{"id":659,"date":"2019-03-07T13:31:41","date_gmt":"2019-03-07T13:31:41","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/graphs-of-the-sine-and-cosine-functions\/"},"modified":"2020-02-04T01:20:12","modified_gmt":"2020-02-04T01:20:12","slug":"graphs-of-the-sine-and-cosine-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/graphs-of-the-sine-and-cosine-functions\/","title":{"raw":"3.4 Graphs of the Sine and Cosine Functions","rendered":"3.4 Graphs of the Sine and Cosine Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Graph\u2009[latex]f\\left(t\\right)=\\text{sin}\\left(t\\right)[\/latex] \u2009and [latex]f\\left(t\\right)=\\text{cos}\\left(t\\right)[\/latex]\u2009.<\/li>\r\n \t<li>Use your knowledge of shifts to transform sine and cosine curves.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"wp-caption-text\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"372\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07131737\/CNX_Precalc_Figure_05_00_001.jpg\" alt=\"Two boats at a dock during low tide.\" width=\"372\" height=\"248\" \/> <strong>Figure 1.\u00a0<\/strong>The tide rises and falls at regular, predictable intervals. (credit: Andrea Schaffer, Flickr)[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165134376081\">Life is dense with phenomena that repeat in regular intervals. Each day, for example, the tides rise and fall in response to the gravitational pull of the moon. Similarly, the progression from day to night occurs as a result of Earth\u2019s rotation, and the pattern of the seasons repeats in response to Earth\u2019s revolution around the sun. Outside of nature, many stocks that mirror a company\u2019s profits are influenced by changes in the economic business cycle.<\/p>\r\n<p id=\"fs-id1165135570435\">In mathematics, a function that repeats its values in regular intervals is known as a <strong>periodic function<\/strong>. The graphs of such functions show a general shape reflective of a pattern that keeps repeating. This means the graph of the function has the same output at exactly the same place in every cycle. And this translates to all the cycles of the function having exactly the same length. So, if we know all the details of one full cycle of a true periodic function, then we know the state of the function\u2019s outputs at all times, future and past. In this chapter, we will investigate various examples of periodic functions.<\/p>\r\n\r\n<h3>Graphing Sine and Cosine Functions<\/h3>\r\nRecall that the cosine and sine functions relate real number values to the [latex]x[\/latex]- and [latex]y[\/latex]-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\">Table 1<\/a> lists some of the values for the sine function on a unit circle.\r\n<div id=\"fs-id1165135169322\" class=\"bc-section section\">\r\n<p style=\"text-align: center\"><strong>Table 1<\/strong><\/p>\r\n\r\n<table id=\"Table_06_01_01\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]t[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]0[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{6}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{4}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{3}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{2\\pi }{3}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{3\\pi }{4}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{5\\pi }{6}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\pi [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]\\mathrm{sin}\\left(t\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]0[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]1[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165137694159\">Plotting the points from the table and continuing along the horizontal axis gives the shape of the sine function. See <a class=\"autogenerated-content\" href=\"#Figure_06_01_002\">Figure 2<\/a>.<\/p>\r\n\r\n<div id=\"Figure_06_01_002\" class=\"small\">\r\n\r\n[caption id=\"attachment_3194\" align=\"aligncenter\" width=\"400\"]<img class=\"wp-image-3194 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/22153024\/34Fig2.png\" alt=\"\" width=\"400\" height=\"255\" \/> <strong>Figure 2:<\/strong>\u00a0\u00a0A graph of sin(t).[\/caption]\r\n\r\n<\/div>\r\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 3<\/a>.<\/p>\r\n\r\n<div id=\"Figure_06_01_003\" class=\"small\">\r\n\r\n[caption id=\"attachment_3195\" align=\"aligncenter\" width=\"664\"]<img class=\"wp-image-3195 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/22153349\/34Fig3.png\" alt=\"A side by side picture of the unit circle and the sine graph that relates points on the circle to points on the graph.\" width=\"664\" height=\"291\" \/> <strong>Figure 3:\u00a0<\/strong>\u00a0Plotting values of the sine function and relating them to the unit circle.[\/caption]\r\n\r\n<\/div>\r\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\">Table 2<\/a>\u00a0lists some of the values for the cosine function on a unit circle.<\/p>\r\n<p style=\"text-align: center\"><strong>Table 2<\/strong><\/p>\r\n\r\n<table id=\"Table_06_01_02\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td class=\"border\">[latex]t[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: left\">[latex]0[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{6}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{4}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{3}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{2\\pi }{3}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{3\\pi }{4}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{5\\pi }{6}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\pi [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">[latex]\\mathrm{cos}\\left(t\\right)[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]1[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]0[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]-\\frac{1}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]-1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\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 4<\/a>.<\/p>\r\n\r\n<div id=\"Figure_06_01_004\" class=\"medium\">\r\n<div class=\"wp-caption-text\"><\/div>\r\n\r\n[caption id=\"attachment_3196\" align=\"aligncenter\" width=\"404\"]<img class=\"wp-image-3196 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/22153540\/34Fig4.png\" alt=\"\" width=\"404\" height=\"248\" \/> <strong>Figure 4:<\/strong> A graph of cos(t).[\/caption]\r\n\r\n<\/div>\r\n<h4>Domain and Range<\/h4>\r\n<p id=\"fs-id1165135628668\">We have created the graphs of the sine and cosine functions by following a point from the positive x axis one complete rotation counterclockwise around the unit circle.\u00a0 This rotation generates\u00a0 input values between 0 and [latex]2\\pi. [\/latex] We know that we can continue to travel counterclockwise around the unit circle over and over again, and generate larger and larger positive input values.\u00a0 That means that we can continue to generate sine and cosine values for any positive input value.\u00a0 Likewise, we can start at the positive x axis, and travel clockwise around the circle.\u00a0 One revolution in the clockwise direction generates inputs starting at 0 and decreasing to [latex]-2\\pi.[\/latex]\u00a0 Again, since we can continue to travel around the circle over and over again in the clockwise direction, we can generate sine and cosine values for any negative input value.\u00a0 Because we can evaluate the sine and cosine of any real number, both of these functions have a domain of all real numbers.<\/p>\r\nWhat are the ranges of the sine and cosine functions?\u00a0 By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that since the x and y coordinates of points on the circle must each be in the interval [latex]\\left[-1,1\\right][\/latex], the range of both functions must be the interval [latex]\\left[-1,1\\right].[\/latex]\r\n<h4>Period<\/h4>\r\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(t+P\\right)=f\\left(t\\right)[\/latex] for all values of [latex]t[\/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 <strong><span class=\"no-emphasis\">period<\/span><\/strong> of the function. <a class=\"autogenerated-content\" href=\"#Figure_06_01_005\">Figure 5a and 5b<\/a> show several periods of the sine and cosine functions.<\/p>\r\n\r\n<div id=\"Figure_06_01_005\" class=\"small\">\r\n\r\n[caption id=\"attachment_3197\" align=\"aligncenter\" width=\"383\"]<img class=\"wp-image-3197 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/22153822\/34fig5a.png\" alt=\"Sine and cosine graphs with one period highlighted.\" width=\"383\" height=\"236\" \/> Figure 5a: Sine graph demonstrating a period.[\/caption]\r\n\r\n[caption id=\"attachment_3198\" align=\"aligncenter\" width=\"427\"]<img class=\"wp-image-3198 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/22153827\/34fig5b1.png\" alt=\"The cosine graph with one period highlighted.\" width=\"427\" height=\"250\" \/> Figure 5b: Cosine graph demonstrating a period.[\/caption]\r\n\r\n<\/div>\r\n<h4>Symmetries<\/h4>\r\n<p id=\"fs-id1165137447360\">Looking again at the sine and cosine functions on a domain centered at the vertical axis helps reveal symmetries. As we can see in <a class=\"autogenerated-content\" href=\"#Figure_06_01_006\">Figure 6<\/a>, the <span class=\"no-emphasis\">sine function<\/span> is symmetric about the origin. Recall that in Section 3.3, we determined from the unit circle that the sine function is an odd function because [latex]\\mathrm{sin}\\left(-t\\right)=-\\mathrm{sin}\\left(t\\right).[\/latex] Now we can clearly see this property from the graph.<\/p>\r\n\r\n<div id=\"Figure_06_01_006\" class=\"small\">\r\n\r\n[caption id=\"attachment_3200\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-3200\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/22172536\/34Fig6-300x184.png\" alt=\"Graph of sin(t) showing the odd symmetry.\" width=\"300\" height=\"184\" \/> <strong>Figure 6:<\/strong>\u00a0 Odd symmetry of the sine function.[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165135484164\"><a class=\"autogenerated-content\" href=\"#Figure_06_01_007\">Figure 7<\/a> shows that the cosine function is symmetric about the <em>y<\/em>-axis. Again, we determined from the unit circle that the cosine function is an even function. Now we can see from the graph that [latex]\\mathrm{cos}\\left(-t\\right)=\\mathrm{cos}\\left(t\\right).[\/latex]<\/p>\r\n\r\n<div id=\"Figure_06_01_007\" class=\"small\">\r\n\r\n[caption id=\"attachment_3201\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-3201\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/22172649\/34Fig7-300x197.png\" alt=\"A graph of the cosine function showing symmetry of an even function across the vertical axis.\" width=\"300\" height=\"197\" \/> <strong>Figure 7:\u00a0<\/strong>\u00a0Even symmetry of the cosine function.[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135187674\">\r\n<div class=\"textbox\">\r\n<h3>Characteristics of Sine and Cosine Functions<\/h3>\r\n<p id=\"fs-id1165137628764\">The sine and cosine functions have several distinct characteristics:<\/p>\r\n\r\n<ul id=\"fs-id1165137662423\">\r\n \t<li>They are periodic functions with a period of [latex]2\\pi .[\/latex]<\/li>\r\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>\r\n \t<li>The graph of [latex]f\\left(t\\right)=\\mathrm{sin}\\left(t\\right)[\/latex] is symmetric about the origin, because it is an odd function.<\/li>\r\n \t<li>The graph of [latex]f\\left(t\\right)=\\mathrm{cos}\\left(t\\right)[\/latex] is symmetric about the vertical axis, because it is an even function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3><span style=\"color: #6c64ad;font-size: 1em;font-weight: 600\">Investigating Sinusoidal Functions<\/span><\/h3>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134032223\" class=\"bc-section section\">\r\n<p id=\"fs-id1165137410921\">Recall when we first defined sine and cosine, we referenced an acute angle [latex]t[\/latex] in standard position as the input and associated each function with either the [latex]x[\/latex] or [latex]y[\/latex] coordinate of a point on the unit circle.\u00a0 In the material that follows, we switch back to typical function notation in the coordinate plane where [latex]y=f\\left(x\\right)[\/latex] and [latex]x[\/latex] will represent the angle and not the coordinate.<\/p>\r\nAs 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 <strong>sinusoidal function<\/strong>. The general forms of sinusoidal functions are:\r\n<div id=\"fs-id1165135512530\" style=\"text-align: center\">[latex]\\text{}y=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)+k [\/latex] and [latex] y=A\\mathrm{cos}\\left(B\\left(x-h\\right)\\right)+k.[\/latex]<\/div>\r\n<\/div>\r\n<div>[latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\n<h3>Determining the Period of Sinusoidal Functions<\/h3>\r\n<div id=\"fs-id1165134032223\" class=\"bc-section section\">\r\n<div id=\"fs-id1165135458566\" class=\"bc-section section\">\r\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>\r\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]\u00a0 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,\u00a0 [latex]f\\left(x\\right)=\\mathrm{sin}\\left(x\\right),[\/latex] [latex]B=1,[\/latex] so the period is [latex]2\\pi,[\/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] since the graph is compressed horizontally by a factor of 1\/2. 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] since the graph is stretched horizontally by a factor of 2. Notice in <a class=\"autogenerated-content\" href=\"#Figure_06_01_008\">Figure 8<\/a> how the period is indirectly related to [latex]|B|.[\/latex]<\/p>\r\n\r\n<div id=\"Figure_06_01_008\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"370\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132942\/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=\"370\" height=\"208\" \/> <strong>Figure 8:\u00a0<\/strong>\u00a0Three sine functions with different periods.[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137766762\">If we let [latex]h=0[\/latex] and [latex]k=0[\/latex] in the general form equations of the sine and cosine functions, we obtain the forms<\/p>\r\n\r\n<div id=\"fs-id1165137855068\" class=\"unnumbered\" style=\"text-align: center\">[latex]y=A\\mathrm{sin}\\left(Bx\\right)[\/latex]<\/div>\r\n<div id=\"fs-id1165134371173\" class=\"unnumbered\" style=\"text-align: center\">[latex]y=A\\mathrm{cos}\\left(Bx\\right)[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165137413926\">The period is [latex]\\frac{2\\pi }{|B|}.[\/latex]<\/p>\r\n\r\n<div id=\"Example_06_01_01\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137772370\">\r\n<div id=\"fs-id1165137772372\">\r\n<h3>Example 1:\u00a0 Identifying the Period of a Sine or Cosine Function<\/h3>\r\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>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137434852\">[reveal-answer q=\"fs-id1165137434852\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137434852\"]\r\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>\r\n<p id=\"fs-id1165134482743\">In the given equation, [latex]B=\\frac{\\pi }{6},[\/latex] so the period will be<\/p>\r\n\r\n<div id=\"fs-id1165137646911\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}P&amp;=\\frac{2\\pi }{|B|}\\\\ \\text{ }&amp;=\\frac{2\\pi }{\\frac{\\pi }{6}}\\hfill \\\\ \\text{ }&amp;=2\\pi \\cdot \\frac{6}{\\pi }\\hfill \\\\ \\text{ }&amp;=12.\\hfill \\end{align*}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137465427\" class=\"precalculus tryit\">\r\n<h3>Try it #1<\/h3>\r\n<div id=\"ti_06_01_01\">\r\n<div id=\"fs-id1165135187453\">\r\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>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137507692\">[reveal-answer q=\"fs-id1165137507692\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137507692\"]\r\n<p id=\"fs-id1165137675634\">[latex]6\\pi [\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135699141\" class=\"bc-section section\">\r\n<h3>Determining Amplitude<\/h3>\r\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 or compression 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=k;[\/latex] because [latex]k=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 vertically by a factor of [latex]|A|[\/latex]. For example, the amplitude of [latex]f\\left(x\\right)=4\\text{ }\\mathrm{sin}\\left(x\\right)[\/latex] is twice the amplitude of [latex]f\\left(x\\right)=2\\text{ }\\mathrm{sin}\\left(x\\right).[\/latex] If [latex]|A|&lt;1,[\/latex] the function is compressed vertically by a factor of [latex]|A|[\/latex]. <a class=\"autogenerated-content\" href=\"#Figure_06_01_009\">Figure 9<\/a> compares several sine functions with different amplitudes.<\/p>\r\n\r\n<div id=\"Figure_06_01_009\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"740\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132945\/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=\"740\" height=\"240\" \/> <strong>Figure 9:\u00a0<\/strong>\u00a0Four sine graphs with different amplitudes.[\/caption]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134284498\">\r\n<p id=\"fs-id1165134497141\">If we let [latex]h=0[\/latex] and [latex]k=0[\/latex] in the general form equations of the sine and cosine functions, we obtain the forms:<\/p>\r\n\r\n<div id=\"fs-id1165135177658\" class=\"unnumbered\" style=\"text-align: center\">[latex]y=A\\mathrm{sin}\\left(Bx\\right)[\/latex] and [latex]y=A\\mathrm{cos}\\left(Bx\\right).[\/latex][latex]\\\\[\/latex]<\/div>\r\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>\r\n\r\n<div id=\"fs-id1165135460914\" class=\"unnumbered\" style=\"text-align: center\">[latex]|A|\\text{ = amplitude = }\\frac{1}{2}|\\text{maximum }-\\text{ minimum}|.[\/latex][latex]\\\\[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"Example_06_01_02\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137653911\">\r\n<div id=\"fs-id1165134377968\">\r\n<h3>Example 2:\u00a0 Identifying the Amplitude of a Sine or Cosine Function<\/h3>\r\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>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135195832\">[reveal-answer q=\"fs-id1165135195832\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135195832\"]\r\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>\r\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 vertically by a factor of 4.<\/p>\r\nThe negative value of [latex]A[\/latex] results in a reflection across the [latex]x[\/latex]-axis of the <span class=\"no-emphasis\">sine function<\/span>, as shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_010\">Figure 10<\/a>.\r\n<div id=\"Figure_06_01_010\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"369\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132949\/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=\"369\" height=\"242\" \/> <strong>Figure 10:\u00a0<\/strong> A graph of -4sin(x).[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135471236\" class=\"precalculus tryit\">\r\n<div id=\"ti_06_01_02\">\r\n<div id=\"fs-id1165137771980\">\r\n<h3>Try it #2<\/h3>\r\n<p id=\"fs-id1165137771982\">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>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137531120\">[reveal-answer q=\"fs-id1165137531120\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137531120\"]\r\n<p id=\"fs-id1165137531121\">The amplitude is [latex]\\frac{1}{2}[\/latex] so there is a vertical compression.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137834807\" class=\"bc-section section\">\r\n<h3>Analyzing Shifts of Sinusoidal Functions<\/h3>\r\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]h[\/latex] and [latex]k.[\/latex] Recall the general form:<\/p>\r\n\r\n<div id=\"fs-id1165134122886\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\text{}y=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)+k [\/latex] and [latex]y=A\\mathrm{cos}\\left(B\\left(x-h\\right)\\right)+k\\text{}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\n<p id=\"fs-id1165137663752\">The value [latex]k[\/latex] is the vertical shift of the function and for sinusoidal functions [latex]k[\/latex] shifts the midline up or down [latex]k[\/latex] units. The function [latex]y=A\\mathrm{sin}\\left(x\\right)+k[\/latex] has a midline of [latex]y=k[\/latex]. See <a class=\"autogenerated-content\" href=\"#Figure_06_01_012\">\u00a0Figure 11<\/a>.<\/p>\r\n\r\n<div id=\"Figure_06_01_012\" class=\"small\">[caption id=\"attachment_3212\" align=\"aligncenter\" width=\"394\"]<img class=\"wp-image-3212 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/29014437\/34Fig11.png\" alt=\"Figure 11: A graph of y=Asin(x)+k with the midline highlighted\" width=\"394\" height=\"233\" \/> Figure 11: A graph of [latex]y=A\\mathrm{sin}\\left(x\\right)+k.[\/latex][\/caption]<\/div>\r\n<p id=\"fs-id1165135242867\">In the equation [latex]y=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)+k,[\/latex] any value of [latex]k[\/latex] other than zero shifts the graph of\u00a0[latex]y=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)[\/latex] up or down. <a class=\"autogenerated-content\" href=\"#Figure_06_01_013\">Figure 12\u00a0<\/a>compares [latex]f\\left(x\\right)=\\mathrm{sin}\\left(x\\right)[\/latex] with [latex]f\\left(x\\right)=\\mathrm{sin}\\left(x\\right)+2,[\/latex] which is shifted 2 units up on a graph.<\/p>\r\n\r\n<div id=\"Figure_06_01_013\" class=\"small\">\r\n\r\n[caption id=\"attachment_3213\" align=\"aligncenter\" width=\"388\"]<img class=\"wp-image-3213 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/29014720\/34Fig12.png\" alt=\"\" width=\"388\" height=\"229\" \/> Figure 12: Vertically shifted sine function.[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165134187254\">The value [latex]h[\/latex] for a sinusoidal function is called the the horizontal displacement of the basic sine or <span class=\"no-emphasis\">cosine function<\/span>. If [latex]h&gt;0,[\/latex] the graph shifts to the right. If [latex]h&lt;0,[\/latex] the graph shifts to the left. The greater the value of [latex]|h|,[\/latex] the more the graph is shifted. <a class=\"autogenerated-content\" href=\"#Figure_06_01_011\">Figure 13<\/a> shows that the graph of [latex]g\\left(x\\right)=\\mathrm{sin}\\left(x-\\pi \\right)[\/latex] shifts [latex]f\\left(x\\right)=\\mathrm{sin}\\left(x\\right)[\/latex] to the right by [latex]\\pi [\/latex] units, which is more than we see in the graph of [latex]p\\left(x\\right)=\\mathrm{sin}\\left(x-\\frac{\\pi }{4}\\right),[\/latex] which shifts [latex]f\\left(x\\right)=\\mathrm{sin}\\left(x\\right)[\/latex] to the right by [latex]\\frac{\\pi }{4}[\/latex] units.<\/p>\r\n\r\n<div id=\"Figure_06_01_011\" class=\"small\">\r\n\r\n[caption id=\"attachment_3215\" align=\"aligncenter\" width=\"451\"]<img class=\"wp-image-3215 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/29021055\/34Fig13.png\" alt=\"\" width=\"451\" height=\"273\" \/> Figure 13: Horizontally shifted sine functions.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135571809\">\r\n<p id=\"fs-id1165133201875\"><strong>Important Note:\u00a0<\/strong> Some books and videos use the form [latex]\\text{}y=A\\mathrm{cos}\\left(Bx-C\\right)+D\\text{}[\/latex] and then factor out B leaving the form\u00a0 [latex]\\text{}y=A\\mathrm{cos}\\left(B\\left(x-\\frac{C}{B}\\right)\\right)+D\\text{}[\/latex] .\u00a0 In this case, the value of [latex]\\frac{C}{B}[\/latex] represents what\u00a0 is shown previously as h (the horizontal shift). We also see the D replaces what we have previously referred to as k. \u00a0It is simply a matter of getting familiar with the form that is being used in a particular problem. \u00a0For the most part, we will prefer the form that uses h and k. \u00a0The same pattern can be used with the sine function.<\/p>\r\n\r\n<div id=\"Example_06_01_04\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137410966\">\r\n<div id=\"fs-id1165137410968\">\r\n<h3>Example 3:\u00a0 Identifying the Vertical Shift of a Function<\/h3>\r\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>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137427502\">[reveal-answer q=\"fs-id1165137427502\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137427502\"]\r\n<p id=\"fs-id1165137427504\">Let\u2019s begin by comparing the equation to the general form [latex]y=A\\mathrm{cos}\\left(B\\left(x-h\\right)\\right)+k,.[\/latex]<\/p>\r\n<p id=\"fs-id1165135503692\">In the given equation, [latex]k=-3[\/latex] so the shift is 3 units downward.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137742086\" class=\"precalculus tryit\">\r\n<div id=\"ti_06_01_04\">\r\n<h3>Try it #3<\/h3>\r\n<div id=\"fs-id1165137410879\">\r\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>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137432579\">[reveal-answer q=\"fs-id1165137432579\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137432579\"]\r\n<p id=\"fs-id1165137432580\">2 units up<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_06_01_03\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137696922\">\r\n<div id=\"fs-id1165137696924\">\r\n<h3>Example 4:\u00a0 Identifying the Horizontal Shift of a Function<\/h3>\r\n<p id=\"fs-id1165137804482\">Determine the direction and magnitude of the horizontal shift for [latex]f\\left(x\\right)=\\mathrm{sin}\\left(x+\\frac{\\pi }{6}\\right)-2.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134483435\">[reveal-answer q=\"fs-id1165134483435\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134483435\"]\r\n<p id=\"fs-id1165134483437\">Let\u2019s begin by comparing the equation to the general form [latex]y=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)+k.[\/latex]<\/p>\r\n<p id=\"fs-id1165137461008\">In the given equation, notice that [latex]B=1[\/latex] and [latex]h=-\\frac{\\pi }{6}.[\/latex] So the horizontal shift is [latex]\\frac{\\pi }{6}[\/latex] units to the left.<\/p>\r\n<p id=\"fs-id1165137664617\">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]h.[\/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]h[\/latex] is negative, the shift is to the left.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137806038\" class=\"precalculus tryit\">\r\n<div id=\"ti_06_01_03\">\r\n<h3>Try it #4<\/h3>\r\n<div id=\"fs-id1165137461117\">\r\n<p id=\"fs-id1165137461118\">Determine the direction and magnitude of the horizontal shift for [latex]f\\left(x\\right)=3\\mathrm{cos}\\left(x-\\frac{\\pi }{2}\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165131959464\">[reveal-answer q=\"fs-id1165131959464\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165131959464\"]\r\n<p id=\"fs-id1165131959465\">[latex]\\frac{\\pi }{2};[\/latex] right<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h3>Sine and Cosine are Co-Functions<\/h3>\r\nYou should recall that in Section 3.1, we described sine and cosine as co-functions.\u00a0 This was in relationship to the acute angles in a right triangle.\u00a0 We can also see this relationship when we consider the functions as functions of real numbers now that we understand how to horizontally shift these functions.\u00a0 See Figure 14 below.\r\n\r\nWe know that [latex]\\mathrm{sin}\\left(x+\\frac{\\pi}{2}\\right)[\/latex] shifts the sine function [latex]\\frac{\\pi}{2}[\/latex] units to the left.\u00a0 Notice that this shift makes the graph of the sine curve (dotted curve) overlap the graph of the cosine curve (solid curve).\r\n\r\n[caption id=\"attachment_2553\" align=\"aligncenter\" width=\"741\"]<img class=\"wp-image-2553\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/27190404\/34CoFunctionGraphic.png\" alt=\"\" width=\"741\" height=\"258\" \/> <strong>Figure 14:<\/strong> Shifted Sine function yields the Cosine function[\/caption]\r\n\r\nLikewise, we know that\u00a0[latex]\\mathrm{cos}\\left(x-\\frac{\\pi}{2}\\right)[\/latex] shifts the cosine function [latex]\\frac{\\pi}{2}[\/latex] units to the right.\u00a0 Looking at the diagram above, you can see that this shift would make the cosine curve (solid\u00a0 curve) align with the sine curve. (dotted curve)\r\n<h3>Putting It All Together<\/h3>\r\nGiven a sinusoidal function in the form [latex]f\\left(x\\right)=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)+k,[\/latex] we can\u00a0identify the midline, the amplitude, the period, and the horizontal shift. Recall, the amplitude is [latex]|A|,[\/latex] the midline is\u00a0[latex]y=k,[\/latex] and the horizontal shift is\u00a0[latex]h.[\/latex]\u00a0 The period is related to the horizontal stretch or compression and must be calculated using the formula\u00a0[latex]P=\\frac{2\\pi }{|B|}.[\/latex]\r\n<div id=\"Example_06_01_05\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137457010\">\r\n<div id=\"fs-id1165137457013\">\r\n<h3>Example 5:\u00a0 Identifying the Variations of a Sinusoidal Function from an Equation<\/h3>\r\n<p id=\"fs-id1165137416718\">Determine the midline, amplitude, period, and horizontal shift of the function [latex]y=3\\mathrm{sin}\\left(2x\\right)+1.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137454382\">[reveal-answer q=\"fs-id1165137454382\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137454382\"]\r\n<p id=\"fs-id1165137454384\">Let\u2019s begin by comparing the equation to the general form [latex]y=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)+k.[\/latex]<\/p>\r\n<p id=\"fs-id1165137408417\">[latex]A=3,[\/latex] so the amplitude is [latex]|A|=3.[\/latex]<\/p>\r\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>\r\n<p id=\"fs-id1165137637576\">There is no subtracted constant inside the parentheses, so [latex]h=0[\/latex] and the horizontal shift is [latex]0.[\/latex]<\/p>\r\n<p id=\"fs-id1165137697063\">Finally, [latex]k=1,[\/latex] so the midline is [latex]y=1.[\/latex]<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\n<div id=\"fs-id1165137701755\">\r\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 15<\/a>.<\/p>\r\n\r\n<div id=\"Figure_06_01_014\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"401\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133001\/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=\"401\" height=\"217\" \/> <strong>Figure 15:\u00a0<\/strong> A graph of y=3sin(2x)+1.[\/caption]\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137447405\" class=\"precalculus tryit\">\r\n<div id=\"ti_06_01_05\">\r\n<h3>Try it #5<\/h3>\r\n<div id=\"fs-id1165137447552\">\r\n<p id=\"fs-id1165137447553\">Determine the midline, amplitude, period, and horizontal shift of the function [latex]y=\\frac{1}{2}\\mathrm{cos}\\left(\\frac{x}{3}-\\frac{\\pi }{3}\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134042357\">[reveal-answer q=\"fs-id1165134042357\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134042357\"]\r\n<div><span style=\"font-size: 1rem;text-align: initial\">First rewrite the equation in the form:\u00a0\u00a0 [latex]y=\\frac{1}{2}\\mathrm{cos}\\left(\\frac{1}{3}\\left(x-\\pi \\right)\\right).[\/latex] <\/span><\/div>\r\n<div>\r\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] horizontal shift: [latex]\\pi [\/latex] units right<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_06_01_06\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137659478\">\r\n<div id=\"fs-id1165134040573\">\r\n<h3>Example 6:\u00a0 Identifying the Equation for a Sinusoidal Function from a Graph<\/h3>\r\n<p id=\"fs-id1165137871008\">Determine the formula for the cosine function in <a class=\"autogenerated-content\" href=\"#Figure_06_01_015\">Figure 16<\/a>.<\/p>\r\n\r\n<div id=\"Figure_06_01_015\" class=\"small\">[caption id=\"\" align=\"aligncenter\" width=\"400\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133003\/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=\"400\" height=\"134\" \/> <strong>Figure 16:\u00a0<\/strong> A graph of -0.5cos(x)+0.5.[\/caption]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135329784\">[reveal-answer q=\"fs-id1165135329784\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135329784\"]\r\n<p id=\"fs-id1165137726017\">To determine the equation, we need to identify each value in the general form of a sinusoidal function.<\/p>\r\n\r\n<div id=\"fs-id1165137726021\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{array}{l}y=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)+k\\hfill \\\\ y=A\\mathrm{cos}\\left(B\\left(x-h\\right)\\right)+k\\hfill \\end{array}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165137704661\">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>\r\nNotice that the extreme point for\u00a0[latex]x=0,[\/latex] is a minimum. \u00a0That would indicate that the cosine function has been reflected. \u00a0Let\u2019s consider the amplitude and then incorporate the reflection as part of our value for A.\r\n\r\nWe can determine the amplitude by recognizing that the difference between the height of local maxima and minima is 1, so [latex]|A|=\\frac{1}{2}=0.5.[\/latex] \u00a0Remember, we determined the graph is reflected about the <em>x<\/em>-axis so that [latex]A=-0.5.[\/latex]\r\n<p id=\"fs-id1165135536557\">Now let\u2019s consider 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]k[\/latex] in the equation, so [latex]k=0.5.[\/latex]<\/p>\r\n<p id=\"fs-id1165134204425\">The graph is not horizontally stretched or compressed since we can see that we have a minimum, maximum and minimum within [latex]2\\pi[\/latex] units, so [latex]B=1;[\/latex] and the graph is not shifted horizontally, so [latex]h=0.[\/latex]<\/p>\r\n<p id=\"fs-id1165135347312\">Putting this all together,<\/p>\r\n\r\n<div id=\"fs-id1165137401884\" class=\"unnumbered\" style=\"text-align: center\">[latex]g\\left(x\\right)=-0.5\\mathrm{cos}\\left(x\\right)+0.5.[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137702221\" class=\"precalculus tryit\">\r\n<div id=\"ti_06_01_06\">\r\n<h3>Try it #6<\/h3>\r\n<div id=\"fs-id1165135582221\">\r\n<p id=\"fs-id1165135582222\">Determine the formula for the sine function in <a class=\"autogenerated-content\" href=\"#Figure_06_01_016\">Figure 17<\/a>.<\/p>\r\n\r\n<div id=\"Figure_06_01_016\" class=\"small\">[caption id=\"\" align=\"aligncenter\" width=\"403\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133006\/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=\"403\" height=\"143\" \/> <strong>Figure 17:\u00a0<\/strong>Determine the function for this graph.[\/caption]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137526465\">[reveal-answer q=\"fs-id1165137526465\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137526465\"]\r\n<p id=\"fs-id1165137526466\">[latex]f\\left(x\\right)=\\mathrm{sin}\\left(x\\right)+2[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_06_01_07\" class=\"textbox examples\">\r\n<div id=\"fs-id1165134164950\">\r\n<div id=\"fs-id1165134058400\">\r\n<h3>Example 7:\u00a0 Identifying the Equation for a Sinusoidal Function from a Graph<\/h3>\r\n<p id=\"fs-id1165134059763\">Determine the equation for the sinusoidal function in <a class=\"autogenerated-content\" href=\"#Figure_06_01_017\">Figure 18<\/a>.<\/p>\r\n\r\n<div id=\"Figure_06_01_017\" class=\"medium\">[caption id=\"\" align=\"aligncenter\" width=\"370\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133009\/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=\"370\" height=\"286\" \/> <strong>Figure 18:<\/strong>\u00a0Determine the function for this graph.[\/caption]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137598813\">[reveal-answer q=\"fs-id1165137598813\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137598813\"]\r\n<p id=\"fs-id1165137598815\">With the highest value at 1 and the lowest value at [latex]-5,[\/latex] the amplitude will be [latex]|A|=3,[\/latex] and the midline will be halfway between at [latex]-2.[\/latex] So [latex]k=-2.[\/latex]<\/p>\r\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>\r\n\r\n<div id=\"fs-id1165135196958\" class=\"unnumbered\" style=\"text-align: center\">[latex]B=\\frac{2\\pi }{P}=\\frac{2\\pi }{6}=\\frac{\\pi }{3}.[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165137611526\">So far, our equation is either [latex]y=3\\mathrm{sin}\\left(\\frac{\\pi }{3}\\left(x-h\\right)\\right)-2[\/latex] or [latex]y=3\\mathrm{cos}\\left(\\frac{\\pi }{3}\\left(x-h\\right)\\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>\r\n\r\n<ul id=\"fs-id1165137466148\">\r\n \t<li>a cosine shifted to the right<\/li>\r\n \t<li>a negative cosine shifted to the left<\/li>\r\n \t<li>a sine shifted to the left<\/li>\r\n \t<li>a negative sine shifted to the right<\/li>\r\n<\/ul>\r\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>\r\n\r\n<div id=\"fs-id1165137619402\" class=\"unnumbered\" style=\"text-align: center\">[latex]y=3\\mathrm{cos}\\left(\\frac{\\pi }{3}\\left(x-1\\right)\\right)-2\\text{ or }y=-3\\mathrm{cos}\\left(\\frac{\\pi }{3}\\left(x+2\\right)\\right)-2.[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165135704043\">Again, these functions are equivalent, so both yield the same graph.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137805588\" class=\"precalculus tryit\">\r\n<div id=\"ti_06_01_07\">\r\n<h3>Try it #7<\/h3>\r\n<div id=\"fs-id1165137436869\">\r\n<p id=\"fs-id1165137436870\">Write a formula for the function graphed in <a class=\"autogenerated-content\" href=\"#Figure_06_01_018\">Figure 19<\/a>.<\/p>\r\n\r\n<div id=\"Figure_06_01_018\" class=\"medium\">[caption id=\"\" align=\"aligncenter\" width=\"370\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133012\/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=\"370\" height=\"223\" \/> <strong>Figure 19:<\/strong>\u00a0 Determine the function for this graph.[\/caption]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135173772\">[reveal-answer q=\"fs-id1165135173772\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135173772\"]\r\n<p id=\"fs-id1165135173773\">two possibilities: [latex]y=4\\mathrm{sin}\\left(\\frac{\\pi }{5}\\left(x-1\\right)\\right)+4[\/latex] or [latex]y=-4\\mathrm{sin}\\left(\\frac{\\pi }{5}\\left(x+4\\right)\\right)+4[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137735424\" class=\"bc-section section\">\r\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>\r\n<p id=\"fs-id1165137456137\">Instead of focusing on the general form equations<\/p>\r\n\r\n<div id=\"fs-id1165137456140\" class=\"unnumbered\">\r\n<p id=\"fs-id1165137454384\" style=\"text-align: center\">[latex]y=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)+k[\/latex] <span style=\"font-size: 1em\">\u00a0and [latex]y=A\\mathrm{cos}\\left(B\\left(x-h\\right)\\right)+k,[\/latex]<\/span><\/p>\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137807234\">we will let [latex]h=0[\/latex] and [latex]k=0[\/latex] and work with a simplified form of the equations in the following examples.<\/p>\r\n\r\n<div id=\"fs-id1165135380117\" class=\"precalculus howto\">\r\n<div class=\"textbox examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165135329942\"><strong>Given the function [latex]y=A\\mathrm{sin}\\left(Bx\\right),[\/latex] sketch its graph.\u00a0 Note that [latex]h=k=0.[\/latex]<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137542466\" type=\"1\">\r\n \t<li>Identify the amplitude, [latex]|A|.[\/latex]<\/li>\r\n \t<li>Identify the period, [latex]P=\\frac{2\\pi }{|B|}.[\/latex]<\/li>\r\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>\r\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>\r\n \t<li>The curve returns to the <em>x<\/em>-axis at [latex]x=\\frac{\\pi }{|B|}.[\/latex]<\/li>\r\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>\r\n \t<li>The curve returns again to the <em>x<\/em>-axis at [latex]x=\\frac{2\\pi }{|B|}.[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_06_01_08\" class=\"textbox examples\">\r\n<div id=\"fs-id1165134156046\">\r\n<div id=\"fs-id1165137565145\">\r\n<h3>Example 8:\u00a0 Graphing a Function and Identifying the Amplitude and Period<\/h3>\r\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>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134190732\">[reveal-answer q=\"fs-id1165134190732\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134190732\"]\r\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>\r\n\r\n<ul>\r\n \t<li style=\"list-style-type: none\">\r\n<ul>\r\n \t<li><em>Step 1.<\/em> We can see from the equation that [latex]A=-2,[\/latex] so the amplitude is 2.\r\n<div id=\"fs-id1165135400292\" class=\"unnumbered\" style=\"text-align: center\">[latex]|A|=2.[\/latex]<\/div><\/li>\r\n \t<li><em>Step 2.<\/em> The equation shows that [latex]B=\\frac{\\pi }{2},[\/latex] so the period is\r\n<div id=\"fs-id1165134178538\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}P&amp;=\\frac{2\\pi }{\\frac{\\pi }{2}} \\\\ &amp;=2\\pi\\cdot \\frac{2}{\\pi } \\\\ &amp;=4. \\end{align*}[\/latex]<\/div><\/li>\r\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>\r\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>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\nThe quarter points include the minimum at [latex]x=1[\/latex] and the maximum at [latex]x=3.[\/latex]\u00a0 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]\u00a0 Figure 20 shows the graph of the function.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"369\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133015\/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=\"369\" height=\"191\" \/> <strong>Figure 20:<\/strong> A graph of [latex]y=2\\mathrm{sin}\\left(\\frac{\\pi x}{2}\\right).[\/latex][\/caption]The graph has range of [-2,2], period of 4, and amplitude of 2.\r\n[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137539724\" class=\"precalculus tryit\">\r\n<div id=\"ti_06_01_08\">\r\n<h3>Try it #8<\/h3>\r\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 horizontal shift.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135342790\">[reveal-answer q=\"fs-id1165135342790\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135342790\"][caption id=\"\" align=\"aligncenter\" width=\"372\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133018\/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).\" width=\"372\" height=\"245\" \/> <strong>Figure 21:<\/strong>\u00a0 A graph of [latex]y=-0.8\\mathrm{cos}\\left(2x\\right).[\/latex][\/caption]\r\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] horizontal shift: [latex]h=0[\/latex] or none<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137425929\" class=\"precalculus howto\">\r\n<div class=\"textbox examples\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137661914\"><strong>Given a sinusoidal function with a horizontal shift and a vertical shift, sketch its graph.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165135503706\" type=\"1\">\r\n \t<li>\u00a0Express the function in the general form [latex]\\begin{align*}y&amp;=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)+k\\text{ or} \\\\ y&amp;=A\\mathrm{cos}\\left(B\\left(x-h\\right)\\right)+k\\hfill \\end{align*}[\/latex]<\/li>\r\n \t<li>\u00a0Identify the amplitude, [latex]|A|.[\/latex]<\/li>\r\n \t<li>\u00a0Identify the period, [latex]P=\\frac{2\\pi }{|B|}.[\/latex]<\/li>\r\n \t<li>\u00a0Identify the horizontal shift, [latex]h,[\/latex] and the vertical shift, [latex]k.[\/latex]<\/li>\r\n \t<li>\u00a0Draw the graph of [latex]f\\left(x\\right)=A\\mathrm{sin}\\left(Bx\\right)[\/latex] or [latex]f\\left(x\\right)=A\\mathrm{cos}\\left(Bx\\right)[\/latex] shifted to the right or left by [latex]h[\/latex] and up or down by [latex]k.[\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_06_01_09\" class=\"textbox examples\">\r\n<div>\r\n<h3>Example 9:\u00a0 Graphing a Transformed Sinusoid<\/h3>\r\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>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135209894\">[reveal-answer q=\"fs-id1165135209894\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135209894\"]\r\n<ul id=\"eip-id1165137474346\">\r\n \t<li><em>Step 1.<\/em> <span style=\"font-size: 1rem;text-align: initial\">First rewrite the equation in the form:\u00a0\u00a0 [latex]y=3\\mathrm{sin}\\left(\\frac{\\pi}{4}\\left(x-1 \\right)\\right).[\/latex] <\/span>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>\r\n \t<li><em>Step 2.<\/em> [latex]|A|=|3|=3.[\/latex] The amplitude is 3.<\/li>\r\n \t<li><em>Step 3.<\/em> Since [latex]|B|=|\\frac{\\pi }{4}|=\\frac{\\pi }{4},[\/latex] we determine the period as follows.\r\n<div id=\"fs-id1165137572143\" class=\"unnumbered\" style=\"text-align: center\">[latex]P=\\frac{2\\pi }{|B|}=\\frac{2\\pi }{\\frac{\\pi }{4}}=2\\pi \\cdot \\frac{4}{\\pi }=8[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165137757960\">The period is 8.<\/p>\r\n<\/li>\r\n \t<li><em>Step 4.<\/em> Since [latex]h=1,[\/latex] the horizontal shift is [latex]1[\/latex] unit right.\u00a0 Since [latex]k=0[\/latex] there is no vertical shift.\r\n<p id=\"fs-id1165137634941\"><\/p>\r\n<\/li>\r\n \t<li><em>Step 5.<\/em><a class=\"autogenerated-content\" href=\"#Figure_06_01_021\">Figure 22<\/a> shows the graph of the function.\r\n<div id=\"Figure_06_01_021\" class=\"small\">\r\n<div class=\"wp-caption-text\"><\/div>\r\n[caption id=\"\" align=\"aligncenter\" width=\"360\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133021\/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=\"360\" height=\"236\" \/> <strong>Figure 22:<\/strong> A horizontally stretched, vertically stretched, and horizontally shifted sinusoid graph of [latex]y=3\\mathrm{sin}\\left(\\frac{\\pi}{4}x-\\frac{\\pi}{4}\\right).[\/latex][\/caption]\r\n[\/hidden-answer]<\/div><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135181399\" class=\"precalculus tryit\">\r\n<div id=\"ti_06_01_09\">\r\n<div id=\"fs-id1165137638347\">\r\n<h3>Try it #9<\/h3>\r\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 horizontal shift.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137480594\">[reveal-answer q=\"fs-id1165137480594\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137480594\"][caption id=\"\" align=\"aligncenter\" width=\"741\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133024\/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.\" width=\"741\" height=\"176\" \/> <strong>Figure 23:\u00a0<\/strong> A graph of [latex]y=-2\\mathrm{cos}\\left(\\left(\\frac{\\pi}{3}\\right)x+\\left(\\frac{\\pi}{6}\\right)\\right).[\/latex][\/caption]First rewrite the equation in the form\u00a0 [latex]g\\left(x\\right)=-2\\mathrm{cos}\\left(\\frac{\\pi }{3}\\left(x+\\frac{1 }{2}\\right)\\right).[\/latex]\r\n<p id=\"fs-id1165137627836\">midline: [latex]y=0;[\/latex] amplitude: [latex]|A|=2;[\/latex] period: [latex]P=\\frac{2\\pi }{|B|}=6;[\/latex] horizontal shift: [latex]h=-\\frac{1}{2}[\/latex] or [latex]\\frac{1}{2}[\/latex] units left<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_06_01_10\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137749524\">\r\n<div id=\"fs-id1165137749526\">\r\n<h3>Example 10:\u00a0 Identifying the Properties of a Sinusoidal Function<\/h3>\r\n<p id=\"fs-id1165137406791\">Given [latex]y=-2\\mathrm{cos}\\left(\\frac{\\pi }{2}x+\\pi \\right)+3,[\/latex] determine the amplitude, period, horizontal shift and vertical shift. Then graph the function.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135487183\">[reveal-answer q=\"fs-id1165135487183\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135487183\"]\r\n<div id=\"fs-id1165134225658\" class=\"unnumbered\"><\/div>\r\n<ul id=\"eip-id1165134311998\">\r\n \t<li><em>Step 1.<\/em>\u00a0\u00a0<span style=\"font-size: 1rem;text-align: initial\">First rewrite the equation in the form:\u00a0\u00a0[latex]y=-2\\mathrm{cos}\\left(\\frac{\\pi}{2}\\left(x + 2 \\right)\\right)+3.[\/latex]<\/span><\/li>\r\n \t<li><em>Step 2.<\/em>\u00a0 Since [latex]A=-2,[\/latex] the amplitude is [latex]|A|=2.[\/latex]<\/li>\r\n \t<li><em>Step 3.\u00a0\u00a0<\/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>\r\n \t<li><em>Step 4.\u00a0\u00a0<\/em>[latex]h=- 2 ,[\/latex] so the horizontal shift\u00a0 is 2 units to the left.<em>\u00a0<\/em>[latex]k=3,[\/latex] so the midline is [latex]y=3,\u2009[\/latex] and the vertical shift is up 3.<\/li>\r\n<\/ul>\r\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 before the vertical shift is done.<\/p>\r\n<p id=\"fs-id1165137761033\"><a class=\"autogenerated-content\" href=\"#Figure_06_01_028\">Figure 24<\/a> shows one cycle of the graph of the function.<\/p>\r\n\r\n<div id=\"Figure_06_01_028\" class=\"small\">[caption id=\"\" align=\"aligncenter\" width=\"400\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133027\/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=\"400\" height=\"260\" \/> <strong>Figure 24:<\/strong> A graph of\u00a0[latex]\\mathrm{cos}\\left(\\frac{\\pi }{2}x+\\pi \\right)+3.[\/latex][\/caption]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137939840\" class=\"bc-section section\">\r\n<div class=\"bc-section section\">\r\n<h3>Using Five Key Points to Graph Sinusoidal Functions<\/h3>\r\n<p id=\"fs-id1415219\">One method of graphing sinusoidal functions is to find five key points. These points will correspond to intervals of equal length representing [latex]\\frac{1}{4}[\/latex] of the period. The key points will indicate the location of maximum and minimum values. If there is no vertical shift, they will also indicate <em>x<\/em>-intercepts. For example, suppose we want to graph the function [latex]y=\\mathrm{cos}\\text{ }\\theta .[\/latex] We know that the period is [latex]2\\pi ,[\/latex] so we find the interval between key points as follows.<\/p>\r\n\r\n<div id=\"fs-id2102442\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\frac{2\\pi }{4}=\\frac{\\pi }{2}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id2290195\">Starting with [latex]\\theta =0,[\/latex] we calculate the first <em>y-<\/em>value, add the length of the interval [latex]\\frac{\\pi }{2}[\/latex] to 0, and calculate the second <em>y<\/em>-value. We then add [latex]\\frac{\\pi }{2}[\/latex] repeatedly until the five key points are determined. The last value should equal the first value, as the calculations cover one full period. Making a table similar to Table 3, we can see these key points clearly on the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_07_07_002\">\u00a0Figure 25<\/a>.<\/p>\r\n\r\n<table id=\"Table_07_07_01\" summary=\"Two rows, six columns. The table has ordered pairs of these column values: (theta, y=cos(theta)), (0,1), (i\/2, 0), (pi,-1), (3pi\/2, 0), (2pi,1).\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup><caption><strong>Table 3<\/strong><\/caption>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]\\theta [\/latex]<\/strong><\/td>\r\n<td class=\"border\">[latex]0[\/latex]<\/td>\r\n<td class=\"border\">[latex]\\frac{\\pi }{2}[\/latex]<\/td>\r\n<td class=\"border\">[latex]\\pi [\/latex]<\/td>\r\n<td class=\"border\">[latex]\\frac{3\\pi }{2}[\/latex]<\/td>\r\n<td class=\"border\">[latex]2\\pi [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]y=\\mathrm{cos}\\text{ }\\theta [\/latex]<\/strong><\/td>\r\n<td class=\"border\">[latex]1[\/latex]<\/td>\r\n<td class=\"border\">[latex]0[\/latex]<\/td>\r\n<td class=\"border\">[latex]-1[\/latex]<\/td>\r\n<td class=\"border\">[latex]0[\/latex]<\/td>\r\n<td class=\"border\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"Figure_07_07_002\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144634\/CNX_Precalc_Figure_07_07_002.jpg\" alt=\"Graph of y=cos(x) from -pi\/2 to 5pi\/2.\" width=\"487\" height=\"217\" \/> <b>Figure 25<\/b>[\/caption]\r\n\r\n<\/div>\r\n<div id=\"Example_07_06_03\" class=\"textbox examples\">\r\n<div id=\"fs-id2478628\">\r\n<div id=\"fs-id1173784\">\r\n<h3>Example 11: \u00a0Graphing Sinusoidal Functions Using Key Points<\/h3>\r\n<p id=\"fs-id1506097\">Graph the function [latex]y=-4\\mathrm{cos}\\left(\\pi x\\right)[\/latex] using amplitude, period, and key points.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1372153\">[reveal-answer q=\"fs-id1372153\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1372153\"]\r\n<p id=\"fs-id1846805\">The amplitude is[latex] |-4|=4. [\/latex] The period is [latex]\\frac{2\\pi }{B}=\\frac{2\\pi }{\\pi }=2.[\/latex] One cycle of the graph can be drawn over the interval [latex]\\left[0,2\\right].[\/latex] To find the key points, we divide the period by 4 which means we will increment [latex]x[\/latex] by [latex]\\frac{2}{4}=\\frac{1}{2}[\/latex]. Make a table similar to Table 4, starting with [latex]x=0[\/latex] and then adding [latex]\\frac{1}{2}[\/latex] successively to [latex]x[\/latex] and calculate [latex]y.[\/latex] See the graph in <a class=\"autogenerated-content\" href=\"#Figure_07_07_003\">Figure 26<\/a>.<\/p>\r\n\r\n<table id=\"Table_07_07_02\" summary=\"Two rows, six columns. The table has ordered pairs of these column values: (theta, y=-4cos(pi*x)), (0,-4), (1\/2, 0), (1,4), (3\/2, 0), (2, -4).\"><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup><caption><strong>Table 4<\/strong><\/caption>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td class=\"border\">[latex]0[\/latex]<\/td>\r\n<td class=\"border\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td class=\"border\">[latex]1[\/latex]<\/td>\r\n<td class=\"border\">[latex]\\frac{3}{2}[\/latex]<\/td>\r\n<td class=\"border\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><strong>[latex]y=-4\\text{ }\\mathrm{cos}\\left(\\pi x\\right)[\/latex]<\/strong><\/td>\r\n<td class=\"border\">[latex]-4[\/latex]<\/td>\r\n<td class=\"border\">[latex]0[\/latex]<\/td>\r\n<td class=\"border\">[latex]4[\/latex]<\/td>\r\n<td class=\"border\">[latex]0[\/latex]<\/td>\r\n<td class=\"border\">[latex]-4[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"Figure_07_07_003\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"330\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144637\/CNX_Precalc_Figure_07_07_003.jpg\" alt=\"Graph of y=-4cos(pi*x) using the five key points: intervals of equal length representing 1\/4 of the period. Here, the points are at 0, 1\/2, 1, 3\/2, and 2.\" width=\"330\" height=\"254\" \/> Figure 26[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id2571729\" class=\"precalculus tryit\">\r\n<h3>TRY IT #10<\/h3>\r\n<div id=\"ti_07_06_02\">\r\n<div id=\"fs-id2102282\">\r\n<p id=\"fs-id1333064\">Graph the function [latex]y=3\\mathrm{sin}\\left(3x\\right)[\/latex] using the amplitude, period, and five key points.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id2303325\">[reveal-answer q=\"fs-id2303325\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id2303325\"]\r\n<table id=\"fs-id2353042\" class=\"unnumbered\" summary=\"Graph of y=3sin(3x) using the five key points: intervals of equal length representing 1\/4 of the period. Here, the points are at 0, pi\/6, pi\/3, pi\/2, and 2pi\/3.\">\r\n<thead>\r\n<tr>\r\n<th class=\"border\" style=\"text-align: center\">x<\/th>\r\n<th class=\"border\" style=\"text-align: center\">[latex]y=3\\mathrm{sin}\\left(3x\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<caption><strong>Table 5<\/strong><\/caption>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center\">0<\/td>\r\n<td class=\"border\" style=\"text-align: center\">0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{6}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{3}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{2}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">[latex]-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{2\\pi }{3}[\/latex]<\/td>\r\n<td class=\"border\" style=\"text-align: center\">0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<span id=\"fs-id2710496\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144640\/CNX_Precalc_Figure_07_07_004.jpg\" alt=\"Graph of y=3sin(3x) using the five key points: intervals of equal length representing 1\/4 of the period. Here, the points are at 0, pi\/6, pi\/3, pi\/2, and 2pi\/3.\" width=\"398\" height=\"299\" \/><\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h3>Using Transformations of Sine and Cosine Functions<\/h3>\r\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>\r\n\r\n<div id=\"Example_06_01_11\" class=\"textbox examples\">\r\n<div id=\"fs-id1165137612101\">\r\n<div id=\"fs-id1165137612103\">\r\n<h3>Example 12:\u00a0 Finding the Vertical Component of Circular Motion<\/h3>\r\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>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137552985\">[reveal-answer q=\"fs-id1165137552985\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137552985\"]\r\n<p id=\"fs-id1165137552987\">Recall that we discussed in Section 3.3 that a point on a circle of radius r will have a y coordinate\u00a0 of\u00a0\u00a0[latex]y=r\\mathrm{sin}\\left(\\theta\\right).[\/latex] In this case, we get the equation [latex]y\\left(\\theta\\right)=3\\mathrm{sin}\\left(\\theta\\right).[\/latex]<\/p>\r\n\r\n<div id=\"Figure_06_01_023\" class=\"small\">[caption id=\"\" align=\"aligncenter\" width=\"370\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133031\/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=\"370\" height=\"242\" \/> <strong>Figure 27:<\/strong> A graph of 3sin([latex]\\theta[\/latex]).[\/caption]\r\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]<\/p>\r\nFrom our study of transformations of trigonometric functions, we also know that the constant 3 causes a vertical stretch of\u00a0the sine function by a factor of 3, which we can see in the graph in <a class=\"autogenerated-content\" href=\"#Figure_06_01_023\">Figure 27<\/a>.\r\n\r\nBecause the outputs of the graph will now oscillate between [latex]-3[\/latex] and [latex]3,[\/latex] the amplitude of the sine wave is [latex]3.[\/latex]\u00a0 This means that the radius of the circle centered at the origin will correspond to the amplitude of the sine function.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135319496\" class=\"precalculus tryit\">\r\n<div id=\"ti_06_01_10\">\r\n<h3>Try it #11<\/h3>\r\n<div id=\"fs-id1165135403587\">\r\n<p id=\"fs-id1165135403588\">What is the radius of the circle whose y-coordinate corresponds to the function [latex]f\\left(x\\right)=7\\mathrm{cos}\\left(x\\right)?[\/latex] Sketch a graph of this function.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137534006\">[reveal-answer q=\"fs-id1165137534006\"]Show Solution[\/reveal-answer][hidden-answer a =\"fs-id1165137534006\"]\r\nThe radius and amplitude are 7.\r\n[caption id=\"\" align=\"aligncenter\" width=\"266\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133034\/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].\" width=\"266\" height=\"294\" \/> <strong>Figure 26<\/strong>: A graph of [latex]y=7\\mathrm{cos}\\left(x\\right).[\/latex][\/caption]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_06_01_12\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135190936\">\r\n<div id=\"fs-id1165135190938\">\r\n<h3>Example 13:\u00a0 Finding the Vertical Component of Circular Motion<\/h3>\r\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 27<\/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>\r\n\r\n<div id=\"Figure_06_01_025\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"370\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133038\/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=\"370\" height=\"228\" \/> <strong>Figure 27:<\/strong> An illustration of a circle lifted 4 feet off the ground.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137863854\">[reveal-answer q=\"fs-id1165137863854\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137863854\"]\r\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 28<\/a>.<\/p>\r\n\r\n<div id=\"Figure_06_01_026\" class=\"small\">[caption id=\"\" align=\"aligncenter\" width=\"299\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133041\/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=\"299\" height=\"320\" \/> <strong>Figure 28:<\/strong> A graph of [latex]y=-3\\mathrm{cos}\\left(x\\right)+4.[\/latex][\/caption]<\/div>\r\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>\r\n<p id=\"fs-id1165137601522\">Second, we see that the graph oscillates 3 feet 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>\r\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>\r\n\r\n<div id=\"fs-id1165133047569\" class=\"unnumbered\" style=\"text-align: center\">[latex]y=-3\\mathrm{cos}\\left(x\\right)+4.[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It #12<\/h3>\r\nA 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 29<\/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]\r\n\r\n&nbsp;\r\n<div id=\"Figure_06_01_029\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"267\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133044\/CNX_Precalc_Figure_06_01_029.jpg\" alt=\"An illustration of a spring with length y.\" width=\"267\" height=\"193\" \/> <strong>Figure 29<\/strong>: An illustration of a spring with length y.[\/caption]\r\n\r\n<\/div>\r\n[reveal-answer q=\"292974\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"292974\"]\r\n\r\n[latex]y=3\\mathrm{cos}\\left(x\\right)-4[\/latex]\r\n\r\n[caption id=\"attachment_640\" align=\"aligncenter\" width=\"281\"]<img class=\"wp-image-640 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133047\/CNX_Precalc_Figure_06_01_027-244x300.jpg\" alt=\"\" width=\"281\" height=\"345\" \/> Figure 30[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 14:\u00a0Determining a Rider\u2019s Height on a Ferris Wheel<\/h3>\r\nThe 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.\r\n\r\n[reveal-answer q=\"91947\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"91947\"]\r\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>\r\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>\r\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>\r\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>\r\n\r\n<ul id=\"fs-id1165137529537\">\r\n \t<li>Amplitude: [latex]\\text{67}\\text{.5,}[\/latex] so [latex]A=67.5[\/latex]<\/li>\r\n \t<li>Midline: [latex]\\text{69}\\text{.5,}[\/latex] so [latex]k=69.5[\/latex]<\/li>\r\n \t<li>Period: [latex]\\text{30,}[\/latex] so [latex]B=\\frac{2\\pi }{30}=\\frac{\\pi }{15}[\/latex]<\/li>\r\n \t<li>Shape: [latex]\\mathrm{-cos}\\left(t\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137767318\">An equation for the rider\u2019s height would be<\/p>\r\n\r\n<div id=\"fs-id1165135403551\" class=\"unnumbered\" style=\"text-align: center\">[latex]y=-67.5\\mathrm{cos}\\left(\\frac{\\pi }{15}t\\right)+69.5[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1165137634889\">where [latex]t[\/latex] is in minutes and [latex]y[\/latex] is measured in meters.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<span style=\"font-size: 1rem;text-align: initial\">Access these online resources for additional instruction and practice with graphs of sine and cosine functions.<\/span>\r\n<div id=\"fs-id1165137540365\" class=\"precalculus media\">\r\n<ul id=\"fs-id1165137761692\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/ampperiod\">Amplitude and Period of Sine and Cosine<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/translasincos\">Translations of Sine and Cosine<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/transformsincos\">Graphing Sine and Cosine Transformations<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/graphsinefunc\">Graphing the Sine Function<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165137574576\" class=\"key-equations\">\r\n<h3>Key Equations<\/h3>\r\n<table id=\"eip-id1165133087385\" summary=\"..\"><caption>\u00a0<\/caption>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\">Sinusoidal functions<\/td>\r\n<td class=\"border\">[latex]\\begin{array}{l}y=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)+k\\text{ or }\\hfill \\\\ y=A\\mathrm{cos}\\left(B\\left(x-h\\right)\\right)+k\\hfill \\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1165137540392\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1165137762207\">\r\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>\r\n \t<li>The function [latex]\\mathrm{sin}\\text{ }x[\/latex] is odd, so its graph is symmetric about the origin. The function [latex]\\mathrm{cos}\\text{ }x[\/latex] is even, so its graph is symmetric about the <em>y<\/em>-axis.<\/li>\r\n \t<li>The graph of a sinusoidal function has the same general shape as a sine or cosine function.<\/li>\r\n \t<li>In the general formula for a sinusoidal function, the period is [latex]P=\\frac{2\\pi }{|B|}.[\/latex]<\/li>\r\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.<\/li>\r\n \t<li>The value [latex]h[\/latex] in the general formula for a sinusoidal function indicates the horizontal shift.<\/li>\r\n \t<li>The value [latex]k[\/latex] in the general formula for a sinusoidal function indicates the vertical shift.<\/li>\r\n \t<li>Combinations of variations of sinusoidal functions can be detected from an equation.<\/li>\r\n \t<li>The equation for a sinusoidal function can be determined from a graph.<\/li>\r\n \t<li>A function can also be graphed by identifying its amplitude, period, vertical shift, and horizontal shift.<\/li>\r\n \t<li>Sinusoidal functions can be used to solve real-world problems.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Glossary<\/h3>\r\n<dl id=\"fs-id1165137414167\">\r\n \t<dt>amplitude<\/dt>\r\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>\r\n<\/dl>\r\n<dl id=\"fs-id1165137602032\">\r\n \t<dt>midline<\/dt>\r\n \t<dd id=\"fs-id1165137602037\">the horizontal line [latex]y=k,[\/latex] where [latex]k[\/latex] appears in the general form of a sinusoidal function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137678058\">\r\n \t<dt>periodic function<\/dt>\r\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>\r\n<\/dl>\r\n<dl id=\"fs-id1165135160153\">\r\n \t<dt>sinusoidal function<\/dt>\r\n \t<dd id=\"fs-id1165137737500\">any function that can be expressed in the form [latex]f\\left(x\\right)=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)+k[\/latex] or [latex]f\\left(x\\right)=A\\mathrm{cos}\\left(B\\left(x-h\\right)\\right)+k[\/latex]<\/dd>\r\n<\/dl>\r\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\u2009[latex]f\\left(t\\right)=\\text{sin}\\left(t\\right)[\/latex] \u2009and [latex]f\\left(t\\right)=\\text{cos}\\left(t\\right)[\/latex]\u2009.<\/li>\n<li>Use your knowledge of shifts to transform sine and cosine curves.<\/li>\n<\/ul>\n<\/div>\n<div class=\"wp-caption-text\">\n<div style=\"width: 382px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07131737\/CNX_Precalc_Figure_05_00_001.jpg\" alt=\"Two boats at a dock during low tide.\" width=\"372\" height=\"248\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 1.\u00a0<\/strong>The tide rises and falls at regular, predictable intervals. (credit: Andrea Schaffer, Flickr)<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165134376081\">Life is dense with phenomena that repeat in regular intervals. Each day, for example, the tides rise and fall in response to the gravitational pull of the moon. Similarly, the progression from day to night occurs as a result of Earth\u2019s rotation, and the pattern of the seasons repeats in response to Earth\u2019s revolution around the sun. Outside of nature, many stocks that mirror a company\u2019s profits are influenced by changes in the economic business cycle.<\/p>\n<p id=\"fs-id1165135570435\">In mathematics, a function that repeats its values in regular intervals is known as a <strong>periodic function<\/strong>. The graphs of such functions show a general shape reflective of a pattern that keeps repeating. This means the graph of the function has the same output at exactly the same place in every cycle. And this translates to all the cycles of the function having exactly the same length. So, if we know all the details of one full cycle of a true periodic function, then we know the state of the function\u2019s outputs at all times, future and past. In this chapter, we will investigate various examples of periodic functions.<\/p>\n<h3>Graphing Sine and Cosine Functions<\/h3>\n<p>Recall that the cosine and sine functions relate real number values to the [latex]x[\/latex]&#8211; and [latex]y[\/latex]-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\">Table 1<\/a> lists some of the values for the sine function on a unit circle.<\/p>\n<div id=\"fs-id1165135169322\" class=\"bc-section section\">\n<p style=\"text-align: center\"><strong>Table 1<\/strong><\/p>\n<table id=\"Table_06_01_01\" summary=\"..\">\n<tbody>\n<tr>\n<td class=\"border\"><strong>[latex]t[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]0[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{6}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{4}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{3}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{2\\pi }{3}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{3\\pi }{4}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{5\\pi }{6}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\pi[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><strong>[latex]\\mathrm{sin}\\left(t\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]0[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]1[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[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 horizontal axis gives the shape of the sine function. See <a class=\"autogenerated-content\" href=\"#Figure_06_01_002\">Figure 2<\/a>.<\/p>\n<div id=\"Figure_06_01_002\" class=\"small\">\n<div id=\"attachment_3194\" style=\"width: 410px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3194\" class=\"wp-image-3194 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/22153024\/34Fig2.png\" alt=\"\" width=\"400\" height=\"255\" \/><\/p>\n<p id=\"caption-attachment-3194\" class=\"wp-caption-text\"><strong>Figure 2:<\/strong>\u00a0\u00a0A graph of sin(t).<\/p>\n<\/div>\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 3<\/a>.<\/p>\n<div id=\"Figure_06_01_003\" class=\"small\">\n<div id=\"attachment_3195\" style=\"width: 674px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3195\" class=\"wp-image-3195 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/22153349\/34Fig3.png\" alt=\"A side by side picture of the unit circle and the sine graph that relates points on the circle to points on the graph.\" width=\"664\" height=\"291\" \/><\/p>\n<p id=\"caption-attachment-3195\" class=\"wp-caption-text\"><strong>Figure 3:\u00a0<\/strong>\u00a0Plotting values of the sine function and relating them to the unit circle.<\/p>\n<\/div>\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\">Table 2<\/a>\u00a0lists some of the values for the cosine function on a unit circle.<\/p>\n<p style=\"text-align: center\"><strong>Table 2<\/strong><\/p>\n<table id=\"Table_06_01_02\" summary=\"..\">\n<tbody>\n<tr>\n<td class=\"border\">[latex]t[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: left\">[latex]0[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{6}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{4}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{3}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{2\\pi }{3}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{3\\pi }{4}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{5\\pi }{6}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\pi[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">[latex]\\mathrm{cos}\\left(t\\right)[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]1[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]0[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]-\\frac{1}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{2}}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]-\\frac{\\sqrt[\\leftroot{1}\\uproot{2} ]{3}}{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[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 4<\/a>.<\/p>\n<div id=\"Figure_06_01_004\" class=\"medium\">\n<div class=\"wp-caption-text\"><\/div>\n<div id=\"attachment_3196\" style=\"width: 414px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3196\" class=\"wp-image-3196 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/22153540\/34Fig4.png\" alt=\"\" width=\"404\" height=\"248\" \/><\/p>\n<p id=\"caption-attachment-3196\" class=\"wp-caption-text\"><strong>Figure 4:<\/strong> A graph of cos(t).<\/p>\n<\/div>\n<\/div>\n<h4>Domain and Range<\/h4>\n<p id=\"fs-id1165135628668\">We have created the graphs of the sine and cosine functions by following a point from the positive x axis one complete rotation counterclockwise around the unit circle.\u00a0 This rotation generates\u00a0 input values between 0 and [latex]2\\pi.[\/latex] We know that we can continue to travel counterclockwise around the unit circle over and over again, and generate larger and larger positive input values.\u00a0 That means that we can continue to generate sine and cosine values for any positive input value.\u00a0 Likewise, we can start at the positive x axis, and travel clockwise around the circle.\u00a0 One revolution in the clockwise direction generates inputs starting at 0 and decreasing to [latex]-2\\pi.[\/latex]\u00a0 Again, since we can continue to travel around the circle over and over again in the clockwise direction, we can generate sine and cosine values for any negative input value.\u00a0 Because we can evaluate the sine and cosine of any real number, both of these functions have a domain of all real numbers.<\/p>\n<p>What are the ranges of the sine and cosine functions?\u00a0 By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that since the x and y coordinates of points on the circle must each be in the interval [latex]\\left[-1,1\\right][\/latex], the range of both functions must be the interval [latex]\\left[-1,1\\right].[\/latex]<\/p>\n<h4>Period<\/h4>\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(t+P\\right)=f\\left(t\\right)[\/latex] for all values of [latex]t[\/latex] in the domain of [latex]f.[\/latex] When this occurs, we call the smallest such horizontal shift with [latex]P>0[\/latex] the <strong><span class=\"no-emphasis\">period<\/span><\/strong> of the function. <a class=\"autogenerated-content\" href=\"#Figure_06_01_005\">Figure 5a and 5b<\/a> show several periods of the sine and cosine functions.<\/p>\n<div id=\"Figure_06_01_005\" class=\"small\">\n<div id=\"attachment_3197\" style=\"width: 393px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3197\" class=\"wp-image-3197 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/22153822\/34fig5a.png\" alt=\"Sine and cosine graphs with one period highlighted.\" width=\"383\" height=\"236\" \/><\/p>\n<p id=\"caption-attachment-3197\" class=\"wp-caption-text\">Figure 5a: Sine graph demonstrating a period.<\/p>\n<\/div>\n<div id=\"attachment_3198\" style=\"width: 437px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3198\" class=\"wp-image-3198 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/22153827\/34fig5b1.png\" alt=\"The cosine graph with one period highlighted.\" width=\"427\" height=\"250\" \/><\/p>\n<p id=\"caption-attachment-3198\" class=\"wp-caption-text\">Figure 5b: Cosine graph demonstrating a period.<\/p>\n<\/div>\n<\/div>\n<h4>Symmetries<\/h4>\n<p id=\"fs-id1165137447360\">Looking again at the sine and cosine functions on a domain centered at the vertical axis helps reveal symmetries. As we can see in <a class=\"autogenerated-content\" href=\"#Figure_06_01_006\">Figure 6<\/a>, the <span class=\"no-emphasis\">sine function<\/span> is symmetric about the origin. Recall that in Section 3.3, we determined from the unit circle that the sine function is an odd function because [latex]\\mathrm{sin}\\left(-t\\right)=-\\mathrm{sin}\\left(t\\right).[\/latex] Now we can clearly see this property from the graph.<\/p>\n<div id=\"Figure_06_01_006\" class=\"small\">\n<div id=\"attachment_3200\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3200\" class=\"size-medium wp-image-3200\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/22172536\/34Fig6-300x184.png\" alt=\"Graph of sin(t) showing the odd symmetry.\" width=\"300\" height=\"184\" \/><\/p>\n<p id=\"caption-attachment-3200\" class=\"wp-caption-text\"><strong>Figure 6:<\/strong>\u00a0 Odd symmetry of the sine function.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135484164\"><a class=\"autogenerated-content\" href=\"#Figure_06_01_007\">Figure 7<\/a> shows that the cosine function is symmetric about the <em>y<\/em>-axis. Again, we determined from the unit circle that the cosine function is an even function. Now we can see from the graph that [latex]\\mathrm{cos}\\left(-t\\right)=\\mathrm{cos}\\left(t\\right).[\/latex]<\/p>\n<div id=\"Figure_06_01_007\" class=\"small\">\n<div id=\"attachment_3201\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3201\" class=\"size-medium wp-image-3201\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/22172649\/34Fig7-300x197.png\" alt=\"A graph of the cosine function showing symmetry of an even function across the vertical axis.\" width=\"300\" height=\"197\" \/><\/p>\n<p id=\"caption-attachment-3201\" class=\"wp-caption-text\"><strong>Figure 7:\u00a0<\/strong>\u00a0Even symmetry of the cosine function.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135187674\">\n<div class=\"textbox\">\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]f\\left(t\\right)=\\mathrm{sin}\\left(t\\right)[\/latex] is symmetric about the origin, because it is an odd function.<\/li>\n<li>The graph of [latex]f\\left(t\\right)=\\mathrm{cos}\\left(t\\right)[\/latex] is symmetric about the vertical axis, because it is an even function.<\/li>\n<\/ul>\n<\/div>\n<h3><span style=\"color: #6c64ad;font-size: 1em;font-weight: 600\">Investigating Sinusoidal Functions<\/span><\/h3>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134032223\" class=\"bc-section section\">\n<p id=\"fs-id1165137410921\">Recall when we first defined sine and cosine, we referenced an acute angle [latex]t[\/latex] in standard position as the input and associated each function with either the [latex]x[\/latex] or [latex]y[\/latex] coordinate of a point on the unit circle.\u00a0 In the material that follows, we switch back to typical function notation in the coordinate plane where [latex]y=f\\left(x\\right)[\/latex] and [latex]x[\/latex] will represent the angle and not the coordinate.<\/p>\n<p>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 <strong>sinusoidal function<\/strong>. The general forms of sinusoidal functions are:<\/p>\n<div id=\"fs-id1165135512530\" style=\"text-align: center\">[latex]\\text{}y=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)+k[\/latex] and [latex]y=A\\mathrm{cos}\\left(B\\left(x-h\\right)\\right)+k.[\/latex]<\/div>\n<\/div>\n<div>[latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<h3>Determining the Period of Sinusoidal Functions<\/h3>\n<div id=\"fs-id1165134032223\" class=\"bc-section section\">\n<div id=\"fs-id1165135458566\" class=\"bc-section section\">\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]\u00a0 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,\u00a0 [latex]f\\left(x\\right)=\\mathrm{sin}\\left(x\\right),[\/latex] [latex]B=1,[\/latex] so the period is [latex]2\\pi,[\/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] since the graph is compressed horizontally by a factor of 1\/2. 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] since the graph is stretched horizontally by a factor of 2. Notice in <a class=\"autogenerated-content\" href=\"#Figure_06_01_008\">Figure 8<\/a> how the period is indirectly related to [latex]|B|.[\/latex]<\/p>\n<div id=\"Figure_06_01_008\" class=\"small\">\n<div style=\"width: 380px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132942\/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=\"370\" height=\"208\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 8:\u00a0<\/strong>\u00a0Three sine functions with different periods.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137766762\">If we let [latex]h=0[\/latex] and [latex]k=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\" style=\"text-align: center\">[latex]y=A\\mathrm{sin}\\left(Bx\\right)[\/latex]<\/div>\n<div id=\"fs-id1165134371173\" class=\"unnumbered\" style=\"text-align: center\">[latex]y=A\\mathrm{cos}\\left(Bx\\right)[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137413926\">The period is [latex]\\frac{2\\pi }{|B|}.[\/latex]<\/p>\n<div id=\"Example_06_01_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137772370\">\n<div id=\"fs-id1165137772372\">\n<h3>Example 1:\u00a0 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\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137434852\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137434852\" class=\"hidden-answer\" style=\"display: none\">\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 id=\"fs-id1165137646911\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}P&=\\frac{2\\pi }{|B|}\\\\ \\text{ }&=\\frac{2\\pi }{\\frac{\\pi }{6}}\\hfill \\\\ \\text{ }&=2\\pi \\cdot \\frac{6}{\\pi }\\hfill \\\\ \\text{ }&=12.\\hfill \\end{align*}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137465427\" class=\"precalculus tryit\">\n<h3>Try it #1<\/h3>\n<div id=\"ti_06_01_01\">\n<div id=\"fs-id1165135187453\">\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\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137507692\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137507692\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137675634\">[latex]6\\pi[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135699141\" class=\"bc-section section\">\n<h3>Determining Amplitude<\/h3>\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 or compression 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=k;[\/latex] because [latex]k=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 vertically by a factor of [latex]|A|[\/latex]. For example, the amplitude of [latex]f\\left(x\\right)=4\\text{ }\\mathrm{sin}\\left(x\\right)[\/latex] is twice the amplitude of [latex]f\\left(x\\right)=2\\text{ }\\mathrm{sin}\\left(x\\right).[\/latex] If [latex]|A|<1,[\/latex] the function is compressed vertically by a factor of [latex]|A|[\/latex]. <a class=\"autogenerated-content\" href=\"#Figure_06_01_009\">Figure 9<\/a> compares several sine functions with different amplitudes.<\/p>\n<div id=\"Figure_06_01_009\" class=\"wp-caption aligncenter\">\n<div style=\"width: 750px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132945\/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=\"740\" height=\"240\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 9:\u00a0<\/strong>\u00a0Four sine graphs with different amplitudes.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134284498\">\n<p id=\"fs-id1165134497141\">If we let [latex]h=0[\/latex] and [latex]k=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\" style=\"text-align: center\">[latex]y=A\\mathrm{sin}\\left(Bx\\right)[\/latex] and [latex]y=A\\mathrm{cos}\\left(Bx\\right).[\/latex][latex]\\\\[\/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\" style=\"text-align: center\">[latex]|A|\\text{ = amplitude = }\\frac{1}{2}|\\text{maximum }-\\text{ minimum}|.[\/latex][latex]\\\\[\/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>Example 2:\u00a0 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\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135195832\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135195832\" class=\"hidden-answer\" style=\"display: none\">\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 vertically by a factor of 4.<\/p>\n<p>The negative value of [latex]A[\/latex] results in a reflection across the [latex]x[\/latex]-axis of the <span class=\"no-emphasis\">sine function<\/span>, as shown in <a class=\"autogenerated-content\" href=\"#Figure_06_01_010\">Figure 10<\/a>.<\/p>\n<div id=\"Figure_06_01_010\" class=\"small\">\n<div style=\"width: 379px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07132949\/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=\"369\" height=\"242\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 10:\u00a0<\/strong> A graph of -4sin(x).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135471236\" class=\"precalculus tryit\">\n<div id=\"ti_06_01_02\">\n<div id=\"fs-id1165137771980\">\n<h3>Try it #2<\/h3>\n<p id=\"fs-id1165137771982\">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 id=\"fs-id1165137531120\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137531120\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137531120\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137531121\">The amplitude is [latex]\\frac{1}{2}[\/latex] so there is a vertical compression.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137834807\" class=\"bc-section section\">\n<h3>Analyzing Shifts of Sinusoidal Functions<\/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]h[\/latex] and [latex]k.[\/latex] Recall the general form:<\/p>\n<div id=\"fs-id1165134122886\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\text{}y=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)+k[\/latex] and [latex]y=A\\mathrm{cos}\\left(B\\left(x-h\\right)\\right)+k\\text{}[\/latex][latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<p id=\"fs-id1165137663752\">The value [latex]k[\/latex] is the vertical shift of the function and for sinusoidal functions [latex]k[\/latex] shifts the midline up or down [latex]k[\/latex] units. The function [latex]y=A\\mathrm{sin}\\left(x\\right)+k[\/latex] has a midline of [latex]y=k[\/latex]. See <a class=\"autogenerated-content\" href=\"#Figure_06_01_012\">\u00a0Figure 11<\/a>.<\/p>\n<div id=\"Figure_06_01_012\" class=\"small\">\n<div id=\"attachment_3212\" style=\"width: 404px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3212\" class=\"wp-image-3212 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/29014437\/34Fig11.png\" alt=\"Figure 11: A graph of y=Asin(x)+k with the midline highlighted\" width=\"394\" height=\"233\" \/><\/p>\n<p id=\"caption-attachment-3212\" class=\"wp-caption-text\">Figure 11: A graph of [latex]y=A\\mathrm{sin}\\left(x\\right)+k.[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135242867\">In the equation [latex]y=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)+k,[\/latex] any value of [latex]k[\/latex] other than zero shifts the graph of\u00a0[latex]y=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)[\/latex] up or down. <a class=\"autogenerated-content\" href=\"#Figure_06_01_013\">Figure 12\u00a0<\/a>compares [latex]f\\left(x\\right)=\\mathrm{sin}\\left(x\\right)[\/latex] with [latex]f\\left(x\\right)=\\mathrm{sin}\\left(x\\right)+2,[\/latex] which is shifted 2 units up on a graph.<\/p>\n<div id=\"Figure_06_01_013\" class=\"small\">\n<div id=\"attachment_3213\" style=\"width: 398px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3213\" class=\"wp-image-3213 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/29014720\/34Fig12.png\" alt=\"\" width=\"388\" height=\"229\" \/><\/p>\n<p id=\"caption-attachment-3213\" class=\"wp-caption-text\">Figure 12: Vertically shifted sine function.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165134187254\">The value [latex]h[\/latex] for a sinusoidal function is called the the horizontal displacement of the basic sine or <span class=\"no-emphasis\">cosine function<\/span>. If [latex]h>0,[\/latex] the graph shifts to the right. If [latex]h<0,[\/latex] the graph shifts to the left. The greater the value of [latex]|h|,[\/latex] the more the graph is shifted. <a class=\"autogenerated-content\" href=\"#Figure_06_01_011\">Figure 13<\/a> shows that the graph of [latex]g\\left(x\\right)=\\mathrm{sin}\\left(x-\\pi \\right)[\/latex] shifts [latex]f\\left(x\\right)=\\mathrm{sin}\\left(x\\right)[\/latex] to the right by [latex]\\pi[\/latex] units, which is more than we see in the graph of [latex]p\\left(x\\right)=\\mathrm{sin}\\left(x-\\frac{\\pi }{4}\\right),[\/latex] which shifts [latex]f\\left(x\\right)=\\mathrm{sin}\\left(x\\right)[\/latex] to the right by [latex]\\frac{\\pi }{4}[\/latex] units.<\/p>\n<div id=\"Figure_06_01_011\" class=\"small\">\n<div id=\"attachment_3215\" style=\"width: 461px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3215\" class=\"wp-image-3215 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/29021055\/34Fig13.png\" alt=\"\" width=\"451\" height=\"273\" \/><\/p>\n<p id=\"caption-attachment-3215\" class=\"wp-caption-text\">Figure 13: Horizontally shifted sine functions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135571809\">\n<p id=\"fs-id1165133201875\"><strong>Important Note:\u00a0<\/strong> Some books and videos use the form [latex]\\text{}y=A\\mathrm{cos}\\left(Bx-C\\right)+D\\text{}[\/latex] and then factor out B leaving the form\u00a0 [latex]\\text{}y=A\\mathrm{cos}\\left(B\\left(x-\\frac{C}{B}\\right)\\right)+D\\text{}[\/latex] .\u00a0 In this case, the value of [latex]\\frac{C}{B}[\/latex] represents what\u00a0 is shown previously as h (the horizontal shift). We also see the D replaces what we have previously referred to as k. \u00a0It is simply a matter of getting familiar with the form that is being used in a particular problem. \u00a0For the most part, we will prefer the form that uses h and k. \u00a0The same pattern can be used with the sine function.<\/p>\n<div id=\"Example_06_01_04\" class=\"textbox examples\">\n<div id=\"fs-id1165137410966\">\n<div id=\"fs-id1165137410968\">\n<h3>Example 3:\u00a0 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\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137427502\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137427502\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137427504\">Let\u2019s begin by comparing the equation to the general form [latex]y=A\\mathrm{cos}\\left(B\\left(x-h\\right)\\right)+k,.[\/latex]<\/p>\n<p id=\"fs-id1165135503692\">In the given equation, [latex]k=-3[\/latex] so the shift is 3 units downward.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137742086\" class=\"precalculus tryit\">\n<div id=\"ti_06_01_04\">\n<h3>Try it #3<\/h3>\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\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137432579\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137432579\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137432580\">2 units up<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_06_01_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137696922\">\n<div id=\"fs-id1165137696924\">\n<h3>Example 4:\u00a0 Identifying the Horizontal Shift of a Function<\/h3>\n<p id=\"fs-id1165137804482\">Determine the direction and magnitude of the horizontal 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\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134483435\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134483435\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134483437\">Let\u2019s begin by comparing the equation to the general form [latex]y=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)+k.[\/latex]<\/p>\n<p id=\"fs-id1165137461008\">In the given equation, notice that [latex]B=1[\/latex] and [latex]h=-\\frac{\\pi }{6}.[\/latex] So the horizontal shift is [latex]\\frac{\\pi }{6}[\/latex] units to the left.<\/p>\n<p id=\"fs-id1165137664617\">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]h.[\/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]h[\/latex] is negative, the shift is to the left.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137806038\" class=\"precalculus tryit\">\n<div id=\"ti_06_01_03\">\n<h3>Try it #4<\/h3>\n<div id=\"fs-id1165137461117\">\n<p id=\"fs-id1165137461118\">Determine the direction and magnitude of the horizontal 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\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165131959464\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165131959464\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165131959465\">[latex]\\frac{\\pi }{2};[\/latex] right<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h3>Sine and Cosine are Co-Functions<\/h3>\n<p>You should recall that in Section 3.1, we described sine and cosine as co-functions.\u00a0 This was in relationship to the acute angles in a right triangle.\u00a0 We can also see this relationship when we consider the functions as functions of real numbers now that we understand how to horizontally shift these functions.\u00a0 See Figure 14 below.<\/p>\n<p>We know that [latex]\\mathrm{sin}\\left(x+\\frac{\\pi}{2}\\right)[\/latex] shifts the sine function [latex]\\frac{\\pi}{2}[\/latex] units to the left.\u00a0 Notice that this shift makes the graph of the sine curve (dotted curve) overlap the graph of the cosine curve (solid curve).<\/p>\n<div id=\"attachment_2553\" style=\"width: 751px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2553\" class=\"wp-image-2553\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/27190404\/34CoFunctionGraphic.png\" alt=\"\" width=\"741\" height=\"258\" \/><\/p>\n<p id=\"caption-attachment-2553\" class=\"wp-caption-text\"><strong>Figure 14:<\/strong> Shifted Sine function yields the Cosine function<\/p>\n<\/div>\n<p>Likewise, we know that\u00a0[latex]\\mathrm{cos}\\left(x-\\frac{\\pi}{2}\\right)[\/latex] shifts the cosine function [latex]\\frac{\\pi}{2}[\/latex] units to the right.\u00a0 Looking at the diagram above, you can see that this shift would make the cosine curve (solid\u00a0 curve) align with the sine curve. (dotted curve)<\/p>\n<h3>Putting It All Together<\/h3>\n<p>Given a sinusoidal function in the form [latex]f\\left(x\\right)=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)+k,[\/latex] we can\u00a0identify the midline, the amplitude, the period, and the horizontal shift. Recall, the amplitude is [latex]|A|,[\/latex] the midline is\u00a0[latex]y=k,[\/latex] and the horizontal shift is\u00a0[latex]h.[\/latex]\u00a0 The period is related to the horizontal stretch or compression and must be calculated using the formula\u00a0[latex]P=\\frac{2\\pi }{|B|}.[\/latex]<\/p>\n<div id=\"Example_06_01_05\" class=\"textbox examples\">\n<div id=\"fs-id1165137457010\">\n<div id=\"fs-id1165137457013\">\n<h3>Example 5:\u00a0 Identifying the Variations of a Sinusoidal Function from an Equation<\/h3>\n<p id=\"fs-id1165137416718\">Determine the midline, amplitude, period, and horizontal shift of the function [latex]y=3\\mathrm{sin}\\left(2x\\right)+1.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137454382\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137454382\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137454382\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137454384\">Let\u2019s begin by comparing the equation to the general form [latex]y=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)+k.[\/latex]<\/p>\n<p id=\"fs-id1165137408417\">[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 id=\"fs-id1165137637576\">There is no subtracted constant inside the parentheses, so [latex]h=0[\/latex] and the horizontal shift is [latex]0.[\/latex]<\/p>\n<p id=\"fs-id1165137697063\">Finally, [latex]k=1,[\/latex] so the midline is [latex]y=1.[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<div id=\"fs-id1165137701755\">\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 15<\/a>.<\/p>\n<div id=\"Figure_06_01_014\" class=\"small\">\n<div style=\"width: 411px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133001\/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=\"401\" height=\"217\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 15:\u00a0<\/strong> A graph of y=3sin(2x)+1.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137447405\" class=\"precalculus tryit\">\n<div id=\"ti_06_01_05\">\n<h3>Try it #5<\/h3>\n<div id=\"fs-id1165137447552\">\n<p id=\"fs-id1165137447553\">Determine the midline, amplitude, period, and horizontal 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 id=\"fs-id1165134042357\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134042357\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134042357\" class=\"hidden-answer\" style=\"display: none\">\n<div><span style=\"font-size: 1rem;text-align: initial\">First rewrite the equation in the form:\u00a0\u00a0 [latex]y=\\frac{1}{2}\\mathrm{cos}\\left(\\frac{1}{3}\\left(x-\\pi \\right)\\right).[\/latex] <\/span><\/div>\n<div>\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] horizontal shift: [latex]\\pi[\/latex] units right<\/p>\n<\/div>\n<\/div>\n<\/div>\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>Example 6:\u00a0 Identifying the Equation for a Sinusoidal Function from a Graph<\/h3>\n<p id=\"fs-id1165137871008\">Determine the formula for the cosine function in <a class=\"autogenerated-content\" href=\"#Figure_06_01_015\">Figure 16<\/a>.<\/p>\n<div id=\"Figure_06_01_015\" class=\"small\">\n<div style=\"width: 410px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133003\/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=\"400\" height=\"134\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 16:\u00a0<\/strong> A graph of -0.5cos(x)+0.5.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135329784\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135329784\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135329784\" class=\"hidden-answer\" style=\"display: none\">\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\" style=\"text-align: center\">[latex]\\begin{array}{l}y=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)+k\\hfill \\\\ y=A\\mathrm{cos}\\left(B\\left(x-h\\right)\\right)+k\\hfill \\end{array}[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137704661\">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>Notice that the extreme point for\u00a0[latex]x=0,[\/latex] is a minimum. \u00a0That would indicate that the cosine function has been reflected. \u00a0Let\u2019s consider the amplitude and then incorporate the reflection as part of our value for A.<\/p>\n<p>We can determine the amplitude by recognizing that the difference between the height of local maxima and minima is 1, so [latex]|A|=\\frac{1}{2}=0.5.[\/latex] \u00a0Remember, we determined the graph is reflected about the <em>x<\/em>-axis so that [latex]A=-0.5.[\/latex]<\/p>\n<p id=\"fs-id1165135536557\">Now let\u2019s consider 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]k[\/latex] in the equation, so [latex]k=0.5.[\/latex]<\/p>\n<p id=\"fs-id1165134204425\">The graph is not horizontally stretched or compressed since we can see that we have a minimum, maximum and minimum within [latex]2\\pi[\/latex] units, so [latex]B=1;[\/latex] and the graph is not shifted horizontally, so [latex]h=0.[\/latex]<\/p>\n<p id=\"fs-id1165135347312\">Putting this all together,<\/p>\n<div id=\"fs-id1165137401884\" class=\"unnumbered\" style=\"text-align: center\">[latex]g\\left(x\\right)=-0.5\\mathrm{cos}\\left(x\\right)+0.5.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137702221\" class=\"precalculus tryit\">\n<div id=\"ti_06_01_06\">\n<h3>Try it #6<\/h3>\n<div id=\"fs-id1165135582221\">\n<p id=\"fs-id1165135582222\">Determine the formula for the sine function in <a class=\"autogenerated-content\" href=\"#Figure_06_01_016\">Figure 17<\/a>.<\/p>\n<div id=\"Figure_06_01_016\" class=\"small\">\n<div style=\"width: 413px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133006\/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=\"403\" height=\"143\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 17:\u00a0<\/strong>Determine the function for this graph.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137526465\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137526465\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137526465\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137526466\">[latex]f\\left(x\\right)=\\mathrm{sin}\\left(x\\right)+2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_06_01_07\" class=\"textbox examples\">\n<div id=\"fs-id1165134164950\">\n<div id=\"fs-id1165134058400\">\n<h3>Example 7:\u00a0 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 18<\/a>.<\/p>\n<div id=\"Figure_06_01_017\" class=\"medium\">\n<div style=\"width: 380px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133009\/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=\"370\" height=\"286\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 18:<\/strong>\u00a0Determine the function for this graph.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137598813\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137598813\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137598813\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137598815\">With the highest value at 1 and the lowest value at [latex]-5,[\/latex] the amplitude will be [latex]|A|=3,[\/latex] and the midline will be halfway between at [latex]-2.[\/latex] So [latex]k=-2.[\/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\" style=\"text-align: center\">[latex]B=\\frac{2\\pi }{P}=\\frac{2\\pi }{6}=\\frac{\\pi }{3}.[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137611526\">So far, our equation is either [latex]y=3\\mathrm{sin}\\left(\\frac{\\pi }{3}\\left(x-h\\right)\\right)-2[\/latex] or [latex]y=3\\mathrm{cos}\\left(\\frac{\\pi }{3}\\left(x-h\\right)\\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\" style=\"text-align: center\">[latex]y=3\\mathrm{cos}\\left(\\frac{\\pi }{3}\\left(x-1\\right)\\right)-2\\text{ or }y=-3\\mathrm{cos}\\left(\\frac{\\pi }{3}\\left(x+2\\right)\\right)-2.[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165135704043\">Again, these functions are equivalent, so both yield the same graph.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137805588\" class=\"precalculus tryit\">\n<div id=\"ti_06_01_07\">\n<h3>Try it #7<\/h3>\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 19<\/a>.<\/p>\n<div id=\"Figure_06_01_018\" class=\"medium\">\n<div style=\"width: 380px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133012\/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=\"370\" height=\"223\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 19:<\/strong>\u00a0 Determine the function for this graph.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135173772\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135173772\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135173772\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135173773\">two possibilities: [latex]y=4\\mathrm{sin}\\left(\\frac{\\pi }{5}\\left(x-1\\right)\\right)+4[\/latex] or [latex]y=-4\\mathrm{sin}\\left(\\frac{\\pi }{5}\\left(x+4\\right)\\right)+4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137735424\" class=\"bc-section section\">\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\">\n<p id=\"fs-id1165137454384\" style=\"text-align: center\">[latex]y=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)+k[\/latex] <span style=\"font-size: 1em\">\u00a0and [latex]y=A\\mathrm{cos}\\left(B\\left(x-h\\right)\\right)+k,[\/latex]<\/span><\/p>\n<\/div>\n<p id=\"fs-id1165137807234\">we will let [latex]h=0[\/latex] and [latex]k=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<div class=\"textbox examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135329942\"><strong>Given the function [latex]y=A\\mathrm{sin}\\left(Bx\\right),[\/latex] sketch its graph.\u00a0 Note that [latex]h=k=0.[\/latex]<\/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>\n<div id=\"Example_06_01_08\" class=\"textbox examples\">\n<div id=\"fs-id1165134156046\">\n<div id=\"fs-id1165137565145\">\n<h3>Example 8:\u00a0 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\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134190732\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134190732\" class=\"hidden-answer\" style=\"display: none\">\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>\n<li style=\"list-style-type: none\">\n<ul>\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\" style=\"text-align: center\">[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\" style=\"text-align: center\">[latex]\\begin{align*}P&=\\frac{2\\pi }{\\frac{\\pi }{2}} \\\\ &=2\\pi\\cdot \\frac{2}{\\pi } \\\\ &=4. \\end{align*}[\/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<\/li>\n<\/ul>\n<p>The quarter points include the minimum at [latex]x=1[\/latex] and the maximum at [latex]x=3.[\/latex]\u00a0 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]\u00a0 Figure 20 shows the graph of the function.<\/p>\n<div style=\"width: 379px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133015\/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=\"369\" height=\"191\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 20:<\/strong> A graph of [latex]y=2\\mathrm{sin}\\left(\\frac{\\pi x}{2}\\right).[\/latex]<\/p>\n<\/div>\n<p>The graph has range of [-2,2], period of 4, and amplitude of 2.\n<\/p><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137539724\" class=\"precalculus tryit\">\n<div id=\"ti_06_01_08\">\n<h3>Try it #8<\/h3>\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 horizontal shift.<\/p>\n<\/div>\n<div id=\"fs-id1165135342790\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135342790\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135342790\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"width: 382px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133018\/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).\" width=\"372\" height=\"245\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 21:<\/strong>\u00a0 A graph of [latex]y=-0.8\\mathrm{cos}\\left(2x\\right).[\/latex]<\/p>\n<\/div>\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] horizontal shift: [latex]h=0[\/latex] or none<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137425929\" class=\"precalculus howto\">\n<div class=\"textbox examples\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137661914\"><strong>Given a sinusoidal function with a horizontal shift and a vertical shift, sketch its graph.<\/strong><\/p>\n<ol id=\"fs-id1165135503706\" type=\"1\">\n<li>\u00a0Express the function in the general form [latex]\\begin{align*}y&=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)+k\\text{ or} \\\\ y&=A\\mathrm{cos}\\left(B\\left(x-h\\right)\\right)+k\\hfill \\end{align*}[\/latex]<\/li>\n<li>\u00a0Identify the amplitude, [latex]|A|.[\/latex]<\/li>\n<li>\u00a0Identify the period, [latex]P=\\frac{2\\pi }{|B|}.[\/latex]<\/li>\n<li>\u00a0Identify the horizontal shift, [latex]h,[\/latex] and the vertical shift, [latex]k.[\/latex]<\/li>\n<li>\u00a0Draw the graph of [latex]f\\left(x\\right)=A\\mathrm{sin}\\left(Bx\\right)[\/latex] or [latex]f\\left(x\\right)=A\\mathrm{cos}\\left(Bx\\right)[\/latex] shifted to the right or left by [latex]h[\/latex] and up or down by [latex]k.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div id=\"Example_06_01_09\" class=\"textbox examples\">\n<div>\n<h3>Example 9:\u00a0 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\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135209894\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135209894\" class=\"hidden-answer\" style=\"display: none\">\n<ul id=\"eip-id1165137474346\">\n<li><em>Step 1.<\/em> <span style=\"font-size: 1rem;text-align: initial\">First rewrite the equation in the form:\u00a0\u00a0 [latex]y=3\\mathrm{sin}\\left(\\frac{\\pi}{4}\\left(x-1 \\right)\\right).[\/latex] <\/span>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\" style=\"text-align: center\">[latex]P=\\frac{2\\pi }{|B|}=\\frac{2\\pi }{\\frac{\\pi }{4}}=2\\pi \\cdot \\frac{4}{\\pi }=8[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137757960\">The period is 8.<\/p>\n<\/li>\n<li><em>Step 4.<\/em> Since [latex]h=1,[\/latex] the horizontal shift is [latex]1[\/latex] unit right.\u00a0 Since [latex]k=0[\/latex] there is no vertical shift.\n<p id=\"fs-id1165137634941\">\n<\/li>\n<li><em>Step 5.<\/em><a class=\"autogenerated-content\" href=\"#Figure_06_01_021\">Figure 22<\/a> shows the graph of the function.\n<div id=\"Figure_06_01_021\" class=\"small\">\n<div class=\"wp-caption-text\"><\/div>\n<div style=\"width: 370px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133021\/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=\"360\" height=\"236\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 22:<\/strong> A horizontally stretched, vertically stretched, and horizontally shifted sinusoid graph of [latex]y=3\\mathrm{sin}\\left(\\frac{\\pi}{4}x-\\frac{\\pi}{4}\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135181399\" class=\"precalculus tryit\">\n<div id=\"ti_06_01_09\">\n<div id=\"fs-id1165137638347\">\n<h3>Try it #9<\/h3>\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 horizontal shift.<\/p>\n<\/div>\n<div id=\"fs-id1165137480594\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137480594\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137480594\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"width: 751px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133024\/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.\" width=\"741\" height=\"176\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 23:\u00a0<\/strong> A graph of [latex]y=-2\\mathrm{cos}\\left(\\left(\\frac{\\pi}{3}\\right)x+\\left(\\frac{\\pi}{6}\\right)\\right).[\/latex]<\/p>\n<\/div>\n<p>First rewrite the equation in the form\u00a0 [latex]g\\left(x\\right)=-2\\mathrm{cos}\\left(\\frac{\\pi }{3}\\left(x+\\frac{1 }{2}\\right)\\right).[\/latex]<\/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] horizontal shift: [latex]h=-\\frac{1}{2}[\/latex] or [latex]\\frac{1}{2}[\/latex] units left<\/p>\n<\/div>\n<\/div>\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>Example 10:\u00a0 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, horizontal shift and vertical shift. Then graph the function.<\/p>\n<\/div>\n<div id=\"fs-id1165135487183\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135487183\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135487183\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165134225658\" class=\"unnumbered\"><\/div>\n<ul id=\"eip-id1165134311998\">\n<li><em>Step 1.<\/em>\u00a0\u00a0<span style=\"font-size: 1rem;text-align: initial\">First rewrite the equation in the form:\u00a0\u00a0[latex]y=-2\\mathrm{cos}\\left(\\frac{\\pi}{2}\\left(x + 2 \\right)\\right)+3.[\/latex]<\/span><\/li>\n<li><em>Step 2.<\/em>\u00a0 Since [latex]A=-2,[\/latex] the amplitude is [latex]|A|=2.[\/latex]<\/li>\n<li><em>Step 3.\u00a0\u00a0<\/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.\u00a0\u00a0<\/em>[latex]h=- 2 ,[\/latex] so the horizontal shift\u00a0 is 2 units to the left.<em>\u00a0<\/em>[latex]k=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 before the vertical shift is done.<\/p>\n<p id=\"fs-id1165137761033\"><a class=\"autogenerated-content\" href=\"#Figure_06_01_028\">Figure 24<\/a> shows one cycle of the graph of the function.<\/p>\n<div id=\"Figure_06_01_028\" class=\"small\">\n<div style=\"width: 410px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133027\/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=\"400\" height=\"260\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 24:<\/strong> A graph of\u00a0[latex]\\mathrm{cos}\\left(\\frac{\\pi }{2}x+\\pi \\right)+3.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137939840\" class=\"bc-section section\">\n<div class=\"bc-section section\">\n<h3>Using Five Key Points to Graph Sinusoidal Functions<\/h3>\n<p id=\"fs-id1415219\">One method of graphing sinusoidal functions is to find five key points. These points will correspond to intervals of equal length representing [latex]\\frac{1}{4}[\/latex] of the period. The key points will indicate the location of maximum and minimum values. If there is no vertical shift, they will also indicate <em>x<\/em>-intercepts. For example, suppose we want to graph the function [latex]y=\\mathrm{cos}\\text{ }\\theta .[\/latex] We know that the period is [latex]2\\pi ,[\/latex] so we find the interval between key points as follows.<\/p>\n<div id=\"fs-id2102442\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\frac{2\\pi }{4}=\\frac{\\pi }{2}[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id2290195\">Starting with [latex]\\theta =0,[\/latex] we calculate the first <em>y-<\/em>value, add the length of the interval [latex]\\frac{\\pi }{2}[\/latex] to 0, and calculate the second <em>y<\/em>-value. We then add [latex]\\frac{\\pi }{2}[\/latex] repeatedly until the five key points are determined. The last value should equal the first value, as the calculations cover one full period. Making a table similar to Table 3, we can see these key points clearly on the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_07_07_002\">\u00a0Figure 25<\/a>.<\/p>\n<table id=\"Table_07_07_01\" summary=\"Two rows, six columns. The table has ordered pairs of these column values: (theta, y=cos(theta)), (0,1), (i\/2, 0), (pi,-1), (3pi\/2, 0), (2pi,1).\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<caption><strong>Table 3<\/strong><\/caption>\n<tbody>\n<tr>\n<td class=\"border\"><strong>[latex]\\theta[\/latex]<\/strong><\/td>\n<td class=\"border\">[latex]0[\/latex]<\/td>\n<td class=\"border\">[latex]\\frac{\\pi }{2}[\/latex]<\/td>\n<td class=\"border\">[latex]\\pi[\/latex]<\/td>\n<td class=\"border\">[latex]\\frac{3\\pi }{2}[\/latex]<\/td>\n<td class=\"border\">[latex]2\\pi[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><strong>[latex]y=\\mathrm{cos}\\text{ }\\theta[\/latex]<\/strong><\/td>\n<td class=\"border\">[latex]1[\/latex]<\/td>\n<td class=\"border\">[latex]0[\/latex]<\/td>\n<td class=\"border\">[latex]-1[\/latex]<\/td>\n<td class=\"border\">[latex]0[\/latex]<\/td>\n<td class=\"border\">[latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"Figure_07_07_002\" class=\"small\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144634\/CNX_Precalc_Figure_07_07_002.jpg\" alt=\"Graph of y=cos(x) from -pi\/2 to 5pi\/2.\" width=\"487\" height=\"217\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 25<\/b><\/p>\n<\/div>\n<\/div>\n<div id=\"Example_07_06_03\" class=\"textbox examples\">\n<div id=\"fs-id2478628\">\n<div id=\"fs-id1173784\">\n<h3>Example 11: \u00a0Graphing Sinusoidal Functions Using Key Points<\/h3>\n<p id=\"fs-id1506097\">Graph the function [latex]y=-4\\mathrm{cos}\\left(\\pi x\\right)[\/latex] using amplitude, period, and key points.<\/p>\n<\/div>\n<div id=\"fs-id1372153\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1372153\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1372153\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1846805\">The amplitude is[latex]|-4|=4.[\/latex] The period is [latex]\\frac{2\\pi }{B}=\\frac{2\\pi }{\\pi }=2.[\/latex] One cycle of the graph can be drawn over the interval [latex]\\left[0,2\\right].[\/latex] To find the key points, we divide the period by 4 which means we will increment [latex]x[\/latex] by [latex]\\frac{2}{4}=\\frac{1}{2}[\/latex]. Make a table similar to Table 4, starting with [latex]x=0[\/latex] and then adding [latex]\\frac{1}{2}[\/latex] successively to [latex]x[\/latex] and calculate [latex]y.[\/latex] See the graph in <a class=\"autogenerated-content\" href=\"#Figure_07_07_003\">Figure 26<\/a>.<\/p>\n<table id=\"Table_07_07_02\" summary=\"Two rows, six columns. The table has ordered pairs of these column values: (theta, y=-4cos(pi*x)), (0,-4), (1\/2, 0), (1,4), (3\/2, 0), (2, -4).\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<caption><strong>Table 4<\/strong><\/caption>\n<tbody>\n<tr>\n<td class=\"border\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td class=\"border\">[latex]0[\/latex]<\/td>\n<td class=\"border\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td class=\"border\">[latex]1[\/latex]<\/td>\n<td class=\"border\">[latex]\\frac{3}{2}[\/latex]<\/td>\n<td class=\"border\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><strong>[latex]y=-4\\text{ }\\mathrm{cos}\\left(\\pi x\\right)[\/latex]<\/strong><\/td>\n<td class=\"border\">[latex]-4[\/latex]<\/td>\n<td class=\"border\">[latex]0[\/latex]<\/td>\n<td class=\"border\">[latex]4[\/latex]<\/td>\n<td class=\"border\">[latex]0[\/latex]<\/td>\n<td class=\"border\">[latex]-4[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"Figure_07_07_003\" class=\"small\">\n<div style=\"width: 340px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144637\/CNX_Precalc_Figure_07_07_003.jpg\" alt=\"Graph of y=-4cos(pi*x) using the five key points: intervals of equal length representing 1\/4 of the period. Here, the points are at 0, 1\/2, 1, 3\/2, and 2.\" width=\"330\" height=\"254\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 26<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2571729\" class=\"precalculus tryit\">\n<h3>TRY IT #10<\/h3>\n<div id=\"ti_07_06_02\">\n<div id=\"fs-id2102282\">\n<p id=\"fs-id1333064\">Graph the function [latex]y=3\\mathrm{sin}\\left(3x\\right)[\/latex] using the amplitude, period, and five key points.<\/p>\n<\/div>\n<div id=\"fs-id2303325\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id2303325\">Show Solution<\/span><\/p>\n<div id=\"qfs-id2303325\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"fs-id2353042\" class=\"unnumbered\" summary=\"Graph of y=3sin(3x) using the five key points: intervals of equal length representing 1\/4 of the period. Here, the points are at 0, pi\/6, pi\/3, pi\/2, and 2pi\/3.\">\n<thead>\n<tr>\n<th class=\"border\" style=\"text-align: center\">x<\/th>\n<th class=\"border\" style=\"text-align: center\">[latex]y=3\\mathrm{sin}\\left(3x\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<caption><strong>Table 5<\/strong><\/caption>\n<tbody>\n<tr>\n<td class=\"border\" style=\"text-align: center\">0<\/td>\n<td class=\"border\" style=\"text-align: center\">0<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{6}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">3<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{3}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">0<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{\\pi }{2}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">[latex]-3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"text-align: center\">[latex]\\frac{2\\pi }{3}[\/latex]<\/td>\n<td class=\"border\" style=\"text-align: center\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span id=\"fs-id2710496\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144640\/CNX_Precalc_Figure_07_07_004.jpg\" alt=\"Graph of y=3sin(3x) using the five key points: intervals of equal length representing 1\/4 of the period. Here, the points are at 0, pi\/6, pi\/3, pi\/2, and 2pi\/3.\" width=\"398\" height=\"299\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\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>Example 12:\u00a0 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\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137552985\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137552985\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137552987\">Recall that we discussed in Section 3.3 that a point on a circle of radius r will have a y coordinate\u00a0 of\u00a0\u00a0[latex]y=r\\mathrm{sin}\\left(\\theta\\right).[\/latex] In this case, we get the equation [latex]y\\left(\\theta\\right)=3\\mathrm{sin}\\left(\\theta\\right).[\/latex]<\/p>\n<div id=\"Figure_06_01_023\" class=\"small\">\n<div style=\"width: 380px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133031\/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=\"370\" height=\"242\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 27:<\/strong> A graph of 3sin([latex]\\theta[\/latex]).<\/p>\n<\/div>\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]<\/p>\n<p>From our study of transformations of trigonometric functions, we also know that the constant 3 causes a vertical stretch of\u00a0the sine function by a factor of 3, which we can see in the graph in <a class=\"autogenerated-content\" href=\"#Figure_06_01_023\">Figure 27<\/a>.<\/p>\n<p>Because the outputs of the graph will now oscillate between [latex]-3[\/latex] and [latex]3,[\/latex] the amplitude of the sine wave is [latex]3.[\/latex]\u00a0 This means that the radius of the circle centered at the origin will correspond to the amplitude of the sine function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135319496\" class=\"precalculus tryit\">\n<div id=\"ti_06_01_10\">\n<h3>Try it #11<\/h3>\n<div id=\"fs-id1165135403587\">\n<p id=\"fs-id1165135403588\">What is the radius of the circle whose y-coordinate corresponds to 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\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137534006\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137534006\" class=\"hidden-answer\" style=\"display: none\">\nThe radius and amplitude are 7.<\/p>\n<div style=\"width: 276px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133034\/CNX_Precalc_Figure_06_01_024.jpg\" alt=\"A graph of 7cos(x). Graph has amplitude of 7, period of 2pi, and range of &#091;-7,7&#093;.\" width=\"266\" height=\"294\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 26<\/strong>: A graph of [latex]y=7\\mathrm{cos}\\left(x\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\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>Example 13:\u00a0 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 27<\/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\">\n<div style=\"width: 380px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133038\/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=\"370\" height=\"228\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 27:<\/strong> An illustration of a circle lifted 4 feet off the ground.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137863854\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137863854\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137863854\" class=\"hidden-answer\" style=\"display: none\">\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 28<\/a>.<\/p>\n<div id=\"Figure_06_01_026\" class=\"small\">\n<div style=\"width: 309px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133041\/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=\"299\" height=\"320\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 28:<\/strong> A graph of [latex]y=-3\\mathrm{cos}\\left(x\\right)+4.[\/latex]<\/p>\n<\/div>\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 feet 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\" style=\"text-align: center\">[latex]y=-3\\mathrm{cos}\\left(x\\right)+4.[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It #12<\/h3>\n<p>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 29<\/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<p>&nbsp;<\/p>\n<div id=\"Figure_06_01_029\" class=\"small\">\n<div style=\"width: 277px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133044\/CNX_Precalc_Figure_06_01_029.jpg\" alt=\"An illustration of a spring with length y.\" width=\"267\" height=\"193\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 29<\/strong>: An illustration of a spring with length y.<\/p>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q292974\">Show Solution<\/span><\/p>\n<div id=\"q292974\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]y=3\\mathrm{cos}\\left(x\\right)-4[\/latex]<\/p>\n<div id=\"attachment_640\" style=\"width: 291px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-640\" class=\"wp-image-640\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07133047\/CNX_Precalc_Figure_06_01_027-244x300.jpg\" alt=\"\" width=\"281\" height=\"345\" \/><\/p>\n<p id=\"caption-attachment-640\" class=\"wp-caption-text\">Figure 30<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 14:\u00a0Determining a Rider\u2019s Height on a Ferris Wheel<\/h3>\n<p>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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q91947\">Show Solution<\/span><\/p>\n<div id=\"q91947\" class=\"hidden-answer\" style=\"display: none\">\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]k=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\" style=\"text-align: center\">[latex]y=-67.5\\mathrm{cos}\\left(\\frac{\\pi }{15}t\\right)+69.5[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1165137634889\">where [latex]t[\/latex] is in minutes and [latex]y[\/latex] is measured in meters.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><span style=\"font-size: 1rem;text-align: initial\">Access these online resources for additional instruction and practice with graphs of sine and cosine functions.<\/span><\/p>\n<div id=\"fs-id1165137540365\" class=\"precalculus media\">\n<ul id=\"fs-id1165137761692\">\n<li><a href=\"http:\/\/openstax.org\/l\/ampperiod\">Amplitude and Period of Sine and Cosine<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/translasincos\">Translations of Sine and Cosine<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/transformsincos\">Graphing Sine and Cosine Transformations<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/graphsinefunc\">Graphing the Sine Function<\/a><\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137574576\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165133087385\" summary=\"..\">\n<caption>\u00a0<\/caption>\n<tbody>\n<tr>\n<td class=\"border\">Sinusoidal functions<\/td>\n<td class=\"border\">[latex]\\begin{array}{l}y=A\\mathrm{sin}\\left(B\\left(x-h\\right)\\right)+k\\text{ or }\\hfill \\\\ y=A\\mathrm{cos}\\left(B\\left(x-h\\right)\\right)+k\\hfill \\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 id=\"fs-id1165137762207\">\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}\\text{ }x[\/latex] is odd, so its graph is symmetric about the origin. The function [latex]\\mathrm{cos}\\text{ }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]<\/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.<\/li>\n<li>The value [latex]h[\/latex] in the general formula for a sinusoidal function indicates the horizontal shift.<\/li>\n<li>The value [latex]k[\/latex] in the general formula for a sinusoidal function indicates the vertical shift.<\/li>\n<li>Combinations of variations of sinusoidal functions can be detected from an equation.<\/li>\n<li>The equation for a sinusoidal function can be determined from a graph.<\/li>\n<li>A function can also be graphed by identifying its amplitude, period, vertical shift, and horizontal shift.<\/li>\n<li>Sinusoidal functions can be used to solve real-world problems.<\/li>\n<\/ul>\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=k,[\/latex] where [latex]k[\/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-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(B\\left(x-h\\right)\\right)+k[\/latex] or [latex]f\\left(x\\right)=A\\mathrm{cos}\\left(B\\left(x-h\\right)\\right)+k[\/latex]<\/dd>\n<\/dl>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-659\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Graphs of the Sine and Cosine Functions. <strong>Authored by<\/strong>: Douglas Hoffman. <strong>Provided by<\/strong>: Openstax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/cnx.org\/contents\/8si1Yf2B@2.21:7IfuGdYn@6\/Graphs-of-the-Sine-and-Cosine-Functions\">https:\/\/cnx.org\/contents\/8si1Yf2B@2.21:7IfuGdYn@6\/Graphs-of-the-Sine-and-Cosine-Functions<\/a>. <strong>Project<\/strong>: Essential Precalcus, Part 2. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":311,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Graphs of the Sine and Cosine Functions\",\"author\":\"Douglas Hoffman\",\"organization\":\"Openstax\",\"url\":\"https:\/\/cnx.org\/contents\/8si1Yf2B@2.21:7IfuGdYn@6\/Graphs-of-the-Sine-and-Cosine-Functions\",\"project\":\"Essential Precalcus, Part 2\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-659","chapter","type-chapter","status-publish","hentry"],"part":478,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/659","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":70,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/659\/revisions"}],"predecessor-version":[{"id":3219,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/659\/revisions\/3219"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/parts\/478"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/659\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/media?parent=659"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=659"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/contributor?post=659"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/license?post=659"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}