1 | 2 |
[latex]h\left(x\right)[/latex] | −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.
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34.
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36.
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38.
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40.
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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.
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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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44.
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46.
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For the following exercises, determine whether the function is odd, even, or neither.
47. [latex]f\left(x\right)=3{x}^{4}[/latex]
48. [latex]g\left(x\right)=\sqrt{x}[/latex]
49. [latex]h\left(x\right)=\frac{1}{x}+3x[/latex]
50. [latex]f\left(x\right)={\left(x - 2\right)}^{2}[/latex]
51. [latex]g\left(x\right)=2{x}^{4}[/latex]
52. [latex]h\left(x\right)=2x-{x}^{3}[/latex]
For the following exercises, describe how the graph of each function is a transformation of the graph of the original function [latex]f[/latex].
53. [latex]g\left(x\right)=-f\left(x\right)[/latex]
54. [latex]g\left(x\right)=f\left(-x\right)[/latex]
55. [latex]g\left(x\right)=4f\left(x\right)[/latex]
56. [latex]g\left(x\right)=6f\left(x\right)[/latex]
57. [latex]g\left(x\right)=f\left(5x\right)[/latex]
58. [latex]g\left(x\right)=f\left(2x\right)[/latex]
59. [latex]g\left(x\right)=f\left(\frac{1}{3}x\right)[/latex]
60. [latex]g\left(x\right)=f\left(\frac{1}{5}x\right)[/latex]
61. [latex]g\left(x\right)=3f\left(-x\right)[/latex]
62. [latex]g\left(x\right)=-f\left(3x\right)[/latex]
For the following exercises, write a formula for the function [latex]g[/latex] that results when the graph of a given toolkit function is transformed as described.
63. The graph of [latex]f\left(x\right)=|x|[/latex] is reflected over the [latex]y[/latex] -axis and horizontally compressed by a factor of [latex]\frac{1}{4}[/latex] .
64. The graph of [latex]f\left(x\right)=\sqrt{x}[/latex] is reflected over the [latex]x[/latex] -axis and horizontally stretched by a factor of 2.65. The graph of [latex]f\left(x\right)=\frac{1}{{x}^{2}}[/latex] is vertically compressed by a factor of [latex]\frac{1}{3}[/latex], then shifted to the left 2 units and down 3 units.
66. The graph of [latex]f\left(x\right)=\frac{1}{x}[/latex] is vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.
67. The graph of [latex]f\left(x\right)={x}^{2}[/latex] is vertically compressed by a factor of [latex]\frac{1}{2}[/latex], then shifted to the right 5 units and up 1 unit.
68. The graph of [latex]f\left(x\right)={x}^{2}[/latex] 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. [latex]g\left(x\right)=4{\left(x+1\right)}^{2}-5[/latex]
70. [latex]g\left(x\right)=5{\left(x+3\right)}^{2}-2[/latex]
71. [latex]h\left(x\right)=-2|x - 4|+3[/latex]
72. [latex]k\left(x\right)=-3\sqrt{x}-1[/latex]
73. [latex]m\left(x\right)=\frac{1}{2}{x}^{3}[/latex]
74. [latex]n\left(x\right)=\frac{1}{3}|x - 2|[/latex]
75. [latex]p\left(x\right)={\left(\frac{1}{3}x\right)}^{3}-3[/latex]
76. [latex]q\left(x\right)={\left(\frac{1}{4}x\right)}^{3}+1[/latex]
77. [latex]a\left(x\right)=\sqrt{-x+4}[/latex]
For the following exercises, use the graph below to sketch the given transformations.
78. [latex]g\left(x\right)=f\left(x\right)-2[/latex]
79. [latex]g\left(x\right)=-f\left(x\right)[/latex]
80. [latex]g\left(x\right)=f\left(x+1\right)[/latex]
81. [latex]g\left(x\right)=f\left(x - 2\right)[/latex]
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