{"id":2302,"date":"2019-05-09T15:58:50","date_gmt":"2019-05-09T15:58:50","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&p=2302"},"modified":"2019-10-09T23:15:05","modified_gmt":"2019-10-09T23:15:05","slug":"1-6-section-exercises","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/1-6-section-exercises\/","title":{"raw":"1.6 Section Exercises","rendered":"1.6 Section Exercises"},"content":{"raw":"
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1.6 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 shift from a vertical shift?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165133297372\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133297372\"]\r\n

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

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<\/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 reflection with respect to the x<\/em>-axis from a reflection with respect to the y<\/em>-axis?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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3. How can you determine whether a function is odd or even from the formula of the function?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137644805\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137644805\"]\r\n

For a function[latex]\\text{ }f,\\text{ }[\/latex]substitute[latex]\\text{ }\\left(-x\\right)\\text{ }[\/latex]for[latex]\\text{ }\\left(x\\right)\\text{ }[\/latex]in[latex]\\text{ }f\\left(x\\right).\\text{ }[\/latex]Simplify. If the resulting function is the same as the original function,[latex]\\text{ }f\\left(-x\\right)=f\\left(x\\right),\\text{ }[\/latex]then the function is even. If the resulting function is the opposite of the original function,[latex]\\text{ }f\\left(-x\\right)=-f\\left(x\\right),\\text{ }[\/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>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Algebraic<\/h4>\r\n
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4. Write a formula for the function obtained when the graph of[latex]\\text{ }f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x}\\text{ }[\/latex]is shifted up 1 unit and to the left 2 units.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n5. Write a formula for the function obtained when the graph of[latex]\\text{ }f\\left(x\\right)=|x|\\text{ }[\/latex] is shifted down 3 units and to the right 1 unit.\r\n\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165133334343\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133334343\"]\r\n

[latex]g\\left(x\\right)=|x-1|-3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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6. Write a formula for the function obtained when the graph of[latex]\\text{ }f\\left(x\\right)=\\frac{1}{x}\\text{ }[\/latex]is shifted down 4 units and to the right 3 units.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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7. Write a formula for the function obtained when the graph of[latex]\\text{ }f\\left(x\\right)=\\frac{1}{{x}^{2}}\\text{ }[\/latex]is shifted up 2 units and to the left 4 units.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135524467\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135524467\"]\r\n

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

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8. [latex]y=f\\left(x-49\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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9. [latex]y=f\\left(x+43\\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]\\text{ }f\\left(x+43\\right)\\text{ }[\/latex]is a horizontal shift to the left 43 units 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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10. [latex]y=f\\left(x+3\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165137400039\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137400039\"]\r\n

The graph of[latex]\\text{ }f\\left(x-4\\right)\\text{ }[\/latex]is a horizontal shift to the right 4 units 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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12. [latex]y=f\\left(x\\right)+5[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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13. [latex]y=f\\left(x\\right)+8[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137704820\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137704820\"]\r\n

The graph of[latex]\\text{ }f\\left(x\\right)+8\\text{ }[\/latex]is a vertical shift up 8 units 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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14. [latex]y=f\\left(x\\right)-2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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15.[latex]y=f\\left(x\\right)-7[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135545762\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135545762\"]\r\n

The graph of[latex]\\text{ }f\\left(x\\right)-7\\text{ }[\/latex]is a vertical shift down 7 units 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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16. [latex]y=f\\left(x-2\\right)+3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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17. [latex]y=f\\left(x+4\\right)-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137782282\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137782282\"]\r\n

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>\r\n[\/hidden-answer]\r\n\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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18. [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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19. [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\"]\r\n

decreasing on[latex]\\text{ }\\left(-\\infty ,-3\\right)\\text{ }[\/latex]and increasing on[latex]\\text{ }\\left(-3,\\infty \\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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20. [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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21. [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)<\/a> to sketch a graph of each transformation of[latex]\\text{ }f\\left(x\\right).[\/latex]<\/strong><\/p>\r\n\r\n

\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"360\"]\"Graph Figure 33.[\/caption]\r\n\r\n<\/div>\r\n
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\r\n\r\n22. [latex]g\\left(x\\right)={2}^{x}+1[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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23. [latex]h\\left(x\\right)={2}^{x}-3[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137644136\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137644136\"]\"Graph<\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n
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24. [latex]w\\left(x\\right)={2}^{x-1}[\/latex]<\/p>\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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25. [latex]f\\left(t\\right)={\\left(t+1\\right)}^{2}-3[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135631538\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135631538\"]\"Graph<\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n
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26. [latex]h\\left(x\\right)=|x-1|+4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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

[reveal-answer q=\"fs-id1165134234193\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134234193\"]\"Graph<\/span>[\/hidden-answer]<\/div>\r\n<\/div>\r\n
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28. [latex]m\\left(t\\right)=3+\\sqrt[\\leftroot{1}\\uproot{2} ]{t+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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29. 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\n0<\/td>\r\n1<\/td>\r\n2<\/td>\r\n3<\/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\u22121<\/td>\r\n0<\/td>\r\n\u22122<\/td>\r\n2<\/td>\r\n3<\/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(x-1\\right),\\text{ }h\\left(x\\right)=f\\left(x\\right)+1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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30. 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\u22123<\/td>\r\n\u22122<\/td>\r\n\u22121<\/td>\r\n0<\/td>\r\n1<\/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\n\u22122<\/td>\r\n\u22124<\/td>\r\n3<\/td>\r\n1<\/td>\r\n0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\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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31.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165135516945\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135516945\"]\r\n

[latex]f\\left(x\\right)=|x-3|-2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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32.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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33.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165135190190\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135190190\"]\r\n

[latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x+3}-1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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34.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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35.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165135560630\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135560630\"]\r\n

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

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36.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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37.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165133030812\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133030812\"]\r\n

[latex]f\\left(x\\right)=|x+3|-2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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38.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n

For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.<\/strong><\/p>\r\n\r\n

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39.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165134190724\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134190724\"]\r\n

[latex]f\\left(x\\right)=-\\sqrt[\\leftroot{1}\\uproot{2} ]{x}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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40.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n

For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions<\/strong>.<\/p>\r\n\r\n

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41.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165137889877\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137889877\"]\r\n

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

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42.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n
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43.\u00a0\"Graph<\/span><\/div>\r\n
[reveal-answer q=\"fs-id1165134159665\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134159665\"]\r\n

[latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{-x}+1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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44.\u00a0\"Graph<\/span><\/div>\r\n<\/div>\r\n

For the following exercises, determine whether the function is odd, even, or neither.<\/strong><\/p>\r\n\r\n

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45. [latex]f\\left(x\\right)=3{x}^{4}[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135671506\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135671506\"]\r\n

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

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

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

[reveal-answer q=\"fs-id1165137610991\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137610991\"]\r\n

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

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

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

[reveal-answer q=\"fs-id1165137679220\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137679220\"]\r\n

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

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

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

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53. [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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54. [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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55. [latex]g\\left(x\\right)=-f\\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 and a horizontal reflection of the graph of[latex]\\text{ }f.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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

\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"466\"]\"Graph Figure 34.[\/caption]\r\n\r\n<\/div>\r\n
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56. [latex]g\\left(x\\right)=f\\left(x\\right)-2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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57. [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
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58. [latex]g\\left(x\\right)=f\\left(x+1\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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59. [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<\/div>\r\n<\/div>\r\n<\/div>","rendered":"
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1.6 Section Exercises<\/h3>\n
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Verbal<\/h4>\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 shift from a vertical shift?<\/p>\n<\/div>\n

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

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<\/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 reflection with respect to the x<\/em>-axis from a reflection with respect to the y<\/em>-axis?<\/p>\n<\/div>\n<\/div>\n

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3. How can you determine whether a function is odd or even from the formula of the function?<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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For a function[latex]\\text{ }f,\\text{ }[\/latex]substitute[latex]\\text{ }\\left(-x\\right)\\text{ }[\/latex]for[latex]\\text{ }\\left(x\\right)\\text{ }[\/latex]in[latex]\\text{ }f\\left(x\\right).\\text{ }[\/latex]Simplify. If the resulting function is the same as the original function,[latex]\\text{ }f\\left(-x\\right)=f\\left(x\\right),\\text{ }[\/latex]then the function is even. If the resulting function is the opposite of the original function,[latex]\\text{ }f\\left(-x\\right)=-f\\left(x\\right),\\text{ }[\/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<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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Algebraic<\/h4>\n
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4. Write a formula for the function obtained when the graph of[latex]\\text{ }f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x}\\text{ }[\/latex]is shifted up 1 unit and to the left 2 units.<\/p>\n<\/div>\n<\/div>\n

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5. Write a formula for the function obtained when the graph of[latex]\\text{ }f\\left(x\\right)=|x|\\text{ }[\/latex] is shifted down 3 units and to the right 1 unit.<\/p>\n<\/div>\n

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

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6. Write a formula for the function obtained when the graph of[latex]\\text{ }f\\left(x\\right)=\\frac{1}{x}\\text{ }[\/latex]is shifted down 4 units and to the right 3 units.<\/p>\n<\/div>\n<\/div>\n

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7. Write a formula for the function obtained when the graph of[latex]\\text{ }f\\left(x\\right)=\\frac{1}{{x}^{2}}\\text{ }[\/latex]is shifted up 2 units and to the left 4 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}{{\\left(x+4\\right)}^{2}}+2[\/latex]<\/p>\n<\/div>\n<\/div>\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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8. [latex]y=f\\left(x-49\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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9. [latex]y=f\\left(x+43\\right)[\/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+43\\right)\\text{ }[\/latex]is a horizontal shift to the left 43 units of the graph of[latex]\\text{ }f.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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10. [latex]y=f\\left(x+3\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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11. [latex]y=f\\left(x-4\\right)[\/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-4\\right)\\text{ }[\/latex]is a horizontal shift to the right 4 units of the graph of[latex]\\text{ }f.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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12. [latex]y=f\\left(x\\right)+5[\/latex]<\/p>\n<\/div>\n<\/div>\n

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13. [latex]y=f\\left(x\\right)+8[\/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)+8\\text{ }[\/latex]is a vertical shift up 8 units of the graph of[latex]\\text{ }f.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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15.[latex]y=f\\left(x\\right)-7[\/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)-7\\text{ }[\/latex]is a vertical shift down 7 units of the graph of[latex]\\text{ }f.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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16. [latex]y=f\\left(x-2\\right)+3[\/latex]<\/p>\n<\/div>\n<\/div>\n

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17. [latex]y=f\\left(x+4\\right)-1[\/latex]<\/p>\n<\/div>\n

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Show Solution<\/span><\/p>\n
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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<\/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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18. [latex]f\\left(x\\right)=4{\\left(x+1\\right)}^{2}-5[\/latex]<\/p>\n<\/div>\n<\/div>\n

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19. [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
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decreasing on[latex]\\text{ }\\left(-\\infty ,-3\\right)\\text{ }[\/latex]and increasing on[latex]\\text{ }\\left(-3,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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21. [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
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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)<\/a> to sketch a graph of each transformation of[latex]\\text{ }f\\left(x\\right).[\/latex]<\/strong><\/p>\n

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\"Graph<\/p>\n

Figure 33.<\/p>\n<\/div>\n<\/div>\n

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

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23. [latex]h\\left(x\\right)={2}^{x}-3[\/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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24. [latex]w\\left(x\\right)={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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25. [latex]f\\left(t\\right)={\\left(t+1\\right)}^{2}-3[\/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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26. [latex]h\\left(x\\right)=|x-1|+4[\/latex]<\/p>\n<\/div>\n<\/div>\n

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27. [latex]k\\left(x\\right)={\\left(x-2\\right)}^{3}-1[\/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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28. [latex]m\\left(t\\right)=3+\\sqrt[\\leftroot{1}\\uproot{2} ]{t+2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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Numeric<\/h4>\n
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29. 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>\n0<\/td>\n1<\/td>\n2<\/td>\n3<\/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\u22121<\/td>\n0<\/td>\n\u22122<\/td>\n2<\/td>\n3<\/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(x-1\\right),\\text{ }h\\left(x\\right)=f\\left(x\\right)+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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30. 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\u22123<\/td>\n\u22122<\/td>\n\u22121<\/td>\n0<\/td>\n1<\/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>\n\u22122<\/td>\n\u22124<\/td>\n3<\/td>\n1<\/td>\n0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\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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31.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
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[latex]f\\left(x\\right)=|x-3|-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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32.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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33.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
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[latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{x+3}-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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34.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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35.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
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[latex]f\\left(x\\right)={\\left(x-2\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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36.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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37.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
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[latex]f\\left(x\\right)=|x+3|-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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38.\u00a0\"Graph<\/span><\/div>\n<\/div>\n

For the following exercises, use the graphs of transformations of the square root function to find a formula for each of the functions.<\/strong><\/p>\n

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39.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
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[latex]f\\left(x\\right)=-\\sqrt[\\leftroot{1}\\uproot{2} ]{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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40.\u00a0\"Graph<\/span><\/div>\n<\/div>\n

For the following exercises, use the graphs of the transformed toolkit functions to write a formula for each of the resulting functions<\/strong>.<\/p>\n

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41.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
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[latex]f\\left(x\\right)=-{\\left(x+1\\right)}^{2}+2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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42.\u00a0\"Graph<\/span><\/div>\n<\/div>\n
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43.\u00a0\"Graph<\/span><\/div>\n
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Show Solution<\/span><\/p>\n
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[latex]f\\left(x\\right)=\\sqrt[\\leftroot{1}\\uproot{2} ]{-x}+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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44.\u00a0\"Graph<\/span><\/div>\n<\/div>\n

For the following exercises, determine whether the function is odd, even, or neither.<\/strong><\/p>\n

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45. [latex]f\\left(x\\right)=3{x}^{4}[\/latex]<\/p>\n<\/div>\n

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

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

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

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

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

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49. [latex]g\\left(x\\right)=2{x}^{4}[\/latex]<\/p>\n<\/div>\n

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

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50. [latex]h\\left(x\\right)=2x-{x}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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53. [latex]g\\left(x\\right)=-f\\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]\\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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54. [latex]g\\left(x\\right)=f\\left(-x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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55. [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 and a horizontal reflection of the graph of[latex]\\text{ }f.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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

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\"Graph<\/p>\n

Figure 34.<\/p>\n<\/div>\n<\/div>\n

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

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57. [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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58. [latex]g\\left(x\\right)=f\\left(x+1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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

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