{"id":2329,"date":"2019-05-12T21:11:45","date_gmt":"2019-05-12T21:11:45","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/?post_type=chapter&p=2329"},"modified":"2019-05-12T21:14:41","modified_gmt":"2019-05-12T21:14:41","slug":"3-4-section-exercises","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/3-4-section-exercises\/","title":{"raw":"3.4 Section Exercises","rendered":"3.4 Section Exercises"},"content":{"raw":"

Section Exercises<\/h3>\r\n
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Verbal<\/h4>\r\n
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1. Why are the sine and cosine functions called periodic functions?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137415637\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137415637\"]\r\n

The sine and cosine functions have the property that[latex]\\text{ }f\\left(x+P\\right)=f\\left(x\\right)\\text{ }[\/latex]for a certain[latex]\\text{ }P.\\text{ }[\/latex]This means that the function values repeat for every[latex]\\text{ }P\\text{ }[\/latex]units on the x<\/em>-axis.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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2. How does the graph of[latex]\\text{ }y=\\mathrm{sin}\\text{ }x\\text{ }[\/latex] compare with the graph of[latex]\\text{ }y=\\mathrm{cos}\\text{ }x?\\text{ }[\/latex] Explain how you could horizontally translate the graph of[latex]\\text{ }y=\\mathrm{sin}\\text{ }x\\text{ }[\/latex] to obtain[latex]\\text{ }y=\\mathrm{cos}\\text{ }x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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3. For the equation[latex]\\text{ }A\\text{ }\\mathrm{cos}\\left(Bx+C\\right)+D,[\/latex]what constants affect the range of the function and how do they affect the range?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137811265\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137811265\"]\r\n

The absolute value of the constant[latex]\\text{ }A\\text{ }[\/latex](amplitude) increases the total range and the constant[latex]\\text{ }D\\text{ }[\/latex](vertical shift) shifts the graph vertically.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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4. How does the range of a translated sine function relate to the equation[latex]\\text{ }y=A\\text{ }\\mathrm{sin}\\left(Bx+C\\right)+D?[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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5. How can the unit circle be used to construct the graph of[latex]\\text{ }f\\left(t\\right)=\\mathrm{sin}\\text{ }t?[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137407584\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137407584\"]\r\n

At the point where the terminal side of[latex]\\text{ }t\\text{ }[\/latex]intersects the unit circle, you can determine that the[latex]\\text{ }\\mathrm{sin}\\text{ }t\\text{ }[\/latex]equals the y<\/em>-coordinate of the point.<\/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, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y<\/em>-values and their corresponding x<\/em>-values on one period for[latex]\\text{ }x>0.\\text{ }[\/latex]Round answers to two decimal places if necessary.<\/p>\r\n\r\n

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7. [latex]f\\left(x\\right)=2\\mathrm{sin}\\text{ }x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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8. [latex]f\\left(x\\right)=\\frac{2}{3}\\mathrm{cos}\\text{ }x[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135456747\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135456747\"]\"A<\/span>\r\n

amplitude:[latex]\\text{ }\\frac{2}{3};\\text{ }[\/latex]period:[latex]\\text{ }2\\pi ;\\text{ }[\/latex]midline:[latex]\\text{ }y=0;\\text{ }[\/latex]maximum:[latex]\\text{ }y=\\frac{2}{3}\\text{ }[\/latex]occurs at[latex]\\text{ }x=0;\\text{ }[\/latex]minimum:[latex]\\text{ }y=-\\frac{2}{3}\\text{ }[\/latex]occurs at[latex]\\text{ }x=\\pi ;\\text{ }[\/latex]for one period, the graph starts at 0 and ends at[latex]\\text{ }2\\pi [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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\r\n\r\n9. [latex]f\\left(x\\right)=-3\\mathrm{sin}\\text{ }x[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n
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10. [latex]f\\left(x\\right)=4\\mathrm{sin}\\text{ }x[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137407212\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137407212\"]\"A<\/span>\r\n

amplitude: 4; period:[latex]\\text{ }2\\pi ;\\text{ }[\/latex]midline:[latex]\\text{ }y=0;\\text{ }[\/latex]maximum[latex]\\text{ }y=4\\text{ }[\/latex]occurs at[latex]\\text{ }x=\\frac{\\pi }{2};\\text{ }[\/latex]minimum:[latex]\\text{ }y=-4\\text{ }[\/latex]occurs at[latex]\\text{ }x=\\frac{3\\pi }{2};\\text{ }[\/latex]one full period occurs from[latex]\\text{ }x=0\\text{ }[\/latex]to[latex]\\text{ }x=2\\pi [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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11. [latex]f\\left(x\\right)=2\\mathrm{cos}\\text{ }x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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12. [latex]f\\left(x\\right)=\\mathrm{cos}\\left(2x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137871346\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137871346\"]\"A<\/span>\r\n

amplitude: 1; period:[latex]\\text{ }\\pi ;\\text{ }[\/latex]midline:[latex]\\text{ }y=0;\\text{ }[\/latex]maximum:[latex]\\text{ }y=1\\text{ }[\/latex]occurs at[latex]\\text{ }x=\\pi ;\\text{ }[\/latex]minimum:[latex]\\text{ }y=-1\\text{ }[\/latex]occurs at[latex]\\text{ }x=\\frac{\\pi }{2};\\text{ }[\/latex]one full period is graphed from[latex]\\text{ }x=0\\text{ }[\/latex]to[latex]\\text{ }x=\\pi [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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13. [latex]f\\left(x\\right)=2\\text{ }\\mathrm{sin}\\left(\\frac{1}{2}x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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14. [latex]f\\left(x\\right)=4\\text{ }\\mathrm{cos}\\left(\\pi x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137804069\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137804069\"]\"A<\/span>\r\n

amplitude: 4; period: 2; midline:[latex]\\text{ }y=0;\\text{ }[\/latex]maximum:[latex]\\text{ }y=4\\text{ }[\/latex]occurs at[latex]\\text{ }x=0;\\text{ }[\/latex]minimum:[latex]\\text{ }y=-4\\text{ }[\/latex]occurs at[latex]\\text{ }x=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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15. [latex]f\\left(x\\right)=3\\text{ }\\mathrm{cos}\\left(\\frac{6}{5}x\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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16. [latex]y=3\\text{ }\\mathrm{sin}\\left(8\\left(x+4\\right)\\right)+5[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137843946\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137843946\"]\"A<\/span>\r\n

amplitude: 3; period:[latex]\\text{ }\\frac{\\pi }{4};\\text{ }[\/latex]midline:[latex]\\text{ }y=5;\\text{ }[\/latex]maximum:[latex]\\text{ }y=8\\text{ }[\/latex]occurs at[latex]\\text{ }x=0.12;\\text{ }[\/latex]minimum:[latex]\\text{ }y=2\\text{ }[\/latex]occurs at[latex]\\text{ }x=0.516;\\text{ }[\/latex]horizontal shift:[latex]\\text{ }-4;\\text{ }[\/latex]vertical translation 5; one period occurs from[latex]\\text{ }x=0\\text{ }[\/latex]to[latex]\\text{ }x=\\frac{\\pi }{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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17. [latex]y=2\\text{ }\\mathrm{sin}\\left(3x-21\\right)+4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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18. [latex]y=5\\text{ }\\mathrm{sin}\\left(5x+20\\right)-2[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134284471\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134284471\"]\"A<\/span>\r\n

amplitude: 5; period:[latex]\\text{ }\\frac{2\\pi }{5};\\text{ }[\/latex]midline:[latex]\\text{ }y=-2;\\text{ }[\/latex]maximum:[latex]\\text{ }y=3\\text{ }[\/latex]occurs at[latex]\\text{ }x=0.08;\\text{ }[\/latex]minimum:[latex]\\text{ }y=-7\\text{ }[\/latex]occurs at[latex]\\text{ }x=0.71;\\text{ }[\/latex]phase shift:[latex]\\text{ }-4;\\text{ }[\/latex]vertical translation:[latex]\\text{ }-2;\\text{ }[\/latex]one full period can be graphed on[latex]\\text{ }x=0\\text{ }[\/latex]to[latex]\\text{ }x=\\frac{2\\pi }{5}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, graph one full period of each function, starting at[latex]\\text{ }x=0.\\text{ }[\/latex]For each function, state the amplitude, period, and midline. State the maximum and minimum y<\/em>-values and their corresponding x<\/em>-values on one period for[latex]\\text{ }x>0.\\text{ }[\/latex]State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.<\/p>\r\n\r\n

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19. [latex]f\\left(t\\right)=2\\mathrm{sin}\\left(t-\\frac{5\\pi }{6}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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20. [latex]f\\left(t\\right)=-\\mathrm{cos}\\left(t+\\frac{\\pi }{3}\\right)+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134541171\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134541171\"]\"A<\/span>\r\n

amplitude: 1 ; period:[latex]\\text{ }2\\pi ;\\text{ }[\/latex]midline:[latex]\\text{ }y=1;\\text{ }[\/latex]maximum:[latex]\\text{ }y=2\\text{ }[\/latex]occurs at[latex]\\text{ }x=2.09;\\text{ }[\/latex]maximum:[latex]\\text{ }y=2\\text{ }[\/latex]occurs at[latex]\\text{ }t=2.09;\\text{ }[\/latex]minimum:[latex]\\text{ }y=0\\text{ }[\/latex]occurs at[latex]\\text{ }t=5.24;\\text{ }[\/latex]phase shift:[latex]\\text{ }-\\frac{\\pi }{3};\\text{ }[\/latex]vertical translation: 1; one full period is from[latex]\\text{ }t=0\\text{ }[\/latex]to[latex]\\text{ }t=2\\pi [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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21. [latex]f\\left(t\\right)=4\\mathrm{cos}\\left(2\\left(t+\\frac{\\pi }{4}\\right)\\right)-3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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22. [latex]f\\left(t\\right)=-\\mathrm{sin}\\left(\\frac{1}{2}t+\\frac{5\\pi }{3}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137541180\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137541180\"]\"A<\/span>\r\n

amplitude: 1; period:[latex]\\text{ }4\\pi ;\\text{ }[\/latex]midline:[latex]\\text{ }y=0;\\text{ }[\/latex]maximum:[latex]\\text{ }y=1\\text{ }[\/latex]occurs at[latex]\\text{ }t=11.52;\\text{ }[\/latex]minimum:[latex]\\text{ }y=-1\\text{ }[\/latex]occurs at[latex]\\text{ }t=5.24;\\text{ }[\/latex]phase shift:[latex]\\text{ }-\\frac{10\\pi }{3};\\text{ }[\/latex]vertical shift: 0<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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23. [latex]f\\left(x\\right)=4\\mathrm{sin}\\left(\\frac{\\pi }{2}\\left(x-3\\right)\\right)+7[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown in (Figure)<\/a>.<\/p>\r\n\r\n

\"A<\/span><\/div>\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165135708054\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135708054\"]\r\n

amplitude: 2; midline:[latex]\\text{ }y=-3;\\text{ }[\/latex]period: 4; equation:[latex]\\text{ }f\\left(x\\right)=2\\mathrm{sin}\\left(\\frac{\\pi }{2}x\\right)-3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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24. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in (Figure)<\/a>.<\/p>\r\n\r\n

\"A<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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25. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in (Figure)<\/a>.<\/p>\r\n\r\n

\"A<\/span><\/div>\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165134378700\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134378700\"]\r\n

amplitude: 2; period: 5; midline:[latex]\\text{ }y=3;\\text{ }[\/latex]equation:[latex]\\text{ }f\\left(x\\right)=-2\\mathrm{cos}\\left(\\frac{2\\pi }{5}x\\right)+3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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26. Determine the amplitude, period, midline, and an equation involving sine for the graph shown in (Figure)<\/a>.<\/p>\r\n\r\n

\"A<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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27. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in (Figure)<\/a>.<\/p>\r\n\r\n

\"A<\/span><\/div>\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165135534972\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135534972\"]\r\n

amplitude: 4; period: 2; midline:[latex]\\text{ }y=0;\\text{ }[\/latex]equation:[latex]\\text{ }f\\left(x\\right)=-4\\mathrm{cos}\\left(\\pi \\left(x-\\frac{\\pi }{2}\\right)\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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28. Determine the amplitude, period, midline, and an equation involving sine for the graph shown in (Figure)<\/a>.<\/p>\r\n\r\n

\"A<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n
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29. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in (Figure)<\/a>.<\/p>\r\n\r\n

\"A<\/span><\/div>\r\n<\/div>\r\n
[reveal-answer q=\"fs-id1165137600948\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137600948\"]amplitude: 2; period: 2; midline[latex]\\text{ }y=1;\\text{ }[\/latex]equation:[latex]\\text{ }f\\left(x\\right)=2\\mathrm{cos}\\left(\\pi x\\right)+1[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n
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30. Determine the amplitude, period, midline, and an equation involving sine for the graph shown in (Figure)<\/a>.<\/p>\r\n\r\n

\"A<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n
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Algebraic<\/h4>\r\n

For the following exercises, let[latex]\\text{ }f\\left(x\\right)=\\mathrm{sin}\\text{ }x.[\/latex]<\/p>\r\n\r\n

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31. On[latex]\\text{ }\\left[0,2\\pi \\right),[\/latex]solve[latex]\\text{ }f\\left(x\\right)=0.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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32. On[latex]\\text{ }\\left[0,2\\pi \\right),[\/latex]solve[latex]\\text{ }f\\left(x\\right)=\\frac{1}{2}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137832261\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137832261\"]\r\n

[latex]\\frac{\\pi }{6},\\frac{5\\pi }{6}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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33.Evaluate[latex]\\text{ }f\\left(\\frac{\\pi }{2}\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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34. On[latex]\\text{ }\\left[0,2\\pi \\right),f\\left(x\\right)=\\frac{\\sqrt{2}}{2}.\\text{ }[\/latex]Find all values of[latex]\\text{ }x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137837133\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137837133\"]\r\n

[latex]\\frac{\\pi }{4},\\frac{3\\pi }{4}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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35. On[latex]\\text{ }\\left[0,2\\pi \\right),[\/latex]the maximum value(s) of the function occur(s) at what x<\/em>-value(s)?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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36. On[latex]\\text{ }\\left[0,2\\pi \\right),[\/latex]the minimum value(s) of the function occur(s) at what x<\/em>-value(s)?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134042136\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134042136\"]\r\n

[latex]\\frac{3\\pi }{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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37. Show that[latex]\\text{ }f\\left(-x\\right)=-f\\left(x\\right).\\text{ }[\/latex]This means that[latex]\\text{ }f\\left(x\\right)=\\mathrm{sin}\\text{ }x\\text{ }[\/latex]is an odd function and possesses symmetry with respect to ________________.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, let[latex]\\text{ }f\\left(x\\right)=\\mathrm{cos}\\text{ }x.[\/latex]<\/p>\r\n\r\n

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38. On[latex]\\text{ }\\left[0,2\\pi \\right),[\/latex]solve the equation[latex]\\text{ }f\\left(x\\right)=\\mathrm{cos}\\text{ }x=0.[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165134129955\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134129955\"]\r\n

[latex]\\frac{\\pi }{2},\\frac{3\\pi }{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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39. On[latex]\\text{ }\\left[0,2\\pi \\right),[\/latex]solve[latex]\\text{ }f\\left(x\\right)=\\frac{1}{2}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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40. On[latex]\\text{ }\\left[0,2\\pi \\right),[\/latex]find the x<\/em>-intercepts of[latex]\\text{ }f\\left(x\\right)=\\mathrm{cos}\\text{ }x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135440505\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135440505\"]\r\n

[latex]\\frac{\\pi }{2},\\frac{3\\pi }{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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41. On[latex]\\text{ }\\left[0,2\\pi \\right),[\/latex]find the x<\/em>-values at which the function has a maximum or minimum value.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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42. On[latex]\\text{ }\\left[0,2\\pi \\right),[\/latex]solve the equation[latex]\\text{ }f\\left(x\\right)=\\frac{\\sqrt{3}}{2}.[\/latex]<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137933103\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137933103\"]\r\n

[latex]\\frac{\\pi }{6},\\frac{11\\pi }{6}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Technology<\/h4>\r\n
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43. Graph[latex]\\text{ }h\\left(x\\right)=x+\\mathrm{sin}\\text{ }x\\text{ }[\/latex]on[latex]\\text{ }\\left[0,2\\pi \\right].\\text{ }[\/latex]Explain why the graph appears as it does.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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44. Graph[latex]\\text{ }h\\left(x\\right)=x+\\mathrm{sin}\\text{ }x\\text{ }[\/latex]on[latex]\\text{ }\\left[-100,100\\right].\\text{ }[\/latex]Did the graph appear as predicted in the previous exercise?<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165137433807\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137433807\"]\r\n

The graph appears linear. The linear functions dominate the shape of the graph for large values of[latex]\\text{ }x.[\/latex]<\/p>\r\n\"A<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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45. Graph[latex]\\text{ }f\\left(x\\right)=x\\text{ }\\mathrm{sin}\\text{ }x\\text{ }[\/latex]on[latex]\\text{ }\\left[0,2\\pi \\right]\\text{ }[\/latex]and verbalize how the graph varies from the graph of[latex]\\text{ }f\\left(x\\right)=\\mathrm{sin}\\text{ }x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

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46. Graph[latex]\\text{ }f\\left(x\\right)=x\\text{ }\\mathrm{sin}\\text{ }x\\text{ }[\/latex]on the window[latex]\\text{ }\\left[-10,10\\right]\\text{ }[\/latex]and explain what the graph shows.<\/p>\r\n\r\n<\/div>\r\n

[reveal-answer q=\"fs-id1165135322029\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135322029\"]\r\n

The graph is symmetric with respect to the y<\/em>-axis and there is no amplitude because the function is not periodic.<\/p>\r\n\"A<\/span>[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n

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47. Graph[latex]\\text{ }f\\left(x\\right)=\\frac{\\mathrm{sin}\\text{ }x}{x}\\text{ }[\/latex]on the window[latex]\\text{ }\\left[-5\\pi ,5\\pi \\right]\\text{ }[\/latex]and explain what the graph shows.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Real-World Applications<\/h4>\r\n
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48. A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o\u2019clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The function[latex]\\text{ }h\\left(t\\right)\\text{ }[\/latex]gives a person\u2019s height in meters above the ground t<\/em> minutes after the wheel begins to turn.<\/p>\r\n\r\n

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  1. Find the amplitude, midline, and period of[latex]\\text{ }h\\left(t\\right).[\/latex]<\/li>\r\n \t
  2. Find a formula for the height function[latex]\\text{ }h\\left(t\\right).[\/latex]<\/li>\r\n \t
  3. How high off the ground is a person after 5 minutes?<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    [reveal-answer q=\"fs-id1165135205671\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135205671\"]\r\n
      \r\n \t
    1. Amplitude: 12.5; period: 10; midline:[latex]\\text{ }y=13.5;[\/latex]<\/li>\r\n \t
    2. [latex]h\\left(t\\right)=12.5\\mathrm{sin}\\left(\\frac{\\pi }{5}\\left(t-2.5\\right)\\right)+13.5;[\/latex]<\/li>\r\n \t
    3. 26 ft<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

      Section Exercises<\/h3>\n
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      Verbal<\/h4>\n
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      1. Why are the sine and cosine functions called periodic functions?<\/p>\n<\/div>\n

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

      The sine and cosine functions have the property that[latex]\\text{ }f\\left(x+P\\right)=f\\left(x\\right)\\text{ }[\/latex]for a certain[latex]\\text{ }P.\\text{ }[\/latex]This means that the function values repeat for every[latex]\\text{ }P\\text{ }[\/latex]units on the x<\/em>-axis.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      2. How does the graph of[latex]\\text{ }y=\\mathrm{sin}\\text{ }x\\text{ }[\/latex] compare with the graph of[latex]\\text{ }y=\\mathrm{cos}\\text{ }x?\\text{ }[\/latex] Explain how you could horizontally translate the graph of[latex]\\text{ }y=\\mathrm{sin}\\text{ }x\\text{ }[\/latex] to obtain[latex]\\text{ }y=\\mathrm{cos}\\text{ }x.[\/latex]<\/p>\n<\/div>\n<\/div>\n

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      3. For the equation[latex]\\text{ }A\\text{ }\\mathrm{cos}\\left(Bx+C\\right)+D,[\/latex]what constants affect the range of the function and how do they affect the range?<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
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      The absolute value of the constant[latex]\\text{ }A\\text{ }[\/latex](amplitude) increases the total range and the constant[latex]\\text{ }D\\text{ }[\/latex](vertical shift) shifts the graph vertically.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      4. How does the range of a translated sine function relate to the equation[latex]\\text{ }y=A\\text{ }\\mathrm{sin}\\left(Bx+C\\right)+D?[\/latex]<\/p>\n<\/div>\n<\/div>\n

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      5. How can the unit circle be used to construct the graph of[latex]\\text{ }f\\left(t\\right)=\\mathrm{sin}\\text{ }t?[\/latex]<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
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      At the point where the terminal side of[latex]\\text{ }t\\text{ }[\/latex]intersects the unit circle, you can determine that the[latex]\\text{ }\\mathrm{sin}\\text{ }t\\text{ }[\/latex]equals the y<\/em>-coordinate of the point.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      Graphical<\/h4>\n

      For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y<\/em>-values and their corresponding x<\/em>-values on one period for[latex]\\text{ }x>0.\\text{ }[\/latex]Round answers to two decimal places if necessary.<\/p>\n

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      7. [latex]f\\left(x\\right)=2\\mathrm{sin}\\text{ }x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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      8. [latex]f\\left(x\\right)=\\frac{2}{3}\\mathrm{cos}\\text{ }x[\/latex]<\/p>\n<\/div>\n

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

      amplitude:[latex]\\text{ }\\frac{2}{3};\\text{ }[\/latex]period:[latex]\\text{ }2\\pi ;\\text{ }[\/latex]midline:[latex]\\text{ }y=0;\\text{ }[\/latex]maximum:[latex]\\text{ }y=\\frac{2}{3}\\text{ }[\/latex]occurs at[latex]\\text{ }x=0;\\text{ }[\/latex]minimum:[latex]\\text{ }y=-\\frac{2}{3}\\text{ }[\/latex]occurs at[latex]\\text{ }x=\\pi ;\\text{ }[\/latex]for one period, the graph starts at 0 and ends at[latex]\\text{ }2\\pi[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      9. [latex]f\\left(x\\right)=-3\\mathrm{sin}\\text{ }x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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      10. [latex]f\\left(x\\right)=4\\mathrm{sin}\\text{ }x[\/latex]<\/p>\n<\/div>\n

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      \"A<\/span><\/p>\n

      amplitude: 4; period:[latex]\\text{ }2\\pi ;\\text{ }[\/latex]midline:[latex]\\text{ }y=0;\\text{ }[\/latex]maximum[latex]\\text{ }y=4\\text{ }[\/latex]occurs at[latex]\\text{ }x=\\frac{\\pi }{2};\\text{ }[\/latex]minimum:[latex]\\text{ }y=-4\\text{ }[\/latex]occurs at[latex]\\text{ }x=\\frac{3\\pi }{2};\\text{ }[\/latex]one full period occurs from[latex]\\text{ }x=0\\text{ }[\/latex]to[latex]\\text{ }x=2\\pi[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      11. [latex]f\\left(x\\right)=2\\mathrm{cos}\\text{ }x[\/latex]<\/p>\n<\/div>\n<\/div>\n

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      12. [latex]f\\left(x\\right)=\\mathrm{cos}\\left(2x\\right)[\/latex]<\/p>\n<\/div>\n

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

      amplitude: 1; period:[latex]\\text{ }\\pi ;\\text{ }[\/latex]midline:[latex]\\text{ }y=0;\\text{ }[\/latex]maximum:[latex]\\text{ }y=1\\text{ }[\/latex]occurs at[latex]\\text{ }x=\\pi ;\\text{ }[\/latex]minimum:[latex]\\text{ }y=-1\\text{ }[\/latex]occurs at[latex]\\text{ }x=\\frac{\\pi }{2};\\text{ }[\/latex]one full period is graphed from[latex]\\text{ }x=0\\text{ }[\/latex]to[latex]\\text{ }x=\\pi[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      13. [latex]f\\left(x\\right)=2\\text{ }\\mathrm{sin}\\left(\\frac{1}{2}x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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      14. [latex]f\\left(x\\right)=4\\text{ }\\mathrm{cos}\\left(\\pi x\\right)[\/latex]<\/p>\n<\/div>\n

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

      amplitude: 4; period: 2; midline:[latex]\\text{ }y=0;\\text{ }[\/latex]maximum:[latex]\\text{ }y=4\\text{ }[\/latex]occurs at[latex]\\text{ }x=0;\\text{ }[\/latex]minimum:[latex]\\text{ }y=-4\\text{ }[\/latex]occurs at[latex]\\text{ }x=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      15. [latex]f\\left(x\\right)=3\\text{ }\\mathrm{cos}\\left(\\frac{6}{5}x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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      16. [latex]y=3\\text{ }\\mathrm{sin}\\left(8\\left(x+4\\right)\\right)+5[\/latex]<\/p>\n<\/div>\n

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

      amplitude: 3; period:[latex]\\text{ }\\frac{\\pi }{4};\\text{ }[\/latex]midline:[latex]\\text{ }y=5;\\text{ }[\/latex]maximum:[latex]\\text{ }y=8\\text{ }[\/latex]occurs at[latex]\\text{ }x=0.12;\\text{ }[\/latex]minimum:[latex]\\text{ }y=2\\text{ }[\/latex]occurs at[latex]\\text{ }x=0.516;\\text{ }[\/latex]horizontal shift:[latex]\\text{ }-4;\\text{ }[\/latex]vertical translation 5; one period occurs from[latex]\\text{ }x=0\\text{ }[\/latex]to[latex]\\text{ }x=\\frac{\\pi }{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      17. [latex]y=2\\text{ }\\mathrm{sin}\\left(3x-21\\right)+4[\/latex]<\/p>\n<\/div>\n<\/div>\n

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      18. [latex]y=5\\text{ }\\mathrm{sin}\\left(5x+20\\right)-2[\/latex]<\/p>\n<\/div>\n

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

      amplitude: 5; period:[latex]\\text{ }\\frac{2\\pi }{5};\\text{ }[\/latex]midline:[latex]\\text{ }y=-2;\\text{ }[\/latex]maximum:[latex]\\text{ }y=3\\text{ }[\/latex]occurs at[latex]\\text{ }x=0.08;\\text{ }[\/latex]minimum:[latex]\\text{ }y=-7\\text{ }[\/latex]occurs at[latex]\\text{ }x=0.71;\\text{ }[\/latex]phase shift:[latex]\\text{ }-4;\\text{ }[\/latex]vertical translation:[latex]\\text{ }-2;\\text{ }[\/latex]one full period can be graphed on[latex]\\text{ }x=0\\text{ }[\/latex]to[latex]\\text{ }x=\\frac{2\\pi }{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

      For the following exercises, graph one full period of each function, starting at[latex]\\text{ }x=0.\\text{ }[\/latex]For each function, state the amplitude, period, and midline. State the maximum and minimum y<\/em>-values and their corresponding x<\/em>-values on one period for[latex]\\text{ }x>0.\\text{ }[\/latex]State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.<\/p>\n

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      19. [latex]f\\left(t\\right)=2\\mathrm{sin}\\left(t-\\frac{5\\pi }{6}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n

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      20. [latex]f\\left(t\\right)=-\\mathrm{cos}\\left(t+\\frac{\\pi }{3}\\right)+1[\/latex]<\/p>\n<\/div>\n

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

      amplitude: 1 ; period:[latex]\\text{ }2\\pi ;\\text{ }[\/latex]midline:[latex]\\text{ }y=1;\\text{ }[\/latex]maximum:[latex]\\text{ }y=2\\text{ }[\/latex]occurs at[latex]\\text{ }x=2.09;\\text{ }[\/latex]maximum:[latex]\\text{ }y=2\\text{ }[\/latex]occurs at[latex]\\text{ }t=2.09;\\text{ }[\/latex]minimum:[latex]\\text{ }y=0\\text{ }[\/latex]occurs at[latex]\\text{ }t=5.24;\\text{ }[\/latex]phase shift:[latex]\\text{ }-\\frac{\\pi }{3};\\text{ }[\/latex]vertical translation: 1; one full period is from[latex]\\text{ }t=0\\text{ }[\/latex]to[latex]\\text{ }t=2\\pi[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      21. [latex]f\\left(t\\right)=4\\mathrm{cos}\\left(2\\left(t+\\frac{\\pi }{4}\\right)\\right)-3[\/latex]<\/p>\n<\/div>\n<\/div>\n

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      22. [latex]f\\left(t\\right)=-\\mathrm{sin}\\left(\\frac{1}{2}t+\\frac{5\\pi }{3}\\right)[\/latex]<\/p>\n<\/div>\n

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

      amplitude: 1; period:[latex]\\text{ }4\\pi ;\\text{ }[\/latex]midline:[latex]\\text{ }y=0;\\text{ }[\/latex]maximum:[latex]\\text{ }y=1\\text{ }[\/latex]occurs at[latex]\\text{ }t=11.52;\\text{ }[\/latex]minimum:[latex]\\text{ }y=-1\\text{ }[\/latex]occurs at[latex]\\text{ }t=5.24;\\text{ }[\/latex]phase shift:[latex]\\text{ }-\\frac{10\\pi }{3};\\text{ }[\/latex]vertical shift: 0<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      23. [latex]f\\left(x\\right)=4\\mathrm{sin}\\left(\\frac{\\pi }{2}\\left(x-3\\right)\\right)+7[\/latex]<\/p>\n<\/div>\n<\/div>\n

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      Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown in (Figure)<\/a>.<\/p>\n

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

      amplitude: 2; midline:[latex]\\text{ }y=-3;\\text{ }[\/latex]period: 4; equation:[latex]\\text{ }f\\left(x\\right)=2\\mathrm{sin}\\left(\\frac{\\pi }{2}x\\right)-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      24. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in (Figure)<\/a>.<\/p>\n

      \"A<\/span><\/div>\n<\/div>\n<\/div>\n
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      25. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in (Figure)<\/a>.<\/p>\n

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

      amplitude: 2; period: 5; midline:[latex]\\text{ }y=3;\\text{ }[\/latex]equation:[latex]\\text{ }f\\left(x\\right)=-2\\mathrm{cos}\\left(\\frac{2\\pi }{5}x\\right)+3[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      26. Determine the amplitude, period, midline, and an equation involving sine for the graph shown in (Figure)<\/a>.<\/p>\n

      \"A<\/span><\/div>\n<\/div>\n<\/div>\n
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      27. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in (Figure)<\/a>.<\/p>\n

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

      amplitude: 4; period: 2; midline:[latex]\\text{ }y=0;\\text{ }[\/latex]equation:[latex]\\text{ }f\\left(x\\right)=-4\\mathrm{cos}\\left(\\pi \\left(x-\\frac{\\pi }{2}\\right)\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      28. Determine the amplitude, period, midline, and an equation involving sine for the graph shown in (Figure)<\/a>.<\/p>\n

      \"A<\/span><\/div>\n<\/div>\n<\/div>\n
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      29. Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in (Figure)<\/a>.<\/p>\n

      \"A<\/span><\/div>\n<\/div>\n
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      Show Solution<\/span><\/p>\n
      amplitude: 2; period: 2; midline[latex]\\text{ }y=1;\\text{ }[\/latex]equation:[latex]\\text{ }f\\left(x\\right)=2\\mathrm{cos}\\left(\\pi x\\right)+1[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      30. Determine the amplitude, period, midline, and an equation involving sine for the graph shown in (Figure)<\/a>.<\/p>\n

      \"A<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n
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      Algebraic<\/h4>\n

      For the following exercises, let[latex]\\text{ }f\\left(x\\right)=\\mathrm{sin}\\text{ }x.[\/latex]<\/p>\n

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      31. On[latex]\\text{ }\\left[0,2\\pi \\right),[\/latex]solve[latex]\\text{ }f\\left(x\\right)=0.[\/latex]<\/p>\n<\/div>\n<\/div>\n

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      32. On[latex]\\text{ }\\left[0,2\\pi \\right),[\/latex]solve[latex]\\text{ }f\\left(x\\right)=\\frac{1}{2}.[\/latex]<\/p>\n<\/div>\n

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      [latex]\\frac{\\pi }{6},\\frac{5\\pi }{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      33.Evaluate[latex]\\text{ }f\\left(\\frac{\\pi }{2}\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n

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      34. On[latex]\\text{ }\\left[0,2\\pi \\right),f\\left(x\\right)=\\frac{\\sqrt{2}}{2}.\\text{ }[\/latex]Find all values of[latex]\\text{ }x.[\/latex]<\/p>\n<\/div>\n

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      [latex]\\frac{\\pi }{4},\\frac{3\\pi }{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      35. On[latex]\\text{ }\\left[0,2\\pi \\right),[\/latex]the maximum value(s) of the function occur(s) at what x<\/em>-value(s)?<\/p>\n<\/div>\n<\/div>\n

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      36. On[latex]\\text{ }\\left[0,2\\pi \\right),[\/latex]the minimum value(s) of the function occur(s) at what x<\/em>-value(s)?<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
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      [latex]\\frac{3\\pi }{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      37. Show that[latex]\\text{ }f\\left(-x\\right)=-f\\left(x\\right).\\text{ }[\/latex]This means that[latex]\\text{ }f\\left(x\\right)=\\mathrm{sin}\\text{ }x\\text{ }[\/latex]is an odd function and possesses symmetry with respect to ________________.<\/p>\n<\/div>\n<\/div>\n

      For the following exercises, let[latex]\\text{ }f\\left(x\\right)=\\mathrm{cos}\\text{ }x.[\/latex]<\/p>\n

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      38. On[latex]\\text{ }\\left[0,2\\pi \\right),[\/latex]solve the equation[latex]\\text{ }f\\left(x\\right)=\\mathrm{cos}\\text{ }x=0.[\/latex]<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
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      [latex]\\frac{\\pi }{2},\\frac{3\\pi }{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      39. On[latex]\\text{ }\\left[0,2\\pi \\right),[\/latex]solve[latex]\\text{ }f\\left(x\\right)=\\frac{1}{2}.[\/latex]<\/p>\n<\/div>\n<\/div>\n

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      40. On[latex]\\text{ }\\left[0,2\\pi \\right),[\/latex]find the x<\/em>-intercepts of[latex]\\text{ }f\\left(x\\right)=\\mathrm{cos}\\text{ }x.[\/latex]<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
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      [latex]\\frac{\\pi }{2},\\frac{3\\pi }{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      41. On[latex]\\text{ }\\left[0,2\\pi \\right),[\/latex]find the x<\/em>-values at which the function has a maximum or minimum value.<\/p>\n<\/div>\n<\/div>\n

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      42. On[latex]\\text{ }\\left[0,2\\pi \\right),[\/latex]solve the equation[latex]\\text{ }f\\left(x\\right)=\\frac{\\sqrt{3}}{2}.[\/latex]<\/p>\n<\/div>\n

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      Show Solution<\/span><\/p>\n
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      [latex]\\frac{\\pi }{6},\\frac{11\\pi }{6}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n

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      Technology<\/h4>\n
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      43. Graph[latex]\\text{ }h\\left(x\\right)=x+\\mathrm{sin}\\text{ }x\\text{ }[\/latex]on[latex]\\text{ }\\left[0,2\\pi \\right].\\text{ }[\/latex]Explain why the graph appears as it does.<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      44. Graph[latex]\\text{ }h\\left(x\\right)=x+\\mathrm{sin}\\text{ }x\\text{ }[\/latex]on[latex]\\text{ }\\left[-100,100\\right].\\text{ }[\/latex]Did the graph appear as predicted in the previous exercise?<\/p>\n<\/div>\n

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

      The graph appears linear. The linear functions dominate the shape of the graph for large values of[latex]\\text{ }x.[\/latex]<\/p>\n

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

      \n
      \n

      45. Graph[latex]\\text{ }f\\left(x\\right)=x\\text{ }\\mathrm{sin}\\text{ }x\\text{ }[\/latex]on[latex]\\text{ }\\left[0,2\\pi \\right]\\text{ }[\/latex]and verbalize how the graph varies from the graph of[latex]\\text{ }f\\left(x\\right)=\\mathrm{sin}\\text{ }x.[\/latex]<\/p>\n<\/div>\n<\/div>\n

      \n
      \n

      46. Graph[latex]\\text{ }f\\left(x\\right)=x\\text{ }\\mathrm{sin}\\text{ }x\\text{ }[\/latex]on the window[latex]\\text{ }\\left[-10,10\\right]\\text{ }[\/latex]and explain what the graph shows.<\/p>\n<\/div>\n

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

      The graph is symmetric with respect to the y<\/em>-axis and there is no amplitude because the function is not periodic.<\/p>\n

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

      \n
      \n

      47. Graph[latex]\\text{ }f\\left(x\\right)=\\frac{\\mathrm{sin}\\text{ }x}{x}\\text{ }[\/latex]on the window[latex]\\text{ }\\left[-5\\pi ,5\\pi \\right]\\text{ }[\/latex]and explain what the graph shows.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

      \n

      Real-World Applications<\/h4>\n
      \n
      \n

      48. A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o\u2019clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The function[latex]\\text{ }h\\left(t\\right)\\text{ }[\/latex]gives a person\u2019s height in meters above the ground t<\/em> minutes after the wheel begins to turn.<\/p>\n

        \n
      1. Find the amplitude, midline, and period of[latex]\\text{ }h\\left(t\\right).[\/latex]<\/li>\n
      2. Find a formula for the height function[latex]\\text{ }h\\left(t\\right).[\/latex]<\/li>\n
      3. How high off the ground is a person after 5 minutes?<\/li>\n<\/ol>\n<\/div>\n
        \n
        Show Solution<\/span><\/p>\n
        \n
          \n
        1. Amplitude: 12.5; period: 10; midline:[latex]\\text{ }y=13.5;[\/latex]<\/li>\n
        2. [latex]h\\left(t\\right)=12.5\\mathrm{sin}\\left(\\frac{\\pi }{5}\\left(t-2.5\\right)\\right)+13.5;[\/latex]<\/li>\n
        3. 26 ft<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":158103,"menu_order":8,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2329","chapter","type-chapter","status-web-only","hentry"],"part":478,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2329","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/users\/158103"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2329\/revisions"}],"predecessor-version":[{"id":2331,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2329\/revisions\/2331"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/parts\/478"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/2329\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/media?parent=2329"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2329"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/contributor?post=2329"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/license?post=2329"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}