Section Exercises
![Graph of f(x) increasing on (0, oo), approaching y = 0 on (-oo,0), passing through the point (1,1).](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200837/CNX_Precalc_Figure_01_05_201.jpg)
[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.
![Graph of an absolute function with vertex at (3,-2), decreasing on (-oo,3) and increasing on (3,oo).](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200838/CNX_Precalc_Figure_01_05_210.jpg)
34.
![Graph of a parabola with vertex at (1,-3), decreasing on (-oo,1) and increasing on (1,oo).](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200840/CNX_Precalc_Figure_01_05_211.jpg)
![Graph of a square root function originating at (-3,-1), increasing on [-3,oo).](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200841/CNX_Precalc_Figure_01_05_212.jpg)
36.
![Graph of an absolute function with vertex at (-2, 2), decreasing on (-oo,-2) and increasing on (-2,oo).](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200842/CNX_Precalc_Figure_01_05_213.jpg)
![Graph of a parabola with vertex at (2,0), decreasingon (-inf., 2), increasing on (2, inf.)](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200844/CNX_Precalc_Figure_01_05_214.jpg)
38.
![Graph of a square root function originating at (-3,0) increasing on [-3, inf) and passing through (1,2).](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200845/CNX_Precalc_Figure_01_05_215.jpg)
![Graph of an absolute function with vertex at (-3,-2), decreasing on (-inf., -3) and increasing on (-3,inf.), passing through (0,1).](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200847/CNX_Precalc_Figure_01_05_216f.jpg)
40.
![Graph of a square root function originating at (-2, -2) increasing on [-2, inf) and passing through (2,0).](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200848/CNX_Precalc_Figure_01_05_217F.jpg)
For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.
![Graph of a square root function originating at (0,0) decreasing on (0,inf) passing through (4,-2).](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200849/CNX_Precalc_Figure_01_05_218.jpg)
42.
![Graph of a square root function originating at (0,0) decreasing on (-inf., 0) passing through (-1,1).](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200851/CNX_Precalc_Figure_01_05_219.jpg)
For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions.
![Graph of a parabola concave down, vertex at (-1,2).](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200852/CNX_Precalc_Figure_01_05_220.jpg)
44.
![Graph of a cubic function passing through (2,1) .](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200854/CNX_Precalc_Figure_01_05_221.jpg)
![Graph of a square root function concave down, originating at (0,1) decreasing on (-inf., 0).](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200855/CNX_Precalc_Figure_01_05_222.jpg)
46.
![Graph of an absolute function downward facing, vertex at (2,3).](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200856/CNX_Precalc_Figure_01_05_223.jpg)
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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