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Study Guides > College Algebra

Solutions

Solutions to Try Its

1. The graphs of f(x)f\left(x\right) and g(x)g\left(x\right) are shown below. The transformation is a horizontal shift. The function is shifted to the left by 2 units. Graph of a square root function and a horizontally shift square foot function. 2.
Graph of a vertically reflected absolute function. a)
 
Graph of an absolute function translated one unit left. b)
3. g(x)=f(x)g\left(x\right)=-f\left(x\right)
xx -2 0 2 4
g(x)g\left(x\right) 5-5 10-10 15-15 20-20

h(x)=f(x)h\left(x\right)=f\left(-x\right)

xx -2 0 2 4
h(x)h\left(x\right) 15 10 5 unknown
4. even 5.
xx 2 4 6 8
g(x)g\left(x\right) 9 12 15 0
6. g(x)=3x2g\left(x\right)=3x - 2 7. g(x)=f(13x)g\left(x\right)=f\left(\frac{1}{3}x\right) so using the square root function we get g(x)=13xg\left(x\right)=\sqrt{\frac{1}{3}x} 8. Graph of h(x)=|x-2|+4. 9. g(x)=1x1+1g\left(x\right)=\frac{1}{x - 1}+1 10. Notice: g(x)=f(x)g\left(x\right)=f\left(-x\right) looks the same as f(x)f\left(x\right) .Graph of x^2 and its reflections.

Solution to Odd-Numbered Exercises

1. A horizontal shift results when a constant is added to or subtracted from the input. A vertical shifts results when a constant is added to or subtracted from the output. 3. A horizontal compression results when a constant greater than 1 is multiplied by the input. A vertical compression results when a constant between 0 and 1 is multiplied by the output. 5. For a function ff, substitute (x)\left(-x\right) for (x)\left(x\right) in f(x)f\left(x\right). Simplify. If the resulting function is the same as the original function, f(x)=f(x)f\left(-x\right)=f\left(x\right), then the function is even. If the resulting function is the opposite of the original function, f(x)=f(x)f\left(-x\right)=-f\left(x\right), then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even. 7. g(x)=x13g\left(x\right)=|x - 1|-3 9. g(x)=1(x+4)2+2g\left(x\right)=\frac{1}{{\left(x+4\right)}^{2}}+2 11. The graph of f(x+43)f\left(x+43\right) is a horizontal shift to the left 43 units of the graph of ff. 13. The graph of f(x4)f\left(x - 4\right) is a horizontal shift to the right 4 units of the graph of ff. 15. The graph of f(x)+8f\left(x\right)+8 is a vertical shift up 8 units of the graph of ff. 17. The graph of f(x)7f\left(x\right)-7 is a vertical shift down 7 units of the graph of ff. 19. The graph of f(x+4)1f\left(x+4\right)-1 is a horizontal shift to the left 4 units and a vertical shift down 1 unit of the graph of ff. 21. decreasing on (,3)\left(-\infty ,-3\right) and increasing on (3,)\left(-3,\infty \right) 23. decreasing on (0,)\left(0,\infty \right) 25. Graph of k(x). 27. Graph of f(t). 29. Graph of k(x). 31. g(x)=f(x1),h(x)=f(x)+1g\left(x\right)=f\left(x - 1\right),h\left(x\right)=f\left(x\right)+1 33. f(x)=x32f\left(x\right)=|x - 3|-2 35. f(x)=x+31f\left(x\right)=\sqrt{x+3}-1 37. f(x)=(x2)2f\left(x\right)={\left(x - 2\right)}^{2} 39. f(x)=x+32f\left(x\right)=|x+3|-2 41. f(x)=xf\left(x\right)=-\sqrt{x} 43. f(x)=(x+1)2+2f\left(x\right)=-{\left(x+1\right)}^{2}+2 45. f(x)=x+1f\left(x\right)=\sqrt{-x}+1 47. even 49. odd 51. even 53. The graph of gg is a vertical reflection (across the xx -axis) of the graph of ff. 55. The graph of gg is a vertical stretch by a factor of 4 of the graph of ff. 57. The graph of gg is a horizontal compression by a factor of 15\frac{1}{5} of the graph of ff. 59. The graph of gg is a horizontal stretch by a factor of 3 of the graph of ff. 61. The graph of gg is a horizontal reflection across the yy -axis and a vertical stretch by a factor of 3 of the graph of ff. 63. g(x)=4xg\left(x\right)=|-4x| 65. g(x)=13(x+2)23g\left(x\right)=\frac{1}{3{\left(x+2\right)}^{2}}-3 67. g(x)=12(x5)2+1g\left(x\right)=\frac{1}{2}{\left(x - 5\right)}^{2}+1 69. The graph of the function f(x)=x2f\left(x\right)={x}^{2} is shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units. Graph of a parabola. 71. The graph of f(x)=xf\left(x\right)=|x| is stretched vertically by a factor of 2, shifted horizontally 4 units to the right, reflected across the horizontal axis, and then shifted vertically 3 units up. Graph of an absolute function. 73. The graph of the function f(x)=x3f\left(x\right)={x}^{3} is compressed vertically by a factor of 12\frac{1}{2}. Graph of a cubic function. 75. The graph of the function is stretched horizontally by a factor of 3 and then shifted vertically downward by 3 units. Graph of a cubic function. 77. The graph of f(x)=xf\left(x\right)=\sqrt{x} is shifted right 4 units and then reflected across the vertical line x=4x=4. Graph of a square root function. 79.
Graph of a polynomial.
81.
Graph of a polynomial.

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