{"id":128,"date":"2019-08-20T17:02:56","date_gmt":"2019-08-20T21:02:56","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/graphs-of-the-sine-and-cosine-functions\/"},"modified":"2022-06-01T10:39:31","modified_gmt":"2022-06-01T14:39:31","slug":"graphs-of-the-sine-and-cosine-functions","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/graphs-of-the-sine-and-cosine-functions\/","title":{"raw":"Graphs of the Sine and Cosine Functions","rendered":"Graphs of the Sine and Cosine Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section, you will:\n<ul>\n \t<li>Graph variations of \u2009y=sin( x )\u2009 and \u2009y=cos( x ).<\/li>\n \t<li>Use phase shifts of sine and cosine curves.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Figure_06_01_001\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143045\/CNX_Precalc_Figure_06_01_001.jpg\" alt=\"A photo of a rainbow colored beam of light stretching across the floor.\" width=\"487\" height=\"390\"> <strong>Figure 1. <\/strong>Light can be separated into colors because of its wavelike properties. (credit: \"wonderferret\"\/ Flickr)[\/caption]\n\n<\/div>\n<p id=\"fs-id1165135329803\">White light, such as the light from the sun, is not actually white at all. Instead, it is a composition of all the colors of the rainbow in the form of waves. The individual colors can be seen only when white light passes through an optical prism that separates the waves according to their wavelengths to form a rainbow.<\/p>\n<p id=\"fs-id1165134544966\">Light waves can be represented graphically by the sine function. In the chapter on <a class=\"target-chapter\" href=\"\/contents\/5d813d51-ebc0-49c9-96a5-f49768c8bfb3\">Trigonometric Functions<\/a>, we examined trigonometric functions such as the sine function. In this section, we will interpret and create graphs of sine and cosine functions.<\/p>\n\n<div id=\"fs-id1165135169322\" class=\"bc-section section\">\n<h3>Graphing Sine and Cosine Functions<\/h3>\n<p id=\"fs-id1165135241395\">Recall that the sine and cosine functions relate real number values to the <em>x<\/em>- and <em>y<\/em>-coordinates of a point on the unit circle. So what do they look like on a graph on a coordinate plane? Let\u2019s start with the <span class=\"no-emphasis\">sine function<\/span>. We can create a table of values and use them to sketch a graph. <a class=\"autogenerated-content\" href=\"#Table_06_01_01\">(Figure)<\/a> lists some of the values for the sine function on a unit circle.<\/p>\n\n<table id=\"Table_06_01_01\" summary=\"..\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{6}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{4}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{3}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{2}[\/latex]<\/td>\n<td>[latex]\\frac{2\\pi }{3}[\/latex]<\/td>\n<td>[latex]\\frac{3\\pi }{4}[\/latex]<\/td>\n<td>[latex]\\frac{5\\pi }{6}[\/latex]<\/td>\n<td>[latex]\\pi [\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]\\mathrm{sin}\\left(x\\right)[\/latex]<\/strong><\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137694159\">Plotting the points from the table and continuing along the <em>x<\/em>-axis gives the shape of the sine function. See <a class=\"autogenerated-content\" href=\"#Figure_06_01_002\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_06_01_002\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143048\/CNX_Precalc_Figure_06_01_002.jpg\" alt=\"A graph of sin(x). Local maximum at (pi\/2, 1). Local minimum at (3pi\/2, -1). Period of 2pi.\" width=\"487\" height=\"216\"> <strong>Figure 2. <\/strong>The sine function[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137410832\">Notice how the sine values are positive between 0 and[latex]\\,\\pi ,\\,[\/latex]which correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative between[latex]\\,\\pi \\,[\/latex]and[latex]\\,2\\pi ,\\,[\/latex]which correspond to the values of the sine function in quadrants III and IV on the unit circle. See <a class=\"autogenerated-content\" href=\"#Figure_06_01_003\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_06_01_003\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143051\/CNX_Precalc_Figure_06_01_003.jpg\" alt=\"A side-by-side graph of a unit circle and a graph of sin(x). The two graphs show the equivalence of the coordinates.\" width=\"487\" height=\"219\"> <strong>Figure 3. <\/strong>Plotting values of the sine function[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137849285\">Now let\u2019s take a similar look at the <span class=\"no-emphasis\">cosine function<\/span>. Again, we can create a table of values and use them to sketch a graph. <a class=\"autogenerated-content\" href=\"#Table_06_01_02\">(Figure)<\/a> lists some of the values for the cosine function on a unit circle.<\/p>\n\n<table id=\"Table_06_01_02\" summary=\"..\">\n<tbody>\n<tr>\n<td>[latex]\\mathbf{x}[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{6}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{4}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{3}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{2}[\/latex]<\/td>\n<td>[latex]\\frac{2\\pi }{3}[\/latex]<\/td>\n<td>[latex]\\frac{3\\pi }{4}[\/latex]<\/td>\n<td>[latex]\\frac{5\\pi }{6}[\/latex]<\/td>\n<td>[latex]\\pi [\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\mathbf{cos}\\left(\\mathbf{x}\\right)[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]-\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]-\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\n<td>[latex]-\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165135531416\">As with the sine function, we can plots points to create a graph of the cosine function as in <a class=\"autogenerated-content\" href=\"#Figure_06_01_004\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_06_01_004\" class=\"medium\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143056\/CNX_Precalc_Figure_06_01_004.jpg\" alt=\"A graph of cos(x). Local maxima at (0,1) and (2pi, 1). Local minimum at (pi, -1). Period of 2pi.\" width=\"731\" height=\"216\"> <strong>Figure 4. <\/strong>The cosine function[\/caption]\n<p id=\"fs-id1165135628668\">Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval[latex]\\,\\left[-1,1\\right].[\/latex]<\/p>\n<p id=\"fs-id1165137727184\">In both graphs, the shape of the graph repeats after[latex]\\,2\\pi ,\\,[\/latex]which means the functions are periodic with a period of[latex]\\,2\\pi .\\,[\/latex]A periodic function is a function for which a specific <span class=\"no-emphasis\">horizontal shift<\/span>, <em>P<\/em>, results in a function equal to the original function:[latex]\\,f\\left(x+P\\right)=f\\left(x\\right)\\,[\/latex]for all values of[latex]\\,x\\,[\/latex]in the domain of[latex]\\,f.\\,[\/latex]When this occurs, we call the smallest such horizontal shift with[latex]\\,P&gt;0\\,[\/latex]the <span class=\"no-emphasis\">period<\/span> of the function. <a class=\"autogenerated-content\" href=\"#Figure_06_01_005\">(Figure)<\/a> shows several periods of the sine and cosine functions.<\/p>\n\n<div id=\"Figure_06_01_005\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143101\/CNX_Precalc_Figure_06_01_005.jpg\" alt=\"Side-by-side graphs of sin(x) and cos(x). Graphs show period lengths for both functions, which is 2pi.\" width=\"487\" height=\"442\"> <strong>Figure 5.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137447360\">Looking again at the sine and cosine functions on a domain centered at the <em>y<\/em>-axis helps reveal symmetries. As we can see in <a class=\"autogenerated-content\" href=\"#Figure_06_01_006\">(Figure)<\/a>, the <span class=\"no-emphasis\">sine function<\/span> is symmetric about the origin. Recall from <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/chapter\/the-other-trigonometric-functions\/\">The Other Trigonometric Functions<\/a> that we determined from the unit circle that the sine function is an odd function because[latex]\\,\\mathrm{sin}\\left(-x\\right)=-\\mathrm{sin}\\,x.\\,[\/latex]\nNow we can clearly see this property from the graph.<\/p>\n\n<div id=\"Figure_06_01_006\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143106\/CNX_Precalc_Figure_06_01_006.jpg\" alt=\"A graph of sin(x) that shows that sin(x) is an odd function due to the odd symmetry of the graph.\" width=\"487\" height=\"191\"> <strong>Figure 6. <\/strong>Odd symmetry of the sine function[\/caption]\n\n<\/div>\n<p id=\"fs-id1165135484164\"><a class=\"autogenerated-content\" href=\"#Figure_06_01_007\">(Figure)<\/a> shows that the cosine function is symmetric about the <em>y<\/em>-axis. Again, we determined that the cosine function is an even function. Now we can see from the graph that [latex]\\mathrm{cos}\\left(-x\\right)=\\mathrm{cos}\\text{ }x.[\/latex]<\/p>\n\n<div id=\"Figure_06_01_007\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143115\/CNX_Precalc_Figure_06_01_007.jpg\" alt=\"A graph of cos(x) that shows that cos(x) is an even function due to the even symmetry of the graph.\" width=\"487\" height=\"216\"> <strong>Figure 7. <\/strong>Even symmetry of the cosine function[\/caption]\n\n<\/div>\n<div id=\"fs-id1165135187674\">\n<h3>Characteristics of Sine and Cosine Functions<\/h3>\n<p id=\"fs-id1165137628764\">The sine and cosine functions have several distinct characteristics:<\/p>\n\n<ul id=\"fs-id1165137662423\">\n \t<li>They are periodic functions with a period of[latex]\\,2\\pi .[\/latex]<\/li>\n \t<li>The domain of each function is[latex]\\,\\left(-\\infty ,\\infty \\right)\\,[\/latex]and the range is[latex]\\,\\left[-1,1\\right].[\/latex]<\/li>\n \t<li>The graph of[latex]\\,y=\\mathrm{sin}\\text{ }x\\,[\/latex]is symmetric about the origin, because it is an odd function.<\/li>\n \t<li>The graph of[latex]\\,y=\\mathrm{cos}\\text{ }x\\,[\/latex]is symmetric about the[latex]\\,y\\text{-}[\/latex]axis, because it is an even function.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134032223\" class=\"bc-section section\">\n<h3>Investigating Sinusoidal Functions<\/h3>\n<p id=\"fs-id1165137410921\">As we can see, sine and cosine functions have a regular period and range. If we watch ocean waves or ripples on a pond, we will see that they resemble the sine or cosine functions. However, they are not necessarily identical. Some are taller or longer than others. A function that has the same general shape as a sine or <span class=\"no-emphasis\">cosine function<\/span> is known as a sinusoidal function. The general forms of sinusoidal functions are<\/p>\n\n<div id=\"fs-id1165135512530\">[latex]\\begin{array}{l}y=A\\mathrm{sin}\\left(Bx-C\\right)+D\\hfill \\\\ \\text{ and}\\hfill \\\\ y=A\\mathrm{cos}\\left(Bx-C\\right)+D\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"fs-id1165135458566\" class=\"bc-section section\">\n<h4>Determining the Period of Sinusoidal Functions<\/h4>\n<p id=\"fs-id1165135708019\">Looking at the forms of sinusoidal functions, we can see that they are transformations of the sine and cosine functions. We can use what we know about transformations to determine the period.<\/p>\n<p id=\"fs-id1165137639577\">In the general formula,[latex]\\,B\\,[\/latex]is related to the period by[latex]\\,P=\\frac{2\\pi }{|B|}.\\,[\/latex]If[latex]\\,|B|&gt;1,\\,[\/latex]then the period is less than[latex]\\,2\\pi \\,[\/latex]and the function undergoes a horizontal compression, whereas if[latex]\\,|B|&lt;1,\\,[\/latex]then the period is greater than[latex]\\,2\\pi \\,[\/latex]and the function undergoes a horizontal stretch. For example,[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\left(x\\right),\\,[\/latex][latex]B=1,\\,[\/latex]so the period is[latex]\\,2\\pi ,\\text{}[\/latex]which we knew. If[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\left(2x\\right),\\,[\/latex]then[latex]\\,B=2,\\,[\/latex]so the period is[latex]\\,\\pi \\,[\/latex]and the graph is compressed. If[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\left(\\frac{x}{2}\\right),\\,[\/latex]then[latex]\\,B=\\frac{1}{2},\\,[\/latex]so the period is[latex]\\,4\\pi \\,[\/latex]and the graph is stretched. Notice in <a class=\"autogenerated-content\" href=\"#Figure_06_01_008\">(Figure)<\/a> how the period is indirectly related to[latex]\\,|B|.[\/latex]<\/p>\n\n<div id=\"Figure_06_01_008\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143117\/CNX_Precalc_Figure_06_01_008.jpg\" alt=\"A graph with three items. The x-axis ranges from 0 to 2pi. The y-axis ranges from -1 to 1. The first item is the graph of sin(x) for one full period. The second is the graph of sin(2x) over two periods. The third is the graph of sin(x\/2) for one half of a period.\" width=\"487\" height=\"274\"> <strong>Figure 8.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"fs-id1165137697004\">\n<h3>Period of Sinusoidal Functions<\/h3>\n<p id=\"fs-id1165137766762\">If we let[latex]\\,C=0\\,[\/latex]and[latex]\\,D=0\\,[\/latex]in the general form equations of the sine and cosine functions, we obtain the forms<\/p>\n\n<div id=\"fs-id1165137855068\" class=\"unnumbered aligncenter\">[latex]y=A\\mathrm{sin}\\left(Bx\\right)[\/latex]<\/div>\n<div id=\"fs-id1165134371173\" class=\"unnumbered aligncenter\">[latex]y=A\\mathrm{cos}\\left(Bx\\right)[\/latex]<\/div>\n<p id=\"fs-id1165137413926\">The period is[latex]\\,\\frac{2\\pi }{|B|}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"Example_06_01_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137772370\">\n<div id=\"fs-id1165137772372\">\n<h3>Identifying the Period of a Sine or Cosine Function<\/h3>\n<p id=\"fs-id1165137389619\">Determine the period of the function[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\left(\\frac{\\pi }{6}x\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137434852\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137434852\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137434852\"]\n<p id=\"fs-id1165135188750\">Let\u2019s begin by comparing the equation to the general form[latex]\\,y=A\\mathrm{sin}\\left(Bx\\right).[\/latex]<\/p>\n<p id=\"fs-id1165134482743\">In the given equation,[latex]\\,B=\\frac{\\pi }{6},\\,[\/latex]so the period will be<\/p>\n\n<div class=\"unnumbered\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ P=\\frac{2\\pi }{|B|}\\end{array}\\hfill \\\\ \\text{ }=\\frac{2\\pi }{\\frac{\\pi }{6}}\\hfill \\\\ \\text{ }=2\\pi \\cdot \\frac{6}{\\pi }\\hfill \\\\ \\text{ }=12\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137465427\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_01\">\n<div>\n<p id=\"fs-id1165135208858\">Determine the period of the function[latex]\\,g\\left(x\\right)=\\mathrm{cos}\\left(\\frac{x}{3}\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137507692\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137507692\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137507692\"]\n<p id=\"fs-id1165137675634\">[latex]\\,6\\pi \\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135699141\" class=\"bc-section section\">\n<h4>Determining Amplitude<\/h4>\n<p id=\"fs-id1165135207425\">Returning to the general formula for a sinusoidal function, we have analyzed how the variable[latex]\\,B\\,[\/latex]relates to the period. Now let\u2019s turn to the variable[latex]\\,A\\,[\/latex]so we can analyze how it is related to the <strong>amplitude<\/strong>, or greatest distance from rest.[latex]\\,A\\,[\/latex]represents the vertical stretch factor, and its absolute value[latex]\\,|A|\\,[\/latex]is the amplitude. The local maxima will be a distance[latex]\\,|A|\\,[\/latex]above the horizontal <strong>midline<\/strong> of the graph, which is the line[latex]\\,y=D;\\,[\/latex]because[latex]\\,D=0\\,[\/latex]in this case, the midline is the <em>x<\/em>-axis. The local minima will be the same distance below the midline. If[latex]\\,|A|&gt;1,\\,[\/latex]the function is stretched. For example, the amplitude of[latex]\\,f\\left(x\\right)=4\\,\\mathrm{sin}\\,x\\,[\/latex]is twice the amplitude of[latex]\\,f\\left(x\\right)=2\\,\\mathrm{sin}\\,x.\\,[\/latex]If[latex]\\,|A|&lt;1,\\,[\/latex]the function is compressed. <a class=\"autogenerated-content\" href=\"#Figure_06_01_009\">(Figure)<\/a> compares several sine functions with different amplitudes.<\/p>\n\n<div id=\"Figure_06_01_009\" class=\"wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143125\/CNX_Precalc_Figure_06_01_009.jpg\" alt=\"A graph with four items. The x-axis ranges from -6pi to 6pi. The y-axis ranges from -4 to 4. The first item is the graph of sin(x), which has an amplitude of 1. The second is a graph of 2sin(x), which has amplitude of 2. The third is a graph of 3sin(x), which has an amplitude of 3. The fourth is a graph of 4 sin(x) with an amplitude of 4.\" width=\"975\" height=\"316\"> <strong>Figure 9.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"fs-id1165134284498\">\n<h3>Amplitude of Sinusoidal Functions<\/h3>\n<p id=\"fs-id1165134497141\">If we let[latex]\\,C=0\\,[\/latex]and[latex]\\,D=0\\,[\/latex]in the general form equations of the sine and cosine functions, we obtain the forms<\/p>\n\n<div id=\"fs-id1165135177658\" class=\"unnumbered aligncenter\">[latex]y=A\\mathrm{sin}\\left(Bx\\right)\\text{ and }y=A\\mathrm{cos}\\left(Bx\\right)[\/latex]<\/div>\n<p id=\"fs-id1165137464064\">The amplitude is[latex]\\,A,\\,[\/latex]and the vertical height from the midline is[latex]\\,|A|.\\,[\/latex]In addition, notice in the example that<\/p>\n\n<div id=\"fs-id1165135460914\" class=\"unnumbered aligncenter\">[latex]|A|\\text{ = amplitude = }\\frac{1}{2}|\\text{maximum }-\\text{ minimum}|[\/latex]<\/div>\n<\/div>\n<div id=\"Example_06_01_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137653911\">\n<div id=\"fs-id1165134377968\">\n<h3>Identifying the Amplitude of a Sine or Cosine Function<\/h3>\n<p id=\"fs-id1165137932594\">What is the amplitude of the sinusoidal function[latex]\\,f\\left(x\\right)=-4\\mathrm{sin}\\left(x\\right)?\\,[\/latex]Is the function stretched or compressed vertically?<\/p>\n\n<\/div>\n<div id=\"fs-id1165135195832\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135195832\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135195832\"]\n<p id=\"fs-id1165135195834\">Let\u2019s begin by comparing the function to the simplified form[latex]\\,y=A\\mathrm{sin}\\left(Bx\\right).[\/latex]<\/p>\n<p id=\"fs-id1165137930335\">In the given function,[latex]\\,A=-4,\\,[\/latex]so the amplitude is[latex]\\,|A|=|-4|=4.\\,[\/latex]The function is stretched.[\/hidden-answer]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134226786\">\n<h4>Analysis<\/h4>\nThe negative value of[latex]\\,A\\,[\/latex]results in a reflection across the <em>x<\/em>-axis of the <span class=\"no-emphasis\">sine function<\/span>, as shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_010\">(Figure)<\/a>.\n<div id=\"Figure_06_01_010\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143134\/CNX_Precalc_Figure_06_01_010.jpg\" alt=\"A graph of -4sin(x). The function has an amplitude of 4. Local minima at (-3pi\/2, -4) and (pi\/2, -4). Local maxima at (-pi\/2, 4) and (3pi\/2, 4). Period of 2pi.\" width=\"487\" height=\"319\"> <strong>Figure 10.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135471236\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_02\">\n<div id=\"fs-id1165137771980\">\n\nWhat is the amplitude of the sinusoidal function[latex]\\,f\\left(x\\right)=\\frac{1}{2}\\mathrm{sin}\\left(x\\right)?\\,[\/latex]Is the function stretched or compressed vertically?\n\n<\/div>\n<div>\n\n[latex]\\frac{1}{2}\\,[\/latex]compressed\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137834807\" class=\"bc-section section\">\n<h3>Analyzing Graphs of Variations of <em>y<\/em> = sin<em> x<\/em> and <em>y<\/em> = cos <em>x<\/em><\/h3>\n<p id=\"fs-id1165135193998\">Now that we understand how[latex]\\,A\\,[\/latex]and[latex]\\,B\\,[\/latex]relate to the general form equation for the sine and cosine functions, we will explore the variables[latex]\\,C\\,[\/latex]and[latex]\\,D.\\,[\/latex]Recall the general form:<\/p>\n\n<div class=\"unnumbered\">[latex]\\begin{array}{c}y=A\\mathrm{sin}\\left(Bx-C\\right)+D\\text{ and }y=A\\mathrm{cos}\\left(Bx-C\\right)+D\\\\ or\\\\ y=A\\mathrm{sin}\\left(B\\left(x-\\frac{C}{B}\\right)\\right)+D\\text{ and }y=A\\mathrm{cos}\\left(B\\left(x-\\frac{C}{B}\\right)\\right)+D\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134187254\">The value[latex]\\,\\frac{C}{B}\\,[\/latex]for a sinusoidal function is called the <strong>phase shift<\/strong>, or the horizontal displacement of the basic sine or <span class=\"no-emphasis\">cosine function<\/span>. If[latex]\\,C&gt;0,\\,[\/latex]the graph shifts to the right. If[latex]\\,C&lt;0,\\,[\/latex]the graph shifts to the left. The greater the value of[latex]\\,|C|,\\,[\/latex]the more the graph is shifted. <a class=\"autogenerated-content\" href=\"#Figure_06_01_011\">(Figure)<\/a> shows that the graph of[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\left(x-\\pi \\right)\\,[\/latex]shifts to the right by[latex]\\,\\pi \\,[\/latex]units, which is more than we see in the graph of[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\left(x-\\frac{\\pi }{4}\\right),\\,[\/latex]which shifts to the right by[latex]\\,\\frac{\\pi }{4}\\,[\/latex]units.<\/p>\n\n<div id=\"Figure_06_01_011\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143137\/CNX_Precalc_Figure_06_01_011.jpg\" alt=\"A graph with three items. The first item is a graph of sin(x). The second item is a graph of sin(x-pi\/4), which is the same as sin(x) except shifted to the right by pi\/4. The third item is a graph of sin(x-pi), which is the same as sin(x) except shifted to the right by pi.\" width=\"487\" height=\"255\"> <strong>Figure 11.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137663752\">While[latex]\\,C\\,[\/latex]relates to the horizontal shift,[latex]\\,D\\,[\/latex]indicates the vertical shift from the midline in the general formula for a sinusoidal function. See <a class=\"autogenerated-content\" href=\"#Figure_06_01_012\">(Figure)<\/a>. The function[latex]\\,y=\\mathrm{cos}\\left(x\\right)+D\\,[\/latex]has its midline at[latex]\\,y=D.[\/latex]<\/p>\n\n<div id=\"Figure_06_01_012\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143140\/CNX_Precalc_Figure_06_01_012.jpg\" alt=\"A graph of y=Asin(x)+D. Graph shows the midline of the function at y=D.\" width=\"487\" height=\"255\"> <strong>Figure 12.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165135242867\">Any value of[latex]\\,D\\,[\/latex]other than zero shifts the graph up or down. <a class=\"autogenerated-content\" href=\"#Figure_06_01_013\">(Figure)<\/a> compares[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\,x\\,[\/latex]with[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\,x+2,\\,[\/latex]which is shifted 2 units up on a graph.<\/p>\n\n<div id=\"Figure_06_01_013\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143145\/CNX_Precalc_Figure_06_01_013.jpg\" alt=\"A graph with two items. The first item is a graph of sin(x). The second item is a graph of sin(x)+2, which is the same as sin(x) except shifted up by 2.\" width=\"487\" height=\"221\"> <strong>Figure 13.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"fs-id1165135571809\">\n<h3>Variations of Sine and Cosine Functions<\/h3>\n<p id=\"fs-id1165133201875\">Given an equation in the form[latex]\\,f\\left(x\\right)=A\\mathrm{sin}\\left(Bx-C\\right)+D\\,[\/latex]or[latex]\\,f\\left(x\\right)=A\\mathrm{cos}\\left(Bx-C\\right)+D,\\,[\/latex][latex]\\frac{C}{B}\\,[\/latex]is the phase shift and[latex]\\,D\\,[\/latex]is the <span class=\"no-emphasis\">vertical shift<\/span>.<\/p>\n\n<\/div>\n<div id=\"Example_06_01_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137696922\">\n<div>\n<h3>Identifying the Phase Shift of a Function<\/h3>\n<p id=\"fs-id1165137804482\">Determine the direction and magnitude of the phase shift for[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\left(x+\\frac{\\pi }{6}\\right)-2.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134483435\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134483435\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134483435\"]\n<p id=\"fs-id1165134483437\">Let\u2019s begin by comparing the equation to the general form[latex]\\,y=A\\mathrm{sin}\\left(Bx-C\\right)+D.[\/latex]<\/p>\n<p id=\"fs-id1165137461008\">In the given equation, notice that[latex]\\,B=1\\,[\/latex]and[latex]\\,C=-\\frac{\\pi }{6}.\\,[\/latex]So the phase shift is<\/p>\n\n<div id=\"fs-id1165137693976\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{r}\\hfill \\\\ \\hfill \\frac{C}{B}=-\\frac{\\frac{\\pi }{6}}{1}\\\\ \\hfill \\text{ }=-\\frac{\\pi }{6}\\end{array}[\/latex]<\/div>\nor[latex]\\,\\frac{\\pi }{6}\\,[\/latex]units to the left.[\/hidden-answer]\n\n<\/div>\n<div id=\"fs-id1165134156073\">\n<h4>Analysis<\/h4>\nWe must pay attention to the sign in the equation for the general form of a sinusoidal function. The equation shows a minus sign before[latex]\\,C.\\,[\/latex]Therefore[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\left(x+\\frac{\\pi }{6}\\right)-2\\,[\/latex]can be rewritten as[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\left(x-\\left(-\\frac{\\pi }{6}\\right)\\right)-2.\\,[\/latex]If the value of[latex]\\,C\\,[\/latex]is negative, the shift is to the left.\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_03\">\n<div id=\"fs-id1165137461117\">\n<p id=\"fs-id1165137461118\">Determine the direction and magnitude of the phase shift for[latex]\\,f\\left(x\\right)=3\\mathrm{cos}\\left(x-\\frac{\\pi }{2}\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165131959464\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165131959464\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165131959464\"]\n<p id=\"fs-id1165131959465\">[latex]\\frac{\\pi }{2};\\,[\/latex]right<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_06_01_04\" class=\"textbox examples\">\n<div id=\"fs-id1165137410966\">\n<div id=\"fs-id1165137410968\">\n<h3>Identifying the Vertical Shift of a Function<\/h3>\n<p id=\"fs-id1165135186656\">Determine the direction and magnitude of the vertical shift for[latex]\\,f\\left(x\\right)=\\mathrm{cos}\\left(x\\right)-3.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137427502\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137427502\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137427502\"]\n<p id=\"fs-id1165137427504\">Let\u2019s begin by comparing the equation to the general form[latex]\\,y=A\\mathrm{cos}\\left(Bx-C\\right)+D.[\/latex]<\/p>\n<p id=\"fs-id1165135503692\">In the given equation,[latex]\\,D=-3\\,[\/latex]so the shift is 3 units downward.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137742086\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_04\">\n<div id=\"fs-id1165137410879\">\n<p id=\"fs-id1165137410880\">Determine the direction and magnitude of the vertical shift for[latex]\\,f\\left(x\\right)=3\\mathrm{sin}\\left(x\\right)+2.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137432579\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137432579\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137432579\"]\n<p id=\"fs-id1165137432580\">2 units up<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137824883\" class=\"precalculus howto\">\n<p id=\"fs-id1165135417813\"><strong>Given a sinusoidal function in the form<\/strong>[latex]\\,f\\left(x\\right)=A\\mathrm{sin}\\left(Bx-C\\right)+D,\\,[\/latex]<strong>identify the midline, amplitude, period, and phase shift.<\/strong><\/p>\n\n<ol id=\"fs-id1165137805755\" type=\"1\">\n \t<li>Determine the amplitude as[latex]\\,|A|.[\/latex]<\/li>\n \t<li>Determine the period as[latex]\\,P=\\frac{2\\pi }{|B|}.[\/latex]<\/li>\n \t<li>Determine the phase shift as[latex]\\,\\frac{C}{B}.[\/latex]<\/li>\n \t<li>Determine the midline as[latex]\\,y=D.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_06_01_05\" class=\"textbox examples\">\n<div id=\"fs-id1165137457010\">\n<div id=\"fs-id1165137457013\">\n<h3>Identifying the Variations of a Sinusoidal Function from an Equation<\/h3>\n<p id=\"fs-id1165137416718\">Determine the midline, amplitude, period, and phase shift of the function[latex]\\,y=3\\mathrm{sin}\\left(2x\\right)+1.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137454382\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137454382\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137454382\"]\n<p id=\"fs-id1165137454384\">Let\u2019s begin by comparing the equation to the general form[latex]\\,y=A\\mathrm{sin}\\left(Bx-C\\right)+D.[\/latex]<\/p>\n[latex]A=3,\\,[\/latex]so the amplitude is[latex]\\,|A|=3.[\/latex]\n<p id=\"fs-id1165137438431\">Next,[latex]\\,B=2,\\,[\/latex]so the period is[latex]\\,P=\\frac{2\\pi }{|B|}=\\frac{2\\pi }{2}=\\pi .[\/latex]<\/p>\nThere is no added constant inside the parentheses, so[latex]\\,C=0\\,[\/latex]and the phase shift is[latex]\\,\\frac{C}{B}=\\frac{0}{2}=0.[\/latex]\n<p id=\"fs-id1165137697063\">Finally,[latex]\\,D=1,\\,[\/latex]so the midline is[latex]\\,y=1.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137701755\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135414237\">Inspecting the graph, we can determine that the period is[latex]\\,\\pi ,\\,[\/latex]the midline is[latex]\\,y=1,\\,[\/latex]and the amplitude is 3. See <a class=\"autogenerated-content\" href=\"#Figure_06_01_014\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_06_01_014\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143154\/CNX_Precalc_Figure_06_01_014.jpg\" alt=\"A graph of y=3sin(2x)+1. The graph has an amplitude of 3. There is a midline at y=1. There is a period of pi. Local maximum at (pi\/4, 4) and local minimum at (3pi\/4, -2).\" width=\"487\" height=\"263\"> <strong>Figure 14.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137447405\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_05\">\n<div id=\"fs-id1165137447552\">\n<p id=\"fs-id1165137447553\">Determine the midline, amplitude, period, and phase shift of the function[latex]\\,y=\\frac{1}{2}\\mathrm{cos}\\left(\\frac{x}{3}-\\frac{\\pi }{3}\\right).[\/latex]<\/p>\n\n<\/div>\n<div>[reveal-answer q=\"fs-id1165134042358\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134042358\"]\n<p id=\"fs-id1165134042358\">midline:[latex]\\,y=0;\\,[\/latex]amplitude:[latex]\\,|A|=\\frac{1}{2};\\,[\/latex]period:[latex]\\,P=\\frac{2\\pi }{|B|}=6\\pi ;\\,[\/latex]phase shift:[latex]\\,\\frac{C}{B}=\\pi [\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_06_01_06\" class=\"textbox examples\">\n<div id=\"fs-id1165137659478\">\n<div id=\"fs-id1165134040573\">\n<h3>Identifying the Equation for a Sinusoidal Function from a Graph<\/h3>\nDetermine the formula for the cosine function in <a class=\"autogenerated-content\" href=\"#Figure_06_01_015\">(Figure)<\/a>.\n<div id=\"Figure_06_01_015\" class=\"small wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143157\/CNX_Precalc_Figure_06_01_015.jpg\" alt=\"A graph of -0.5cos(x)+0.5. The graph has an amplitude of 0.5. The graph has a period of 2pi. The graph has a range of [0, 1]. The graph is also reflected about the x-axis from the parent function cos(x).\" width=\"487\" height=\"163\"> <strong>Figure 15.<\/strong>[\/caption]<\/div>\n<\/div>\n<div id=\"fs-id1165135329784\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135329784\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135329784\"]\n<p id=\"fs-id1165137726017\">To determine the equation, we need to identify each value in the general form of a sinusoidal function.<\/p>\n\n<div id=\"fs-id1165137726021\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}y=A\\mathrm{sin}\\left(Bx-C\\right)+D\\hfill \\\\ y=A\\mathrm{cos}\\left(Bx-C\\right)+D\\hfill \\end{array}[\/latex]<\/div>\nThe graph could represent either a sine or a <span class=\"no-emphasis\">cosine function<\/span> that is shifted and\/or reflected. When[latex]\\,x=0,\\,[\/latex]the graph has an extreme point,[latex]\\,\\left(0,0\\right).\\,[\/latex]Since the cosine function has an extreme point for[latex]\\,x=0,\\,[\/latex]let us write our equation in terms of a cosine function.\n<p id=\"fs-id1165135536557\">Let\u2019s start with the midline. We can see that the graph rises and falls an equal distance above and below[latex]\\,y=0.5.\\,[\/latex]This value, which is the midline, is[latex]\\,D\\,[\/latex]in the equation, so[latex]\\,D=0.5.[\/latex]<\/p>\nThe greatest distance above and below the midline is the amplitude. The maxima are 0.5 units above the midline and the minima are 0.5 units below the midline. So[latex]\\,|A|=0.5.\\,[\/latex]Another way we could have determined the amplitude is by recognizing that the difference between the height of local maxima and minima is 1, so[latex]\\,|A|=\\frac{1}{2}=0.5.\\,[\/latex]Also, the graph is reflected about the <em>x<\/em>-axis so that[latex]\\,A=-0.5.[\/latex]\n<p id=\"fs-id1165134204425\">The graph is not horizontally stretched or compressed, so[latex]\\,B=1;\\,[\/latex]and the graph is not shifted horizontally, so[latex]\\,C=0.[\/latex]<\/p>\n<p id=\"fs-id1165135347312\">Putting this all together,<\/p>\n\n<div class=\"unnumbered\">[latex]g\\left(x\\right)=-0.5\\mathrm{cos}\\left(x\\right)+0.5[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137702221\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_06\">\n<div>\n<p id=\"fs-id1165135582222\">Determine the formula for the sine function in <a class=\"autogenerated-content\" href=\"#Figure_06_01_016\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_06_01_016\" class=\"small wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143201\/CNX_Precalc_Figure_06_01_016.jpg\" alt=\"A graph of sin(x)+2. Period of 2pi, amplitude of 1, and range of [1, 3].\" width=\"487\" height=\"173\"> <strong>Figure 16.<\/strong>[\/caption]<\/div>\n<\/div>\n<div id=\"fs-id1165137526465\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137526465\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137526465\"]\n<p id=\"fs-id1165137526466\">[latex]f\\left(x\\right)=\\mathrm{sin}\\left(x\\right)+2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_06_01_07\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1165134058400\">\n<h3>Identifying the Equation for a Sinusoidal Function from a Graph<\/h3>\n<p id=\"fs-id1165134059763\">Determine the equation for the sinusoidal function in <a class=\"autogenerated-content\" href=\"#Figure_06_01_017\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_06_01_017\" class=\"medium\">[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143204\/CNX_Precalc_Figure_06_01_017.jpg\" alt=\"A graph of 3cos(pi\/3x-pi\/3)-2. Graph has amplitude of 3, period of 6, range of [-5,1].\" width=\"731\" height=\"565\"> <strong>Figure 17.<\/strong>[\/caption]<\/div>\n<\/div>\n<div id=\"fs-id1165137598813\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137598813\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137598813\"]\n<p id=\"fs-id1165137598815\">With the highest value at 1 and the lowest value at[latex]\\,-5,\\,[\/latex]the midline will be halfway between at[latex]\\,-2.\\,[\/latex]So[latex]\\,D=-2.\\,[\/latex]<\/p>\n<p id=\"fs-id1165133281392\">The distance from the midline to the highest or lowest value gives an amplitude of[latex]\\,|A|=3.[\/latex]<\/p>\n<p id=\"fs-id1165137824298\">The period of the graph is 6, which can be measured from the peak at[latex]\\,x=1\\,[\/latex]to the next peak at[latex]\\,x=7,[\/latex]or from the distance between the lowest points. Therefore,[latex]P=\\frac{2\\pi }{|B|}=6.\\,[\/latex]Using the positive value for[latex]\\,B,[\/latex]we find that<\/p>\n\n<div id=\"fs-id1165135196958\" class=\"unnumbered aligncenter\">[latex]B=\\frac{2\\pi }{P}=\\frac{2\\pi }{6}=\\frac{\\pi }{3}[\/latex]<\/div>\n<p id=\"fs-id1165137611526\">So far, our equation is either[latex]\\,y=3\\mathrm{sin}\\left(\\frac{\\pi }{3}x-C\\right)-2\\,[\/latex]or[latex]\\,y=3\\mathrm{cos}\\left(\\frac{\\pi }{3}x-C\\right)-2.\\,[\/latex]For the shape and shift, we have more than one option. We could write this as any one of the following:<\/p>\n\n<ul id=\"fs-id1165137466148\">\n \t<li>a cosine shifted to the right<\/li>\n \t<li>a negative cosine shifted to the left<\/li>\n \t<li>a sine shifted to the left<\/li>\n \t<li>a negative sine shifted to the right<\/li>\n<\/ul>\n<p id=\"fs-id1165137619397\">While any of these would be correct, the cosine shifts are easier to work with than the sine shifts in this case because they involve integer values. So our function becomes<\/p>\n\n<div id=\"fs-id1165137619402\" class=\"unnumbered aligncenter\">[latex]y=3\\mathrm{cos}\\left(\\frac{\\pi }{3}x-\\frac{\\pi }{3}\\right)-2\\text{ or }y=-3\\mathrm{cos}\\left(\\frac{\\pi }{3}x+\\frac{2\\pi }{3}\\right)-2[\/latex]<\/div>\n<p id=\"fs-id1165135704043\">Again, these functions are equivalent, so both yield the same graph.[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137805588\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_07\">\n<div id=\"fs-id1165137436869\">\n<p id=\"fs-id1165137436870\">Write a formula for the function graphed in <a class=\"autogenerated-content\" href=\"#Figure_06_01_018\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_06_01_018\" class=\"medium\">[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143209\/CNX_Precalc_Figure_06_01_018n.jpg\" alt=\"A graph of 4sin((pi\/5)x-pi\/5)+4. Graph has period of 10, amplitude of 4, range of [0,8].\" width=\"731\" height=\"440\"> <strong>Figure 18.<\/strong>[\/caption]<\/div>\n<\/div>\n<div id=\"fs-id1165135173772\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135173772\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135173772\"]\n<p id=\"fs-id1165135173773\">two possibilities:[latex]\\,y=4\\mathrm{sin}\\left(\\frac{\\pi }{5}x-\\frac{\\pi }{5}\\right)+4\\,[\/latex]or[latex]\\,y=-4\\mathrm{sin}\\left(\\frac{\\pi }{5}x+\\frac{4\\pi }{5}\\right)+4[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137735424\" class=\"bc-section section\">\n<h3>Graphing Variations of <em>y<\/em> = sin <em>x<\/em> and <em>y<\/em> = cos <em>x<\/em><\/h3>\n<p id=\"fs-id1165134148513\">Throughout this section, we have learned about types of variations of sine and cosine functions and used that information to write equations from graphs. Now we can use the same information to create graphs from equations.<\/p>\n<p id=\"fs-id1165137456137\">Instead of focusing on the general form equations<\/p>\n\n<div id=\"fs-id1165137456140\" class=\"unnumbered aligncenter\">[latex]y=A\\mathrm{sin}\\left(Bx-C\\right)+D\\text{ and }y=A\\mathrm{cos}\\left(Bx-C\\right)+D,[\/latex]<\/div>\n<p id=\"fs-id1165137807234\">we will let[latex]\\,C=0\\,[\/latex]and[latex]\\,D=0\\,[\/latex]and work with a simplified form of the equations in the following examples.<\/p>\n\n<div id=\"fs-id1165135380117\" class=\"precalculus howto\">\n<p id=\"fs-id1165135329942\"><strong>Given the function[latex]\\,y=A\\mathrm{sin}\\left(Bx\\right),\\,[\/latex]sketch its graph.<\/strong><\/p>\n\n<ol id=\"fs-id1165137542466\" type=\"1\">\n \t<li>Identify the amplitude,[latex]\\,|A|.[\/latex]<\/li>\n \t<li>Identify the period,[latex]\\,P=\\frac{2\\pi }{|B|}.[\/latex]<\/li>\n \t<li>Start at the origin, with the function increasing to the right if[latex]\\,A\\,[\/latex]is positive or decreasing if[latex]\\,A\\,[\/latex]is negative.<\/li>\n \t<li>At[latex]\\,x=\\frac{\\pi }{2|B|}\\,[\/latex]there is a local maximum for[latex]\\,A&gt;0\\,[\/latex]or a minimum for[latex]\\,A&lt;0,\\,[\/latex]with[latex]\\,y=A.[\/latex]<\/li>\n \t<li>The curve returns to the <em>x<\/em>-axis at[latex]\\,x=\\frac{\\pi }{|B|}.[\/latex]<\/li>\n \t<li>There is a local minimum for[latex]\\,A&gt;0\\,[\/latex](maximum for[latex]\\,A&lt;0[\/latex]) at[latex]\\,x=\\frac{3\\pi }{2|B|}\\,[\/latex]with[latex]\\,y=\u2013A.[\/latex]<\/li>\n \t<li>The curve returns again to the <em>x<\/em>-axis at[latex]\\,x=\\frac{2\\pi }{|B|}.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_06_01_08\" class=\"textbox examples\">\n<div id=\"fs-id1165134156046\">\n<div id=\"fs-id1165137565145\">\n<h3>Graphing a Function and Identifying the Amplitude and Period<\/h3>\n<p id=\"fs-id1165137565150\">Sketch a graph of[latex]\\,f\\left(x\\right)=-2\\mathrm{sin}\\left(\\frac{\\pi x}{2}\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134190732\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134190732\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134190732\"]\n<p id=\"fs-id1165134190734\">Let\u2019s begin by comparing the equation to the form[latex]\\,y=A\\mathrm{sin}\\left(Bx\\right).[\/latex]<\/p>\n\n<ul id=\"eip-id1165135169452\">\n \t<li><em>Step 1.<\/em> We can see from the equation that[latex]\\,A=-2,[\/latex]so the amplitude is 2.\n<div id=\"fs-id1165135400292\" class=\"unnumbered aligncenter\">[latex]|A|=2[\/latex]<\/div><\/li>\n \t<li><em>Step 2.<\/em> The equation shows that[latex]\\,B=\\frac{\\pi }{2},\\,[\/latex]so the period is\n<div id=\"fs-id1165134178538\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}P=\\frac{2\\pi }{\\frac{\\pi }{2}}\\hfill \\\\ \\text{ }=2\\pi \\cdot \\frac{2}{\\pi }\\hfill \\\\ \\text{ }=4\\hfill \\end{array}[\/latex]<\/div><\/li>\n \t<li><em>Step 3.<\/em> Because[latex]\\,A\\,[\/latex]is negative, the graph descends as we move to the right of the origin.<\/li>\n \t<li><em>Step 4\u20137.<\/em> The <em>x<\/em>-intercepts are at the beginning of one period,[latex]\\,x=0,\\,[\/latex]the horizontal midpoints are at[latex]\\,x=2\\,[\/latex]and at the end of one period at[latex]\\,x=4.[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137786248\">The quarter points include the minimum at[latex]\\,x=1\\,[\/latex]and the maximum at[latex]\\,x=3.\\,[\/latex]A local minimum will occur 2 units below the midline, at[latex]\\,x=1,\\,[\/latex]and a local maximum will occur at 2 units above the midline, at[latex]\\,x=3.\\,[\/latex]<a class=\"autogenerated-content\" href=\"#Figure_06_01_019\">(Figure)<\/a> shows the graph of the function.<\/p>\n\n<div id=\"Figure_06_01_019\" class=\"small wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143217\/CNX_Precalc_Figure_06_01_019.jpg\" alt=\"A graph of -2sin((pi\/2)x). Graph has range of [-2,2], period of 4, and amplitude of 2.\" width=\"487\" height=\"252\"> <strong>Figure 19.<\/strong>[\/caption][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137539724\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_08\">\n<div id=\"fs-id1165137409841\">\n<p id=\"fs-id1165137628752\">Sketch a graph of[latex]\\,g\\left(x\\right)=-0.8\\mathrm{cos}\\left(2x\\right).\\,[\/latex]Determine the midline, amplitude, period, and phase shift.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135342790\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135342790\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135342790\"]<span id=\"fs-id1165135397949\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143231\/CNX_Precalc_Figure_06_01_020.jpg\" alt=\"A graph of -0.8cos(2x). Graph has range of [-0.8, 0.8], period of pi, amplitude of 0.8, and is reflected about the x-axis compared to it's parent function cos(x).\"><\/span>\n<p id=\"eip-id1165137938401\">midline:[latex]\\,y=0;\\,[\/latex]amplitude:[latex]\\,|A|=0.8;\\,[\/latex]period:[latex]\\,P=\\frac{2\\pi }{|B|}=\\pi ;\\,[\/latex]phase shift:[latex]\\,\\frac{C}{B}=0\\,[\/latex] or none<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137425929\" class=\"precalculus howto\">\n<p id=\"fs-id1165137661914\"><strong>Given a sinusoidal function with a phase shift and a vertical shift, sketch its graph.<\/strong><\/p>\n\n<ol id=\"fs-id1165135503706\" type=\"1\">\n \t<li>Express the function in the general form[latex]\\,y=A\\mathrm{sin}\\left(Bx-C\\right)+D\\text{ or }y=A\\mathrm{cos}\\left(Bx-C\\right)+D.[\/latex]<\/li>\n \t<li>Identify the amplitude,[latex]\\,|A|.[\/latex]<\/li>\n \t<li>Identify the period,[latex]\\,P=\\frac{2\\pi }{|B|}.[\/latex]<\/li>\n \t<li>Identify the phase shift,[latex]\\,\\frac{C}{B}.[\/latex]<\/li>\n \t<li>Draw the graph of[latex]\\,f\\left(x\\right)=A\\mathrm{sin}\\left(Bx\\right)\\,[\/latex] shifted to the right or left by[latex]\\,\\frac{C}{B}\\,[\/latex]and up or down by[latex]\\,D.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_06_01_09\" class=\"textbox examples\">\n<div id=\"fs-id1165134116767\">\n<div id=\"fs-id1165134116769\">\n<h3>Graphing a Transformed Sinusoid<\/h3>\n<p id=\"fs-id1165137723733\">Sketch a graph of[latex]\\,f\\left(x\\right)=3\\mathrm{sin}\\left(\\frac{\\pi }{4}x-\\frac{\\pi }{4}\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135209894\" class=\"solution textbox shaded\">\n\n[reveal-answer q=\"178576\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"178576\"]\n<ul id=\"eip-id1165137474346\">\n \t<li><em>Step 1.<\/em> The function is already written in general form:[latex]\\,f\\left(x\\right)=3\\mathrm{sin}\\left(\\frac{\\pi }{4}x-\\frac{\\pi }{4}\\right).[\/latex]This graph will have the shape of a <span class=\"no-emphasis\">sine function<\/span>, starting at the midline and increasing to the right.<\/li>\n \t<li><em>Step 2.<\/em>[latex]\\,|A|=|3|=3.\\,[\/latex]The amplitude is 3.<\/li>\n \t<li><em>Step 3.<\/em> Since[latex]\\,|B|=|\\frac{\\pi }{4}|=\\frac{\\pi }{4},\\,[\/latex]we determine the period as follows.\n<div id=\"fs-id1165137572143\" class=\"unnumbered aligncenter\">[latex]P=\\frac{2\\pi }{|B|}=\\frac{2\\pi }{\\frac{\\pi }{4}}=2\\pi \\cdot \\frac{4}{\\pi }=8[\/latex]<\/div>\n<p id=\"fs-id1165137757960\">The period is 8.<\/p>\n<\/li>\n \t<li><em>Step 4.<\/em> Since[latex]\\,C=\\frac{\\pi }{4},\\,[\/latex]the phase shift is\n<div id=\"fs-id1165135684362\" class=\"unnumbered aligncenter\">[latex]\\frac{C}{B}=\\frac{\\frac{\\pi }{4}}{\\frac{\\pi }{4}}=1.[\/latex]<\/div>\n<p id=\"fs-id1165137634941\">The phase shift is 1 unit.<\/p>\n<\/li>\n \t<li><em>Step 5.<\/em><a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2844&amp;action=edit#Figure_06_01_021\">(Figure)<\/a> shows the graph of the function.\n<div id=\"Figure_06_01_021\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143237\/CNX_Precalc_Figure_06_01_021.jpg\" alt=\"A graph of 3sin(*(pi\/4)x-pi\/4). Graph has amplitude of 3, period of 8, and a phase shift of 1 to the right.\" width=\"487\" height=\"319\"> <strong>Figure 20. <\/strong>A horizontally compressed, vertically stretched, and horizontally shifted sinusoid[\/caption]\n\n<\/div><\/li>\n<\/ul>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135181399\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_09\">\n<div id=\"fs-id1165137638347\">\n<p id=\"fs-id1165137638348\">Draw a graph of[latex]\\,g\\left(x\\right)=-2\\mathrm{cos}\\left(\\frac{\\pi }{3}x+\\frac{\\pi }{6}\\right).\\,[\/latex]Determine the midline, amplitude, period, and phase shift.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137480594\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137480594\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137480594\"]<span id=\"fs-id1165137442771\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143239\/CNX_Precalc_Figure_06_01_022.jpg\" alt=\"A graph of -2cos((pi\/3)x+(pi\/6)). Graph has amplitude of 2, period of 6, and has a phase shift of 0.5 to the left.\"><\/span>\n<p id=\"fs-id1165137627836\">midline:[latex]\\,y=0;\\,[\/latex]amplitude:[latex]\\,|A|=2;\\,[\/latex]period:[latex]\\,P=\\frac{2\\pi }{|B|}=6;\\,[\/latex]phase shift:[latex]\\,\\frac{C}{B}=-\\frac{1}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_06_01_10\" class=\"textbox examples\">\n<div id=\"fs-id1165137749524\">\n<div id=\"fs-id1165137749526\">\n<h3>Identifying the Properties of a Sinusoidal Function<\/h3>\n<p id=\"fs-id1165137406791\">Given[latex]\\,y=-2\\mathrm{cos}\\left(\\frac{\\pi }{2}x+\\pi \\right)+3,\\,[\/latex]determine the amplitude, period, phase shift, and horizontal shift. Then graph the function.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135487183\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135487183\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135487183\"]\n<p id=\"fs-id1165137431258\">Begin by comparing the equation to the general form and use the steps outlined in <a class=\"autogenerated-content\" href=\"#Example_06_01_09\">(Figure)<\/a>.<\/p>\n\n<div id=\"fs-id1165134225658\" class=\"unnumbered aligncenter\">[latex]y=A\\mathrm{cos}\\left(Bx-C\\right)+D[\/latex]<\/div>\n<ul id=\"eip-id1165134311998\">\n \t<li><em>Step 1.<\/em> The function is already written in general form.<\/li>\n \t<li><em>Step 2.<\/em> Since[latex]\\,A=-2,\\,[\/latex]the amplitude is[latex]\\,|A|=2.[\/latex]<\/li>\n \t<li><em>Step 3.<\/em>[latex]\\,|B|=\\frac{\\pi }{2},\\,[\/latex]so the period is[latex]\\,P=\\frac{2\\pi }{|B|}=\\frac{2\\pi }{\\frac{\\pi }{2}}=2\\pi \\cdot \\frac{2}{\\pi }=4.\\,[\/latex]The period is 4.<\/li>\n \t<li><em>Step 4.<\/em>[latex]\\,C=-\\pi ,[\/latex]so we calculate the phase shift as[latex]\\,\\frac{C}{B}=\\frac{-\\pi ,}{\\frac{\\pi }{2}}=-\\pi \\cdot \\frac{2}{\\pi }=-2.\\,[\/latex]The phase shift is[latex]\\,-2.[\/latex]<\/li>\n \t<li><em>Step 5.<\/em>[latex]D=3,[\/latex]so the midline is[latex]\\,y=3,\u2009[\/latex]and the vertical shift is up 3.<\/li>\n<\/ul>\n<p id=\"fs-id1165137936633\">Since[latex]\\,A\\,[\/latex]is negative, the graph of the cosine function has been reflected about the <em>x<\/em>-axis.<\/p>\n<a class=\"autogenerated-content\" href=\"#Figure_06_01_028\">(Figure)<\/a> shows one cycle of the graph of the function.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143244\/CNX_Precalc_Figure_06_01_028.jpg\" alt=\"A graph of -2cos((pi\/2)x+pi)+3. Graph shows an amplitude of 2, midline at y=3, and a period of 4.\" width=\"487\" height=\"317\"> <strong>Figure 21.<\/strong>[\/caption]\n<p id=\"fs-id1165137761033\">[\/hidden-answer]<span id=\"fs-id1165137794283\"><\/span><\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137939840\" class=\"bc-section section\">\n<h3>Using Transformations of Sine and Cosine Functions<\/h3>\n<p id=\"fs-id1165137891269\">We can use the transformations of sine and cosine functions in numerous applications. As mentioned at the beginning of the chapter, <span class=\"no-emphasis\">circular motion<\/span> can be modeled using either the sine or <span class=\"no-emphasis\">cosine function<\/span>.<\/p>\n\n<div id=\"Example_06_01_11\" class=\"textbox examples\">\n<div id=\"fs-id1165137612101\">\n<div id=\"fs-id1165137612103\">\n<h3>Finding the Vertical Component of Circular Motion<\/h3>\n<p id=\"fs-id1165137731540\">A point rotates around a circle of radius 3 centered at the origin. Sketch a graph of the <em>y<\/em>-coordinate of the point as a function of the angle of rotation.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137552985\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137552985\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137552985\"]\n<p id=\"fs-id1165137552987\">Recall that, for a point on a circle of radius <em>r<\/em>, the <em>y<\/em>-coordinate of the point is[latex]\\,y=r\\,\\mathrm{sin}\\left(x\\right),\\,[\/latex]\nso in this case, we get the equation[latex]\\,y\\left(x\\right)=3\\,\\mathrm{sin}\\left(x\\right).\\,[\/latex]\nThe constant 3 causes a vertical stretch of the <em>y<\/em>-values of the function by a factor of 3, which we can see in the graph in <a class=\"autogenerated-content\" href=\"#Figure_06_01_023\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_06_01_023\" class=\"small wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143251\/CNX_Precalc_Figure_06_01_023.jpg\" alt=\"A graph of 3sin(x). Graph has period of 2pi, amplitude of 3, and range of [-3,3].\" width=\"487\" height=\"319\"> <strong>Figure 22.<\/strong>[\/caption][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137556901\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137400109\">Notice that the period of the function is still[latex]\\,2\\pi ;\\,[\/latex]as we travel around the circle, we return to the point[latex]\\,\\left(3,0\\right)\\,[\/latex]for[latex]\\,x=2\\pi ,4\\pi ,6\\pi ,....[\/latex]Because the outputs of the graph will now oscillate between[latex]\\,\u20133\\,[\/latex]and[latex]\\,3,\\,[\/latex]the amplitude of the sine wave is[latex]\\,3.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135319496\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_10\">\n<div id=\"fs-id1165135403587\">\n<p id=\"fs-id1165135403588\">What is the amplitude of the function[latex]\\,f\\left(x\\right)=7\\mathrm{cos}\\left(x\\right)?\\,[\/latex]Sketch a graph of this function.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137534006\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137534006\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137534006\"]\n<p id=\"fs-id1165137534008\">7<\/p>\n<span id=\"fs-id1165134187242\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143253\/CNX_Precalc_Figure_06_01_024.jpg\" alt=\"A graph of 7cos(x). Graph has amplitude of 7, period of 2pi, and range of [-7,7].\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_06_01_12\" class=\"textbox examples\">\n<div id=\"fs-id1165135190936\">\n<div id=\"fs-id1165135190938\">\n<h3>Finding the Vertical Component of Circular Motion<\/h3>\n<p id=\"fs-id1165135210138\">A circle with radius 3 ft is mounted with its center 4 ft off the ground. The point closest to the ground is labeled <em>P<\/em>, as shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_025\">(Figure)<\/a>. Sketch a graph of the height above the ground of the point[latex]\\,P\\,[\/latex]as the circle is rotated; then find a function that gives the height in terms of the angle of rotation.<\/p>\n\n<div id=\"Figure_06_01_025\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143304\/CNX_Precalc_Figure_06_01_025.jpg\" alt=\"An illustration of a circle lifted 4 feet off the ground. Circle has radius of 3 ft. There is a point P labeled on the circle's circumference.\" width=\"487\" height=\"300\"> <strong>Figure 23.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137863854\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137863854\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137863854\"]\n<p id=\"fs-id1165137863856\">Sketching the height, we note that it will start 1 ft above the ground, then increase up to 7 ft above the ground, and continue to oscillate 3 ft above and below the center value of 4 ft, as shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_026\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_06_01_026\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143316\/CNX_Precalc_Figure_06_01_026.jpg\" alt=\"A graph of -3cox(x)+4. Graph has midline at y=4, amplitude of 3, and period of 2pi.\" width=\"487\" height=\"521\"> <strong>Figure 24.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137601519\">Although we could use a transformation of either the sine or cosine function, we start by looking for characteristics that would make one function easier to use than the other. Let\u2019s use a cosine function because it starts at the highest or lowest value, while a <span class=\"no-emphasis\">sine function<\/span> starts at the middle value. A standard cosine starts at the highest value, and this graph starts at the lowest value, so we need to incorporate a vertical reflection.<\/p>\n<p id=\"fs-id1165137601522\">Second, we see that the graph oscillates 3 above and below the center, while a basic cosine has an amplitude of 1, so this graph has been vertically stretched by 3, as in the last example.<\/p>\n<p id=\"fs-id1165134401716\">Finally, to move the center of the circle up to a height of 4, the graph has been vertically shifted up by 4. Putting these transformations together, we find that<\/p>\n\n<div id=\"fs-id1165133047569\" class=\"unnumbered aligncenter\">[latex]y=-3\\mathrm{cos}\\left(x\\right)+4[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_11\">\n<div id=\"fs-id1165137447011\">\n<p id=\"fs-id1165137447012\">A weight is attached to a spring that is then hung from a board, as shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_029\">(Figure)<\/a>. As the spring oscillates up and down, the position[latex]\\,y\\,[\/latex]of the weight relative to the board ranges from[latex]\\,\u20131\\,[\/latex]in. (at time[latex]\\,x=0)\\,[\/latex]to[latex]\\,\u20137\\,[\/latex]in. (at time[latex]\\,x=\\pi )\\,[\/latex]below the board. Assume the position of[latex]\\,y\\,[\/latex]is given as a sinusoidal function of[latex]\\,x.\\,[\/latex]Sketch a graph of the function, and then find a cosine function that gives the position[latex]\\,y\\,[\/latex]in terms of[latex]\\,x.[\/latex]<\/p>\n\n<div id=\"Figure_06_01_029\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143327\/CNX_Precalc_Figure_06_01_029.jpg\" alt=\"An illustration of a spring with length y.\" width=\"487\" height=\"351\"> <strong>Figure 25.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736527\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137736527\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137736527\"]\n<p id=\"fs-id1165137736529\">[latex]y=3\\mathrm{cos}\\left(x\\right)-4[\/latex]<\/p>\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143331\/CNX_Precalc_Figure_06_01_027.jpg\" alt=\"A cosine graph with range [-1,-7]. Period is 2 pi. Local maximums at (0,-1), (2pi,-1), and (4pi, -1). Local minimums at (pi,-7) and (3pi, -7).\">[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_06_01_13\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1165137812716\">\n<h3>Determining a Rider\u2019s Height on a Ferris Wheel<\/h3>\n<p id=\"fs-id1165137500935\">The London Eye is a huge Ferris wheel with a diameter of 135 meters (443 feet). It completes one rotation every 30 minutes. Riders board from a platform 2 meters above the ground. Express a rider\u2019s height above ground as a function of time in minutes.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137837117\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137837117\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137837117\"]\n<p id=\"fs-id1165137837120\">With a diameter of 135 m, the wheel has a radius of 67.5 m. The height will oscillate with amplitude 67.5 m above and below the center.<\/p>\n<p id=\"fs-id1165134086000\">Passengers board 2 m above ground level, so the center of the wheel must be located[latex]\\,67.5+2=69.5\\,[\/latex]m above ground level. The midline of the oscillation will be at 69.5 m.<\/p>\n<p id=\"fs-id1165137578349\">The wheel takes 30 minutes to complete 1 revolution, so the height will oscillate with a period of 30 minutes.<\/p>\n<p id=\"fs-id1165137529532\">Lastly, because the rider boards at the lowest point, the height will start at the smallest value and increase, following the shape of a vertically reflected cosine curve.<\/p>\n\n<ul id=\"fs-id1165137529537\">\n \t<li>Amplitude:[latex]\\,\\text{67}\\text{.5,}\\,[\/latex]so[latex]\\,A=67.5[\/latex]<\/li>\n \t<li>Midline:[latex]\\,\\text{69}\\text{.5,}\\,[\/latex]so[latex]\\,D=69.5[\/latex]<\/li>\n \t<li>Period:[latex]\\,\\text{30,}\\,[\/latex]so[latex]\\,B=\\frac{2\\pi }{30}=\\frac{\\pi }{15}[\/latex]<\/li>\n \t<li>Shape:[latex]\\,\\mathrm{-cos}\\left(t\\right)[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137767318\">An equation for the rider\u2019s height would be<\/p>\n\n<div id=\"fs-id1165135403551\" class=\"unnumbered aligncenter\">[latex]y=-67.5\\mathrm{cos}\\left(\\frac{\\pi }{15}t\\right)+69.5[\/latex]<\/div>\n<p id=\"fs-id1165137634889\">where[latex]\\,t\\,[\/latex]is in minutes and[latex]\\,y\\,[\/latex]is measured in meters.[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137540365\" class=\"precalculus media\">\n<p id=\"fs-id1165137761688\">Access these online resources for additional instruction and practice with graphs of sine and cosine functions.<\/p>\n\n<ul id=\"fs-id1165137761692\">\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/ampperiod\">Amplitude and Period of Sine and Cosine<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/translasincos\">Translations of Sine and Cosine<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/transformsincos\">Graphing Sine and Cosine Transformations<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphsinefunc\">Graphing the Sine Function<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137574576\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165133087385\" summary=\"..\"><caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td>Sinusoidal functions<\/td>\n<td>[latex]\\begin{array}{l}f\\left(x\\right)=A\\mathrm{sin}\\left(Bx-C\\right)+D\\\\ f\\left(x\\right)=A\\mathrm{cos}\\left(Bx-C\\right)+D\\end{array}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137540392\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul>\n \t<li>Periodic functions repeat after a given value. The smallest such value is the period. The basic sine and cosine functions have a period of[latex]\\,2\\pi .[\/latex]<\/li>\n \t<li>The function [latex]\\mathrm{sin}\\,x\\,[\/latex]is odd, so its graph is symmetric about the origin. The function [latex]\\,\\mathrm{cos}\\,x\\,[\/latex]is even, so its graph is symmetric about the <em>y<\/em>-axis.<\/li>\n \t<li>The graph of a sinusoidal function has the same general shape as a sine or cosine function.<\/li>\n \t<li>In the general formula for a sinusoidal function, the period is[latex]\\,P=\\frac{2\\pi }{|B|}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_06_01_01\">(Figure)<\/a>.<\/li>\n \t<li>In the general formula for a sinusoidal function,[latex]\\,|A|\\,[\/latex]represents amplitude. If[latex]\\,|A|&gt;1,\\,[\/latex]the function is stretched, whereas if[latex]\\,|A|&lt;1,\\,[\/latex]the function is compressed. See <a class=\"autogenerated-content\" href=\"#Example_06_01_02\">(Figure)<\/a>.<\/li>\n \t<li>The value[latex]\\,\\frac{C}{B}\\,[\/latex]in the general formula for a sinusoidal function indicates the phase shift. See <a class=\"autogenerated-content\" href=\"#Example_06_01_03\">(Figure)<\/a>.<\/li>\n \t<li>The value[latex]\\,D\\,[\/latex]in the general formula for a sinusoidal function indicates the vertical shift from the midline. See <a class=\"autogenerated-content\" href=\"#Example_06_01_04\">(Figure)<\/a>.<\/li>\n \t<li>Combinations of variations of sinusoidal functions can be detected from an equation. See <a class=\"autogenerated-content\" href=\"#Example_06_01_05\">(Figure)<\/a>.<\/li>\n \t<li>The equation for a sinusoidal function can be determined from a graph. See <a class=\"autogenerated-content\" href=\"#Example_06_01_06\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_06_01_07\">(Figure)<\/a>.<\/li>\n \t<li>A function can be graphed by identifying its amplitude and period. See <a class=\"autogenerated-content\" href=\"#Example_06_01_08\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_06_01_09\">(Figure)<\/a>.<\/li>\n \t<li>A function can also be graphed by identifying its amplitude, period, phase shift, and horizontal shift. See <a class=\"autogenerated-content\" href=\"#Example_06_01_10\">(Figure)<\/a>.<\/li>\n \t<li>Sinusoidal functions can be used to solve real-world problems. See <a class=\"autogenerated-content\" href=\"#Example_06_01_11\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_06_01_12\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_06_01_13\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137842570\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165137475812\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165137432049\">\n<div id=\"fs-id1165137415633\">\n<p id=\"fs-id1165137415634\">Why are the sine and cosine functions called periodic functions?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137415637\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137415637\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137415637\"]\n<p id=\"fs-id1165137415638\">The sine and cosine functions have the property that[latex]\\,f\\left(x+P\\right)=f\\left(x\\right)\\,[\/latex]for a certain[latex]\\,P.\\,[\/latex]This means that the function values repeat for every[latex]\\,P\\,[\/latex]units on the <em>x<\/em>-axis.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135305829\">\n<div id=\"fs-id1165135305831\">\n<p id=\"fs-id1165135305832\">How does the graph of[latex]\\,y=\\mathrm{sin}\\,x\\,[\/latex]\ncompare with the graph of[latex]\\,y=\\mathrm{cos}\\,x?\\,[\/latex]\nExplain how you could horizontally translate the graph of[latex]\\,y=\\mathrm{sin}\\,x\\,[\/latex]\nto obtain[latex]\\,y=\\mathrm{cos}\\,x.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134290285\">\n<div id=\"fs-id1165134290287\">\n<p id=\"fs-id1165134290288\">For the equation[latex]\\,A\\,\\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\n<\/div>\n<div id=\"fs-id1165137811265\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137811265\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137811265\"]\n<p id=\"fs-id1165137431114\">The absolute value of the constant[latex]\\,A\\,[\/latex](amplitude) increases the total range and the constant[latex]\\,D\\,[\/latex](vertical shift) shifts the graph vertically.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137810849\">\n<div id=\"fs-id1165137810851\">\n<p id=\"fs-id1165137810852\">How does the range of a translated sine function relate to the equation[latex]\\,y=A\\,\\mathrm{sin}\\left(Bx+C\\right)+D?[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137410268\">\n<div id=\"fs-id1165137432413\">\n<p id=\"fs-id1165137432414\">How can the unit circle be used to construct the graph of[latex]\\,f\\left(t\\right)=\\mathrm{sin}\\,t?[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137407584\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137407584\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137407584\"]\n<p id=\"fs-id1165137407585\">At the point where the terminal side of[latex]\\,t\\,[\/latex]intersects the unit circle, you can determine that the[latex]\\,\\mathrm{sin}\\,t\\,[\/latex]equals the <em>y<\/em>-coordinate of the point.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137663688\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165137742733\">For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum <em>y<\/em>-values and their corresponding <em>x<\/em>-values on one period for[latex]\\,x&gt;0.\\,[\/latex]Round answers to two decimal places if necessary.<\/p>\n\n<div id=\"fs-id1165137570454\">\n<div id=\"fs-id1165137570456\">\n<p id=\"fs-id1165137570457\">[latex]f\\left(x\\right)=2\\mathrm{sin}\\,x[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137423636\">\n<div id=\"fs-id1165137423638\">\n<p id=\"fs-id1165137423639\">[latex]f\\left(x\\right)=\\frac{2}{3}\\mathrm{cos}\\,x[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135456747\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135456747\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135456747\"]<span id=\"fs-id1165137871371\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143339\/CNX_Precalc_Figure_06_01_202.jpg\" alt=\"A graph of (2\/3)cos(x). Graph has amplitude of 2\/3, period of 2pi, and range of [-2\/3, 2\/3].\"><\/span>\n<p id=\"fs-id1165137817435\">amplitude:[latex]\\,\\frac{2}{3};\\,[\/latex]period:[latex]\\,2\\pi ;\\,[\/latex]midline:[latex]\\,y=0;\\,[\/latex]maximum:[latex]\\,y=\\frac{2}{3}\\,[\/latex]occurs at[latex]\\,x=0;\\,[\/latex]minimum:[latex]\\,y=-\\frac{2}{3}\\,[\/latex]occurs at[latex]\\,x=\\pi ;\\,[\/latex]for one period, the graph starts at 0 and ends at[latex]\\,2\\pi [\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135663289\">\n<div id=\"fs-id1165135663292\">\n<p id=\"fs-id1165135663294\">[latex]f\\left(x\\right)=-3\\mathrm{sin}\\,x[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137678150\">\n<div id=\"fs-id1165135160163\">\n<p id=\"fs-id1165135160164\">[latex]f\\left(x\\right)=4\\mathrm{sin}\\,x[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"486349\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"486349\"]<span id=\"fs-id1165135151296\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143349\/CNX_Precalc_Figure_06_01_204.jpg\" alt=\"A graph of 4sin(x). Graph has amplitude of 4, period of 2pi, and range of [-4, 4].\"><\/span>\n<p id=\"fs-id1165137535655\">amplitude: 4; period:[latex]\\,2\\pi ;\\,[\/latex]midline:[latex]\\,y=0;\\,[\/latex]maximum[latex]\\,y=4\\,[\/latex]occurs at[latex]\\,x=\\frac{\\pi }{2};\\,[\/latex]minimum:[latex]\\,y=-4\\,[\/latex]occurs at[latex]\\,x=\\frac{3\\pi }{2};\\,[\/latex]one full period occurs from[latex]\\,x=0\\,[\/latex]to[latex]\\,x=2\\pi [\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137451618\">\n<div id=\"fs-id1165137451620\">\n<p id=\"fs-id1165137451622\">[latex]f\\left(x\\right)=2\\mathrm{cos}\\,x[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137794096\">\n<div id=\"fs-id1165137794099\">\n<p id=\"fs-id1165137585075\">[latex]f\\left(x\\right)=\\mathrm{cos}\\left(2x\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137871346\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137871346\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137871346\"]<span id=\"fs-id1165135386486\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143352\/CNX_Precalc_Figure_06_01_206.jpg\" alt=\"A graph of cos(2x). Graph has amplitude of 1, period of pi, and range of [-1,1].\"><\/span>\n<p id=\"fs-id1165137827174\">amplitude: 1; period:[latex]\\,\\pi ;\\,[\/latex]midline:[latex]\\,y=0;\\,[\/latex]maximum:[latex]\\,y=1\\,[\/latex]occurs at[latex]\\,x=\\pi ;\\,[\/latex]minimum:[latex]\\,y=-1\\,[\/latex]occurs at[latex]\\,x=\\frac{\\pi }{2};\\,[\/latex]one full period is graphed from[latex]\\,x=0\\,[\/latex]to[latex]\\,x=\\pi [\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135241360\">\n<div id=\"fs-id1165137531898\">\n<p id=\"fs-id1165137531899\">[latex]f\\left(x\\right)=2\\,\\mathrm{sin}\\left(\\frac{1}{2}x\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137793956\">\n<div id=\"fs-id1165137793958\">\n<p id=\"fs-id1165137793959\">[latex]f\\left(x\\right)=4\\,\\mathrm{cos}\\left(\\pi x\\right)[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"12808\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"12808\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143354\/CNX_Precalc_Figure_06_01_208.jpg\" alt=\"A graph of 4cos(pi*x). Grpah has amplitude of 4, period of 2, and range of [-4, 4].\">\n<p id=\"fs-id1165137434223\">amplitude: 4; period: 2; midline:[latex]\\,y=0;\\,[\/latex]maximum:[latex]\\,y=4\\,[\/latex]occurs at[latex]\\,x=0;\\,[\/latex]minimum:[latex]\\,y=-4\\,[\/latex]occurs at[latex]\\,x=1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135487094\">\n<div id=\"fs-id1165135181423\">\n<p id=\"fs-id1165135181424\">[latex]f\\left(x\\right)=3\\,\\mathrm{cos}\\left(\\frac{6}{5}x\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134277994\">\n<div>\n<p id=\"fs-id1165134277997\">[latex]y=3\\,\\mathrm{sin}\\left(8\\left(x+4\\right)\\right)+5[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137843946\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137843946\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137843946\"]<span id=\"fs-id1165137424209\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143401\/CNX_Precalc_Figure_06_01_210.jpg\" alt=\"A graph of 3sin(8(x+4))+5. Graph has amplitude of 3, range of [2, 8], and period of pi\/4.\"><\/span>\n<p id=\"fs-id1165134107350\">amplitude: 3; period:[latex]\\,\\frac{\\pi }{4};\\,[\/latex]midline:[latex]\\,y=5;\\,[\/latex]maximum:[latex]\\,y=8\\,[\/latex]occurs at[latex]\\,x=0.12;\\,[\/latex]minimum:[latex]\\,y=2\\,[\/latex]occurs at[latex]\\,x=0.516;\\,[\/latex]horizontal shift:[latex]\\,-4;\\,[\/latex]vertical translation 5; one period occurs from[latex]\\,x=0\\,[\/latex]to[latex]\\,x=\\frac{\\pi }{4}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137870948\">\n<div id=\"fs-id1165135457711\">\n<p id=\"fs-id1165135457712\">[latex]y=2\\,\\mathrm{sin}\\left(3x-21\\right)+4[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135675218\">\n<div id=\"fs-id1165137414495\">\n<p id=\"fs-id1165137414496\">[latex]y=5\\,\\mathrm{sin}\\left(5x+20\\right)-2[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134284471\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134284471\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134284471\"]<span id=\"fs-id1165135487190\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143404\/CNX_Precalc_Figure_06_01_212.jpg\" alt=\"A graph of 5sin(5x+20)-2. Graph has an amplitude of 5, period of 2pi\/5, and range of [-7,3].\"><\/span>\n<p id=\"fs-id1165137862366\">amplitude: 5; period:[latex]\\,\\frac{2\\pi }{5};\\,[\/latex]midline:[latex]\\,y=-2;\\,[\/latex]maximum:[latex]\\,y=3\\,[\/latex]occurs at[latex]\\,x=0.08;\\,[\/latex]minimum:[latex]\\,y=-7\\,[\/latex]occurs at[latex]\\,x=0.71;\\,[\/latex]phase shift:[latex]\\,-4;\\,[\/latex]vertical translation:[latex]\\,-2;\\,[\/latex]one full period can be graphed on[latex]\\,x=0\\,[\/latex]to[latex]\\,x=\\frac{2\\pi }{5}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137803788\">For the following exercises, graph one full period of each function, starting at[latex]\\,x=0.\\,[\/latex]For each function, state the amplitude, period, and midline. State the maximum and minimum <em>y<\/em>-values and their corresponding <em>x<\/em>-values on one period for[latex]\\,x&gt;0.\\,[\/latex]State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.<\/p>\n\n<div id=\"fs-id1165135154308\">\n<div id=\"fs-id1165137694190\">\n<p id=\"fs-id1165137694191\">[latex]f\\left(t\\right)=2\\mathrm{sin}\\left(t-\\frac{5\\pi }{6}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137898084\">\n<div id=\"fs-id1165137898086\">\n<p id=\"fs-id1165137898087\">[latex]f\\left(t\\right)=-\\mathrm{cos}\\left(t+\\frac{\\pi }{3}\\right)+1[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134541171\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134541171\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134541171\"]<span id=\"fs-id1165135411369\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143411\/CNX_Precalc_Figure_06_01_214.jpg\" alt=\"A graph of -cos(t+pi\/3)+1. Graph has amplitude of 1, period of 2pi, and range of [0,2]. Phase shifted pi\/3 to the left.\"><\/span>\n<p id=\"fs-id1165137911368\">amplitude: 1 ; period:[latex]\\,2\\pi ;\\,[\/latex]midline:[latex]\\,y=1;\\,[\/latex]maximum:[latex]\\,y=2\\,[\/latex]occurs at[latex]\\,x=2.09;\\,[\/latex]maximum:[latex]\\,y=2\\,[\/latex]occurs at[latex]\\,t=2.09;\\,[\/latex]minimum:[latex]\\,y=0\\,[\/latex]occurs at[latex]\\,t=5.24;\\,[\/latex]phase shift:[latex]\\,-\\frac{\\pi }{3};\\,[\/latex]vertical translation: 1; one full period is from[latex]\\,t=0\\,[\/latex]to[latex]\\,t=2\\pi [\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135250664\">\n<div id=\"fs-id1165135250666\">\n<p id=\"fs-id1165135250667\">[latex]f\\left(t\\right)=4\\mathrm{cos}\\left(2\\left(t+\\frac{\\pi }{4}\\right)\\right)-3[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137552973\">\n<p id=\"fs-id1165137552974\">[latex]f\\left(t\\right)=-\\mathrm{sin}\\left(\\frac{1}{2}t+\\frac{5\\pi }{3}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137541180\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137541180\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137541180\"]<span id=\"fs-id1165135349231\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143419\/CNX_Precalc_Figure_06_01_216.jpg\" alt=\"A graph of -sin((1\/2)*t + 5pi\/3). Graph has amplitude of 1, range of [-1,1], period of 4pi, and a phase shift of -10pi\/3.\"><\/span>amplitude: 1; period:[latex]\\,4\\pi ;\\,[\/latex]midline:[latex]\\,y=0;\\,[\/latex]maximum:[latex]\\,y=1\\,[\/latex]occurs at[latex]\\,t=11.52;\\,[\/latex]minimum:[latex]\\,y=-1\\,[\/latex]occurs at[latex]\\,t=5.24;\\,[\/latex]phase shift:[latex]\\,-\\frac{10\\pi }{3};\\,[\/latex]vertical shift: 0[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165135247503\">\n<div id=\"fs-id1165135247505\">\n<p id=\"fs-id1165135247506\">[latex]f\\left(x\\right)=4\\mathrm{sin}\\left(\\frac{\\pi }{2}\\left(x-3\\right)\\right)+7[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137658354\">\n<div id=\"fs-id1165137658356\">\n<p id=\"fs-id1165137823542\">Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_218\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_06_01_218\" class=\"small wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"371\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143443\/CNX_Precalc_Figure_06_01_218.jpg\" alt=\"A sinusoidal graph with amplitude of 2, range of [-5, -1], period of 4, and midline at y=-3.\" width=\"371\" height=\"288\"> <strong>Figure 26.<\/strong>[\/caption]<\/div>\n<\/div>\n<div id=\"fs-id1165135708054\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135708054\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135708054\"]\n<p id=\"fs-id1165135708055\">amplitude: 2; midline:[latex]\\,y=-3;\\,[\/latex]period: 4; equation:[latex]\\,f\\left(x\\right)=2\\mathrm{sin}\\left(\\frac{\\pi }{2}x\\right)-3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137705148\">\n<div id=\"fs-id1165135169261\">\n<p id=\"fs-id1165135169262\">Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_219\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_06_01_219\" class=\"small wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"308\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143449\/CNX_Precalc_Figure_06_01_219.jpg\" alt=\"A graph with a cosine parent function, with amplitude of 3, period of pi, midline at y=-1, and range of [-4,2]\" width=\"308\" height=\"322\"> <strong>Figure 27.<\/strong>[\/caption]<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135688798\">\n<div id=\"fs-id1165135688800\">\n<p id=\"fs-id1165135688801\">Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_220\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_06_01_220\" class=\"small wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"432\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143502\/CNX_Precalc_Figure_06_01_220.jpg\" alt=\"A graph with a cosine parent function with an amplitude of 2, period of 5, midline at y=3, and a range of [1,5].\" width=\"432\" height=\"290\"> <strong>Figure 28.<\/strong>[\/caption]<\/div>\n<\/div>\n<div id=\"fs-id1165134378700\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134378700\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134378700\"]\n<p id=\"fs-id1165134378701\">amplitude: 2; period: 5; midline:[latex]\\,y=3;\\,[\/latex]equation:[latex]\\,f\\left(x\\right)=-2\\mathrm{cos}\\left(\\frac{2\\pi }{5}x\\right)+3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135333824\">\n<div id=\"fs-id1165135333826\">\n<p id=\"fs-id1165135333827\">Determine the amplitude, period, midline, and an equation involving sine for the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_221\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_06_01_221\" class=\"small wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"400\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143505\/CNX_Precalc_Figure_06_01_221.jpg\" alt=\"A sinusoidal graph with amplitude of 4, period of 10, midline at y=0, and range [-4,4].\" width=\"400\" height=\"384\"> <strong>Figure 29.<\/strong>[\/caption]<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137473550\">\n<div>\n<p id=\"fs-id1165134389955\">Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_222\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_06_01_222\" class=\"small wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"401\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143516\/CNX_Precalc_Figure_06_01_222.jpg\" alt=\"A graph with cosine parent function, range of function is [-4,4], amplitude of 4, period of 2.\" width=\"401\" height=\"313\"> <strong>Figure 30.<\/strong>[\/caption]<\/div>\n<\/div>\n<div id=\"fs-id1165135534972\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135534972\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135534972\"]\n<p id=\"fs-id1165135534973\">amplitude: 4; period: 2; midline:[latex]\\,y=0;\\,[\/latex]equation:[latex]\\,f\\left(x\\right)=-4\\mathrm{cos}\\left(\\pi \\left(x-\\frac{\\pi }{2}\\right)\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137425688\">\n<p id=\"fs-id1165137425689\">Determine the amplitude, period, midline, and an equation involving sine for the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_223\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_06_01_223\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"307\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143523\/CNX_Precalc_Figure_06_01_223.jpg\" alt=\"A graph with sine parent function. Amplitude 2, period 2, midline y=0\" width=\"307\" height=\"188\"> <strong>Figure 31.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137588535\">\n<div id=\"fs-id1165137588538\">\n\nDetermine the amplitude, period, midline, and an equation involving cosine for the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_224\">(Figure)<\/a>.\n<div id=\"Figure_06_01_224\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"308\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143528\/CNX_Precalc_Figure_06_01_224.jpg\" alt=\"A graph with cosine parent function. Amplitude 2, period 2, midline y=1\" width=\"308\" height=\"188\"> <strong>Figure 32.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137600948\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137600948\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137600948\"]\n<p id=\"fs-id1165137600949\">amplitude: 2; period: 2; midline[latex]\\,y=1;\\,[\/latex]equation:[latex]\\,f\\left(x\\right)=2\\mathrm{cos}\\left(\\pi x\\right)+1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135530674\">\n<div id=\"fs-id1165137823256\">\n<p id=\"fs-id1165137823257\">Determine the amplitude, period, midline, and an equation involving sine for the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_225\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_06_01_225\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"306\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143530\/CNX_Precalc_Figure_06_01_225.jpg\" alt=\"A graph with a sine parent function. Amplitude 1, period 4 and midline y=0.\" width=\"306\" height=\"188\"> <strong>Figure 33.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134043561\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165134043566\">For the following exercises, let[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\,x.[\/latex]<\/p>\n\n<div id=\"fs-id1165135471271\">\n<div id=\"fs-id1165135471273\">\n\nOn[latex]\\,\\left[0,2\\pi \\right),[\/latex]solve[latex]\\,f\\left(x\\right)=0.[\/latex]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137400540\">\n<div id=\"fs-id1165137400542\">\n<p id=\"fs-id1165137400543\">On[latex]\\,\\left[0,2\\pi \\right),[\/latex]solve[latex]\\,f\\left(x\\right)=\\frac{1}{2}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137832261\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137832261\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137832261\"]\n<p id=\"fs-id1165137784896\">[latex]\\frac{\\pi }{6},\\frac{5\\pi }{6}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137637941\">\n<div id=\"fs-id1165137637943\">\n<p id=\"fs-id1165137637944\">Evaluate[latex]\\,f\\left(\\frac{\\pi }{2}\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137804692\">\n<div id=\"fs-id1165137442246\">\n<p id=\"fs-id1165137442247\">On[latex]\\,\\left[0,2\\pi \\right),f\\left(x\\right)=\\frac{\\sqrt{2}}{2}.\\,[\/latex]Find all values of[latex]\\,x.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137837133\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137837133\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137837133\"]\n<p id=\"fs-id1165137837134\">[latex]\\frac{\\pi }{4},\\frac{3\\pi }{4}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137416978\">\n<div id=\"fs-id1165135189866\">\n<p id=\"fs-id1165135189867\">On[latex]\\,\\left[0,2\\pi \\right),[\/latex]the maximum value(s) of the function occur(s) at what <em>x<\/em>-value(s)?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137942464\">\n<div id=\"fs-id1165137942466\">\n<p id=\"fs-id1165137942467\">On[latex]\\,\\left[0,2\\pi \\right),[\/latex]the minimum value(s) of the function occur(s) at what <em>x<\/em>-value(s)?<\/p>\n\n<\/div>\n<div id=\"fs-id1165134042136\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134042136\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134042136\"]\n<p id=\"fs-id1165134042137\">[latex]\\frac{3\\pi }{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137633931\">\n<div id=\"fs-id1165137633933\">\n<p id=\"fs-id1165137633934\">Show that[latex]\\,f\\left(-x\\right)=-f\\left(x\\right).\\,[\/latex]This means that[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\,x\\,[\/latex]is an odd function and possesses symmetry with respect to ________________.<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137806520\">For the following exercises, let[latex]\\,f\\left(x\\right)=\\mathrm{cos}\\,x.[\/latex]<\/p>\n\n<div id=\"fs-id1165137844202\">\n<div id=\"fs-id1165137844204\">\n<p id=\"fs-id1165137844205\">On[latex]\\,\\left[0,2\\pi \\right),[\/latex]solve the equation[latex]\\,f\\left(x\\right)=\\mathrm{cos}\\,x=0.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134129955\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134129955\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134129955\"]\n<p id=\"fs-id1165134129956\">[latex]\\frac{\\pi }{2},\\frac{3\\pi }{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134043804\">\n<div id=\"fs-id1165135191893\">\n<p id=\"fs-id1165135191894\">On[latex]\\,\\left[0,2\\pi \\right),[\/latex]solve[latex]\\,f\\left(x\\right)=\\frac{1}{2}.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137423113\">\n<div id=\"fs-id1165137423115\">\n<p id=\"fs-id1165137447173\">On[latex]\\,\\left[0,2\\pi \\right),[\/latex]find the <em>x<\/em>-intercepts of[latex]\\,f\\left(x\\right)=\\mathrm{cos}\\,x.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135440505\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135440505\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135440505\"]\n<p id=\"fs-id1165135440506\">[latex]\\frac{\\pi }{2},\\frac{3\\pi }{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134394631\">\n<div id=\"fs-id1165134394633\">\n<p id=\"fs-id1165134230442\">On[latex]\\,\\left[0,2\\pi \\right),[\/latex]find the <em>x<\/em>-values at which the function has a maximum or minimum value.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137442136\">\n<div id=\"fs-id1165137442139\">\n<p id=\"fs-id1165137442140\">On[latex]\\,\\left[0,2\\pi \\right),[\/latex]solve the equation[latex]\\,f\\left(x\\right)=\\frac{\\sqrt{3}}{2}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137933103\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137933103\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137933103\"]\n<p id=\"fs-id1165137933104\">[latex]\\frac{\\pi }{6},\\frac{11\\pi }{6}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137638455\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<div id=\"fs-id1165135193041\">\n<div id=\"fs-id1165135193043\">\n<p id=\"fs-id1165137639606\">Graph[latex]\\,h\\left(x\\right)=x+\\mathrm{sin}\\,x\\,[\/latex]on[latex]\\,\\left[0,2\\pi \\right].\\,[\/latex]Explain why the graph appears as it does.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736577\">\n<div id=\"fs-id1165137736579\">\n<p id=\"fs-id1165137637316\">Graph[latex]\\,h\\left(x\\right)=x+\\mathrm{sin}\\,x\\,[\/latex]on[latex]\\,\\left[-100,100\\right].\\,[\/latex]Did the graph appear as predicted in the previous exercise?<\/p>\n\n<\/div>\n<div id=\"fs-id1165137433807\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137433807\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137433807\"]\n<p id=\"fs-id1165137433809\">The graph appears linear. The linear functions dominate the shape of the graph for large values of[latex]\\,x.[\/latex]<\/p>\n<span id=\"fs-id1165137604843\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143536\/CNX_Precalc_Figure_06_01_227.jpg\" alt=\"A sinusoidal graph that increases like the function y=x, shown from 0 to 100.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137891285\">\n<div id=\"fs-id1165137891288\">\n<p id=\"fs-id1165137891290\">Graph[latex]\\,f\\left(x\\right)=x\\,\\mathrm{sin}\\,x\\,[\/latex]on[latex]\\,\\left[0,2\\pi \\right]\\,[\/latex]and verbalize how the graph varies from the graph of[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\,x.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133085093\">\n<div id=\"fs-id1165133085095\">\n<p id=\"fs-id1165133085096\">Graph[latex]\\,f\\left(x\\right)=x\\,\\mathrm{sin}\\,x\\,[\/latex]on the window[latex]\\,\\left[-10,10\\right]\\,[\/latex]and explain what the graph shows.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135322029\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135322029\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135322029\"]\n<p id=\"fs-id1165134283628\">The graph is symmetric with respect to the <em>y<\/em>-axis and there is no amplitude because the function is not periodic.<\/p>\n<span id=\"fs-id1165137570542\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143538\/CNX_Precalc_Figure_06_01_229.jpg\" alt=\"A sinusoidal graph that has increasing peaks and decreasing lows as the absolute value of x increases.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137433574\">\n<div id=\"fs-id1165137433577\">\n<p id=\"fs-id1165137696192\">Graph[latex]\\,f\\left(x\\right)=\\frac{\\mathrm{sin}\\,x}{x}\\,[\/latex]on the window[latex]\\,\\left[-5\\pi ,5\\pi \\right]\\,[\/latex]and explain what the graph shows.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135333770\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id1165135333775\">\n<div id=\"fs-id1165137435670\">\n<p id=\"fs-id1165137435671\">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]\\,h\\left(t\\right)\\,[\/latex]gives a person\u2019s height in meters above the ground <em>t<\/em> minutes after the wheel begins to turn.<\/p>\n\n<ol type=\"a\">\n \t<li>Find the amplitude, midline, and period of[latex]\\,h\\left(t\\right).[\/latex]<\/li>\n \t<li>Find a formula for the height function[latex]\\,h\\left(t\\right).[\/latex]<\/li>\n \t<li>How high off the ground is a person after 5 minutes?<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165135205671\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135205671\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135205671\"]\n<ol id=\"fs-id1165135205673\" type=\"a\">\n \t<li>Amplitude: 12.5; period: 10; midline:[latex]\\,y=13.5;[\/latex]<\/li>\n \t<li>[latex]h\\left(t\\right)=12.5\\mathrm{sin}\\left(\\frac{\\pi }{5}\\left(t-2.5\\right)\\right)+13.5;[\/latex]<\/li>\n \t<li>26 ft<\/li>\n<\/ol>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165137414167\">\n \t<dt>amplitude<\/dt>\n \t<dd id=\"fs-id1165137463141\">the vertical height of a function; the constant[latex]\\,A\\,[\/latex]appearing in the definition of a sinusoidal function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137602032\">\n \t<dt>midline<\/dt>\n \t<dd id=\"fs-id1165137602037\">the horizontal line[latex]\\,y=D,\\,[\/latex]where[latex]\\,D\\,[\/latex]appears in the general form of a sinusoidal function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137678058\">\n \t<dt>periodic function<\/dt>\n \t<dd id=\"fs-id1165137678063\">a function[latex]\\,f\\left(x\\right)\\,[\/latex]that satisfies[latex]\\,f\\left(x+P\\right)=f\\left(x\\right)\\,[\/latex]for a specific constant[latex]\\,P\\,[\/latex]and any value of[latex]\\,x[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137939683\">\n \t<dt>phase shift<\/dt>\n \t<dd id=\"fs-id1165137939688\">the horizontal displacement of the basic sine or cosine function; the constant[latex]\\,\\frac{C}{B}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135160153\">\n \t<dt>sinusoidal function<\/dt>\n \t<dd id=\"fs-id1165137737500\">any function that can be expressed in the form[latex]\\,f\\left(x\\right)=A\\mathrm{sin}\\left(Bx-C\\right)+D\\,[\/latex]or[latex]\\,f\\left(x\\right)=A\\mathrm{cos}\\left(Bx-C\\right)+D[\/latex]<\/dd>\n<\/dl>\n<\/div>\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Graph variations of \u2009y=sin( x )\u2009 and \u2009y=cos( x ).<\/li>\n<li>Use phase shifts of sine and cosine curves.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Figure_06_01_001\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143045\/CNX_Precalc_Figure_06_01_001.jpg\" alt=\"A photo of a rainbow colored beam of light stretching across the floor.\" width=\"487\" height=\"390\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1. <\/strong>Light can be separated into colors because of its wavelike properties. (credit: &#8220;wonderferret&#8221;\/ Flickr)<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165135329803\">White light, such as the light from the sun, is not actually white at all. Instead, it is a composition of all the colors of the rainbow in the form of waves. The individual colors can be seen only when white light passes through an optical prism that separates the waves according to their wavelengths to form a rainbow.<\/p>\n<p id=\"fs-id1165134544966\">Light waves can be represented graphically by the sine function. In the chapter on <a class=\"target-chapter\" href=\"\/contents\/5d813d51-ebc0-49c9-96a5-f49768c8bfb3\">Trigonometric Functions<\/a>, we examined trigonometric functions such as the sine function. In this section, we will interpret and create graphs of sine and cosine functions.<\/p>\n<div id=\"fs-id1165135169322\" class=\"bc-section section\">\n<h3>Graphing Sine and Cosine Functions<\/h3>\n<p id=\"fs-id1165135241395\">Recall that the sine and cosine functions relate real number values to the <em>x<\/em>&#8211; and <em>y<\/em>-coordinates of a point on the unit circle. So what do they look like on a graph on a coordinate plane? Let\u2019s start with the <span class=\"no-emphasis\">sine function<\/span>. We can create a table of values and use them to sketch a graph. <a class=\"autogenerated-content\" href=\"#Table_06_01_01\">(Figure)<\/a> lists some of the values for the sine function on a unit circle.<\/p>\n<table id=\"Table_06_01_01\" summary=\"..\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{6}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{4}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{3}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{2}[\/latex]<\/td>\n<td>[latex]\\frac{2\\pi }{3}[\/latex]<\/td>\n<td>[latex]\\frac{3\\pi }{4}[\/latex]<\/td>\n<td>[latex]\\frac{5\\pi }{6}[\/latex]<\/td>\n<td>[latex]\\pi[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]\\mathrm{sin}\\left(x\\right)[\/latex]<\/strong><\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137694159\">Plotting the points from the table and continuing along the <em>x<\/em>-axis gives the shape of the sine function. See <a class=\"autogenerated-content\" href=\"#Figure_06_01_002\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_01_002\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143048\/CNX_Precalc_Figure_06_01_002.jpg\" alt=\"A graph of sin(x). Local maximum at (pi\/2, 1). Local minimum at (3pi\/2, -1). Period of 2pi.\" width=\"487\" height=\"216\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2. <\/strong>The sine function<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137410832\">Notice how the sine values are positive between 0 and[latex]\\,\\pi ,\\,[\/latex]which correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative between[latex]\\,\\pi \\,[\/latex]and[latex]\\,2\\pi ,\\,[\/latex]which correspond to the values of the sine function in quadrants III and IV on the unit circle. See <a class=\"autogenerated-content\" href=\"#Figure_06_01_003\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_01_003\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143051\/CNX_Precalc_Figure_06_01_003.jpg\" alt=\"A side-by-side graph of a unit circle and a graph of sin(x). The two graphs show the equivalence of the coordinates.\" width=\"487\" height=\"219\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3. <\/strong>Plotting values of the sine function<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137849285\">Now let\u2019s take a similar look at the <span class=\"no-emphasis\">cosine function<\/span>. Again, we can create a table of values and use them to sketch a graph. <a class=\"autogenerated-content\" href=\"#Table_06_01_02\">(Figure)<\/a> lists some of the values for the cosine function on a unit circle.<\/p>\n<table id=\"Table_06_01_02\" summary=\"..\">\n<tbody>\n<tr>\n<td>[latex]\\mathbf{x}[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{6}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{4}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{3}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi }{2}[\/latex]<\/td>\n<td>[latex]\\frac{2\\pi }{3}[\/latex]<\/td>\n<td>[latex]\\frac{3\\pi }{4}[\/latex]<\/td>\n<td>[latex]\\frac{5\\pi }{6}[\/latex]<\/td>\n<td>[latex]\\pi[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\mathbf{cos}\\left(\\mathbf{x}\\right)[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]-\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]-\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\n<td>[latex]-\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165135531416\">As with the sine function, we can plots points to create a graph of the cosine function as in <a class=\"autogenerated-content\" href=\"#Figure_06_01_004\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_01_004\" class=\"medium\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143056\/CNX_Precalc_Figure_06_01_004.jpg\" alt=\"A graph of cos(x). Local maxima at (0,1) and (2pi, 1). Local minimum at (pi, -1). Period of 2pi.\" width=\"731\" height=\"216\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4. <\/strong>The cosine function<\/figcaption><\/figure>\n<p id=\"fs-id1165135628668\">Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval[latex]\\,\\left[-1,1\\right].[\/latex]<\/p>\n<p id=\"fs-id1165137727184\">In both graphs, the shape of the graph repeats after[latex]\\,2\\pi ,\\,[\/latex]which means the functions are periodic with a period of[latex]\\,2\\pi .\\,[\/latex]A periodic function is a function for which a specific <span class=\"no-emphasis\">horizontal shift<\/span>, <em>P<\/em>, results in a function equal to the original function:[latex]\\,f\\left(x+P\\right)=f\\left(x\\right)\\,[\/latex]for all values of[latex]\\,x\\,[\/latex]in the domain of[latex]\\,f.\\,[\/latex]When this occurs, we call the smallest such horizontal shift with[latex]\\,P>0\\,[\/latex]the <span class=\"no-emphasis\">period<\/span> of the function. <a class=\"autogenerated-content\" href=\"#Figure_06_01_005\">(Figure)<\/a> shows several periods of the sine and cosine functions.<\/p>\n<div id=\"Figure_06_01_005\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143101\/CNX_Precalc_Figure_06_01_005.jpg\" alt=\"Side-by-side graphs of sin(x) and cos(x). Graphs show period lengths for both functions, which is 2pi.\" width=\"487\" height=\"442\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 5.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137447360\">Looking again at the sine and cosine functions on a domain centered at the <em>y<\/em>-axis helps reveal symmetries. As we can see in <a class=\"autogenerated-content\" href=\"#Figure_06_01_006\">(Figure)<\/a>, the <span class=\"no-emphasis\">sine function<\/span> is symmetric about the origin. Recall from <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/chapter\/the-other-trigonometric-functions\/\">The Other Trigonometric Functions<\/a> that we determined from the unit circle that the sine function is an odd function because[latex]\\,\\mathrm{sin}\\left(-x\\right)=-\\mathrm{sin}\\,x.\\,[\/latex]<br \/>\nNow we can clearly see this property from the graph.<\/p>\n<div id=\"Figure_06_01_006\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143106\/CNX_Precalc_Figure_06_01_006.jpg\" alt=\"A graph of sin(x) that shows that sin(x) is an odd function due to the odd symmetry of the graph.\" width=\"487\" height=\"191\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 6. <\/strong>Odd symmetry of the sine function<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165135484164\"><a class=\"autogenerated-content\" href=\"#Figure_06_01_007\">(Figure)<\/a> shows that the cosine function is symmetric about the <em>y<\/em>-axis. Again, we determined that the cosine function is an even function. Now we can see from the graph that [latex]\\mathrm{cos}\\left(-x\\right)=\\mathrm{cos}\\text{ }x.[\/latex]<\/p>\n<div id=\"Figure_06_01_007\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143115\/CNX_Precalc_Figure_06_01_007.jpg\" alt=\"A graph of cos(x) that shows that cos(x) is an even function due to the even symmetry of the graph.\" width=\"487\" height=\"216\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 7. <\/strong>Even symmetry of the cosine function<\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id1165135187674\">\n<h3>Characteristics of Sine and Cosine Functions<\/h3>\n<p id=\"fs-id1165137628764\">The sine and cosine functions have several distinct characteristics:<\/p>\n<ul id=\"fs-id1165137662423\">\n<li>They are periodic functions with a period of[latex]\\,2\\pi .[\/latex]<\/li>\n<li>The domain of each function is[latex]\\,\\left(-\\infty ,\\infty \\right)\\,[\/latex]and the range is[latex]\\,\\left[-1,1\\right].[\/latex]<\/li>\n<li>The graph of[latex]\\,y=\\mathrm{sin}\\text{ }x\\,[\/latex]is symmetric about the origin, because it is an odd function.<\/li>\n<li>The graph of[latex]\\,y=\\mathrm{cos}\\text{ }x\\,[\/latex]is symmetric about the[latex]\\,y\\text{-}[\/latex]axis, because it is an even function.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134032223\" class=\"bc-section section\">\n<h3>Investigating Sinusoidal Functions<\/h3>\n<p id=\"fs-id1165137410921\">As we can see, sine and cosine functions have a regular period and range. If we watch ocean waves or ripples on a pond, we will see that they resemble the sine or cosine functions. However, they are not necessarily identical. Some are taller or longer than others. A function that has the same general shape as a sine or <span class=\"no-emphasis\">cosine function<\/span> is known as a sinusoidal function. The general forms of sinusoidal functions are<\/p>\n<div id=\"fs-id1165135512530\">[latex]\\begin{array}{l}y=A\\mathrm{sin}\\left(Bx-C\\right)+D\\hfill \\\\ \\text{ and}\\hfill \\\\ y=A\\mathrm{cos}\\left(Bx-C\\right)+D\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"fs-id1165135458566\" class=\"bc-section section\">\n<h4>Determining the Period of Sinusoidal Functions<\/h4>\n<p id=\"fs-id1165135708019\">Looking at the forms of sinusoidal functions, we can see that they are transformations of the sine and cosine functions. We can use what we know about transformations to determine the period.<\/p>\n<p id=\"fs-id1165137639577\">In the general formula,[latex]\\,B\\,[\/latex]is related to the period by[latex]\\,P=\\frac{2\\pi }{|B|}.\\,[\/latex]If[latex]\\,|B|>1,\\,[\/latex]then the period is less than[latex]\\,2\\pi \\,[\/latex]and the function undergoes a horizontal compression, whereas if[latex]\\,|B|<1,\\,[\/latex]then the period is greater than[latex]\\,2\\pi \\,[\/latex]and the function undergoes a horizontal stretch. For example,[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\left(x\\right),\\,[\/latex][latex]B=1,\\,[\/latex]so the period is[latex]\\,2\\pi ,\\text{}[\/latex]which we knew. If[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\left(2x\\right),\\,[\/latex]then[latex]\\,B=2,\\,[\/latex]so the period is[latex]\\,\\pi \\,[\/latex]and the graph is compressed. If[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\left(\\frac{x}{2}\\right),\\,[\/latex]then[latex]\\,B=\\frac{1}{2},\\,[\/latex]so the period is[latex]\\,4\\pi \\,[\/latex]and the graph is stretched. Notice in <a class=\"autogenerated-content\" href=\"#Figure_06_01_008\">(Figure)<\/a> how the period is indirectly related to[latex]\\,|B|.[\/latex]<\/p>\n<div id=\"Figure_06_01_008\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143117\/CNX_Precalc_Figure_06_01_008.jpg\" alt=\"A graph with three items. The x-axis ranges from 0 to 2pi. The y-axis ranges from -1 to 1. The first item is the graph of sin(x) for one full period. The second is the graph of sin(2x) over two periods. The third is the graph of sin(x\/2) for one half of a period.\" width=\"487\" height=\"274\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 8.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id1165137697004\">\n<h3>Period of Sinusoidal Functions<\/h3>\n<p id=\"fs-id1165137766762\">If we let[latex]\\,C=0\\,[\/latex]and[latex]\\,D=0\\,[\/latex]in the general form equations of the sine and cosine functions, we obtain the forms<\/p>\n<div id=\"fs-id1165137855068\" class=\"unnumbered aligncenter\">[latex]y=A\\mathrm{sin}\\left(Bx\\right)[\/latex]<\/div>\n<div id=\"fs-id1165134371173\" class=\"unnumbered aligncenter\">[latex]y=A\\mathrm{cos}\\left(Bx\\right)[\/latex]<\/div>\n<p id=\"fs-id1165137413926\">The period is[latex]\\,\\frac{2\\pi }{|B|}.[\/latex]<\/p>\n<\/div>\n<div id=\"Example_06_01_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137772370\">\n<div id=\"fs-id1165137772372\">\n<h3>Identifying the Period of a Sine or Cosine Function<\/h3>\n<p id=\"fs-id1165137389619\">Determine the period of the function[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\left(\\frac{\\pi }{6}x\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137434852\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135188750\">Let\u2019s begin by comparing the equation to the general form[latex]\\,y=A\\mathrm{sin}\\left(Bx\\right).[\/latex]<\/p>\n<p id=\"fs-id1165134482743\">In the given equation,[latex]\\,B=\\frac{\\pi }{6},\\,[\/latex]so the period will be<\/p>\n<div class=\"unnumbered\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ P=\\frac{2\\pi }{|B|}\\end{array}\\hfill \\\\ \\text{ }=\\frac{2\\pi }{\\frac{\\pi }{6}}\\hfill \\\\ \\text{ }=2\\pi \\cdot \\frac{6}{\\pi }\\hfill \\\\ \\text{ }=12\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137465427\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_01\">\n<div>\n<p id=\"fs-id1165135208858\">Determine the period of the function[latex]\\,g\\left(x\\right)=\\mathrm{cos}\\left(\\frac{x}{3}\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137507692\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137675634\">[latex]\\,6\\pi \\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135699141\" class=\"bc-section section\">\n<h4>Determining Amplitude<\/h4>\n<p id=\"fs-id1165135207425\">Returning to the general formula for a sinusoidal function, we have analyzed how the variable[latex]\\,B\\,[\/latex]relates to the period. Now let\u2019s turn to the variable[latex]\\,A\\,[\/latex]so we can analyze how it is related to the <strong>amplitude<\/strong>, or greatest distance from rest.[latex]\\,A\\,[\/latex]represents the vertical stretch factor, and its absolute value[latex]\\,|A|\\,[\/latex]is the amplitude. The local maxima will be a distance[latex]\\,|A|\\,[\/latex]above the horizontal <strong>midline<\/strong> of the graph, which is the line[latex]\\,y=D;\\,[\/latex]because[latex]\\,D=0\\,[\/latex]in this case, the midline is the <em>x<\/em>-axis. The local minima will be the same distance below the midline. If[latex]\\,|A|>1,\\,[\/latex]the function is stretched. For example, the amplitude of[latex]\\,f\\left(x\\right)=4\\,\\mathrm{sin}\\,x\\,[\/latex]is twice the amplitude of[latex]\\,f\\left(x\\right)=2\\,\\mathrm{sin}\\,x.\\,[\/latex]If[latex]\\,|A|<1,\\,[\/latex]the function is compressed. <a class=\"autogenerated-content\" href=\"#Figure_06_01_009\">(Figure)<\/a> compares several sine functions with different amplitudes.<\/p>\n<div id=\"Figure_06_01_009\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143125\/CNX_Precalc_Figure_06_01_009.jpg\" alt=\"A graph with four items. The x-axis ranges from -6pi to 6pi. The y-axis ranges from -4 to 4. The first item is the graph of sin(x), which has an amplitude of 1. The second is a graph of 2sin(x), which has amplitude of 2. The third is a graph of 3sin(x), which has an amplitude of 3. The fourth is a graph of 4 sin(x) with an amplitude of 4.\" width=\"975\" height=\"316\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 9.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id1165134284498\">\n<h3>Amplitude of Sinusoidal Functions<\/h3>\n<p id=\"fs-id1165134497141\">If we let[latex]\\,C=0\\,[\/latex]and[latex]\\,D=0\\,[\/latex]in the general form equations of the sine and cosine functions, we obtain the forms<\/p>\n<div id=\"fs-id1165135177658\" class=\"unnumbered aligncenter\">[latex]y=A\\mathrm{sin}\\left(Bx\\right)\\text{ and }y=A\\mathrm{cos}\\left(Bx\\right)[\/latex]<\/div>\n<p id=\"fs-id1165137464064\">The amplitude is[latex]\\,A,\\,[\/latex]and the vertical height from the midline is[latex]\\,|A|.\\,[\/latex]In addition, notice in the example that<\/p>\n<div id=\"fs-id1165135460914\" class=\"unnumbered aligncenter\">[latex]|A|\\text{ = amplitude = }\\frac{1}{2}|\\text{maximum }-\\text{ minimum}|[\/latex]<\/div>\n<\/div>\n<div id=\"Example_06_01_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137653911\">\n<div id=\"fs-id1165134377968\">\n<h3>Identifying the Amplitude of a Sine or Cosine Function<\/h3>\n<p id=\"fs-id1165137932594\">What is the amplitude of the sinusoidal function[latex]\\,f\\left(x\\right)=-4\\mathrm{sin}\\left(x\\right)?\\,[\/latex]Is the function stretched or compressed vertically?<\/p>\n<\/div>\n<div id=\"fs-id1165135195832\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135195834\">Let\u2019s begin by comparing the function to the simplified form[latex]\\,y=A\\mathrm{sin}\\left(Bx\\right).[\/latex]<\/p>\n<p id=\"fs-id1165137930335\">In the given function,[latex]\\,A=-4,\\,[\/latex]so the amplitude is[latex]\\,|A|=|-4|=4.\\,[\/latex]The function is stretched.<\/details>\n<\/p>\n<\/div>\n<div id=\"fs-id1165134226786\">\n<h4>Analysis<\/h4>\n<p>The negative value of[latex]\\,A\\,[\/latex]results in a reflection across the <em>x<\/em>-axis of the <span class=\"no-emphasis\">sine function<\/span>, as shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_010\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_01_010\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143134\/CNX_Precalc_Figure_06_01_010.jpg\" alt=\"A graph of -4sin(x). The function has an amplitude of 4. Local minima at (-3pi\/2, -4) and (pi\/2, -4). Local maxima at (-pi\/2, 4) and (3pi\/2, 4). Period of 2pi.\" width=\"487\" height=\"319\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 10.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135471236\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_02\">\n<div id=\"fs-id1165137771980\">\n<p>What is the amplitude of the sinusoidal function[latex]\\,f\\left(x\\right)=\\frac{1}{2}\\mathrm{sin}\\left(x\\right)?\\,[\/latex]Is the function stretched or compressed vertically?<\/p>\n<\/div>\n<div>\n<p>[latex]\\frac{1}{2}\\,[\/latex]compressed<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137834807\" class=\"bc-section section\">\n<h3>Analyzing Graphs of Variations of <em>y<\/em> = sin<em> x<\/em> and <em>y<\/em> = cos <em>x<\/em><\/h3>\n<p id=\"fs-id1165135193998\">Now that we understand how[latex]\\,A\\,[\/latex]and[latex]\\,B\\,[\/latex]relate to the general form equation for the sine and cosine functions, we will explore the variables[latex]\\,C\\,[\/latex]and[latex]\\,D.\\,[\/latex]Recall the general form:<\/p>\n<div class=\"unnumbered\">[latex]\\begin{array}{c}y=A\\mathrm{sin}\\left(Bx-C\\right)+D\\text{ and }y=A\\mathrm{cos}\\left(Bx-C\\right)+D\\\\ or\\\\ y=A\\mathrm{sin}\\left(B\\left(x-\\frac{C}{B}\\right)\\right)+D\\text{ and }y=A\\mathrm{cos}\\left(B\\left(x-\\frac{C}{B}\\right)\\right)+D\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134187254\">The value[latex]\\,\\frac{C}{B}\\,[\/latex]for a sinusoidal function is called the <strong>phase shift<\/strong>, or the horizontal displacement of the basic sine or <span class=\"no-emphasis\">cosine function<\/span>. If[latex]\\,C>0,\\,[\/latex]the graph shifts to the right. If[latex]\\,C<0,\\,[\/latex]the graph shifts to the left. The greater the value of[latex]\\,|C|,\\,[\/latex]the more the graph is shifted. <a class=\"autogenerated-content\" href=\"#Figure_06_01_011\">(Figure)<\/a> shows that the graph of[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\left(x-\\pi \\right)\\,[\/latex]shifts to the right by[latex]\\,\\pi \\,[\/latex]units, which is more than we see in the graph of[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\left(x-\\frac{\\pi }{4}\\right),\\,[\/latex]which shifts to the right by[latex]\\,\\frac{\\pi }{4}\\,[\/latex]units.<\/p>\n<div id=\"Figure_06_01_011\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143137\/CNX_Precalc_Figure_06_01_011.jpg\" alt=\"A graph with three items. The first item is a graph of sin(x). The second item is a graph of sin(x-pi\/4), which is the same as sin(x) except shifted to the right by pi\/4. The third item is a graph of sin(x-pi), which is the same as sin(x) except shifted to the right by pi.\" width=\"487\" height=\"255\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 11.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137663752\">While[latex]\\,C\\,[\/latex]relates to the horizontal shift,[latex]\\,D\\,[\/latex]indicates the vertical shift from the midline in the general formula for a sinusoidal function. See <a class=\"autogenerated-content\" href=\"#Figure_06_01_012\">(Figure)<\/a>. The function[latex]\\,y=\\mathrm{cos}\\left(x\\right)+D\\,[\/latex]has its midline at[latex]\\,y=D.[\/latex]<\/p>\n<div id=\"Figure_06_01_012\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143140\/CNX_Precalc_Figure_06_01_012.jpg\" alt=\"A graph of y=Asin(x)+D. Graph shows the midline of the function at y=D.\" width=\"487\" height=\"255\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 12.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165135242867\">Any value of[latex]\\,D\\,[\/latex]other than zero shifts the graph up or down. <a class=\"autogenerated-content\" href=\"#Figure_06_01_013\">(Figure)<\/a> compares[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\,x\\,[\/latex]with[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\,x+2,\\,[\/latex]which is shifted 2 units up on a graph.<\/p>\n<div id=\"Figure_06_01_013\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143145\/CNX_Precalc_Figure_06_01_013.jpg\" alt=\"A graph with two items. The first item is a graph of sin(x). The second item is a graph of sin(x)+2, which is the same as sin(x) except shifted up by 2.\" width=\"487\" height=\"221\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 13.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id1165135571809\">\n<h3>Variations of Sine and Cosine Functions<\/h3>\n<p id=\"fs-id1165133201875\">Given an equation in the form[latex]\\,f\\left(x\\right)=A\\mathrm{sin}\\left(Bx-C\\right)+D\\,[\/latex]or[latex]\\,f\\left(x\\right)=A\\mathrm{cos}\\left(Bx-C\\right)+D,\\,[\/latex][latex]\\frac{C}{B}\\,[\/latex]is the phase shift and[latex]\\,D\\,[\/latex]is the <span class=\"no-emphasis\">vertical shift<\/span>.<\/p>\n<\/div>\n<div id=\"Example_06_01_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137696922\">\n<div>\n<h3>Identifying the Phase Shift of a Function<\/h3>\n<p id=\"fs-id1165137804482\">Determine the direction and magnitude of the phase shift for[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\left(x+\\frac{\\pi }{6}\\right)-2.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134483435\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134483437\">Let\u2019s begin by comparing the equation to the general form[latex]\\,y=A\\mathrm{sin}\\left(Bx-C\\right)+D.[\/latex]<\/p>\n<p id=\"fs-id1165137461008\">In the given equation, notice that[latex]\\,B=1\\,[\/latex]and[latex]\\,C=-\\frac{\\pi }{6}.\\,[\/latex]So the phase shift is<\/p>\n<div id=\"fs-id1165137693976\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{r}\\hfill \\\\ \\hfill \\frac{C}{B}=-\\frac{\\frac{\\pi }{6}}{1}\\\\ \\hfill \\text{ }=-\\frac{\\pi }{6}\\end{array}[\/latex]<\/div>\n<p>or[latex]\\,\\frac{\\pi }{6}\\,[\/latex]units to the left.<\/details>\n<\/div>\n<div id=\"fs-id1165134156073\">\n<h4>Analysis<\/h4>\n<p>We must pay attention to the sign in the equation for the general form of a sinusoidal function. The equation shows a minus sign before[latex]\\,C.\\,[\/latex]Therefore[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\left(x+\\frac{\\pi }{6}\\right)-2\\,[\/latex]can be rewritten as[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\left(x-\\left(-\\frac{\\pi }{6}\\right)\\right)-2.\\,[\/latex]If the value of[latex]\\,C\\,[\/latex]is negative, the shift is to the left.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_03\">\n<div id=\"fs-id1165137461117\">\n<p id=\"fs-id1165137461118\">Determine the direction and magnitude of the phase shift for[latex]\\,f\\left(x\\right)=3\\mathrm{cos}\\left(x-\\frac{\\pi }{2}\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165131959464\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165131959465\">[latex]\\frac{\\pi }{2};\\,[\/latex]right<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_06_01_04\" class=\"textbox examples\">\n<div id=\"fs-id1165137410966\">\n<div id=\"fs-id1165137410968\">\n<h3>Identifying the Vertical Shift of a Function<\/h3>\n<p id=\"fs-id1165135186656\">Determine the direction and magnitude of the vertical shift for[latex]\\,f\\left(x\\right)=\\mathrm{cos}\\left(x\\right)-3.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137427502\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137427504\">Let\u2019s begin by comparing the equation to the general form[latex]\\,y=A\\mathrm{cos}\\left(Bx-C\\right)+D.[\/latex]<\/p>\n<p id=\"fs-id1165135503692\">In the given equation,[latex]\\,D=-3\\,[\/latex]so the shift is 3 units downward.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137742086\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_04\">\n<div id=\"fs-id1165137410879\">\n<p id=\"fs-id1165137410880\">Determine the direction and magnitude of the vertical shift for[latex]\\,f\\left(x\\right)=3\\mathrm{sin}\\left(x\\right)+2.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137432579\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137432580\">2 units up<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137824883\" class=\"precalculus howto\">\n<p id=\"fs-id1165135417813\"><strong>Given a sinusoidal function in the form<\/strong>[latex]\\,f\\left(x\\right)=A\\mathrm{sin}\\left(Bx-C\\right)+D,\\,[\/latex]<strong>identify the midline, amplitude, period, and phase shift.<\/strong><\/p>\n<ol id=\"fs-id1165137805755\" type=\"1\">\n<li>Determine the amplitude as[latex]\\,|A|.[\/latex]<\/li>\n<li>Determine the period as[latex]\\,P=\\frac{2\\pi }{|B|}.[\/latex]<\/li>\n<li>Determine the phase shift as[latex]\\,\\frac{C}{B}.[\/latex]<\/li>\n<li>Determine the midline as[latex]\\,y=D.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_06_01_05\" class=\"textbox examples\">\n<div id=\"fs-id1165137457010\">\n<div id=\"fs-id1165137457013\">\n<h3>Identifying the Variations of a Sinusoidal Function from an Equation<\/h3>\n<p id=\"fs-id1165137416718\">Determine the midline, amplitude, period, and phase shift of the function[latex]\\,y=3\\mathrm{sin}\\left(2x\\right)+1.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137454382\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137454384\">Let\u2019s begin by comparing the equation to the general form[latex]\\,y=A\\mathrm{sin}\\left(Bx-C\\right)+D.[\/latex]<\/p>\n<p>[latex]A=3,\\,[\/latex]so the amplitude is[latex]\\,|A|=3.[\/latex]<\/p>\n<p id=\"fs-id1165137438431\">Next,[latex]\\,B=2,\\,[\/latex]so the period is[latex]\\,P=\\frac{2\\pi }{|B|}=\\frac{2\\pi }{2}=\\pi .[\/latex]<\/p>\n<p>There is no added constant inside the parentheses, so[latex]\\,C=0\\,[\/latex]and the phase shift is[latex]\\,\\frac{C}{B}=\\frac{0}{2}=0.[\/latex]<\/p>\n<p id=\"fs-id1165137697063\">Finally,[latex]\\,D=1,\\,[\/latex]so the midline is[latex]\\,y=1.[\/latex]<\/details>\n<\/p>\n<\/div>\n<div id=\"fs-id1165137701755\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135414237\">Inspecting the graph, we can determine that the period is[latex]\\,\\pi ,\\,[\/latex]the midline is[latex]\\,y=1,\\,[\/latex]and the amplitude is 3. See <a class=\"autogenerated-content\" href=\"#Figure_06_01_014\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_01_014\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143154\/CNX_Precalc_Figure_06_01_014.jpg\" alt=\"A graph of y=3sin(2x)+1. The graph has an amplitude of 3. There is a midline at y=1. There is a period of pi. Local maximum at (pi\/4, 4) and local minimum at (3pi\/4, -2).\" width=\"487\" height=\"263\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 14.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137447405\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_05\">\n<div id=\"fs-id1165137447552\">\n<p id=\"fs-id1165137447553\">Determine the midline, amplitude, period, and phase shift of the function[latex]\\,y=\\frac{1}{2}\\mathrm{cos}\\left(\\frac{x}{3}-\\frac{\\pi }{3}\\right).[\/latex]<\/p>\n<\/div>\n<div>\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134042358\">midline:[latex]\\,y=0;\\,[\/latex]amplitude:[latex]\\,|A|=\\frac{1}{2};\\,[\/latex]period:[latex]\\,P=\\frac{2\\pi }{|B|}=6\\pi ;\\,[\/latex]phase shift:[latex]\\,\\frac{C}{B}=\\pi[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_06_01_06\" class=\"textbox examples\">\n<div id=\"fs-id1165137659478\">\n<div id=\"fs-id1165134040573\">\n<h3>Identifying the Equation for a Sinusoidal Function from a Graph<\/h3>\n<p>Determine the formula for the cosine function in <a class=\"autogenerated-content\" href=\"#Figure_06_01_015\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_01_015\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143157\/CNX_Precalc_Figure_06_01_015.jpg\" alt=\"A graph of -0.5cos(x)+0.5. The graph has an amplitude of 0.5. The graph has a period of 2pi. The graph has a range of [0, 1]. The graph is also reflected about the x-axis from the parent function cos(x).\" width=\"487\" height=\"163\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 15.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135329784\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137726017\">To determine the equation, we need to identify each value in the general form of a sinusoidal function.<\/p>\n<div id=\"fs-id1165137726021\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}y=A\\mathrm{sin}\\left(Bx-C\\right)+D\\hfill \\\\ y=A\\mathrm{cos}\\left(Bx-C\\right)+D\\hfill \\end{array}[\/latex]<\/div>\n<p>The graph could represent either a sine or a <span class=\"no-emphasis\">cosine function<\/span> that is shifted and\/or reflected. When[latex]\\,x=0,\\,[\/latex]the graph has an extreme point,[latex]\\,\\left(0,0\\right).\\,[\/latex]Since the cosine function has an extreme point for[latex]\\,x=0,\\,[\/latex]let us write our equation in terms of a cosine function.<\/p>\n<p id=\"fs-id1165135536557\">Let\u2019s start with the midline. We can see that the graph rises and falls an equal distance above and below[latex]\\,y=0.5.\\,[\/latex]This value, which is the midline, is[latex]\\,D\\,[\/latex]in the equation, so[latex]\\,D=0.5.[\/latex]<\/p>\n<p>The greatest distance above and below the midline is the amplitude. The maxima are 0.5 units above the midline and the minima are 0.5 units below the midline. So[latex]\\,|A|=0.5.\\,[\/latex]Another way we could have determined the amplitude is by recognizing that the difference between the height of local maxima and minima is 1, so[latex]\\,|A|=\\frac{1}{2}=0.5.\\,[\/latex]Also, the graph is reflected about the <em>x<\/em>-axis so that[latex]\\,A=-0.5.[\/latex]<\/p>\n<p id=\"fs-id1165134204425\">The graph is not horizontally stretched or compressed, so[latex]\\,B=1;\\,[\/latex]and the graph is not shifted horizontally, so[latex]\\,C=0.[\/latex]<\/p>\n<p id=\"fs-id1165135347312\">Putting this all together,<\/p>\n<div class=\"unnumbered\">[latex]g\\left(x\\right)=-0.5\\mathrm{cos}\\left(x\\right)+0.5[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137702221\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_06\">\n<div>\n<p id=\"fs-id1165135582222\">Determine the formula for the sine function in <a class=\"autogenerated-content\" href=\"#Figure_06_01_016\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_01_016\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143201\/CNX_Precalc_Figure_06_01_016.jpg\" alt=\"A graph of sin(x)+2. Period of 2pi, amplitude of 1, and range of [1, 3].\" width=\"487\" height=\"173\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 16.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137526465\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137526466\">[latex]f\\left(x\\right)=\\mathrm{sin}\\left(x\\right)+2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_06_01_07\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1165134058400\">\n<h3>Identifying the Equation for a Sinusoidal Function from a Graph<\/h3>\n<p id=\"fs-id1165134059763\">Determine the equation for the sinusoidal function in <a class=\"autogenerated-content\" href=\"#Figure_06_01_017\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_01_017\" class=\"medium\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143204\/CNX_Precalc_Figure_06_01_017.jpg\" alt=\"A graph of 3cos(pi\/3x-pi\/3)-2. Graph has amplitude of 3, period of 6, range of [-5,1].\" width=\"731\" height=\"565\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 17.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137598813\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137598815\">With the highest value at 1 and the lowest value at[latex]\\,-5,\\,[\/latex]the midline will be halfway between at[latex]\\,-2.\\,[\/latex]So[latex]\\,D=-2.\\,[\/latex]<\/p>\n<p id=\"fs-id1165133281392\">The distance from the midline to the highest or lowest value gives an amplitude of[latex]\\,|A|=3.[\/latex]<\/p>\n<p id=\"fs-id1165137824298\">The period of the graph is 6, which can be measured from the peak at[latex]\\,x=1\\,[\/latex]to the next peak at[latex]\\,x=7,[\/latex]or from the distance between the lowest points. Therefore,[latex]P=\\frac{2\\pi }{|B|}=6.\\,[\/latex]Using the positive value for[latex]\\,B,[\/latex]we find that<\/p>\n<div id=\"fs-id1165135196958\" class=\"unnumbered aligncenter\">[latex]B=\\frac{2\\pi }{P}=\\frac{2\\pi }{6}=\\frac{\\pi }{3}[\/latex]<\/div>\n<p id=\"fs-id1165137611526\">So far, our equation is either[latex]\\,y=3\\mathrm{sin}\\left(\\frac{\\pi }{3}x-C\\right)-2\\,[\/latex]or[latex]\\,y=3\\mathrm{cos}\\left(\\frac{\\pi }{3}x-C\\right)-2.\\,[\/latex]For the shape and shift, we have more than one option. We could write this as any one of the following:<\/p>\n<ul id=\"fs-id1165137466148\">\n<li>a cosine shifted to the right<\/li>\n<li>a negative cosine shifted to the left<\/li>\n<li>a sine shifted to the left<\/li>\n<li>a negative sine shifted to the right<\/li>\n<\/ul>\n<p id=\"fs-id1165137619397\">While any of these would be correct, the cosine shifts are easier to work with than the sine shifts in this case because they involve integer values. So our function becomes<\/p>\n<div id=\"fs-id1165137619402\" class=\"unnumbered aligncenter\">[latex]y=3\\mathrm{cos}\\left(\\frac{\\pi }{3}x-\\frac{\\pi }{3}\\right)-2\\text{ or }y=-3\\mathrm{cos}\\left(\\frac{\\pi }{3}x+\\frac{2\\pi }{3}\\right)-2[\/latex]<\/div>\n<p id=\"fs-id1165135704043\">Again, these functions are equivalent, so both yield the same graph.<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137805588\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_07\">\n<div id=\"fs-id1165137436869\">\n<p id=\"fs-id1165137436870\">Write a formula for the function graphed in <a class=\"autogenerated-content\" href=\"#Figure_06_01_018\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_01_018\" class=\"medium\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143209\/CNX_Precalc_Figure_06_01_018n.jpg\" alt=\"A graph of 4sin((pi\/5)x-pi\/5)+4. Graph has period of 10, amplitude of 4, range of [0,8].\" width=\"731\" height=\"440\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 18.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135173772\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135173773\">two possibilities:[latex]\\,y=4\\mathrm{sin}\\left(\\frac{\\pi }{5}x-\\frac{\\pi }{5}\\right)+4\\,[\/latex]or[latex]\\,y=-4\\mathrm{sin}\\left(\\frac{\\pi }{5}x+\\frac{4\\pi }{5}\\right)+4[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137735424\" class=\"bc-section section\">\n<h3>Graphing Variations of <em>y<\/em> = sin <em>x<\/em> and <em>y<\/em> = cos <em>x<\/em><\/h3>\n<p id=\"fs-id1165134148513\">Throughout this section, we have learned about types of variations of sine and cosine functions and used that information to write equations from graphs. Now we can use the same information to create graphs from equations.<\/p>\n<p id=\"fs-id1165137456137\">Instead of focusing on the general form equations<\/p>\n<div id=\"fs-id1165137456140\" class=\"unnumbered aligncenter\">[latex]y=A\\mathrm{sin}\\left(Bx-C\\right)+D\\text{ and }y=A\\mathrm{cos}\\left(Bx-C\\right)+D,[\/latex]<\/div>\n<p id=\"fs-id1165137807234\">we will let[latex]\\,C=0\\,[\/latex]and[latex]\\,D=0\\,[\/latex]and work with a simplified form of the equations in the following examples.<\/p>\n<div id=\"fs-id1165135380117\" class=\"precalculus howto\">\n<p id=\"fs-id1165135329942\"><strong>Given the function[latex]\\,y=A\\mathrm{sin}\\left(Bx\\right),\\,[\/latex]sketch its graph.<\/strong><\/p>\n<ol id=\"fs-id1165137542466\" type=\"1\">\n<li>Identify the amplitude,[latex]\\,|A|.[\/latex]<\/li>\n<li>Identify the period,[latex]\\,P=\\frac{2\\pi }{|B|}.[\/latex]<\/li>\n<li>Start at the origin, with the function increasing to the right if[latex]\\,A\\,[\/latex]is positive or decreasing if[latex]\\,A\\,[\/latex]is negative.<\/li>\n<li>At[latex]\\,x=\\frac{\\pi }{2|B|}\\,[\/latex]there is a local maximum for[latex]\\,A>0\\,[\/latex]or a minimum for[latex]\\,A<0,\\,[\/latex]with[latex]\\,y=A.[\/latex]<\/li>\n<li>The curve returns to the <em>x<\/em>-axis at[latex]\\,x=\\frac{\\pi }{|B|}.[\/latex]<\/li>\n<li>There is a local minimum for[latex]\\,A>0\\,[\/latex](maximum for[latex]\\,A<0[\/latex]) at[latex]\\,x=\\frac{3\\pi }{2|B|}\\,[\/latex]with[latex]\\,y=\u2013A.[\/latex]<\/li>\n<li>The curve returns again to the <em>x<\/em>-axis at[latex]\\,x=\\frac{2\\pi }{|B|}.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_06_01_08\" class=\"textbox examples\">\n<div id=\"fs-id1165134156046\">\n<div id=\"fs-id1165137565145\">\n<h3>Graphing a Function and Identifying the Amplitude and Period<\/h3>\n<p id=\"fs-id1165137565150\">Sketch a graph of[latex]\\,f\\left(x\\right)=-2\\mathrm{sin}\\left(\\frac{\\pi x}{2}\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134190732\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134190734\">Let\u2019s begin by comparing the equation to the form[latex]\\,y=A\\mathrm{sin}\\left(Bx\\right).[\/latex]<\/p>\n<ul id=\"eip-id1165135169452\">\n<li><em>Step 1.<\/em> We can see from the equation that[latex]\\,A=-2,[\/latex]so the amplitude is 2.\n<div id=\"fs-id1165135400292\" class=\"unnumbered aligncenter\">[latex]|A|=2[\/latex]<\/div>\n<\/li>\n<li><em>Step 2.<\/em> The equation shows that[latex]\\,B=\\frac{\\pi }{2},\\,[\/latex]so the period is\n<div id=\"fs-id1165134178538\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}P=\\frac{2\\pi }{\\frac{\\pi }{2}}\\hfill \\\\ \\text{ }=2\\pi \\cdot \\frac{2}{\\pi }\\hfill \\\\ \\text{ }=4\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<li><em>Step 3.<\/em> Because[latex]\\,A\\,[\/latex]is negative, the graph descends as we move to the right of the origin.<\/li>\n<li><em>Step 4\u20137.<\/em> The <em>x<\/em>-intercepts are at the beginning of one period,[latex]\\,x=0,\\,[\/latex]the horizontal midpoints are at[latex]\\,x=2\\,[\/latex]and at the end of one period at[latex]\\,x=4.[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137786248\">The quarter points include the minimum at[latex]\\,x=1\\,[\/latex]and the maximum at[latex]\\,x=3.\\,[\/latex]A local minimum will occur 2 units below the midline, at[latex]\\,x=1,\\,[\/latex]and a local maximum will occur at 2 units above the midline, at[latex]\\,x=3.\\,[\/latex]<a class=\"autogenerated-content\" href=\"#Figure_06_01_019\">(Figure)<\/a> shows the graph of the function.<\/p>\n<div id=\"Figure_06_01_019\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143217\/CNX_Precalc_Figure_06_01_019.jpg\" alt=\"A graph of -2sin((pi\/2)x). Graph has range of &#091;-2,2&#093;, period of 4, and amplitude of 2.\" width=\"487\" height=\"252\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 19.<\/strong><\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137539724\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_08\">\n<div id=\"fs-id1165137409841\">\n<p id=\"fs-id1165137628752\">Sketch a graph of[latex]\\,g\\left(x\\right)=-0.8\\mathrm{cos}\\left(2x\\right).\\,[\/latex]Determine the midline, amplitude, period, and phase shift.<\/p>\n<\/div>\n<div id=\"fs-id1165135342790\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135397949\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143231\/CNX_Precalc_Figure_06_01_020.jpg\" alt=\"A graph of -0.8cos(2x). Graph has range of &#091;-0.8, 0.8&#093;, period of pi, amplitude of 0.8, and is reflected about the x-axis compared to it's parent function cos(x).\" \/><\/span><\/p>\n<p id=\"eip-id1165137938401\">midline:[latex]\\,y=0;\\,[\/latex]amplitude:[latex]\\,|A|=0.8;\\,[\/latex]period:[latex]\\,P=\\frac{2\\pi }{|B|}=\\pi ;\\,[\/latex]phase shift:[latex]\\,\\frac{C}{B}=0\\,[\/latex] or none<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137425929\" class=\"precalculus howto\">\n<p id=\"fs-id1165137661914\"><strong>Given a sinusoidal function with a phase shift and a vertical shift, sketch its graph.<\/strong><\/p>\n<ol id=\"fs-id1165135503706\" type=\"1\">\n<li>Express the function in the general form[latex]\\,y=A\\mathrm{sin}\\left(Bx-C\\right)+D\\text{ or }y=A\\mathrm{cos}\\left(Bx-C\\right)+D.[\/latex]<\/li>\n<li>Identify the amplitude,[latex]\\,|A|.[\/latex]<\/li>\n<li>Identify the period,[latex]\\,P=\\frac{2\\pi }{|B|}.[\/latex]<\/li>\n<li>Identify the phase shift,[latex]\\,\\frac{C}{B}.[\/latex]<\/li>\n<li>Draw the graph of[latex]\\,f\\left(x\\right)=A\\mathrm{sin}\\left(Bx\\right)\\,[\/latex] shifted to the right or left by[latex]\\,\\frac{C}{B}\\,[\/latex]and up or down by[latex]\\,D.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_06_01_09\" class=\"textbox examples\">\n<div id=\"fs-id1165134116767\">\n<div id=\"fs-id1165134116769\">\n<h3>Graphing a Transformed Sinusoid<\/h3>\n<p id=\"fs-id1165137723733\">Sketch a graph of[latex]\\,f\\left(x\\right)=3\\mathrm{sin}\\left(\\frac{\\pi }{4}x-\\frac{\\pi }{4}\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135209894\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ul id=\"eip-id1165137474346\">\n<li><em>Step 1.<\/em> The function is already written in general form:[latex]\\,f\\left(x\\right)=3\\mathrm{sin}\\left(\\frac{\\pi }{4}x-\\frac{\\pi }{4}\\right).[\/latex]This graph will have the shape of a <span class=\"no-emphasis\">sine function<\/span>, starting at the midline and increasing to the right.<\/li>\n<li><em>Step 2.<\/em>[latex]\\,|A|=|3|=3.\\,[\/latex]The amplitude is 3.<\/li>\n<li><em>Step 3.<\/em> Since[latex]\\,|B|=|\\frac{\\pi }{4}|=\\frac{\\pi }{4},\\,[\/latex]we determine the period as follows.\n<div id=\"fs-id1165137572143\" class=\"unnumbered aligncenter\">[latex]P=\\frac{2\\pi }{|B|}=\\frac{2\\pi }{\\frac{\\pi }{4}}=2\\pi \\cdot \\frac{4}{\\pi }=8[\/latex]<\/div>\n<p id=\"fs-id1165137757960\">The period is 8.<\/p>\n<\/li>\n<li><em>Step 4.<\/em> Since[latex]\\,C=\\frac{\\pi }{4},\\,[\/latex]the phase shift is\n<div id=\"fs-id1165135684362\" class=\"unnumbered aligncenter\">[latex]\\frac{C}{B}=\\frac{\\frac{\\pi }{4}}{\\frac{\\pi }{4}}=1.[\/latex]<\/div>\n<p id=\"fs-id1165137634941\">The phase shift is 1 unit.<\/p>\n<\/li>\n<li><em>Step 5.<\/em><a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2844&amp;action=edit#Figure_06_01_021\">(Figure)<\/a> shows the graph of the function.\n<div id=\"Figure_06_01_021\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143237\/CNX_Precalc_Figure_06_01_021.jpg\" alt=\"A graph of 3sin(*(pi\/4)x-pi\/4). Graph has amplitude of 3, period of 8, and a phase shift of 1 to the right.\" width=\"487\" height=\"319\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 20. <\/strong>A horizontally compressed, vertically stretched, and horizontally shifted sinusoid<\/figcaption><\/figure>\n<\/div>\n<\/li>\n<\/ul>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135181399\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_09\">\n<div id=\"fs-id1165137638347\">\n<p id=\"fs-id1165137638348\">Draw a graph of[latex]\\,g\\left(x\\right)=-2\\mathrm{cos}\\left(\\frac{\\pi }{3}x+\\frac{\\pi }{6}\\right).\\,[\/latex]Determine the midline, amplitude, period, and phase shift.<\/p>\n<\/div>\n<div id=\"fs-id1165137480594\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137442771\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143239\/CNX_Precalc_Figure_06_01_022.jpg\" alt=\"A graph of -2cos((pi\/3)x+(pi\/6)). Graph has amplitude of 2, period of 6, and has a phase shift of 0.5 to the left.\" \/><\/span><\/p>\n<p id=\"fs-id1165137627836\">midline:[latex]\\,y=0;\\,[\/latex]amplitude:[latex]\\,|A|=2;\\,[\/latex]period:[latex]\\,P=\\frac{2\\pi }{|B|}=6;\\,[\/latex]phase shift:[latex]\\,\\frac{C}{B}=-\\frac{1}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_06_01_10\" class=\"textbox examples\">\n<div id=\"fs-id1165137749524\">\n<div id=\"fs-id1165137749526\">\n<h3>Identifying the Properties of a Sinusoidal Function<\/h3>\n<p id=\"fs-id1165137406791\">Given[latex]\\,y=-2\\mathrm{cos}\\left(\\frac{\\pi }{2}x+\\pi \\right)+3,\\,[\/latex]determine the amplitude, period, phase shift, and horizontal shift. Then graph the function.<\/p>\n<\/div>\n<div id=\"fs-id1165135487183\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137431258\">Begin by comparing the equation to the general form and use the steps outlined in <a class=\"autogenerated-content\" href=\"#Example_06_01_09\">(Figure)<\/a>.<\/p>\n<div id=\"fs-id1165134225658\" class=\"unnumbered aligncenter\">[latex]y=A\\mathrm{cos}\\left(Bx-C\\right)+D[\/latex]<\/div>\n<ul id=\"eip-id1165134311998\">\n<li><em>Step 1.<\/em> The function is already written in general form.<\/li>\n<li><em>Step 2.<\/em> Since[latex]\\,A=-2,\\,[\/latex]the amplitude is[latex]\\,|A|=2.[\/latex]<\/li>\n<li><em>Step 3.<\/em>[latex]\\,|B|=\\frac{\\pi }{2},\\,[\/latex]so the period is[latex]\\,P=\\frac{2\\pi }{|B|}=\\frac{2\\pi }{\\frac{\\pi }{2}}=2\\pi \\cdot \\frac{2}{\\pi }=4.\\,[\/latex]The period is 4.<\/li>\n<li><em>Step 4.<\/em>[latex]\\,C=-\\pi ,[\/latex]so we calculate the phase shift as[latex]\\,\\frac{C}{B}=\\frac{-\\pi ,}{\\frac{\\pi }{2}}=-\\pi \\cdot \\frac{2}{\\pi }=-2.\\,[\/latex]The phase shift is[latex]\\,-2.[\/latex]<\/li>\n<li><em>Step 5.<\/em>[latex]D=3,[\/latex]so the midline is[latex]\\,y=3,\u2009[\/latex]and the vertical shift is up 3.<\/li>\n<\/ul>\n<p id=\"fs-id1165137936633\">Since[latex]\\,A\\,[\/latex]is negative, the graph of the cosine function has been reflected about the <em>x<\/em>-axis.<\/p>\n<p><a class=\"autogenerated-content\" href=\"#Figure_06_01_028\">(Figure)<\/a> shows one cycle of the graph of the function.<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143244\/CNX_Precalc_Figure_06_01_028.jpg\" alt=\"A graph of -2cos((pi\/2)x+pi)+3. Graph shows an amplitude of 2, midline at y=3, and a period of 4.\" width=\"487\" height=\"317\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 21.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id1165137761033\"><\/details>\n<p><span id=\"fs-id1165137794283\"><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137939840\" class=\"bc-section section\">\n<h3>Using Transformations of Sine and Cosine Functions<\/h3>\n<p id=\"fs-id1165137891269\">We can use the transformations of sine and cosine functions in numerous applications. As mentioned at the beginning of the chapter, <span class=\"no-emphasis\">circular motion<\/span> can be modeled using either the sine or <span class=\"no-emphasis\">cosine function<\/span>.<\/p>\n<div id=\"Example_06_01_11\" class=\"textbox examples\">\n<div id=\"fs-id1165137612101\">\n<div id=\"fs-id1165137612103\">\n<h3>Finding the Vertical Component of Circular Motion<\/h3>\n<p id=\"fs-id1165137731540\">A point rotates around a circle of radius 3 centered at the origin. Sketch a graph of the <em>y<\/em>-coordinate of the point as a function of the angle of rotation.<\/p>\n<\/div>\n<div id=\"fs-id1165137552985\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137552987\">Recall that, for a point on a circle of radius <em>r<\/em>, the <em>y<\/em>-coordinate of the point is[latex]\\,y=r\\,\\mathrm{sin}\\left(x\\right),\\,[\/latex]<br \/>\nso in this case, we get the equation[latex]\\,y\\left(x\\right)=3\\,\\mathrm{sin}\\left(x\\right).\\,[\/latex]<br \/>\nThe constant 3 causes a vertical stretch of the <em>y<\/em>-values of the function by a factor of 3, which we can see in the graph in <a class=\"autogenerated-content\" href=\"#Figure_06_01_023\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_01_023\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143251\/CNX_Precalc_Figure_06_01_023.jpg\" alt=\"A graph of 3sin(x). Graph has period of 2pi, amplitude of 3, and range of &#091;-3,3&#093;.\" width=\"487\" height=\"319\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 22.<\/strong><\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137556901\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137400109\">Notice that the period of the function is still[latex]\\,2\\pi ;\\,[\/latex]as we travel around the circle, we return to the point[latex]\\,\\left(3,0\\right)\\,[\/latex]for[latex]\\,x=2\\pi ,4\\pi ,6\\pi ,....[\/latex]Because the outputs of the graph will now oscillate between[latex]\\,\u20133\\,[\/latex]and[latex]\\,3,\\,[\/latex]the amplitude of the sine wave is[latex]\\,3.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135319496\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_10\">\n<div id=\"fs-id1165135403587\">\n<p id=\"fs-id1165135403588\">What is the amplitude of the function[latex]\\,f\\left(x\\right)=7\\mathrm{cos}\\left(x\\right)?\\,[\/latex]Sketch a graph of this function.<\/p>\n<\/div>\n<div id=\"fs-id1165137534006\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137534008\">7<\/p>\n<p><span id=\"fs-id1165134187242\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143253\/CNX_Precalc_Figure_06_01_024.jpg\" alt=\"A graph of 7cos(x). Graph has amplitude of 7, period of 2pi, and range of [-7,7].\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_06_01_12\" class=\"textbox examples\">\n<div id=\"fs-id1165135190936\">\n<div id=\"fs-id1165135190938\">\n<h3>Finding the Vertical Component of Circular Motion<\/h3>\n<p id=\"fs-id1165135210138\">A circle with radius 3 ft is mounted with its center 4 ft off the ground. The point closest to the ground is labeled <em>P<\/em>, as shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_025\">(Figure)<\/a>. Sketch a graph of the height above the ground of the point[latex]\\,P\\,[\/latex]as the circle is rotated; then find a function that gives the height in terms of the angle of rotation.<\/p>\n<div id=\"Figure_06_01_025\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143304\/CNX_Precalc_Figure_06_01_025.jpg\" alt=\"An illustration of a circle lifted 4 feet off the ground. Circle has radius of 3 ft. There is a point P labeled on the circle's circumference.\" width=\"487\" height=\"300\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 23.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137863854\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137863856\">Sketching the height, we note that it will start 1 ft above the ground, then increase up to 7 ft above the ground, and continue to oscillate 3 ft above and below the center value of 4 ft, as shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_026\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_01_026\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143316\/CNX_Precalc_Figure_06_01_026.jpg\" alt=\"A graph of -3cox(x)+4. Graph has midline at y=4, amplitude of 3, and period of 2pi.\" width=\"487\" height=\"521\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 24.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137601519\">Although we could use a transformation of either the sine or cosine function, we start by looking for characteristics that would make one function easier to use than the other. Let\u2019s use a cosine function because it starts at the highest or lowest value, while a <span class=\"no-emphasis\">sine function<\/span> starts at the middle value. A standard cosine starts at the highest value, and this graph starts at the lowest value, so we need to incorporate a vertical reflection.<\/p>\n<p id=\"fs-id1165137601522\">Second, we see that the graph oscillates 3 above and below the center, while a basic cosine has an amplitude of 1, so this graph has been vertically stretched by 3, as in the last example.<\/p>\n<p id=\"fs-id1165134401716\">Finally, to move the center of the circle up to a height of 4, the graph has been vertically shifted up by 4. Putting these transformations together, we find that<\/p>\n<div id=\"fs-id1165133047569\" class=\"unnumbered aligncenter\">[latex]y=-3\\mathrm{cos}\\left(x\\right)+4[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_06_01_11\">\n<div id=\"fs-id1165137447011\">\n<p id=\"fs-id1165137447012\">A weight is attached to a spring that is then hung from a board, as shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_029\">(Figure)<\/a>. As the spring oscillates up and down, the position[latex]\\,y\\,[\/latex]of the weight relative to the board ranges from[latex]\\,\u20131\\,[\/latex]in. (at time[latex]\\,x=0)\\,[\/latex]to[latex]\\,\u20137\\,[\/latex]in. (at time[latex]\\,x=\\pi )\\,[\/latex]below the board. Assume the position of[latex]\\,y\\,[\/latex]is given as a sinusoidal function of[latex]\\,x.\\,[\/latex]Sketch a graph of the function, and then find a cosine function that gives the position[latex]\\,y\\,[\/latex]in terms of[latex]\\,x.[\/latex]<\/p>\n<div id=\"Figure_06_01_029\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143327\/CNX_Precalc_Figure_06_01_029.jpg\" alt=\"An illustration of a spring with length y.\" width=\"487\" height=\"351\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 25.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736527\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137736529\">[latex]y=3\\mathrm{cos}\\left(x\\right)-4[\/latex]<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143331\/CNX_Precalc_Figure_06_01_027.jpg\" alt=\"A cosine graph with range &#091;-1,-7&#093;. Period is 2 pi. Local maximums at (0,-1), (2pi,-1), and (4pi, -1). Local minimums at (pi,-7) and (3pi, -7).\" \/><\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_06_01_13\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1165137812716\">\n<h3>Determining a Rider\u2019s Height on a Ferris Wheel<\/h3>\n<p id=\"fs-id1165137500935\">The London Eye is a huge Ferris wheel with a diameter of 135 meters (443 feet). It completes one rotation every 30 minutes. Riders board from a platform 2 meters above the ground. Express a rider\u2019s height above ground as a function of time in minutes.<\/p>\n<\/div>\n<div id=\"fs-id1165137837117\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137837120\">With a diameter of 135 m, the wheel has a radius of 67.5 m. The height will oscillate with amplitude 67.5 m above and below the center.<\/p>\n<p id=\"fs-id1165134086000\">Passengers board 2 m above ground level, so the center of the wheel must be located[latex]\\,67.5+2=69.5\\,[\/latex]m above ground level. The midline of the oscillation will be at 69.5 m.<\/p>\n<p id=\"fs-id1165137578349\">The wheel takes 30 minutes to complete 1 revolution, so the height will oscillate with a period of 30 minutes.<\/p>\n<p id=\"fs-id1165137529532\">Lastly, because the rider boards at the lowest point, the height will start at the smallest value and increase, following the shape of a vertically reflected cosine curve.<\/p>\n<ul id=\"fs-id1165137529537\">\n<li>Amplitude:[latex]\\,\\text{67}\\text{.5,}\\,[\/latex]so[latex]\\,A=67.5[\/latex]<\/li>\n<li>Midline:[latex]\\,\\text{69}\\text{.5,}\\,[\/latex]so[latex]\\,D=69.5[\/latex]<\/li>\n<li>Period:[latex]\\,\\text{30,}\\,[\/latex]so[latex]\\,B=\\frac{2\\pi }{30}=\\frac{\\pi }{15}[\/latex]<\/li>\n<li>Shape:[latex]\\,\\mathrm{-cos}\\left(t\\right)[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137767318\">An equation for the rider\u2019s height would be<\/p>\n<div id=\"fs-id1165135403551\" class=\"unnumbered aligncenter\">[latex]y=-67.5\\mathrm{cos}\\left(\\frac{\\pi }{15}t\\right)+69.5[\/latex]<\/div>\n<p id=\"fs-id1165137634889\">where[latex]\\,t\\,[\/latex]is in minutes and[latex]\\,y\\,[\/latex]is measured in meters.<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137540365\" class=\"precalculus media\">\n<p id=\"fs-id1165137761688\">Access these online resources for additional instruction and practice with graphs of sine and cosine functions.<\/p>\n<ul id=\"fs-id1165137761692\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/ampperiod\">Amplitude and Period of Sine and Cosine<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/translasincos\">Translations of Sine and Cosine<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/transformsincos\">Graphing Sine and Cosine Transformations<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphsinefunc\">Graphing the Sine Function<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137574576\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165133087385\" summary=\"..\">\n<caption>&nbsp;<\/caption>\n<tbody>\n<tr>\n<td>Sinusoidal functions<\/td>\n<td>[latex]\\begin{array}{l}f\\left(x\\right)=A\\mathrm{sin}\\left(Bx-C\\right)+D\\\\ f\\left(x\\right)=A\\mathrm{cos}\\left(Bx-C\\right)+D\\end{array}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137540392\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul>\n<li>Periodic functions repeat after a given value. The smallest such value is the period. The basic sine and cosine functions have a period of[latex]\\,2\\pi .[\/latex]<\/li>\n<li>The function [latex]\\mathrm{sin}\\,x\\,[\/latex]is odd, so its graph is symmetric about the origin. The function [latex]\\,\\mathrm{cos}\\,x\\,[\/latex]is even, so its graph is symmetric about the <em>y<\/em>-axis.<\/li>\n<li>The graph of a sinusoidal function has the same general shape as a sine or cosine function.<\/li>\n<li>In the general formula for a sinusoidal function, the period is[latex]\\,P=\\frac{2\\pi }{|B|}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_06_01_01\">(Figure)<\/a>.<\/li>\n<li>In the general formula for a sinusoidal function,[latex]\\,|A|\\,[\/latex]represents amplitude. If[latex]\\,|A|>1,\\,[\/latex]the function is stretched, whereas if[latex]\\,|A|<1,\\,[\/latex]the function is compressed. See <a class=\"autogenerated-content\" href=\"#Example_06_01_02\">(Figure)<\/a>.<\/li>\n<li>The value[latex]\\,\\frac{C}{B}\\,[\/latex]in the general formula for a sinusoidal function indicates the phase shift. See <a class=\"autogenerated-content\" href=\"#Example_06_01_03\">(Figure)<\/a>.<\/li>\n<li>The value[latex]\\,D\\,[\/latex]in the general formula for a sinusoidal function indicates the vertical shift from the midline. See <a class=\"autogenerated-content\" href=\"#Example_06_01_04\">(Figure)<\/a>.<\/li>\n<li>Combinations of variations of sinusoidal functions can be detected from an equation. See <a class=\"autogenerated-content\" href=\"#Example_06_01_05\">(Figure)<\/a>.<\/li>\n<li>The equation for a sinusoidal function can be determined from a graph. See <a class=\"autogenerated-content\" href=\"#Example_06_01_06\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_06_01_07\">(Figure)<\/a>.<\/li>\n<li>A function can be graphed by identifying its amplitude and period. See <a class=\"autogenerated-content\" href=\"#Example_06_01_08\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_06_01_09\">(Figure)<\/a>.<\/li>\n<li>A function can also be graphed by identifying its amplitude, period, phase shift, and horizontal shift. See <a class=\"autogenerated-content\" href=\"#Example_06_01_10\">(Figure)<\/a>.<\/li>\n<li>Sinusoidal functions can be used to solve real-world problems. See <a class=\"autogenerated-content\" href=\"#Example_06_01_11\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_06_01_12\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_06_01_13\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137842570\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165137475812\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165137432049\">\n<div id=\"fs-id1165137415633\">\n<p id=\"fs-id1165137415634\">Why are the sine and cosine functions called periodic functions?<\/p>\n<\/div>\n<div id=\"fs-id1165137415637\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137415638\">The sine and cosine functions have the property that[latex]\\,f\\left(x+P\\right)=f\\left(x\\right)\\,[\/latex]for a certain[latex]\\,P.\\,[\/latex]This means that the function values repeat for every[latex]\\,P\\,[\/latex]units on the <em>x<\/em>-axis.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135305829\">\n<div id=\"fs-id1165135305831\">\n<p id=\"fs-id1165135305832\">How does the graph of[latex]\\,y=\\mathrm{sin}\\,x\\,[\/latex]<br \/>\ncompare with the graph of[latex]\\,y=\\mathrm{cos}\\,x?\\,[\/latex]<br \/>\nExplain how you could horizontally translate the graph of[latex]\\,y=\\mathrm{sin}\\,x\\,[\/latex]<br \/>\nto obtain[latex]\\,y=\\mathrm{cos}\\,x.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134290285\">\n<div id=\"fs-id1165134290287\">\n<p id=\"fs-id1165134290288\">For the equation[latex]\\,A\\,\\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<div id=\"fs-id1165137811265\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137431114\">The absolute value of the constant[latex]\\,A\\,[\/latex](amplitude) increases the total range and the constant[latex]\\,D\\,[\/latex](vertical shift) shifts the graph vertically.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137810849\">\n<div id=\"fs-id1165137810851\">\n<p id=\"fs-id1165137810852\">How does the range of a translated sine function relate to the equation[latex]\\,y=A\\,\\mathrm{sin}\\left(Bx+C\\right)+D?[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137410268\">\n<div id=\"fs-id1165137432413\">\n<p id=\"fs-id1165137432414\">How can the unit circle be used to construct the graph of[latex]\\,f\\left(t\\right)=\\mathrm{sin}\\,t?[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137407584\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137407585\">At the point where the terminal side of[latex]\\,t\\,[\/latex]intersects the unit circle, you can determine that the[latex]\\,\\mathrm{sin}\\,t\\,[\/latex]equals the <em>y<\/em>-coordinate of the point.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137663688\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165137742733\">For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum <em>y<\/em>-values and their corresponding <em>x<\/em>-values on one period for[latex]\\,x>0.\\,[\/latex]Round answers to two decimal places if necessary.<\/p>\n<div id=\"fs-id1165137570454\">\n<div id=\"fs-id1165137570456\">\n<p id=\"fs-id1165137570457\">[latex]f\\left(x\\right)=2\\mathrm{sin}\\,x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137423636\">\n<div id=\"fs-id1165137423638\">\n<p id=\"fs-id1165137423639\">[latex]f\\left(x\\right)=\\frac{2}{3}\\mathrm{cos}\\,x[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135456747\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137871371\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143339\/CNX_Precalc_Figure_06_01_202.jpg\" alt=\"A graph of (2\/3)cos(x). Graph has amplitude of 2\/3, period of 2pi, and range of &#091;-2\/3, 2\/3&#093;.\" \/><\/span><\/p>\n<p id=\"fs-id1165137817435\">amplitude:[latex]\\,\\frac{2}{3};\\,[\/latex]period:[latex]\\,2\\pi ;\\,[\/latex]midline:[latex]\\,y=0;\\,[\/latex]maximum:[latex]\\,y=\\frac{2}{3}\\,[\/latex]occurs at[latex]\\,x=0;\\,[\/latex]minimum:[latex]\\,y=-\\frac{2}{3}\\,[\/latex]occurs at[latex]\\,x=\\pi ;\\,[\/latex]for one period, the graph starts at 0 and ends at[latex]\\,2\\pi[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135663289\">\n<div id=\"fs-id1165135663292\">\n<p id=\"fs-id1165135663294\">[latex]f\\left(x\\right)=-3\\mathrm{sin}\\,x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137678150\">\n<div id=\"fs-id1165135160163\">\n<p id=\"fs-id1165135160164\">[latex]f\\left(x\\right)=4\\mathrm{sin}\\,x[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135151296\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143349\/CNX_Precalc_Figure_06_01_204.jpg\" alt=\"A graph of 4sin(x). Graph has amplitude of 4, period of 2pi, and range of &#091;-4, 4&#093;.\" \/><\/span><\/p>\n<p id=\"fs-id1165137535655\">amplitude: 4; period:[latex]\\,2\\pi ;\\,[\/latex]midline:[latex]\\,y=0;\\,[\/latex]maximum[latex]\\,y=4\\,[\/latex]occurs at[latex]\\,x=\\frac{\\pi }{2};\\,[\/latex]minimum:[latex]\\,y=-4\\,[\/latex]occurs at[latex]\\,x=\\frac{3\\pi }{2};\\,[\/latex]one full period occurs from[latex]\\,x=0\\,[\/latex]to[latex]\\,x=2\\pi[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137451618\">\n<div id=\"fs-id1165137451620\">\n<p id=\"fs-id1165137451622\">[latex]f\\left(x\\right)=2\\mathrm{cos}\\,x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137794096\">\n<div id=\"fs-id1165137794099\">\n<p id=\"fs-id1165137585075\">[latex]f\\left(x\\right)=\\mathrm{cos}\\left(2x\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137871346\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135386486\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143352\/CNX_Precalc_Figure_06_01_206.jpg\" alt=\"A graph of cos(2x). Graph has amplitude of 1, period of pi, and range of &#091;-1,1&#093;.\" \/><\/span><\/p>\n<p id=\"fs-id1165137827174\">amplitude: 1; period:[latex]\\,\\pi ;\\,[\/latex]midline:[latex]\\,y=0;\\,[\/latex]maximum:[latex]\\,y=1\\,[\/latex]occurs at[latex]\\,x=\\pi ;\\,[\/latex]minimum:[latex]\\,y=-1\\,[\/latex]occurs at[latex]\\,x=\\frac{\\pi }{2};\\,[\/latex]one full period is graphed from[latex]\\,x=0\\,[\/latex]to[latex]\\,x=\\pi[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135241360\">\n<div id=\"fs-id1165137531898\">\n<p id=\"fs-id1165137531899\">[latex]f\\left(x\\right)=2\\,\\mathrm{sin}\\left(\\frac{1}{2}x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137793956\">\n<div id=\"fs-id1165137793958\">\n<p id=\"fs-id1165137793959\">[latex]f\\left(x\\right)=4\\,\\mathrm{cos}\\left(\\pi x\\right)[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143354\/CNX_Precalc_Figure_06_01_208.jpg\" alt=\"A graph of 4cos(pi*x). Grpah has amplitude of 4, period of 2, and range of &#091;-4, 4&#093;.\" \/><\/p>\n<p id=\"fs-id1165137434223\">amplitude: 4; period: 2; midline:[latex]\\,y=0;\\,[\/latex]maximum:[latex]\\,y=4\\,[\/latex]occurs at[latex]\\,x=0;\\,[\/latex]minimum:[latex]\\,y=-4\\,[\/latex]occurs at[latex]\\,x=1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135487094\">\n<div id=\"fs-id1165135181423\">\n<p id=\"fs-id1165135181424\">[latex]f\\left(x\\right)=3\\,\\mathrm{cos}\\left(\\frac{6}{5}x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134277994\">\n<div>\n<p id=\"fs-id1165134277997\">[latex]y=3\\,\\mathrm{sin}\\left(8\\left(x+4\\right)\\right)+5[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137843946\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137424209\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143401\/CNX_Precalc_Figure_06_01_210.jpg\" alt=\"A graph of 3sin(8(x+4))+5. Graph has amplitude of 3, range of &#091;2, 8&#093;, and period of pi\/4.\" \/><\/span><\/p>\n<p id=\"fs-id1165134107350\">amplitude: 3; period:[latex]\\,\\frac{\\pi }{4};\\,[\/latex]midline:[latex]\\,y=5;\\,[\/latex]maximum:[latex]\\,y=8\\,[\/latex]occurs at[latex]\\,x=0.12;\\,[\/latex]minimum:[latex]\\,y=2\\,[\/latex]occurs at[latex]\\,x=0.516;\\,[\/latex]horizontal shift:[latex]\\,-4;\\,[\/latex]vertical translation 5; one period occurs from[latex]\\,x=0\\,[\/latex]to[latex]\\,x=\\frac{\\pi }{4}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137870948\">\n<div id=\"fs-id1165135457711\">\n<p id=\"fs-id1165135457712\">[latex]y=2\\,\\mathrm{sin}\\left(3x-21\\right)+4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135675218\">\n<div id=\"fs-id1165137414495\">\n<p id=\"fs-id1165137414496\">[latex]y=5\\,\\mathrm{sin}\\left(5x+20\\right)-2[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134284471\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135487190\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143404\/CNX_Precalc_Figure_06_01_212.jpg\" alt=\"A graph of 5sin(5x+20)-2. Graph has an amplitude of 5, period of 2pi\/5, and range of &#091;-7,3&#093;.\" \/><\/span><\/p>\n<p id=\"fs-id1165137862366\">amplitude: 5; period:[latex]\\,\\frac{2\\pi }{5};\\,[\/latex]midline:[latex]\\,y=-2;\\,[\/latex]maximum:[latex]\\,y=3\\,[\/latex]occurs at[latex]\\,x=0.08;\\,[\/latex]minimum:[latex]\\,y=-7\\,[\/latex]occurs at[latex]\\,x=0.71;\\,[\/latex]phase shift:[latex]\\,-4;\\,[\/latex]vertical translation:[latex]\\,-2;\\,[\/latex]one full period can be graphed on[latex]\\,x=0\\,[\/latex]to[latex]\\,x=\\frac{2\\pi }{5}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137803788\">For the following exercises, graph one full period of each function, starting at[latex]\\,x=0.\\,[\/latex]For each function, state the amplitude, period, and midline. State the maximum and minimum <em>y<\/em>-values and their corresponding <em>x<\/em>-values on one period for[latex]\\,x>0.\\,[\/latex]State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.<\/p>\n<div id=\"fs-id1165135154308\">\n<div id=\"fs-id1165137694190\">\n<p id=\"fs-id1165137694191\">[latex]f\\left(t\\right)=2\\mathrm{sin}\\left(t-\\frac{5\\pi }{6}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137898084\">\n<div id=\"fs-id1165137898086\">\n<p id=\"fs-id1165137898087\">[latex]f\\left(t\\right)=-\\mathrm{cos}\\left(t+\\frac{\\pi }{3}\\right)+1[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134541171\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135411369\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143411\/CNX_Precalc_Figure_06_01_214.jpg\" alt=\"A graph of -cos(t+pi\/3)+1. Graph has amplitude of 1, period of 2pi, and range of &#091;0,2&#093;. Phase shifted pi\/3 to the left.\" \/><\/span><\/p>\n<p id=\"fs-id1165137911368\">amplitude: 1 ; period:[latex]\\,2\\pi ;\\,[\/latex]midline:[latex]\\,y=1;\\,[\/latex]maximum:[latex]\\,y=2\\,[\/latex]occurs at[latex]\\,x=2.09;\\,[\/latex]maximum:[latex]\\,y=2\\,[\/latex]occurs at[latex]\\,t=2.09;\\,[\/latex]minimum:[latex]\\,y=0\\,[\/latex]occurs at[latex]\\,t=5.24;\\,[\/latex]phase shift:[latex]\\,-\\frac{\\pi }{3};\\,[\/latex]vertical translation: 1; one full period is from[latex]\\,t=0\\,[\/latex]to[latex]\\,t=2\\pi[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135250664\">\n<div id=\"fs-id1165135250666\">\n<p id=\"fs-id1165135250667\">[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<div>\n<div id=\"fs-id1165137552973\">\n<p id=\"fs-id1165137552974\">[latex]f\\left(t\\right)=-\\mathrm{sin}\\left(\\frac{1}{2}t+\\frac{5\\pi }{3}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137541180\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135349231\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143419\/CNX_Precalc_Figure_06_01_216.jpg\" alt=\"A graph of -sin((1\/2)*t + 5pi\/3). Graph has amplitude of 1, range of &#091;-1,1&#093;, period of 4pi, and a phase shift of -10pi\/3.\" \/><\/span>amplitude: 1; period:[latex]\\,4\\pi ;\\,[\/latex]midline:[latex]\\,y=0;\\,[\/latex]maximum:[latex]\\,y=1\\,[\/latex]occurs at[latex]\\,t=11.52;\\,[\/latex]minimum:[latex]\\,y=-1\\,[\/latex]occurs at[latex]\\,t=5.24;\\,[\/latex]phase shift:[latex]\\,-\\frac{10\\pi }{3};\\,[\/latex]vertical shift: 0<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135247503\">\n<div id=\"fs-id1165135247505\">\n<p id=\"fs-id1165135247506\">[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<div id=\"fs-id1165137658354\">\n<div id=\"fs-id1165137658356\">\n<p id=\"fs-id1165137823542\">Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_218\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_01_218\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 371px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143443\/CNX_Precalc_Figure_06_01_218.jpg\" alt=\"A sinusoidal graph with amplitude of 2, range of [-5, -1], period of 4, and midline at y=-3.\" width=\"371\" height=\"288\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 26.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135708054\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135708055\">amplitude: 2; midline:[latex]\\,y=-3;\\,[\/latex]period: 4; equation:[latex]\\,f\\left(x\\right)=2\\mathrm{sin}\\left(\\frac{\\pi }{2}x\\right)-3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137705148\">\n<div id=\"fs-id1165135169261\">\n<p id=\"fs-id1165135169262\">Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_219\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_01_219\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 308px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143449\/CNX_Precalc_Figure_06_01_219.jpg\" alt=\"A graph with a cosine parent function, with amplitude of 3, period of pi, midline at y=-1, and range of [-4,2]\" width=\"308\" height=\"322\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 27.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135688798\">\n<div id=\"fs-id1165135688800\">\n<p id=\"fs-id1165135688801\">Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_220\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_01_220\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 432px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143502\/CNX_Precalc_Figure_06_01_220.jpg\" alt=\"A graph with a cosine parent function with an amplitude of 2, period of 5, midline at y=3, and a range of [1,5].\" width=\"432\" height=\"290\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 28.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134378700\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134378701\">amplitude: 2; period: 5; midline:[latex]\\,y=3;\\,[\/latex]equation:[latex]\\,f\\left(x\\right)=-2\\mathrm{cos}\\left(\\frac{2\\pi }{5}x\\right)+3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135333824\">\n<div id=\"fs-id1165135333826\">\n<p id=\"fs-id1165135333827\">Determine the amplitude, period, midline, and an equation involving sine for the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_221\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_01_221\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 400px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143505\/CNX_Precalc_Figure_06_01_221.jpg\" alt=\"A sinusoidal graph with amplitude of 4, period of 10, midline at y=0, and range [-4,4].\" width=\"400\" height=\"384\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 29.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137473550\">\n<div>\n<p id=\"fs-id1165134389955\">Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_222\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_01_222\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 401px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143516\/CNX_Precalc_Figure_06_01_222.jpg\" alt=\"A graph with cosine parent function, range of function is [-4,4], amplitude of 4, period of 2.\" width=\"401\" height=\"313\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 30.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135534972\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135534973\">amplitude: 4; period: 2; midline:[latex]\\,y=0;\\,[\/latex]equation:[latex]\\,f\\left(x\\right)=-4\\mathrm{cos}\\left(\\pi \\left(x-\\frac{\\pi }{2}\\right)\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137425688\">\n<p id=\"fs-id1165137425689\">Determine the amplitude, period, midline, and an equation involving sine for the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_223\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_01_223\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 307px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143523\/CNX_Precalc_Figure_06_01_223.jpg\" alt=\"A graph with sine parent function. Amplitude 2, period 2, midline y=0\" width=\"307\" height=\"188\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 31.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137588535\">\n<div id=\"fs-id1165137588538\">\n<p>Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_224\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_01_224\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 308px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143528\/CNX_Precalc_Figure_06_01_224.jpg\" alt=\"A graph with cosine parent function. Amplitude 2, period 2, midline y=1\" width=\"308\" height=\"188\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 32.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137600948\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137600949\">amplitude: 2; period: 2; midline[latex]\\,y=1;\\,[\/latex]equation:[latex]\\,f\\left(x\\right)=2\\mathrm{cos}\\left(\\pi x\\right)+1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135530674\">\n<div id=\"fs-id1165137823256\">\n<p id=\"fs-id1165137823257\">Determine the amplitude, period, midline, and an equation involving sine for the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_225\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_06_01_225\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 306px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143530\/CNX_Precalc_Figure_06_01_225.jpg\" alt=\"A graph with a sine parent function. Amplitude 1, period 4 and midline y=0.\" width=\"306\" height=\"188\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 33.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134043561\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165134043566\">For the following exercises, let[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\,x.[\/latex]<\/p>\n<div id=\"fs-id1165135471271\">\n<div id=\"fs-id1165135471273\">\n<p>On[latex]\\,\\left[0,2\\pi \\right),[\/latex]solve[latex]\\,f\\left(x\\right)=0.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137400540\">\n<div id=\"fs-id1165137400542\">\n<p id=\"fs-id1165137400543\">On[latex]\\,\\left[0,2\\pi \\right),[\/latex]solve[latex]\\,f\\left(x\\right)=\\frac{1}{2}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137832261\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137784896\">[latex]\\frac{\\pi }{6},\\frac{5\\pi }{6}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137637941\">\n<div id=\"fs-id1165137637943\">\n<p id=\"fs-id1165137637944\">Evaluate[latex]\\,f\\left(\\frac{\\pi }{2}\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137804692\">\n<div id=\"fs-id1165137442246\">\n<p id=\"fs-id1165137442247\">On[latex]\\,\\left[0,2\\pi \\right),f\\left(x\\right)=\\frac{\\sqrt{2}}{2}.\\,[\/latex]Find all values of[latex]\\,x.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137837133\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137837134\">[latex]\\frac{\\pi }{4},\\frac{3\\pi }{4}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137416978\">\n<div id=\"fs-id1165135189866\">\n<p id=\"fs-id1165135189867\">On[latex]\\,\\left[0,2\\pi \\right),[\/latex]the maximum value(s) of the function occur(s) at what <em>x<\/em>-value(s)?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137942464\">\n<div id=\"fs-id1165137942466\">\n<p id=\"fs-id1165137942467\">On[latex]\\,\\left[0,2\\pi \\right),[\/latex]the minimum value(s) of the function occur(s) at what <em>x<\/em>-value(s)?<\/p>\n<\/div>\n<div id=\"fs-id1165134042136\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134042137\">[latex]\\frac{3\\pi }{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137633931\">\n<div id=\"fs-id1165137633933\">\n<p id=\"fs-id1165137633934\">Show that[latex]\\,f\\left(-x\\right)=-f\\left(x\\right).\\,[\/latex]This means that[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\,x\\,[\/latex]is an odd function and possesses symmetry with respect to ________________.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137806520\">For the following exercises, let[latex]\\,f\\left(x\\right)=\\mathrm{cos}\\,x.[\/latex]<\/p>\n<div id=\"fs-id1165137844202\">\n<div id=\"fs-id1165137844204\">\n<p id=\"fs-id1165137844205\">On[latex]\\,\\left[0,2\\pi \\right),[\/latex]solve the equation[latex]\\,f\\left(x\\right)=\\mathrm{cos}\\,x=0.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134129955\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134129956\">[latex]\\frac{\\pi }{2},\\frac{3\\pi }{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134043804\">\n<div id=\"fs-id1165135191893\">\n<p id=\"fs-id1165135191894\">On[latex]\\,\\left[0,2\\pi \\right),[\/latex]solve[latex]\\,f\\left(x\\right)=\\frac{1}{2}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137423113\">\n<div id=\"fs-id1165137423115\">\n<p id=\"fs-id1165137447173\">On[latex]\\,\\left[0,2\\pi \\right),[\/latex]find the <em>x<\/em>-intercepts of[latex]\\,f\\left(x\\right)=\\mathrm{cos}\\,x.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135440505\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135440506\">[latex]\\frac{\\pi }{2},\\frac{3\\pi }{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134394631\">\n<div id=\"fs-id1165134394633\">\n<p id=\"fs-id1165134230442\">On[latex]\\,\\left[0,2\\pi \\right),[\/latex]find the <em>x<\/em>-values at which the function has a maximum or minimum value.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137442136\">\n<div id=\"fs-id1165137442139\">\n<p id=\"fs-id1165137442140\">On[latex]\\,\\left[0,2\\pi \\right),[\/latex]solve the equation[latex]\\,f\\left(x\\right)=\\frac{\\sqrt{3}}{2}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137933103\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137933104\">[latex]\\frac{\\pi }{6},\\frac{11\\pi }{6}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137638455\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<div id=\"fs-id1165135193041\">\n<div id=\"fs-id1165135193043\">\n<p id=\"fs-id1165137639606\">Graph[latex]\\,h\\left(x\\right)=x+\\mathrm{sin}\\,x\\,[\/latex]on[latex]\\,\\left[0,2\\pi \\right].\\,[\/latex]Explain why the graph appears as it does.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736577\">\n<div id=\"fs-id1165137736579\">\n<p id=\"fs-id1165137637316\">Graph[latex]\\,h\\left(x\\right)=x+\\mathrm{sin}\\,x\\,[\/latex]on[latex]\\,\\left[-100,100\\right].\\,[\/latex]Did the graph appear as predicted in the previous exercise?<\/p>\n<\/div>\n<div id=\"fs-id1165137433807\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137433809\">The graph appears linear. The linear functions dominate the shape of the graph for large values of[latex]\\,x.[\/latex]<\/p>\n<p><span id=\"fs-id1165137604843\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143536\/CNX_Precalc_Figure_06_01_227.jpg\" alt=\"A sinusoidal graph that increases like the function y=x, shown from 0 to 100.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137891285\">\n<div id=\"fs-id1165137891288\">\n<p id=\"fs-id1165137891290\">Graph[latex]\\,f\\left(x\\right)=x\\,\\mathrm{sin}\\,x\\,[\/latex]on[latex]\\,\\left[0,2\\pi \\right]\\,[\/latex]and verbalize how the graph varies from the graph of[latex]\\,f\\left(x\\right)=\\mathrm{sin}\\,x.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133085093\">\n<div id=\"fs-id1165133085095\">\n<p id=\"fs-id1165133085096\">Graph[latex]\\,f\\left(x\\right)=x\\,\\mathrm{sin}\\,x\\,[\/latex]on the window[latex]\\,\\left[-10,10\\right]\\,[\/latex]and explain what the graph shows.<\/p>\n<\/div>\n<div id=\"fs-id1165135322029\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134283628\">The graph is symmetric with respect to the <em>y<\/em>-axis and there is no amplitude because the function is not periodic.<\/p>\n<p><span id=\"fs-id1165137570542\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19143538\/CNX_Precalc_Figure_06_01_229.jpg\" alt=\"A sinusoidal graph that has increasing peaks and decreasing lows as the absolute value of x increases.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137433574\">\n<div id=\"fs-id1165137433577\">\n<p id=\"fs-id1165137696192\">Graph[latex]\\,f\\left(x\\right)=\\frac{\\mathrm{sin}\\,x}{x}\\,[\/latex]on the window[latex]\\,\\left[-5\\pi ,5\\pi \\right]\\,[\/latex]and explain what the graph shows.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135333770\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id1165135333775\">\n<div id=\"fs-id1165137435670\">\n<p id=\"fs-id1165137435671\">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]\\,h\\left(t\\right)\\,[\/latex]gives a person\u2019s height in meters above the ground <em>t<\/em> minutes after the wheel begins to turn.<\/p>\n<ol type=\"a\">\n<li>Find the amplitude, midline, and period of[latex]\\,h\\left(t\\right).[\/latex]<\/li>\n<li>Find a formula for the height function[latex]\\,h\\left(t\\right).[\/latex]<\/li>\n<li>How high off the ground is a person after 5 minutes?<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165135205671\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ol id=\"fs-id1165135205673\" type=\"a\">\n<li>Amplitude: 12.5; period: 10; midline:[latex]\\,y=13.5;[\/latex]<\/li>\n<li>[latex]h\\left(t\\right)=12.5\\mathrm{sin}\\left(\\frac{\\pi }{5}\\left(t-2.5\\right)\\right)+13.5;[\/latex]<\/li>\n<li>26 ft<\/li>\n<\/ol>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165137414167\">\n<dt>amplitude<\/dt>\n<dd id=\"fs-id1165137463141\">the vertical height of a function; the constant[latex]\\,A\\,[\/latex]appearing in the definition of a sinusoidal function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137602032\">\n<dt>midline<\/dt>\n<dd id=\"fs-id1165137602037\">the horizontal line[latex]\\,y=D,\\,[\/latex]where[latex]\\,D\\,[\/latex]appears in the general form of a sinusoidal function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137678058\">\n<dt>periodic function<\/dt>\n<dd id=\"fs-id1165137678063\">a function[latex]\\,f\\left(x\\right)\\,[\/latex]that satisfies[latex]\\,f\\left(x+P\\right)=f\\left(x\\right)\\,[\/latex]for a specific constant[latex]\\,P\\,[\/latex]and any value of[latex]\\,x[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137939683\">\n<dt>phase shift<\/dt>\n<dd id=\"fs-id1165137939688\">the horizontal displacement of the basic sine or cosine function; the constant[latex]\\,\\frac{C}{B}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135160153\">\n<dt>sinusoidal function<\/dt>\n<dd id=\"fs-id1165137737500\">any function that can be expressed in the form[latex]\\,f\\left(x\\right)=A\\mathrm{sin}\\left(Bx-C\\right)+D\\,[\/latex]or[latex]\\,f\\left(x\\right)=A\\mathrm{cos}\\left(Bx-C\\right)+D[\/latex]<\/dd>\n<\/dl>\n<\/div>\n<\/div>\n","protected":false},"author":291,"menu_order":2,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-128","chapter","type-chapter","status-publish","hentry"],"part":125,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/128","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/users\/291"}],"version-history":[{"count":1,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/128\/revisions"}],"predecessor-version":[{"id":129,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/128\/revisions\/129"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/parts\/125"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/128\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=128"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=128"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=128"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=128"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}