{"id":2957,"date":"2019-10-01T14:36:12","date_gmt":"2019-10-01T14:36:12","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&p=2957"},"modified":"2019-10-09T23:28:10","modified_gmt":"2019-10-09T23:28:10","slug":"1-7-section-exercises","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/1-7-section-exercises\/","title":{"raw":"1.7 Section Exercises","rendered":"1.7 Section Exercises"},"content":{"raw":"

1.7 Section Exercises<\/h3>\r\n
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Verbal<\/h4>\r\n
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1. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal stretch from a vertical stretch?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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2. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal compression from a vertical compression?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135541729\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135541729\"]\r\n

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.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Algebraic<\/h4>\r\n3.\u00a0 Write a formula for the function obtained when the graph of [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x}[\/latex] is vertically stretched by a factor of 4.\r\n\r\n[reveal-answer q=\"653711\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"653711\"][latex]g\\left(x\\right)=4f\\left(x\\right)=4\\sqrt[\\leftroot{1}\\uproot{2} ]{x}[\/latex][latex]\\\\[\/latex][\/hidden-answer]\r\n\r\n4.\u00a0Write a formula for the function obtained when the graph of [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x}[\/latex] is vertically compressed by a factor of 1\/3.\r\n\r\n5.\u00a0Write a formula for the function obtained when the graph of [latex]f\\left(x\\right)=|x|[\/latex] is horizontally stretched by a factor of 5.\r\n\r\n[reveal-answer q=\"231797\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"231797\"][latex]g\\left(x\\right)=\\frac{1}{3}f\\left(x\\right)=\\frac{1}{3}|x|[\/latex][latex]\\\\[\/latex][\/hidden-answer]\r\n\r\n6.\u00a0Write a formula for the function obtained when the graph of [latex]f\\left(x\\right)=|x|[\/latex] is horizontally compressed by a factor of 1\/4.\r\n\r\n7.\u00a0Write a formula for the function obtained when the graph of [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x}[\/latex] is horizontally stretched by a factor of 3 and vertically stretched by a factor of 6.\r\n\r\n[reveal-answer q=\"691205\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"691205\"][latex]g\\left(x\\right)=6f\\left(\\frac{1}{3}x\\right)=6\\sqrt[\\leftroot{1}\\uproot{2} ]{\\frac{1}{3}x}[\/latex][latex]\\\\[\/latex][\/hidden-answer]\r\n

8.\u00a0Write a formula for the function obtained when the graph of [latex]f\\left(x\\right)=|x|[\/latex] is horizontally compressed by a factor of 1\/2 and vertically compressed by a factor of 1\/7.<\/p>\r\n

For the following exercises, write a formula for the function[latex]\\text{ }g\\text{ }[\/latex]that results when the graph of a given toolkit function is transformed as described.<\/strong><\/p>\r\n\r\n

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9. The graph of[latex]\\text{ }f\\left(x\\right)=|x|\\text{ }[\/latex]is reflected over the[latex]\\text{ }y[\/latex]-<\/em>axis and horizontally compressed by a factor of[latex]\\text{ }\\frac{1}{4}[\/latex] .<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137810354\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137810354\"]\r\n

[latex]g\\left(x\\right)=|-4x|[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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10. The graph of[latex]\\text{ }f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x}\\text{ }[\/latex]is reflected over the[latex]\\text{ }x[\/latex]-axis and horizontally stretched by a factor of 2.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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11. The graph of[latex]\\text{ }f\\left(x\\right)=\\frac{1}{{x}^{2}}\\text{ }[\/latex]is vertically compressed by a factor of[latex]\\text{ }\\frac{1}{3},\\text{ }[\/latex]then shifted to the left 2 units and down 3 units.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137812539\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137812539\"]\r\n

[latex]g\\left(x\\right)=\\frac{1}{3{\\left(x+2\\right)}^{2}}-3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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12. The graph of[latex]\\text{ }f\\left(x\\right)=\\frac{1}{x}\\text{ }[\/latex]is vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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13. The graph of[latex]\\text{ }f\\left(x\\right)={x}^{2}\\text{ }[\/latex]is vertically compressed by a factor of[latex]\\text{ }\\frac{1}{2},\\text{ }[\/latex]then shifted to the right 5 units and up 1 unit.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165133047549\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133047549\"]\r\n

[latex]g\\left(x\\right)=\\frac{1}{2}{\\left(x-5\\right)}^{2}+1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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14. The graph of[latex]\\text{ }f\\left(x\\right)={x}^{2}\\text{ }[\/latex]is horizontally stretched by a factor of 3, then shifted to the left 4 units and down 3 units.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\nFor the following exercises, describe how the graph of the function is a transformation of the graph of the original function[latex]\\text{ }f.[\/latex]<\/strong>\r\n

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15. [latex]h\\left(x\\right)=3f\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134038728\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134038728\"]\r\n

The graph of [latex]3f\\left(x\\right)[\/latex] is a vertical stretch by a factor of 3 of the graph of [latex]f.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

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16. [latex]g\\left(x\\right)=0.5f\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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17. [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135208810\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135208810\"]\r\n

The graph of[latex]\\text{ }g\\text{ }[\/latex]is a vertical reflection (across the [latex]\\text{ }x[\/latex]-axis) of the graph of[latex]\\text{ }f.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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18. [latex]g\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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19. [latex]g\\left(x\\right)=4f\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135193808\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135193808\"]\r\n

The graph of[latex]\\text{ }g\\text{ }[\/latex]is a vertical stretch by a factor of 4 of the graph of[latex]\\text{ }f.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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20. [latex]g\\left(x\\right)=6f\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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21. [latex]g\\left(x\\right)=f\\left(5x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135481187\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135481187\"]\r\n

The graph of[latex]\\text{ }g\\text{ }[\/latex]is a horizontal compression by a factor of[latex]\\text{ }\\frac{1}{5}\\text{ }[\/latex]of the graph of[latex]\\text{ }f.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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22. [latex]g\\left(x\\right)=f\\left(2x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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23. [latex]g\\left(x\\right)=f\\left(\\frac{1}{3}x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137749379\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137749379\"]\r\n

The graph of[latex]\\text{ }g\\text{ }[\/latex]is a horizontal stretch by a factor of 3 of the graph of[latex]\\text{ }f.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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24. [latex]g\\left(x\\right)=f\\left(\\frac{1}{5}x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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25. [latex]g\\left(x\\right)=3f\\left(-x\\right)[\/latex]<\/p>\r\n[reveal-answer q=\"784563\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"784563\"]The graph of [latex]g[\/latex] is a horizontal reflection over the y-axis and a vertical stretch by a factor of 3 of [latex]f.[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n

<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, determine the interval(s) on which the function is increasing and decreasing.<\/strong><\/p>\r\n\r\n

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26. [latex]f\\left(x\\right)=4{\\left(x+1\\right)}^{2}-5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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27. [latex]g\\left(x\\right)=5{\\left(x+3\\right)}^{2}-2[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165133086204\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133086204\"]decreasing on[latex]\\text{ }\\left(-\\infty ,-3\\right)\\text{ }[\/latex]and increasing on[latex]\\text{ }\\left(-3,\\infty \\right)[\/latex]\r\n[\/hidden-answer]<\/div>\r\n
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28. [latex]a\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{-x+4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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29. [latex]k\\left(x\\right)=-3\\sqrt[\\leftroot{1}\\uproot{2} ]{x}-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135628497\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135628497\"]\r\n

decreasing on [latex]\\left(0,\\text{ }\\infty \\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Graphical<\/h4>\r\n

For the following exercises, use the graph of[latex]\\text{ }f\\left(x\\right)={2}^{x}\\text{ }[\/latex]shown in Figure 1<\/a> to sketch a graph of each transformation of[latex]\\text{ }f\\left(x\\right).[\/latex]<\/p>\r\n\r\n

\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"360\"]\"Graph Figure 1[\/caption]\r\n\r\n<\/div>\r\n
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\r\n\r\n30. [latex]g\\left(x\\right)=3\\left({2}^{x}\\right)+1[\/latex]\r\n\r\n31. [latex]w\\left(x\\right)=-4{2}^{x-1}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.<\/strong><\/p>\r\n\r\n

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32. [latex]f\\left(t\\right)=4{\\left(t+1\\right)}^{2}-3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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33. [latex]h\\left(x\\right)=-2|x-1|+4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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34. [latex]k\\left(x\\right)={\\left(0.5x-2\\right)}^{3}-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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35. [latex]m\\left(t\\right)=3+\\sqrt[\\leftroot{1}\\uproot{2} ]{0.25t+2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Numeric<\/h4>\r\n
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36. Tabular representations for the functions[latex]\\text{ }f,\\text{ }g,\\text{ }[\/latex]and[latex]\\text{ }h\\text{ }[\/latex]are given below. Write[latex]\\text{ }g\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }h\\left(x\\right)\\text{ }[\/latex]as transformations of[latex]\\text{ }f\\left(x\\right).[\/latex]<\/p>\r\n\r\n<\/colgroup>\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/strong><\/td>\r\n\u22122<\/td>\r\n\u22121<\/td>\r\n0<\/td>\r\n1<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\r\n\u22121<\/td>\r\n\u22123<\/td>\r\n4<\/td>\r\n2<\/td>\r\n1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/colgroup>\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/strong><\/td>\r\n\u22124<\/td>\r\n\u22122<\/td>\r\n0<\/td>\r\n2<\/td>\r\n4<\/td>\r\n<\/tr>\r\n
[latex]g\\left(x\\right)[\/latex]<\/strong><\/td>\r\n\u22121<\/td>\r\n\u22123<\/td>\r\n4<\/td>\r\n2<\/td>\r\n1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/colgroup>\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/strong><\/td>\r\n\u22122<\/td>\r\n\u22121<\/td>\r\n0<\/td>\r\n1<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
[latex]h\\left(x\\right)[\/latex]<\/strong><\/td>\r\n3<\/td>\r\n9<\/td>\r\n-12<\/td>\r\n-6<\/td>\r\n-3<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n

37. Tabular representations for the functions[latex]\\text{ }f,\\text{ }g,\\text{ }[\/latex]and[latex]\\text{ }h\\text{ }[\/latex]are given below. Write[latex]\\text{ }g\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }h\\left(x\\right)\\text{ }[\/latex]as transformations of[latex]\\text{ }f\\left(x\\right).[\/latex]<\/p>\r\n\r\n<\/colgroup>\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/strong><\/td>\r\n\u22122<\/td>\r\n\u22121<\/td>\r\n0<\/td>\r\n1<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\r\n\u22122<\/td>\r\n\u22121<\/td>\r\n\u22123<\/td>\r\n1<\/td>\r\n2<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/colgroup>\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/strong><\/td>\r\n\u22121<\/td>\r\n-1\/2<\/td>\r\n0<\/td>\r\n1\/2<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
[latex]g\\left(x\\right)[\/latex]<\/strong><\/td>\r\n\u22122<\/td>\r\n\u22121<\/td>\r\n\u22123<\/td>\r\n1<\/td>\r\n2<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/colgroup>\r\n\r\n\r\n\r\n
[latex]x[\/latex]<\/strong><\/td>\r\n\u22122<\/td>\r\n\u22121<\/td>\r\n0<\/td>\r\n1<\/td>\r\n2<\/td>\r\n<\/tr>\r\n
[latex]h\\left(x\\right)[\/latex]<\/strong><\/td>\r\n\u22124<\/td>\r\n-2<\/td>\r\n-6<\/td>\r\n2<\/td>\r\n4<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165137443424\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137443424\"]\r\n

[latex]g\\left(x\\right)=f\\left(2x\\right),\\text{ }h\\left(x\\right)=2f\\left(x\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.<\/strong><\/p>\r\n\r\n

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<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.<\/strong><\/p>\r\n\r\n

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41. [latex]g\\left(x\\right)=4{\\left(x+1\\right)}^{2}-5[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137445949\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137445949\"]\r\n

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

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42. [latex]g\\left(x\\right)=5{\\left(x+3\\right)}^{2}-2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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43. [latex]h\\left(x\\right)=-2|x-4|+3[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137762365\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137762365\"]\r\n

The graph of[latex]\\text{ }f\\left(x\\right)=|x|\\text{ }[\/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.<\/p>\r\n\"Graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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44. [latex]k\\left(x\\right)=-3\\sqrt[\\leftroot{1}\\uproot{2} ]{x}-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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45. [latex]m\\left(x\\right)=\\frac{1}{2}{x}^{3}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137817390\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137817390\"]\r\n

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

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46. [latex]n\\left(x\\right)=\\frac{1}{3}|x-2|[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n47. [latex]p\\left(x\\right)={\\left(\\frac{1}{3}x\\right)}^{3}-3[\/latex]\r\n\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165135253220\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135253220\"]\r\n

The graph of the function is stretched horizontally by a factor of 3 and then shifted vertically downward by 3 units.<\/p>\r\n\"Graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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48. [latex]q\\left(x\\right)={\\left(\\frac{1}{4}x\\right)}^{3}+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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49. [latex]a\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{-x+4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135433479\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135433479\"]\r\n

The graph of[latex]\\text{ }f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x}\\text{ }[\/latex]is shifted right 4 units and then reflected across the vertical line[latex]\\text{ }x=4.[\/latex]<\/p>\r\n\"Graph<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, use the graph in Figure 2<\/a> to sketch the given transformations.<\/strong><\/p>\r\n\r\n

\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"466\"]\"Graph Figure 2.[\/caption]\r\n\r\n<\/div>\r\n
\r\n
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50. [latex]g\\left(x\\right)=f\\left(x\\right)-2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
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51. [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137936918\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137936918\"]\"Graph<\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n
\r\n
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52. [latex]g\\left(x\\right)=f\\left(2x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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53. [latex]g\\left(x\\right)=f\\left(x-2\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134211351\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134211351\"]\"Graph<\/span>[\/hidden-answer]<\/div>\r\n
54. [latex]g\\left(x\\right)=\\frac{1}{5}f\\left(x\\right)[\/latex]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

1.7 Section Exercises<\/h3>\n
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Verbal<\/h4>\n
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<\/div>\n<\/div>\n
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1. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal stretch from a vertical stretch?<\/p>\n<\/div>\n<\/div>\n

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2. When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal compression from a vertical compression?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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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.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Algebraic<\/h4>\n

3.\u00a0 Write a formula for the function obtained when the graph of [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x}[\/latex] is vertically stretched by a factor of 4.<\/p>\n

Show Solution<\/span><\/p>\n
[latex]g\\left(x\\right)=4f\\left(x\\right)=4\\sqrt[\\leftroot{1}\\uproot{2} ]{x}[\/latex][latex]\\\\[\/latex]<\/div>\n<\/div>\n

4.\u00a0Write a formula for the function obtained when the graph of [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x}[\/latex] is vertically compressed by a factor of 1\/3.<\/p>\n

5.\u00a0Write a formula for the function obtained when the graph of [latex]f\\left(x\\right)=|x|[\/latex] is horizontally stretched by a factor of 5.<\/p>\n

Show Solution<\/span><\/p>\n
[latex]g\\left(x\\right)=\\frac{1}{3}f\\left(x\\right)=\\frac{1}{3}|x|[\/latex][latex]\\\\[\/latex]<\/div>\n<\/div>\n

6.\u00a0Write a formula for the function obtained when the graph of [latex]f\\left(x\\right)=|x|[\/latex] is horizontally compressed by a factor of 1\/4.<\/p>\n

7.\u00a0Write a formula for the function obtained when the graph of [latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x}[\/latex] is horizontally stretched by a factor of 3 and vertically stretched by a factor of 6.<\/p>\n

Show Solution<\/span><\/p>\n
[latex]g\\left(x\\right)=6f\\left(\\frac{1}{3}x\\right)=6\\sqrt[\\leftroot{1}\\uproot{2} ]{\\frac{1}{3}x}[\/latex][latex]\\\\[\/latex]<\/div>\n<\/div>\n

8.\u00a0Write a formula for the function obtained when the graph of [latex]f\\left(x\\right)=|x|[\/latex] is horizontally compressed by a factor of 1\/2 and vertically compressed by a factor of 1\/7.<\/p>\n

For the following exercises, write a formula for the function[latex]\\text{ }g\\text{ }[\/latex]that results when the graph of a given toolkit function is transformed as described.<\/strong><\/p>\n

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9. The graph of[latex]\\text{ }f\\left(x\\right)=|x|\\text{ }[\/latex]is reflected over the[latex]\\text{ }y[\/latex]–<\/em>axis and horizontally compressed by a factor of[latex]\\text{ }\\frac{1}{4}[\/latex] .<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]g\\left(x\\right)=|-4x|[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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10. The graph of[latex]\\text{ }f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x}\\text{ }[\/latex]is reflected over the[latex]\\text{ }x[\/latex]-axis and horizontally stretched by a factor of 2.<\/p>\n<\/div>\n<\/div>\n

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11. The graph of[latex]\\text{ }f\\left(x\\right)=\\frac{1}{{x}^{2}}\\text{ }[\/latex]is vertically compressed by a factor of[latex]\\text{ }\\frac{1}{3},\\text{ }[\/latex]then shifted to the left 2 units and down 3 units.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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[latex]g\\left(x\\right)=\\frac{1}{3{\\left(x+2\\right)}^{2}}-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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12. The graph of[latex]\\text{ }f\\left(x\\right)=\\frac{1}{x}\\text{ }[\/latex]is vertically stretched by a factor of 8, then shifted to the right 4 units and up 2 units.<\/p>\n<\/div>\n<\/div>\n

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13. The graph of[latex]\\text{ }f\\left(x\\right)={x}^{2}\\text{ }[\/latex]is vertically compressed by a factor of[latex]\\text{ }\\frac{1}{2},\\text{ }[\/latex]then shifted to the right 5 units and up 1 unit.<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

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

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14. The graph of[latex]\\text{ }f\\left(x\\right)={x}^{2}\\text{ }[\/latex]is horizontally stretched by a factor of 3, then shifted to the left 4 units and down 3 units.<\/p>\n<\/div>\n<\/div>\n

For the following exercises, describe how the graph of the function is a transformation of the graph of the original function[latex]\\text{ }f.[\/latex]<\/strong><\/p>\n

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15. [latex]h\\left(x\\right)=3f\\left(x\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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The graph of [latex]3f\\left(x\\right)[\/latex] is a vertical stretch by a factor of 3 of the graph of [latex]f.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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16. [latex]g\\left(x\\right)=0.5f\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

<\/div>\n
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17. [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

The graph of[latex]\\text{ }g\\text{ }[\/latex]is a vertical reflection (across the [latex]\\text{ }x[\/latex]-axis) of the graph of[latex]\\text{ }f.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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18. [latex]g\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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19. [latex]g\\left(x\\right)=4f\\left(x\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

The graph of[latex]\\text{ }g\\text{ }[\/latex]is a vertical stretch by a factor of 4 of the graph of[latex]\\text{ }f.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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20. [latex]g\\left(x\\right)=6f\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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21. [latex]g\\left(x\\right)=f\\left(5x\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

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

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22. [latex]g\\left(x\\right)=f\\left(2x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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23. [latex]g\\left(x\\right)=f\\left(\\frac{1}{3}x\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

The graph of[latex]\\text{ }g\\text{ }[\/latex]is a horizontal stretch by a factor of 3 of the graph of[latex]\\text{ }f.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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24. [latex]g\\left(x\\right)=f\\left(\\frac{1}{5}x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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25. [latex]g\\left(x\\right)=3f\\left(-x\\right)[\/latex]<\/p>\n

Show Solution<\/span><\/p>\n
The graph of [latex]g[\/latex] is a horizontal reflection over the y-axis and a vertical stretch by a factor of 3 of [latex]f.[\/latex]<\/div>\n<\/div>\n<\/div>\n
<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, determine the interval(s) on which the function is increasing and decreasing.<\/strong><\/p>\n

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26. [latex]f\\left(x\\right)=4{\\left(x+1\\right)}^{2}-5[\/latex]<\/p>\n<\/div>\n<\/div>\n

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27. [latex]g\\left(x\\right)=5{\\left(x+3\\right)}^{2}-2[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
decreasing on[latex]\\text{ }\\left(-\\infty ,-3\\right)\\text{ }[\/latex]and increasing on[latex]\\text{ }\\left(-3,\\infty \\right)[\/latex]\n<\/div>\n<\/div>\n<\/div>\n
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28. [latex]a\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{-x+4}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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29. [latex]k\\left(x\\right)=-3\\sqrt[\\leftroot{1}\\uproot{2} ]{x}-1[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

decreasing on [latex]\\left(0,\\text{ }\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Graphical<\/h4>\n

For the following exercises, use the graph of[latex]\\text{ }f\\left(x\\right)={2}^{x}\\text{ }[\/latex]shown in Figure 1<\/a> to sketch a graph of each transformation of[latex]\\text{ }f\\left(x\\right).[\/latex]<\/p>\n

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\"Graph<\/p>\n

Figure 1<\/p>\n<\/div>\n<\/div>\n

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

31. [latex]w\\left(x\\right)=-4{2}^{x-1}[\/latex]<\/p>\n<\/div>\n<\/div>\n

For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.<\/strong><\/p>\n

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32. [latex]f\\left(t\\right)=4{\\left(t+1\\right)}^{2}-3[\/latex]<\/p>\n<\/div>\n<\/div>\n

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33. [latex]h\\left(x\\right)=-2|x-1|+4[\/latex]<\/p>\n<\/div>\n<\/div>\n

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34. [latex]k\\left(x\\right)={\\left(0.5x-2\\right)}^{3}-1[\/latex]<\/p>\n<\/div>\n<\/div>\n

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35. [latex]m\\left(t\\right)=3+\\sqrt[\\leftroot{1}\\uproot{2} ]{0.25t+2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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Numeric<\/h4>\n
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36. Tabular representations for the functions[latex]\\text{ }f,\\text{ }g,\\text{ }[\/latex]and[latex]\\text{ }h\\text{ }[\/latex]are given below. Write[latex]\\text{ }g\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }h\\left(x\\right)\\text{ }[\/latex]as transformations of[latex]\\text{ }f\\left(x\\right).[\/latex]<\/p>\n\n\n\n\n\n\n\n<\/colgroup>\n\n\n\n
[latex]x[\/latex]<\/strong><\/td>\n\u22122<\/td>\n\u22121<\/td>\n0<\/td>\n1<\/td>\n2<\/td>\n<\/tr>\n
[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n\u22121<\/td>\n\u22123<\/td>\n4<\/td>\n2<\/td>\n1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n\n<\/colgroup>\n\n\n\n
[latex]x[\/latex]<\/strong><\/td>\n\u22124<\/td>\n\u22122<\/td>\n0<\/td>\n2<\/td>\n4<\/td>\n<\/tr>\n
[latex]g\\left(x\\right)[\/latex]<\/strong><\/td>\n\u22121<\/td>\n\u22123<\/td>\n4<\/td>\n2<\/td>\n1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n\n<\/colgroup>\n\n\n\n
[latex]x[\/latex]<\/strong><\/td>\n\u22122<\/td>\n\u22121<\/td>\n0<\/td>\n1<\/td>\n2<\/td>\n<\/tr>\n
[latex]h\\left(x\\right)[\/latex]<\/strong><\/td>\n3<\/td>\n9<\/td>\n-12<\/td>\n-6<\/td>\n-3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n

37. Tabular representations for the functions[latex]\\text{ }f,\\text{ }g,\\text{ }[\/latex]and[latex]\\text{ }h\\text{ }[\/latex]are given below. Write[latex]\\text{ }g\\left(x\\right)\\text{ }[\/latex]and[latex]\\text{ }h\\left(x\\right)\\text{ }[\/latex]as transformations of[latex]\\text{ }f\\left(x\\right).[\/latex]<\/p>\n\n\n\n\n\n\n\n<\/colgroup>\n\n\n\n
[latex]x[\/latex]<\/strong><\/td>\n\u22122<\/td>\n\u22121<\/td>\n0<\/td>\n1<\/td>\n2<\/td>\n<\/tr>\n
[latex]f\\left(x\\right)[\/latex]<\/strong><\/td>\n\u22122<\/td>\n\u22121<\/td>\n\u22123<\/td>\n1<\/td>\n2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n\n<\/colgroup>\n\n\n\n
[latex]x[\/latex]<\/strong><\/td>\n\u22121<\/td>\n-1\/2<\/td>\n0<\/td>\n1\/2<\/td>\n1<\/td>\n<\/tr>\n
[latex]g\\left(x\\right)[\/latex]<\/strong><\/td>\n\u22122<\/td>\n\u22121<\/td>\n\u22123<\/td>\n1<\/td>\n2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\n\n\n\n\n\n\n<\/colgroup>\n\n\n\n
[latex]x[\/latex]<\/strong><\/td>\n\u22122<\/td>\n\u22121<\/td>\n0<\/td>\n1<\/td>\n2<\/td>\n<\/tr>\n
[latex]h\\left(x\\right)[\/latex]<\/strong><\/td>\n\u22124<\/td>\n-2<\/td>\n-6<\/td>\n2<\/td>\n4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n
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Show Solution<\/span><\/p>\n
\n

[latex]g\\left(x\\right)=f\\left(2x\\right),\\text{ }h\\left(x\\right)=2f\\left(x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

For the following exercises, write an equation for each graphed function by using transformations of the graphs of one of the toolkit functions.<\/strong><\/p>\n

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<\/div>\n
<\/div>\n<\/div>\n<\/div>\n
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\n<\/div>\n<\/div>\n

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<\/div>\n<\/div>\n
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\n<\/div>\n<\/div>\n

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation.<\/strong><\/p>\n

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41. [latex]g\\left(x\\right)=4{\\left(x+1\\right)}^{2}-5[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

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

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

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42. [latex]g\\left(x\\right)=5{\\left(x+3\\right)}^{2}-2[\/latex]<\/p>\n<\/div>\n<\/div>\n

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43. [latex]h\\left(x\\right)=-2|x-4|+3[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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The graph of[latex]\\text{ }f\\left(x\\right)=|x|\\text{ }[\/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.<\/p>\n

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

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44. [latex]k\\left(x\\right)=-3\\sqrt[\\leftroot{1}\\uproot{2} ]{x}-1[\/latex]<\/p>\n<\/div>\n<\/div>\n

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45. [latex]m\\left(x\\right)=\\frac{1}{2}{x}^{3}[\/latex]<\/p>\n<\/div>\n

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

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

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46. [latex]n\\left(x\\right)=\\frac{1}{3}|x-2|[\/latex]<\/p>\n<\/div>\n<\/div>\n

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47. [latex]p\\left(x\\right)={\\left(\\frac{1}{3}x\\right)}^{3}-3[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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The graph of the function is stretched horizontally by a factor of 3 and then shifted vertically downward by 3 units.<\/p>\n

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

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48. [latex]q\\left(x\\right)={\\left(\\frac{1}{4}x\\right)}^{3}+1[\/latex]<\/p>\n<\/div>\n<\/div>\n

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49. [latex]a\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{-x+4}[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\n

The graph of[latex]\\text{ }f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x}\\text{ }[\/latex]is shifted right 4 units and then reflected across the vertical line[latex]\\text{ }x=4.[\/latex]<\/p>\n

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

For the following exercises, use the graph in Figure 2<\/a> to sketch the given transformations.<\/strong><\/p>\n

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\"Graph<\/p>\n

Figure 2.<\/p>\n<\/div>\n<\/div>\n

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50. [latex]g\\left(x\\right)=f\\left(x\\right)-2[\/latex]<\/p>\n<\/div>\n<\/div>\n

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51. [latex]g\\left(x\\right)=-f\\left(x\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n
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52. [latex]g\\left(x\\right)=f\\left(2x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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53. [latex]g\\left(x\\right)=f\\left(x-2\\right)[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
\"Graph<\/span><\/div>\n<\/div>\n<\/div>\n
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