{"id":991,"date":"2015-11-12T18:35:32","date_gmt":"2015-11-12T18:35:32","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=991"},"modified":"2015-11-12T18:35:32","modified_gmt":"2015-11-12T18:35:32","slug":"solutions-52","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/atd-sanjac-collegealgebra\/chapter\/solutions-52\/","title":{"raw":"Solutions","rendered":"Solutions"},"content":{"raw":"

Solutions to Try Its<\/h2>\n1.\u00a0The graphs of [latex]f\\left(x\\right)\\\\[\/latex] and [latex]g\\left(x\\right)\\\\[\/latex] are shown below. The transformation is a horizontal shift. The function is shifted to the left by 2 units.\n\n\"Graph<\/span>\n\n2.\n\n[caption id=\"\" align=\"alignnone\" width=\"487\"]\"Graph a)[\/caption]\n\n\u00a0\n\n[caption id=\"\" align=\"alignnone\" width=\"487\"]\"Graph b)[\/caption]\n\n3.\u00a0[latex]g\\left(x\\right)=-f\\left(x\\right)\\\\[\/latex]\n<\/colgroup>
[latex]x\\\\[\/latex]<\/td>\n-2<\/td>\n0<\/td>\n2<\/td>\n4<\/td>\n<\/tr>
[latex]g\\left(x\\right)\\\\[\/latex]<\/td>\n[latex]-5\\\\[\/latex]<\/td>\n[latex]-10\\\\[\/latex]<\/td>\n[latex]-15\\\\[\/latex]<\/td>\n[latex]-20\\\\[\/latex]<\/td>\n<\/tr><\/tbody><\/table>

[latex]h\\left(x\\right)=f\\left(-x\\right)\\\\[\/latex]<\/p>\n\n<\/colgroup>
[latex]x\\\\[\/latex]<\/td>\n-2<\/td>\n0<\/td>\n2<\/td>\n4<\/td>\n<\/tr>
[latex]h\\left(x\\right)\\\\[\/latex]<\/td>\n15<\/td>\n10<\/td>\n5<\/td>\nunknown<\/td>\n<\/tr><\/tbody><\/table>\n4. even\n\n5.\n
[latex]x\\\\[\/latex]<\/td>\n2<\/td>\n4<\/td>\n6<\/td>\n8<\/td>\n<\/tr>
[latex]g\\left(x\\right)\\\\[\/latex]<\/td>\n9<\/td>\n12<\/td>\n15<\/td>\n0<\/td>\n<\/tr><\/tbody><\/table>\n6.\u00a0[latex]g\\left(x\\right)=3x - 2\\\\[\/latex]\n\n7.\u00a0[latex]g\\left(x\\right)=f\\left(\\frac{1}{3}x\\right)\\\\[\/latex] so using the square root function we get [latex]g\\left(x\\right)=\\sqrt{\\frac{1}{3}x}\\\\[\/latex]\n\n8.\n\"Graph\n\n9.\u00a0[latex]g\\left(x\\right)=\\frac{1}{x - 1}+1\\\\[\/latex]\n\n10.\u00a0Notice: [latex]g\\left(x\\right)=f\\left(-x\\right)\\\\[\/latex] looks the same as [latex]f\\left(x\\right)\\\\[\/latex] .\"Graph<\/span>\n

Solution to Odd-Numbered Exercises<\/h2>\n1.\u00a0A 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.\n\n3.\u00a0A 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.\n\n5.\u00a0For a function [latex]f[\/latex], substitute [latex]\\left(-x\\right)\\\\[\/latex] for [latex]\\left(x\\right)\\\\[\/latex] in [latex]f\\left(x\\right)\\\\[\/latex]. Simplify. If the resulting function is the same as the original function, [latex]f\\left(-x\\right)=f\\left(x\\right)\\\\[\/latex], then the function is even. If the resulting function is the opposite of the original function, [latex]f\\left(-x\\right)=-f\\left(x\\right)\\\\[\/latex], then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even.\n\n7.\u00a0[latex]g\\left(x\\right)=|x - 1|-3\\\\[\/latex]\n\n9.\u00a0[latex]g\\left(x\\right)=\\frac{1}{{\\left(x+4\\right)}^{2}}+2\\\\[\/latex]\n\n11.\u00a0The graph of [latex]f\\left(x+43\\right)\\\\[\/latex] is a horizontal shift to the left 43 units of the graph of [latex]f\\\\[\/latex].\n\n13.\u00a0The graph of [latex]f\\left(x - 4\\right)\\\\[\/latex] is a horizontal shift to the right 4 units of the graph of [latex]f\\\\[\/latex].\n\n15.\u00a0The graph of [latex]f\\left(x\\right)+8\\\\[\/latex] is a vertical shift up 8 units of the graph of [latex]f\\\\[\/latex].\n\n17.\u00a0The graph of [latex]f\\left(x\\right)-7\\\\[\/latex] is a vertical shift down 7 units of the graph of [latex]f\\\\[\/latex].\n\n19. The graph of [latex]f\\left(x+4\\right)-1\\\\[\/latex] is a horizontal shift to the left 4 units and a vertical shift down 1 unit of the graph of [latex]f\\\\[\/latex].\n\n21.\u00a0decreasing on [latex]\\left(-\\infty ,-3\\right)\\\\[\/latex] and increasing on [latex]\\left(-3,\\infty \\right)\\\\[\/latex]\n\n23.\u00a0decreasing on [latex]\\left(0,\\infty \\right)\\\\[\/latex]\n\n25.\n\"Graph\n\n27.\n\"Graph\n\n29.\n\"Graph\n\n31.\u00a0[latex]g\\left(x\\right)=f\\left(x - 1\\right),h\\left(x\\right)=f\\left(x\\right)+1\\\\[\/latex]\n\n33.\u00a0[latex]f\\left(x\\right)=|x - 3|-2\\\\[\/latex]\n\n35.\u00a0[latex]f\\left(x\\right)=\\sqrt{x+3}-1\\\\[\/latex]\n\n37.\u00a0[latex]f\\left(x\\right)={\\left(x - 2\\right)}^{2}\\\\[\/latex]\n\n39.\u00a0[latex]f\\left(x\\right)=|x+3|-2\\\\[\/latex]\n\n41.\u00a0[latex]f\\left(x\\right)=-\\sqrt{x}\\\\[\/latex]\n\n43.\u00a0[latex]f\\left(x\\right)=-{\\left(x+1\\right)}^{2}+2\\\\[\/latex]\n\n45.\u00a0[latex]f\\left(x\\right)=\\sqrt{-x}+1\\\\[\/latex]\n\n47. even\n\n49. odd\n\n51. even\n\n53.\u00a0The graph of [latex]g\\\\[\/latex] is a vertical reflection (across the [latex]x\\\\[\/latex] -axis) of the graph of [latex]f\\\\[\/latex].\n\n55.\u00a0The graph of [latex]g\\\\[\/latex] is a vertical stretch by a factor of 4 of the graph of [latex]f\\\\[\/latex].\n\n57.\u00a0The graph of [latex]g\\\\[\/latex] is a horizontal compression by a factor of [latex]\\frac{1}{5}\\\\[\/latex] of the graph of [latex]f\\\\[\/latex].\n\n59.\u00a0The graph of [latex]g\\\\[\/latex] is a horizontal stretch by a factor of 3 of the graph of [latex]f\\\\[\/latex].\n\n61.\u00a0The graph of [latex]g\\\\[\/latex] is a horizontal reflection across the [latex]y\\\\[\/latex] -axis and a vertical stretch by a factor of 3 of the graph of [latex]f\\\\[\/latex].\n\n63.\u00a0[latex]g\\left(x\\right)=|-4x|\\\\[\/latex]\n\n65.\u00a0[latex]g\\left(x\\right)=\\frac{1}{3{\\left(x+2\\right)}^{2}}-3\\\\[\/latex]\n\n67.\u00a0[latex]g\\left(x\\right)=\\frac{1}{2}{\\left(x - 5\\right)}^{2}+1\\\\[\/latex]\n\n69.\u00a0The graph of the function [latex]f\\left(x\\right)={x}^{2}\\\\[\/latex] is shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units.\n\"Graph<\/span>\n\n71.\u00a0The graph of [latex]f\\left(x\\right)=|x|\\\\[\/latex] 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.\n\"Graph<\/span>\n\n73.\u00a0The graph of the function [latex]f\\left(x\\right)={x}^{3}\\\\[\/latex] is compressed vertically by a factor of [latex]\\frac{1}{2}\\\\[\/latex].\n\"Graph<\/span>\n\n75.\u00a0The graph of the function is stretched horizontally by a factor of 3 and then shifted vertically downward by 3 units.\n\"Graph<\/span>\n\n77.\u00a0The graph of [latex]f\\left(x\\right)=\\sqrt{x}\\\\[\/latex] is shifted right 4 units and then reflected across the vertical line [latex]x=4\\\\[\/latex].\n\"Graph<\/span>\n\n79.\n
\"Graph<\/span><\/div>\n81.\n
\"Graph<\/span><\/div>","rendered":"

Solutions to Try Its<\/h2>\n

1.\u00a0The graphs of [latex]f\\left(x\\right)\\\\[\/latex] and [latex]g\\left(x\\right)\\\\[\/latex] are shown below. The transformation is a horizontal shift. The function is shifted to the left by 2 units.
\n
\n\"Graph<\/span><\/p>\n

2.<\/p>\n

\"Graph<\/p>\n

a)<\/p>\n<\/div>\n

\u00a0<\/p>\n

\"Graph<\/p>\n

b)<\/p>\n<\/div>\n

3.\u00a0[latex]g\\left(x\\right)=-f\\left(x\\right)\\\\[\/latex]<\/p>\n\n\n\n\n\n\n<\/colgroup>\n\n\n\n
[latex]x\\\\[\/latex]<\/td>\n-2<\/td>\n0<\/td>\n2<\/td>\n4<\/td>\n<\/tr>\n
[latex]g\\left(x\\right)\\\\[\/latex]<\/td>\n[latex]-5\\\\[\/latex]<\/td>\n[latex]-10\\\\[\/latex]<\/td>\n[latex]-15\\\\[\/latex]<\/td>\n[latex]-20\\\\[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

[latex]h\\left(x\\right)=f\\left(-x\\right)\\\\[\/latex]<\/p>\n\n\n\n\n\n\n<\/colgroup>\n\n\n\n
[latex]x\\\\[\/latex]<\/td>\n-2<\/td>\n0<\/td>\n2<\/td>\n4<\/td>\n<\/tr>\n
[latex]h\\left(x\\right)\\\\[\/latex]<\/td>\n15<\/td>\n10<\/td>\n5<\/td>\nunknown<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

4. even<\/p>\n

5.<\/p>\n\n<\/colgroup>\n\n\n\n
[latex]x\\\\[\/latex]<\/td>\n2<\/td>\n4<\/td>\n6<\/td>\n8<\/td>\n<\/tr>\n
[latex]g\\left(x\\right)\\\\[\/latex]<\/td>\n9<\/td>\n12<\/td>\n15<\/td>\n0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

6.\u00a0[latex]g\\left(x\\right)=3x - 2\\\\[\/latex]<\/p>\n

7.\u00a0[latex]g\\left(x\\right)=f\\left(\\frac{1}{3}x\\right)\\\\[\/latex] so using the square root function we get [latex]g\\left(x\\right)=\\sqrt{\\frac{1}{3}x}\\\\[\/latex]<\/p>\n

8.
\n\"Graph<\/p>\n

9.\u00a0[latex]g\\left(x\\right)=\\frac{1}{x - 1}+1\\\\[\/latex]<\/p>\n

10.\u00a0Notice: [latex]g\\left(x\\right)=f\\left(-x\\right)\\\\[\/latex] looks the same as [latex]f\\left(x\\right)\\\\[\/latex] .\"Graph<\/span><\/p>\n

Solution to Odd-Numbered Exercises<\/h2>\n

1.\u00a0A 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.<\/p>\n

3.\u00a0A 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.<\/p>\n

5.\u00a0For a function [latex]f[\/latex], substitute [latex]\\left(-x\\right)\\\\[\/latex] for [latex]\\left(x\\right)\\\\[\/latex] in [latex]f\\left(x\\right)\\\\[\/latex]. Simplify. If the resulting function is the same as the original function, [latex]f\\left(-x\\right)=f\\left(x\\right)\\\\[\/latex], then the function is even. If the resulting function is the opposite of the original function, [latex]f\\left(-x\\right)=-f\\left(x\\right)\\\\[\/latex], then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even.<\/p>\n

7.\u00a0[latex]g\\left(x\\right)=|x - 1|-3\\\\[\/latex]<\/p>\n

9.\u00a0[latex]g\\left(x\\right)=\\frac{1}{{\\left(x+4\\right)}^{2}}+2\\\\[\/latex]<\/p>\n

11.\u00a0The graph of [latex]f\\left(x+43\\right)\\\\[\/latex] is a horizontal shift to the left 43 units of the graph of [latex]f\\\\[\/latex].<\/p>\n

13.\u00a0The graph of [latex]f\\left(x - 4\\right)\\\\[\/latex] is a horizontal shift to the right 4 units of the graph of [latex]f\\\\[\/latex].<\/p>\n

15.\u00a0The graph of [latex]f\\left(x\\right)+8\\\\[\/latex] is a vertical shift up 8 units of the graph of [latex]f\\\\[\/latex].<\/p>\n

17.\u00a0The graph of [latex]f\\left(x\\right)-7\\\\[\/latex] is a vertical shift down 7 units of the graph of [latex]f\\\\[\/latex].<\/p>\n

19. The graph of [latex]f\\left(x+4\\right)-1\\\\[\/latex] is a horizontal shift to the left 4 units and a vertical shift down 1 unit of the graph of [latex]f\\\\[\/latex].<\/p>\n

21.\u00a0decreasing on [latex]\\left(-\\infty ,-3\\right)\\\\[\/latex] and increasing on [latex]\\left(-3,\\infty \\right)\\\\[\/latex]<\/p>\n

23.\u00a0decreasing on [latex]\\left(0,\\infty \\right)\\\\[\/latex]<\/p>\n

25.
\n\"Graph<\/p>\n

27.
\n\"Graph<\/p>\n

29.
\n\"Graph<\/p>\n

31.\u00a0[latex]g\\left(x\\right)=f\\left(x - 1\\right),h\\left(x\\right)=f\\left(x\\right)+1\\\\[\/latex]<\/p>\n

33.\u00a0[latex]f\\left(x\\right)=|x - 3|-2\\\\[\/latex]<\/p>\n

35.\u00a0[latex]f\\left(x\\right)=\\sqrt{x+3}-1\\\\[\/latex]<\/p>\n

37.\u00a0[latex]f\\left(x\\right)={\\left(x - 2\\right)}^{2}\\\\[\/latex]<\/p>\n

39.\u00a0[latex]f\\left(x\\right)=|x+3|-2\\\\[\/latex]<\/p>\n

41.\u00a0[latex]f\\left(x\\right)=-\\sqrt{x}\\\\[\/latex]<\/p>\n

43.\u00a0[latex]f\\left(x\\right)=-{\\left(x+1\\right)}^{2}+2\\\\[\/latex]<\/p>\n

45.\u00a0[latex]f\\left(x\\right)=\\sqrt{-x}+1\\\\[\/latex]<\/p>\n

47. even<\/p>\n

49. odd<\/p>\n

51. even<\/p>\n

53.\u00a0The graph of [latex]g\\\\[\/latex] is a vertical reflection (across the [latex]x\\\\[\/latex] -axis) of the graph of [latex]f\\\\[\/latex].<\/p>\n

55.\u00a0The graph of [latex]g\\\\[\/latex] is a vertical stretch by a factor of 4 of the graph of [latex]f\\\\[\/latex].<\/p>\n

57.\u00a0The graph of [latex]g\\\\[\/latex] is a horizontal compression by a factor of [latex]\\frac{1}{5}\\\\[\/latex] of the graph of [latex]f\\\\[\/latex].<\/p>\n

59.\u00a0The graph of [latex]g\\\\[\/latex] is a horizontal stretch by a factor of 3 of the graph of [latex]f\\\\[\/latex].<\/p>\n

61.\u00a0The graph of [latex]g\\\\[\/latex] is a horizontal reflection across the [latex]y\\\\[\/latex] -axis and a vertical stretch by a factor of 3 of the graph of [latex]f\\\\[\/latex].<\/p>\n

63.\u00a0[latex]g\\left(x\\right)=|-4x|\\\\[\/latex]<\/p>\n

65.\u00a0[latex]g\\left(x\\right)=\\frac{1}{3{\\left(x+2\\right)}^{2}}-3\\\\[\/latex]<\/p>\n

67.\u00a0[latex]g\\left(x\\right)=\\frac{1}{2}{\\left(x - 5\\right)}^{2}+1\\\\[\/latex]<\/p>\n

69.\u00a0The graph of the function [latex]f\\left(x\\right)={x}^{2}\\\\[\/latex] is shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units.
\n\"Graph<\/span><\/p>\n

71.\u00a0The graph of [latex]f\\left(x\\right)=|x|\\\\[\/latex] 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.
\n\"Graph<\/span><\/p>\n

73.\u00a0The graph of the function [latex]f\\left(x\\right)={x}^{3}\\\\[\/latex] is compressed vertically by a factor of [latex]\\frac{1}{2}\\\\[\/latex].
\n\"Graph<\/span><\/p>\n

75.\u00a0The graph of the function is stretched horizontally by a factor of 3 and then shifted vertically downward by 3 units.
\n\"Graph<\/span><\/p>\n

77.\u00a0The graph of [latex]f\\left(x\\right)=\\sqrt{x}\\\\[\/latex] is shifted right 4 units and then reflected across the vertical line [latex]x=4\\\\[\/latex].
\n\"Graph<\/span><\/p>\n

79.<\/p>\n

\"Graph<\/span><\/div>\n

81.<\/p>\n

\"Graph<\/span><\/div>\n\n\t\t\t
\n\t\t\t

Candela Citations<\/h3>\n\t\t\t\t\t
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CC licensed content, Shared previously<\/div>