{"id":14093,"date":"2018-09-27T00:40:23","date_gmt":"2018-09-27T00:40:23","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalculus\/chapter\/graphs-of-the-sine-and-cosine-function\/"},"modified":"2021-02-04T22:37:27","modified_gmt":"2021-02-04T22:37:27","slug":"graphs-of-the-sine-and-cosine-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/pdx-precalculus\/chapter\/graphs-of-the-sine-and-cosine-function\/","title":{"raw":"Walkthrough of Unit 4: Graphs of Trig Functions","rendered":"Walkthrough of Unit 4: Graphs of Trig Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Determine amplitude, period, phase shift, and vertical shift of a sine or cosine graph from its equation.<\/li>\r\n \t<li style=\"font-weight: 400\">Graph variations of\u2009y=cos x\u2009and y=sin x\u2009.<\/li>\r\n \t<li>Determine a function formula that would have a given sinusoidal graph.<\/li>\r\n \t<li style=\"font-weight: 400\">Determine functions that model circular and periodic motion.<\/li>\r\n \t<li style=\"font-weight: 400\">Analyze the graph of \u2009y=tan\u2009x and y=cot x.<\/li>\r\n \t<li style=\"font-weight: 400\">Graph variations of \u2009y=tan\u2009x and y=cot x.<\/li>\r\n \t<li>Determine a function formula from a tangent or cotangent graph.<\/li>\r\n \t<li style=\"font-weight: 400\">Analyze the graphs of \u2009y=sec\u2009x\u2009 and \u2009y=csc\u2009x.<\/li>\r\n \t<li style=\"font-weight: 400\">Graph variations of \u2009y=sec\u2009x\u2009 and \u2009y=csc\u2009x.<\/li>\r\n \t<li>Determine a function formula from a secant or cosecant graph.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Graph variations of \u2009y=sin( x )\u2009 and \u2009y=cos( x )<\/h2>\r\nRecall that the sine and cosine functions relate real number values to the <em>x<\/em>- and <em>y<\/em>-coordinates of a point on the unit circle. So what do they look like on a graph on a coordinate plane? Let\u2019s start with the <strong>sine function.<\/strong> We can create a table of values and use them to sketch a graph. The table below\u00a0lists some of the values for the sine function on a unit circle.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>x<\/strong><\/td>\r\n<td>0<\/td>\r\n<td>[latex]\\frac{\\pi}{6}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\pi}{4}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\pi}{3}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\pi}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{2\\pi}{3}[\/latex]<\/td>\r\n<td>[latex]\\frac{3\\pi}{4}[\/latex]<\/td>\r\n<td>[latex]\\frac{5\\pi}{6}[\/latex]<\/td>\r\n<td>[latex]\\pi[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>[latex]\\sin(x)[\/latex]<\/strong><\/td>\r\n<td>0<\/td>\r\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\r\n<td>1<\/td>\r\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nPlotting the points from the table and continuing along the <em>x<\/em>-axis gives the shape of the sine function. See Figure 2.\r\n<figure id=\"Figure_06_01_002\" class=\"small ui-has-child-figcaption\">\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\/3675\/2018\/09\/27003914\/CNX_Precalc_Figure_06_01_002.jpg\" alt=\"A graph of sin(x). Local maximum at (pi\/2, 1). Local minimum at (3pi\/2, -1). Period of 2pi.\" width=\"487\" height=\"216\" \/> <b>Figure 2.<\/b> The sine function[\/caption]<\/figure>\r\nNotice how the sine values are positive between 0 and \u03c0, 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 \u03c0 and 2\u03c0, which correspond to the values of the sine function in quadrants III and IV on the unit circle. See Figure 3.\r\n<figure id=\"Figure_06_01_003\" class=\"small ui-has-child-figcaption\">\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\/3675\/2018\/09\/27003916\/CNX_Precalc_Figure_06_01_003.jpg\" alt=\"A side-by-side graph of a unit circle and a graph of sin(x). The two graphs show the equivalence of the coordinates.\" width=\"487\" height=\"219\" \/> <b>Figure 3.<\/b> Plotting values of the sine function[\/caption]<\/figure>\r\nNow let\u2019s take a similar look at the <strong>cosine function<\/strong>. Again, we can create a table of values and use them to sketch a graph. The table below\u00a0lists some of the values for the cosine function on a unit circle.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><strong>x<\/strong><\/td>\r\n<td>0<\/td>\r\n<td>[latex]\\frac{\\pi}{6}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\pi}{4}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\pi}{3}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\pi}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{2\\pi}{3}[\/latex]<\/td>\r\n<td>[latex]\\frac{3\\pi}{4}[\/latex]<\/td>\r\n<td>[latex]\\frac{5\\pi}{6}[\/latex]<\/td>\r\n<td>[latex]\\pi[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\cos(x)[\/latex]<\/td>\r\n<td>1<\/td>\r\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\r\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td>0<\/td>\r\n<td>[latex]-\\frac{1}{2}[\/latex]<\/td>\r\n<td>[latex]-\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\r\n<td>[latex]-\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\r\n<td>\u22121<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAs with the sine function, we can plots points to create a graph of the cosine function as in\u00a0Figure 4.\r\n<figure id=\"Figure_06_01_004\" class=\"medium ui-has-child-figcaption\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003918\/CNX_Precalc_Figure_06_01_004.jpg\" alt=\"A graph of cos(x). Local maxima at (0,1) and (2pi, 1). Local minimum at (pi, -1). Period of 2pi.\" width=\"731\" height=\"216\" \/> <b>Figure 4.<\/b> The cosine function[\/caption]<\/figure>\r\nBecause we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval [\u22121,1].\r\n\r\nIn both graphs, the shape of the graph repeats after 2\u03c0,which means the functions are periodic with a period of [latex]2\u03c0[\/latex]. A <strong>periodic function<\/strong> is a function for which a specific <strong>horizontal shift<\/strong>, <em>P<\/em>, results in a function equal to the original function: [latex]f (x + P) = f(x)[\/latex] for all values of <em>x<\/em> in the domain of <em>f<\/em>. When this occurs, we call the smallest such horizontal shift with [latex]P &gt; 0[\/latex] the <strong>period<\/strong> of the function. Figure 5\u00a0shows several periods of the sine and cosine functions.\r\n<figure id=\"Figure_06_01_005\" 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\/3675\/2018\/09\/27003920\/CNX_Precalc_Figure_06_01_005.jpg\" alt=\"Side-by-side graphs of sin(x) and cos(x). Graphs show period lengths for both functions, which is 2pi.\" width=\"487\" height=\"442\" \/> <b>Figure 5<\/b>[\/caption]<\/figure>\r\nLooking again at the sine and cosine functions on a domain centered at the <em>y<\/em>-axis helps reveal symmetries. As we can see in Figure 6, the <strong>sine function<\/strong> is symmetric about the origin. Recall from <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/precalctwoxmaster\/chapter\/introduction-to-the-other-trigonometric-functions\/\" target=\"_blank\" rel=\"noopener\">The Other Trigonometric Functions<\/a> that we determined from the unit circle that the sine function is an odd function because [latex]\\sin(\u2212x)=\u2212\\sin x[\/latex]. Now we can clearly see this property from the graph.\r\n<figure id=\"Figure_06_01_006\" class=\"small ui-has-child-figcaption\">\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\/3675\/2018\/09\/27003922\/CNX_Precalc_Figure_06_01_006.jpg\" alt=\"A graph of sin(x) that shows that sin(x) is an odd function due to the odd symmetry of the graph.\" width=\"487\" height=\"191\" \/> <b>Figure 6.<\/b> Odd symmetry of the sine function[\/caption]<\/figure>\r\nFigure 7\u00a0shows that the cosine function is symmetric about the <em>y<\/em>-axis. Again, we determined that the cosine function is an even function. Now we can see from the graph that [latex]\\cos(\u2212x)=\\cos x[\/latex].\r\n<figure id=\"Figure_06_01_007\" class=\"small ui-has-child-figcaption\">\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\/3675\/2018\/09\/27003925\/CNX_Precalc_Figure_06_01_007.jpg\" alt=\"A graph of cos(x) that shows that cos(x) is an even function due to the even symmetry of the graph.\" width=\"487\" height=\"216\" \/> <b>Figure 7.<\/b> Even symmetry of the cosine function[\/caption]<\/figure>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Characteristics of Sine and Cosine Functions<\/h3>\r\nThe sine and cosine functions have several distinct characteristics:\r\n<ul>\r\n \t<li>They are periodic functions with a period of 2\u03c0.<\/li>\r\n \t<li>The domain of each function is\u00a0[latex]\\left(-\\infty,\\infty\\right)[\/latex] and the range is [latex]\\left[\u22121,1\\right][\/latex].<\/li>\r\n \t<li>The graph of [latex]y=\\sin x[\/latex] is symmetric about the origin, because it is an odd function.<\/li>\r\n \t<li>The graph of [latex]y=\\cos x[\/latex] is symmetric about the <em>y<\/em>-axis, because it is an even function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Investigating Sinusoidal Functions<\/h2>\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 <strong>cosine function<\/strong> is known as a <strong>sinusoidal function<\/strong>. The general forms of sinusoidal functions are\r\n<div style=\"text-align: center\">[latex]y = A\\sin (Bx\u2212C) + D[\/latex]<\/div>\r\nand\r\n<div style=\"text-align: center\">[latex]y = A\\cos (Bx\u2212C) + D[\/latex]<\/div>\r\n<h3>Determining the Period of Sinusoidal Functions<\/h3>\r\nLooking 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.\r\n\r\nIn the general formula, <em>B<\/em> is related to the period by [latex]P=\\frac{2\u03c0}{|B|}[\/latex]. If [latex]|B| &gt; 1[\/latex], then the period is less than [latex]2\u03c0[\/latex] and the function undergoes a horizontal compression, whereas if [latex]|B| &lt; 1[\/latex], then the period is greater than [latex]2\u03c0[\/latex] and the function undergoes a horizontal stretch. For example, [latex]f(x) = \\sin(x), B= 1[\/latex], so the period is [latex]2\u03c0[\/latex], which we knew. If [latex]f(x) =\\sin (2x)[\/latex], then [latex]B= 2[\/latex], so the period is [latex]\u03c0[\/latex] and the graph is compressed. If [latex]f(x) = \\sin\\left(\\frac{x}{2} \\right)[\/latex], then [latex]B=\\frac{1}{2}[\/latex], so the period is [latex]4\u03c0[\/latex] and the graph is stretched. Notice in Figure 8\u00a0how the period is indirectly related to [latex]|B|[\/latex].\r\n<figure id=\"Figure_06_01_008\" 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\/3675\/2018\/09\/27003927\/CNX_Precalc_Figure_06_01_008.jpg\" alt=\"A graph with three items. The x-axis ranges from 0 to 2pi. The y-axis ranges from -1 to 1. The first item is the graph of sin(x) for one full period. The second is the graph of sin(2x) over two periods. The third is the graph of sin(x\/2) for one half of a period.\" width=\"487\" height=\"274\" \/> <b>Figure 8<\/b>[\/caption]<\/figure>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Period of Sinusoidal Functions<\/h3>\r\nIf we let <em>C<\/em> = 0 and <em>D<\/em> = 0 in the general form equations of the sine and cosine functions, we obtain the forms\r\n<p style=\"text-align: center\"><span style=\"text-align: center;background-color: initial;font-size: 0.9em\">[latex]y=A\\sin\\left(Bx\\right)[\/latex]<\/span><\/p>\r\n<p style=\"text-align: center\"><span style=\"background-color: initial;font-size: 0.9em\">[latex]y=A\\cos\\left(Bx\\right)[\/latex]<\/span><\/p>\r\nThe period is [latex]\\frac{2\u03c0}{|B|}[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 1: Identifying the Period of a Sine or Cosine Function<\/h3>\r\nDetermine the period of the function [latex]f(x) = \\sin\\left(\\frac{\u03c0}{6}x\\right)[\/latex].\r\n\r\n[reveal-answer q=\"616023\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"616023\"]\r\n\r\nLet's begin by comparing the equation to the general form [latex]y=A\\sin(Bx)[\/latex].\r\n\r\nIn the given equation, [latex]B =\\frac{\u03c0}{6}[\/latex], so the period will be\r\n<p style=\"text-align: center\">[latex]\\begin{align}P&amp;=\\frac{\\frac{2}{\\pi}}{|B|} \\\\ &amp;=\\frac{2\\pi}{\\frac{x}{6}} \\\\ &amp;=2\\pi\\times \\frac{6}{\\pi} \\\\ &amp;=12 \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nDetermine the period of the function [latex]g(x)=\\cos\\left(\\frac{x}{3}\\right)[\/latex].\r\n\r\n[reveal-answer q=\"111765\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"111765\"]\r\n\r\n[latex]6 \\pi[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Determining Amplitude<\/h3>\r\nReturning to the general formula for a sinusoidal function, we have analyzed how the variable <em>B<\/em> relates to the period. Now let\u2019s turn to the variable <em>A<\/em> so we can analyze how it is related to the <strong>amplitude<\/strong>, or greatest distance from rest. <em>A<\/em> represents the vertical stretch factor, and its absolute value |<em>A<\/em>| is the amplitude. The local maxima will be a distance |<em>A<\/em>| above the vertical <strong>midline<\/strong> of the graph, which is the line <em>x\u00a0<\/em>= <em>D<\/em>; because <em>D<\/em> = 0 in this case, the midline is the <em>x<\/em>-axis. The local minima will be the same distance below the midline. If |<em>A<\/em>| &gt; 1, the function is stretched. For example, the amplitude of [latex]f(x)=4\\sin\\left(x\\right)[\/latex] is twice the amplitude of\r\n<div style=\"text-align: center\">[latex]f(x)=2\\sin\\left(x\\right)[\/latex]<\/div>\r\nIf [latex]|<em>A<\/em>| &lt; 1[\/latex], the function is compressed. Figure 9\u00a0compares several sine functions with different amplitudes.\r\n<figure id=\"Figure_06_01_009\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003929\/CNX_Precalc_Figure_06_01_009.jpg\" alt=\"A graph with four items. The x-axis ranges from -6pi to 6pi. The y-axis ranges from -4 to 4. The first item is the graph of sin(x), which has an amplitude of 1. The second is a graph of 2sin(x), which has amplitude of 2. The third is a graph of 3sin(x), which has an amplitude of 3. The fourth is a graph of 4 sin(x) with an amplitude of 4.\" width=\"975\" height=\"316\" \/> <b>Figure 9<\/b>[\/caption]<\/figure>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Amplitude of Sinusoidal Functions<\/h3>\r\nIf we let <em>C\u00a0<\/em>= 0 and <em>D<\/em> = 0 in the general form equations of the sine and cosine functions, we obtain the forms\r\n<p style=\"text-align: center\">[latex]y=A\\sin(Bx)[\/latex] and [latex]y=A\\cos(Bx)[\/latex]<\/p>\r\nThe <strong>amplitude<\/strong> is A, and the vertical height from the <strong>midline<\/strong> is |A|. In addition, notice in the example that\r\n<p style=\"text-align: center\"><span style=\"text-align: center;background-color: initial;font-size: 0.9em\">[latex]|A|=\\text{amplitude}=\\frac{1}{2}|\\text{maximum}\u2212\\text{minimum}|[\/latex]<\/span><\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 2: Identifying the Amplitude of a Sine or Cosine Function<\/h3>\r\nWhat is the amplitude of the sinusoidal function\u00a0[latex]f(x)=\u22124\\sin(x)[\/latex]? Is the function stretched or compressed vertically?\r\n\r\n[reveal-answer q=\"207317\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"207317\"]\r\n\r\nLet\u2019s begin by comparing the function to the simplified form [latex]y=A\\sin(Bx)[\/latex].\r\n\r\nIn the given function, <em>A\u00a0<\/em>= \u22124, so the amplitude is |<em>A<\/em>|=|\u22124| = 4. The function is stretched.\r\n<h4>Analysis of the Solution<\/h4>\r\nThe negative value of <em>A<\/em> results in a reflection across the <em>x<\/em>-axis of the <strong>sine function<\/strong>, as shown in Figure 10.\r\n<figure id=\"Figure_06_01_010\" 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\/3675\/2018\/09\/27003932\/CNX_Precalc_Figure_06_01_010.jpg\" alt=\"A graph of -4sin(x). The function has an amplitude of 4. Local minima at (-3pi\/2, -4) and (pi\/2, -4). Local maxima at (-pi\/2, 4) and (3pi\/2, 4). Period of 2pi.\" width=\"487\" height=\"319\" \/> <b>Figure 10<\/b>[\/caption]<\/figure>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nWhat is the amplitude of the sinusoidal function [latex]f(x)=12\\sin (x)[\/latex]? Is the function stretched or compressed vertically?\r\n\r\n[reveal-answer q=\"525586\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"525586\"]\r\n\r\n[latex]\\frac{1}{2}[\/latex] compressed\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Analyzing Graphs of Variations of <em>y<\/em> = sin<em> x<\/em> and <em>y<\/em> = cos <em>x<\/em><\/h2>\r\nNow that we understand how <em>A<\/em> and <em>B<\/em> relate to the general form equation for the sine and cosine functions, we will explore the variables <em>C\u00a0<\/em>and <em>D<\/em>. Recall the general form:\r\n<div>\r\n<div style=\"text-align: center\">[latex]y = A \\sin(Bx\u2212C)+D[\/latex] and [latex]y=A\\cos(Bx\u2212C)+D[\/latex]<\/div>\r\n<div>or<\/div>\r\n<div style=\"text-align: center\">[latex]y=A\\sin(B(x\u2212\\frac{C}{B}))+D[\/latex] and [latex]y=A\\cos(B(x\u2212\\frac{C}{B}))+D[\/latex]<\/div>\r\n<\/div>\r\nThe value [latex]\\frac{C}{B}[\/latex] for a sinusoidal function is called the <strong>phase shift<\/strong>, or the horizontal displacement of the basic sine or <strong>cosine function<\/strong>. If C &gt; 0, the graph shifts to the right. If C &lt; 0,the graph shifts to the left. The greater the value of |<em>C<\/em>|, the more the graph is shifted. Figure 11\u00a0shows that the graph of [latex]f(x)=\\sin(x\u2212\u03c0)[\/latex] shifts to the right by \u03c0 units, which is more than we see in the graph of [latex]f(x)=\\sin(x\u2212\\frac{\u03c0}{4})[\/latex], which shifts to the right by [latex]\\frac{\u03c0}{4}[\/latex]units.\r\n<figure id=\"Figure_06_01_011\" 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\/3675\/2018\/09\/27003934\/CNX_Precalc_Figure_06_01_011.jpg\" alt=\"A graph with three items. The first item is a graph of sin(x). The second item is a graph of sin(x-pi\/4), which is the same as sin(x) except shifted to the right by pi\/4. The third item is a graph of sin(x-pi), which is the same as sin(x) except shifted to the right by pi.\" width=\"487\" height=\"255\" \/> <b>Figure 11<\/b>[\/caption]<\/figure>\r\nWhile <em>C<\/em> relates to the horizontal shift, <em>D<\/em> indicates the vertical shift from the midline in the general formula for a sinusoidal function. The function [latex]y=\\cos(x)+D[\/latex] has its midline at [latex]y=D[\/latex].\r\n<figure id=\"Figure_06_01_012\" 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\/3675\/2018\/09\/27003936\/CNX_Precalc_Figure_06_01_012.jpg\" alt=\"A graph of y=Asin(x)+D. Graph shows the midline of the function at y=D.\" width=\"487\" height=\"255\" \/> <b>Figure 12<\/b>[\/caption]<\/figure>\r\nAny value of <em>D<\/em> other than zero shifts the graph up or down. Figure 13\u00a0compares [latex]f(x)=\\sin x[\/latex] with [latex]f(x)=\\sin (x)+2[\/latex], which is shifted 2 units up on a graph.\r\n<figure id=\"Figure_06_01_013\" 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\/3675\/2018\/09\/27003938\/CNX_Precalc_Figure_06_01_013.jpg\" alt=\"A graph with two items. The first item is a graph of sin(x). The second item is a graph of sin(x)+2, which is the same as sin(x) except shifted up by 2.\" width=\"487\" height=\"221\" \/> <b>Figure 13<\/b>[\/caption]<\/figure>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Variations of Sine and Cosine Functions<\/h3>\r\nGiven an equation in the form [latex]f(x)=A\\sin(Bx\u2212C)+D[\/latex] or [latex]f(x)=A\\cos(Bx\u2212C)+D[\/latex], [latex]\\frac{C}{B}[\/latex]is the <strong>phase shift<\/strong> and <em>D<\/em> is the <strong>vertical shift<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 3: Identifying the Phase Shift of a Function<\/h3>\r\nDetermine the direction and magnitude of the phase shift for [latex]f(x)=\\sin(x+\\frac{\u03c0}{6})\u22122[\/latex].\r\n\r\n[reveal-answer q=\"673346\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"673346\"]\r\n\r\nLet\u2019s begin by comparing the equation to the general form [latex]y=A\\sin(Bx\u2212C)+D[\/latex].\r\n\r\nIn the given equation, notice that <em>B<\/em> = 1 and [latex]C=\u2212\\frac{\u03c0}{6}[\/latex]. So the phase shift is\r\n<p style=\"text-align: center\">[latex]\\begin{align}\\frac{C}{B}&amp;=\u2212\\frac{\\frac{x}{6}}{1} \\\\ &amp;=\u2212\\frac{\\pi}{6} \\end{align}[\/latex]<\/p>\r\nor [latex]\\frac{\\pi}{6}[\/latex] units to the left.\r\n<h4>Analysis of the Solution<\/h4>\r\nWe must pay attention to the sign in the equation for the general form of a sinusoidal function. The equation shows a minus sign before <em>C<\/em>. Therefore [latex]f(x)=\\sin(x+\\frac{\u03c0}{6})\u22122[\/latex] can be rewritten as [latex]f(x)=\\sin(x\u2212(\u2212\\frac{\u03c0}{6}))\u22122[\/latex]. If the value of <em>C<\/em> is negative, the shift is to the left.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nDetermine the direction and magnitude of the phase shift for [latex]f(x)=3\\cos(x\u2212\\frac{\\pi}{2})[\/latex].\r\n\r\n[reveal-answer q=\"739051\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"739051\"]\r\n\r\n[latex]\\frac{\u03c0}{2}[\/latex]; right\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 4: Identifying the Vertical Shift of a Function<\/h3>\r\nDetermine the direction and magnitude of the vertical shift for [latex]f(x)=\\cos(x)\u22123[\/latex].\r\n\r\n[reveal-answer q=\"920585\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"920585\"]\r\n\r\nLet's begin by comparing the equation to the general form [latex]y=A\\cos(Bx\u2212C)+D[\/latex]. In the given equation, [latex]D=-3[\/latex], so the shift is 3 units downward.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nDetermine the direction and magnitude of the vertical shift for [latex]f(x)=3\\sin(x)+2[\/latex].\r\n\r\n[reveal-answer q=\"558661\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"558661\"]\r\n\r\n2 units up\r\n\r\n<span style=\"font-size: 1rem;text-align: initial\">[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a sinusoidal function in the form [latex]f(x)=A\\sin(Bx\u2212C)+D[\/latex],\u00a0identify the midline, amplitude, period, and phase shift.<\/h3>\r\n<ol>\r\n \t<li>Determine the amplitude as |A|.<\/li>\r\n \t<li>Determine the period as [latex]P=\\frac{2\u03c0}{|B|}[\/latex].<\/li>\r\n \t<li>Determine the phase shift as [latex]\\frac{C}{B}[\/latex].<\/li>\r\n \t<li>Determine the midline as <em>y\u00a0<\/em>= D.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 5: Identifying the Variations of a Sinusoidal Function from an Equation<\/h3>\r\nDetermine the midline, amplitude, period, and phase shift of the function [latex]y=3\\sin(2x)+1[\/latex].\r\n\r\n[reveal-answer q=\"622405\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"622405\"]\r\n\r\nLet\u2019s begin by comparing the equation to the general form [latex]y=A\\sin(Bx\u2212C)+D[\/latex].\u00a0<em>A<\/em> = 3, so the amplitude is |<em>A<\/em>| = 3.\r\n\r\nNext, <em>B<\/em> = 2, so the period is [latex]P=\\frac{2\u03c0}{|B|}=\\frac{2\u03c0}{2}=\u03c0[\/latex].\r\n\r\nThere is no added constant inside the parentheses, so <em>C<\/em> = 0 and the phase shift is [latex]\\frac{C}{B}=\\frac{0}{2}=0[\/latex].\r\n\r\nFinally, <em>D<\/em> = 1, so the midline is <em>y<\/em> = 1.\r\n<h4>Analysis of the Solution<\/h4>\r\nInspecting the graph, we can determine that the period is \u03c0, the midline is <em>y<\/em> = 1,and the amplitude is 3. See Figure 14.\r\n<figure id=\"Figure_06_01_014\" 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\/3675\/2018\/09\/27003941\/CNX_Precalc_Figure_06_01_014.jpg\" alt=\"A graph of y=3sin(2x)+1. The graph has an amplitude of 3. There is a midline at y=1. There is a period of pi. Local maximum at (pi\/4, 4) and local minimum at (3pi\/4, -2).\" width=\"487\" height=\"263\" \/> <b>Figure 14<\/b>[\/caption]<\/figure>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nDetermine the midline, amplitude, period, and phase shift of the function [latex]y=\\frac{1}{2}\\cos(\\frac{x}{3}\u2212\\frac{\u03c0}{3})[\/latex].\r\n\r\n[reveal-answer q=\"453453\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"453453\"]\r\n\r\nmidline: [latex]y=0[\/latex]; amplitude: |<em>A<\/em>|=[latex]\\frac{1}{2}[\/latex]; period: <em>P<\/em>=[latex]\\frac{2\u03c0}{|B|}=6\\pi[\/latex]; phase shift:[latex]\\frac{C}{B}=\\pi[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]105947[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 6: Identifying the Equation for a Sinusoidal Function from a Graph<\/h3>\r\nDetermine the formula for the cosine function in Figure 15.\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\/3675\/2018\/09\/27003943\/CNX_Precalc_Figure_06_01_015.jpg\" alt=\"A graph of -0.5cos(x)+0.5. The graph has an amplitude of 0.5. The graph has a period of 2pi. The graph has a range of [0, 1]. The graph is also reflected about the x-axis from the parent function cos(x).\" width=\"487\" height=\"163\" \/> <b>Figure 15<\/b>[\/caption][reveal-answer q=\"509662\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"509662\"]\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<p style=\"text-align: center\">[latex]y=A\\sin\\left(Bx-C\\right)+D[\/latex]<span id=\"MathJax-Element-411-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: center;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><\/span><\/p>\r\n<p style=\"text-align: center\">[latex]y=A\\cos\\left(Bx-C\\right)+D[\/latex]<\/p>\r\n<p id=\"fs-id1165137704661\">The graph could represent either a sine or a\u00a0<span class=\"no-emphasis\">cosine function<\/span>\u00a0that is shifted and\/or reflected. When [latex]x=0[\/latex], the graph has an extreme point, [latex](0,0)[\/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\n<p id=\"fs-id1165135536557\">Let\u2019s start with the midline. We can see that the graph rises and falls an equal distance above and below [latex]y=0.5[\/latex]. This value, which is the midline, is\u00a0<em>D <\/em>in\u00a0the equation, so <em>D<\/em>=0.5.<\/p>\r\n<p id=\"fs-id1165137938642\">The greatest distance above and below the midline is the amplitude. The maxima are 0.5 units above the midline and the minima are 0.5 units below the midline. So |<em>A<\/em>|=0.5. Another way we could have determined the amplitude is by recognizing that the difference between the height of local maxima and minima is 1, so |<em>A<\/em>|=[latex]\\frac{1}{2}[\/latex]. Also, the graph is reflected about the\u00a0<em>x<\/em>-axis so that\u00a0<em>A<\/em>=0.5.<\/p>\r\n<p id=\"fs-id1165134204425\">The graph is not horizontally stretched or compressed, so\u00a0<em>B<\/em>=0 and the graph is not shifted horizontally, so\u00a0<em>C<\/em>=0.<\/p>\r\n<p id=\"fs-id1165135347312\">Putting this all together,<\/p>\r\n<p style=\"text-align: center\">[latex]g(x)=0.5\\cos\\left(x\\right)+0.5[\/latex]<\/p>\r\n<p style=\"text-align: left\"><span style=\"font-size: 1rem;text-align: initial\">[\/hidden-answer]<\/span><\/p>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nDetermine the formula for the sine function in Figure 16.\r\n<figure id=\"Figure_06_01_016\" class=\"small\">[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003945\/CNX_Precalc_Figure_06_01_016.jpg\" alt=\"A graph of sin(x)+2. Period of 2pi, amplitude of 1, and range of [1, 3].\" width=\"487\" height=\"173\" \/> <b>Figure 16<\/b>[\/caption]<\/figure>\r\n[reveal-answer q=\"448760\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"448760\"]\r\n\r\n[latex]f(x)=\\sin(x)+2[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]126732[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 7: Identifying the Equation for a Sinusoidal Function from a Graph<\/h3>\r\nDetermine the equation for the sinusoidal function in Figure 17.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003947\/CNX_Precalc_Figure_06_01_017.jpg\" alt=\"A graph of 3cos(pi\/3x-pi\/3)-2. Graph has amplitude of 3, period of 6, range of [-5,1].\" width=\"731\" height=\"565\" \/> <b>Figure 17<\/b>[\/caption][reveal-answer q=\"680521\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"680521\"]With the highest value at 1 and the lowest value at\u22125, the midline will be halfway between at \u22122. So <em>D<\/em> = \u22122.The distance from the midline to the highest or lowest value gives an amplitude of |A|=3.The period of the graph is 6, which can be measured from the peak at <em>x\u00a0<\/em>= 1 to the next peak at <em>x<\/em> = 7,\u00a0or\u00a0from the distance between the lowest points. Therefore, [latex]\\text{P}=\\frac{2\\pi}{|B|}=6[\/latex]. Using the positive value for <em>B<\/em>, we find that\r\n<p style=\"text-align: center\">[latex]B=\\frac{2\u03c0}{P}=\\frac{2\u03c0}{6}=\\frac{\u03c0}{3}[\/latex]<\/p>\r\nSo far, our equation is either [latex]y=3\\sin(\\frac{\\pi}{3}x\u2212C)\u22122[\/latex] or [latex]y=3\\cos(\\frac{\\pi}{3}x\u2212C)\u22122[\/latex]. For the shape and shift, we have more than one option. We could write this as any one of the following:\r\n<ul>\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\nWhile 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\r\n<p style=\"text-align: center\">[latex]y=3\\cos(\\frac{\u03c0}{3}x\u2212\\frac{\u03c0}{3})\u22122[\/latex] or [latex]y=\u22123\\cos(\\frac{\u03c0}{3}x+\\frac{2\u03c0}{3})\u22122[\/latex]<\/p>\r\nAgain, these functions are equivalent, so both yield the same graph.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nWrite a formula for the function graphed in Figure 18.\r\n<figure id=\"Figure_06_01_018\" class=\"medium\">[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003949\/CNX_Precalc_Figure_06_01_018n.jpg\" alt=\"A graph of 4sin((pi\/5)x-pi\/5)+4. Graph has period of 10, amplitude of 4, range of [0,8].\" width=\"731\" height=\"440\" \/> <b>Figure 18<\/b>[\/caption]<\/figure>\r\n[reveal-answer q=\"993227\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"993227\"]\r\n\r\ntwo possibilities are: [latex]y=4\\sin(\\frac{\u03c0}{5}x\u2212\\frac{\u03c0}{5})+4[\/latex] or [latex]y=\u22124sin(\\frac{\u03c0}{5}x+4\\frac{\u03c0}{5})+4[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]126749[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Graphing Variations of <em>y<\/em> = sin <em>x<\/em> and <em>y<\/em> = cos <em>x<\/em><\/h2>\r\nThroughout 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.\r\n\r\nInstead of focusing on the general form equations\r\n<div>\r\n<div style=\"text-align: center\">[latex]y=A\\sin(Bx\u2212C)+D[\/latex] and [latex]y=A\\cos(Bx\u2212C)+D[\/latex],<\/div>\r\n<\/div>\r\nwe will let <em>C<\/em> = 0 and <em>D<\/em> = 0 and work with a simplified form of the equations in the following examples.\r\n<div class=\"textbox\">\r\n<h3>How To: Given the function [latex]y=Asin(Bx)[\/latex], sketch its graph.<\/h3>\r\n<ol>\r\n \t<li>Identify the amplitude,|<em>A<\/em>|.<\/li>\r\n \t<li>Identify the period, [latex]P=\\frac{2\u03c0}{|B|}[\/latex].<\/li>\r\n \t<li>Start at the origin, with the function increasing to the right if <em>A<\/em> is positive or decreasing if <em>A<\/em> is negative.<\/li>\r\n \t<li>At [latex]x=\\frac{\u03c0}{2|B|}[\/latex] there is a local maximum for <em>A<\/em> &gt; 0 or a minimum for <em>A<\/em> &lt; 0, with <em>y<\/em> = <em>A<\/em>.<\/li>\r\n \t<li>The curve returns to the <em>x<\/em>-axis at [latex]x=\\frac{\u03c0}{|B|}[\/latex].<\/li>\r\n \t<li>There is a local minimum for <em>A<\/em> &gt; 0 (maximum for <em>A\u00a0<\/em>&lt; 0) at [latex]x=\\frac{3\u03c0}{2|B|}[\/latex] with <em>y\u00a0<\/em>= \u2013<em>A<\/em>.<\/li>\r\n \t<li>The curve returns again to the <em>x<\/em>-axis at [latex]x=\\frac{\u03c0}{2|B|}[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 8: Graphing a Function and Identifying the Amplitude and Period<\/h3>\r\nSketch a graph of [latex]f(x)=\u22122\\sin(\\frac{\u03c0x}{2})[\/latex].\r\n\r\n[reveal-answer q=\"699067\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"699067\"]\r\n\r\nLet\u2019s begin by comparing the equation to the form [latex]y=A\\sin(Bx)[\/latex].\r\n\r\n<strong>Step 1.<\/strong> We can see from the equation that A=\u22122,so the amplitude is 2.\r\n<p style=\"text-align: center\">|<em>A<\/em>| = 2<\/p>\r\n<strong>Step 2.<\/strong> The equation shows that [latex]B=\\frac{\u03c0}{2}[\/latex], so the period is\r\n<p style=\"text-align: center\">[latex] \\begin{align}P&amp;=\\frac{2\\pi}{\\frac{\\pi}{2}}\\\\&amp;=2\\pi\\times\\frac{2}{\\pi}\\\\&amp;=4 \\end{align}[\/latex]<\/p>\r\n<strong>Step 3.<\/strong> Because <em>A<\/em> is negative, the graph descends as we move to the right of the origin.\r\n\r\n<strong>Step 4\u20137.<\/strong> The <em>x<\/em>-intercepts are at the beginning of one period, <em>x\u00a0<\/em>= 0, the horizontal midpoints are at <em>x\u00a0<\/em>= 2 and at the end of one period at <em>x<\/em> = 4.\r\n\r\nThe quarter points include the minimum at <em>x<\/em> = 1 and the maximum at <em>x<\/em> = 3. A local minimum will occur 2 units below the midline, at <em>x<\/em> = 1, and a local maximum will occur at 2 units above the midline, at <em>x<\/em> = 3. Figure 19 shows the graph of the function.\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\/3675\/2018\/09\/27003952\/CNX_Precalc_Figure_06_01_019.jpg\" alt=\"A graph of -2sin((pi\/2)x). Graph has range of [-2,2], period of 4, and amplitude of 2.\" width=\"487\" height=\"252\" \/> <b>Figure 19<\/b>[\/caption][\/hidden-answer]<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nSketch a graph of [latex]g(x)=\u22120.8\\cos(2x)[\/latex]. Determine the midline, amplitude, period, and phase shift.\r\n\r\n[reveal-answer q=\"862743\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"862743\"]\r\n\r\nmidline: y=0; amplitude: |<em>A<\/em>|=0.8; period: P=[latex]\\frac{2\u03c0}{|B|}=\\pi[\/latex]; phase shift: [latex]\\frac{C}{B}=0[\/latex] or none\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004024\/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).\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a sinusoidal function with a phase shift and a vertical shift, sketch its graph.<\/h3>\r\n<ol>\r\n \t<li>Express the function in the general form [latex]y=A\\sin(Bx\u2212C)+D[\/latex] or [latex]y=A\\cos(Bx\u2212C)+D[\/latex].<\/li>\r\n \t<li>Identify the amplitude, |<em>A<\/em>|.<\/li>\r\n \t<li>Identify the period, [latex]P=2\u03c0|B|[\/latex].<\/li>\r\n \t<li>Identify the phase shift, [latex]\\frac{C}{B}[\/latex].<\/li>\r\n \t<li>Draw the graph of [latex]f(x)=A\\sin(Bx)[\/latex] shifted to the right or left by [latex]\\frac{C}{B}[\/latex] and up or down by <em>D<\/em>.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 9: Graphing a Transformed Sinusoid<\/h3>\r\nSketch a graph of [latex]f(x)=3\\sin\\left(\\frac{\u03c0}{4}x\u2212\\frac{\u03c0}{4}\\right)[\/latex].\r\n\r\n[reveal-answer q=\"39590\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"39590\"]\r\n\r\n<strong>Step 1.<\/strong> The function is already written in general form: [latex]f(x)=3\\sin\\left(\\frac{\u03c0}{4}x\u2212\\frac{\u03c0}{4}\\right)[\/latex]. This graph will have the shape of a <strong>sine function<\/strong>, starting at the midline and increasing to the right.\r\n\r\n<strong>Step 2.<\/strong> |<em>A<\/em>|=|3|=3. The amplitude is 3.\r\n\r\n<strong>Step 3.<\/strong> Since [latex]|B|=|\\frac{\u03c0}{4}|=\\frac{\u03c0}{4}[\/latex], we determine the period as follows.\r\n<p style=\"text-align: center\">[latex]P=\\frac{2\u03c0}{|B|}=\\frac{2\u03c0}{\\frac{\u03c0}{4}}=2\u03c0\\times\\frac{4}{\u03c0}=8[\/latex]<\/p>\r\nThe period is 8.\r\n\r\n<strong>Step 4.<\/strong> Since [latex]\\text{C}=\\frac{\u03c0}{4}[\/latex], the phase shift is\r\n<p style=\"text-align: center\">[latex]\\frac{C}{B}=\\frac{\\frac{\\pi}{4}}{\\frac{\\pi}{4}}=1[\/latex].<\/p>\r\nThe phase shift is 1 unit.\r\n\r\n<strong>Step 5.<\/strong> Figure 20 shows the graph of the function.\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\/3675\/2018\/09\/27003954\/CNX_Precalc_Figure_06_01_021.jpg\" alt=\"A graph of 3sin(*(pi\/4)x-pi\/4). Graph has amplitude of 3, period of 8, and a phase shift of 1 to the right.\" width=\"487\" height=\"319\" \/> <b>Figure 20.<\/b> A horizontally compressed, vertically stretched, and horizontally shifted sinusoid[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>\u00a0Try It<\/h3>\r\nDraw a graph of [latex]g(x)=\u22122\\cos(\\frac{\\pi}{3}x+\\frac{\\pi}{6})[\/latex]. Determine the midline, amplitude, period, and phase shift.\r\n\r\n[reveal-answer q=\"145991\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"145991\"]\r\n\r\n[latex]\\text{midline:}y=0;\\text{amplitude:}|A|=2;\\text{period:}\\text{P}=\\frac{2\\pi}{|B|}=6;\\text{phase shift:}\\frac{C}{B}=\u2212\\frac{1}{2}[\/latex]\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004026\/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.\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]173422[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 10: Identifying the Properties of a Sinusoidal Function<\/h3>\r\nGiven [latex]y=\u22122\\cos\\left(\\frac{\\pi}{2}x+\\pi\\right)+3[\/latex], determine the amplitude, period, phase shift, and horizontal shift. Then graph the function.\r\n\r\n[reveal-answer q=\"603659\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"603659\"]\r\n\r\nBegin by comparing the equation to the general form and use the steps outlined in Example 9.\r\n\r\n[latex]y=A\\cos(Bx\u2212C)+D[\/latex]\r\n\r\n<strong>Step 1.<\/strong> The function is already written in general form.\r\n\r\n<strong>Step 2.<\/strong> Since <em>A\u00a0<\/em>= \u22122, the amplitude is|<em>A<\/em>| = 2.\r\n\r\n<strong>Step 3.<\/strong>\u00a0[latex]|B|=\\frac{\\pi}{2}[\/latex], so the period is [latex]P=\\frac{2\u03c0}{|B|}=\\frac{2\\pi}{\\frac{\\pi}{2}}\\times2\\pi=4[\/latex]. The period is 4.\r\n\r\n<strong>Step 4.<\/strong>\u00a0[latex]C=\u2212\\pi[\/latex], so we calculate the phase shift as [latex]\\frac{C}{B}=\\frac{\u2212\\pi}{\\frac{\\pi}{2}}=\u2212\\pi\\times\\frac{2}{\\pi}=\u22122[\/latex]. The phase shift is \u22122.\r\n\r\n<strong>Step 5.<\/strong> <em>D\u00a0<\/em>= 3, so the midline is <em>y\u00a0<\/em>= 3, and the vertical shift is up 3.\r\n\r\nSince <em>A<\/em> is negative, the graph of the cosine function has been reflected about the x-axis.\r\n\r\nFigure 21 shows one cycle of the graph of the function.\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\/3675\/2018\/09\/27003956\/CNX_Precalc_Figure_06_01_028.jpg\" alt=\"A graph of -2cos((pi\/2)x+pi)+3. Graph shows an amplitude of 2, midline at y=3, and a period of 4.\" width=\"487\" height=\"317\" \/> <b>Figure 21<\/b>[\/caption]\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\nhttps:\/\/youtu.be\/QNQAkUUHNxo\r\nhttps:\/\/youtu.be\/tcjZOGaeoeo\r\n<h2>Using Transformations of Sine and Cosine Functions<\/h2>\r\nWe can use the transformations of sine and cosine functions in numerous applications. As mentioned at the beginning of the chapter, <strong>circular motion<\/strong> can be modeled using either the sine or <strong>cosine function<\/strong>.\r\n<div class=\"textbox shaded\">\r\n<h3>Example 11: Finding the Vertical Component of Circular Motion<\/h3>\r\nA 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.\r\n\r\n[reveal-answer q=\"140255\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"140255\"]\r\n\r\nRecall that, for a point on a circle of radius r, the y-coordinate of the point is [latex]y=r\\sin(x)[\/latex], so in this case, we get the equation [latex]y(x)=3\\sin(x)[\/latex]. The constant 3 causes a vertical stretch of the y-values of the function by a factor of 3, which we can see in the graph in Figure 22.\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\/3675\/2018\/09\/27003959\/CNX_Precalc_Figure_06_01_023.jpg\" alt=\"A graph of 3sin(x). Graph has period of 2pi, amplitude of 3, and range of [-3,3].\" width=\"487\" height=\"319\" \/> <b>Figure 22<\/b>[\/caption]\r\n<h4>Analysis of the Solution<\/h4>\r\nNotice that the period of the function is still 2\u03c0; as we travel around the circle, we return to the point (3,0) for [latex]x=2\\pi,4\\pi,6\\pi,\\dots[\/latex] Because the outputs of the graph will now oscillate between \u20133 and 3, the amplitude of the sine wave is 3.\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nWhat is the amplitude of the function [latex]f(x)=7\\cos(x)[\/latex]? Sketch a graph of this function.\r\n\r\n[reveal-answer q=\"317443\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"317443\"]\r\n\r\n7\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004029\/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].\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 12: Finding the Vertical Component of Circular Motion<\/h3>\r\nA 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 Figure 23. Sketch a graph of the height above the ground of the point <em>P<\/em> as the circle is rotated; then find a function that gives the height in terms of the angle of rotation.\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\/3675\/2018\/09\/27004001\/CNX_Precalc_Figure_06_01_025.jpg\" alt=\"An illustration of a circle lifted 4 feet off the ground. Circle has radius of 3 ft. There is a point P labeled on the circle's circumference.\" width=\"487\" height=\"300\" \/> <b>Figure 23<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"367979\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"367979\"]\r\n\r\nSketching 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 Figure 24.\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\/3675\/2018\/09\/27004004\/CNX_Precalc_Figure_06_01_026.jpg\" alt=\"A graph of -3cox(x)+4. Graph has midline at y=4, amplitude of 3, and period of 2pi.\" width=\"487\" height=\"521\" \/> <b>Figure 24<\/b>[\/caption]\r\n\r\nAlthough 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 <strong>sine function<\/strong> 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.\r\n\r\nSecond, we see that the graph oscillates 3 above and below the center, while a basic cosine has an amplitude of 1, so this graph has been vertically stretched by 3, as in the last example.\r\n\r\nFinally, 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\r\n<p style=\"text-align: center\">[latex]y=\u22123\\cos(x)+4[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nA weight is attached to a spring that is then hung from a board, as shown in Figure 25. As the spring oscillates up and down, the position <em>y<\/em> of the weight relative to the board ranges from \u20131 in. (at time <em>x<\/em> = 0) to \u20137in. (at time <em>x<\/em> = \u03c0) below the board. Assume the position of <em>y<\/em> is given as a sinusoidal function of <em>x<\/em>. Sketch a graph of the function, and then find a cosine function that gives the position <em>y<\/em> in terms of x.\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\/3675\/2018\/09\/27004006\/CNX_Precalc_Figure_06_01_029.jpg\" alt=\"An illustration of a spring with length y.\" width=\"487\" height=\"351\" \/> <b>Figure 25<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"518116\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"518116\"]\r\n\r\n[latex]y=3\\cos(x)\u22124[\/latex]\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004032\/CNX_Precalc_Figure_06_01_027.jpg\" alt=\"A cosine graph with range [-1,-7]. Period is 2 pi. Local maximums at (0,-1), (2pi,-1), and (4pi, -1). Local minimums at (pi,-7) and (3pi, -7).\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 13: Determining 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=\"304167\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"304167\"]\r\n\r\nWith 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.\r\n\r\nPassengers board 2 m above ground level, so the center of the wheel must be located 67.5 + 2 = 69.5 m above ground level. The midline of the oscillation will be at 69.5 m.\r\n\r\nThe wheel takes 30 minutes to complete 1 revolution, so the height will oscillate with a period of 30 minutes.\r\n\r\nLastly, 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.\r\n<ul>\r\n \t<li>Amplitude: 67.5, so <em>A\u00a0<\/em>= 67.5<\/li>\r\n \t<li>Midline: 69.5, so <em>D<\/em> = 69.5<\/li>\r\n \t<li>Period: 30, so [latex]B=\\frac{2\\pi}{30}=\\frac{\\pi}{15}[\/latex]<\/li>\r\n \t<li>Shape: \u2212cos(<em>t<\/em>)<\/li>\r\n<\/ul>\r\nAn equation for the rider\u2019s height would be\r\n<p style=\"text-align: center\">[latex]y=\u221267.5\\cos\\left(\\frac{\\pi}{15}t\\right)+69.5[\/latex]<\/p>\r\nwhere <em>t<\/em> is in minutes and <em>y<\/em> is measured in meters.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]127257[\/ohm_question]\r\n\r\n<\/div>\r\n<section id=\"fs-id1165137574576\" class=\"key-equations\">\r\n<h2>Analyzing the Graph of y = tan x and Its Variations<\/h2>\r\nWe will begin with the graph of the <strong>tangent<\/strong> function, plotting points as we did for the sine and cosine functions. Recall that\r\n<div>\r\n<div style=\"text-align: center\">[latex]\\tan x=\\frac{\\sin x}{\\cos x}[\/latex]<\/div>\r\n<\/div>\r\nThe <strong>period<\/strong> of the tangent function is <em>\u03c0<\/em> because the graph repeats itself on intervals of <em>k\u03c0<\/em> where <em>k<\/em> is a constant. If we graph the tangent function on [latex]\u2212\\dfrac{\\pi}{2}\\text{ to }\\dfrac{\\pi}{2}[\/latex], we can see the behavior of the graph on one complete cycle. If we look at any larger interval, we will see that the characteristics of the graph repeat.\r\n\r\nWe can determine whether tangent is an odd or even function by using the definition of tangent.\r\n<p style=\"text-align: center\">[latex]\\begin{align}\\tan(\u2212x)&amp;=\\frac{\\sin(\u2212x)}{\\cos(\u2212x)} &amp;&amp; \\text{Definition of tangent.} \\\\ &amp;=\\frac{\u2212\\sin x}{\\cos x} &amp;&amp; \\text{Sine is an odd function, cosine is even.} \\\\ &amp;=\u2212\\frac{\\sin x}{\\cos x} &amp;&amp; \\text{The quotient of an odd and an even function is odd.} \\\\ &amp;=\u2212\\tan x &amp;&amp; \\text{Definition of tangent.} \\end{align}[\/latex]<\/p>\r\nTherefore, tangent is an odd function. We can further analyze the graphical behavior of the tangent function by looking at values for some of the special angles, as listed in the table below.\r\n<table id=\"Table_06_02_00\" style=\"width: 1035px\" summary=\"Two rows and 10 columns. First row is labeled x and second row is labeled tangent of x. The table has ordered pairs of these column values: (-pi\/2,undefined), (-pi\/3, negative square root of 3), (-pi\/4, -1), (-pi\/6, negative square root of 3 over 3), (0, 0), (pi\/6, square root of 3 over 3), (pi\/4, 1), (pi\/3, square root of 3), (pi\/2, undefined).\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 30px\"><em><strong> x <\/strong><\/em><\/td>\r\n<td style=\"width: 80px\">[latex]-\\frac{\\pi}{6}[\/latex]<\/td>\r\n<td style=\"width: 80px\">[latex]-\\frac{\\pi}{3}[\/latex]<\/td>\r\n<td style=\"width: 80px\">[latex]-\\frac{\\pi}{4}[\/latex]<\/td>\r\n<td style=\"width: 80px\">[latex]-\\frac{\\pi}{6}[\/latex]<\/td>\r\n<td style=\"width: 80px\">0<\/td>\r\n<td style=\"width: 80px\">[latex]\\frac{\\pi}{6}[\/latex]<\/td>\r\n<td style=\"width: 80px\">[latex]\\frac{\\pi}{4}[\/latex]<\/td>\r\n<td style=\"width: 80px\">[latex]\\frac{\\pi}{3}[\/latex]<\/td>\r\n<td style=\"width: 80px\">[latex]\\frac{\\pi}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 30px\"><strong> tan (<em>x<\/em>) <\/strong><\/td>\r\n<td style=\"width: 80px\">undefined<\/td>\r\n<td style=\"width: 80px\">[latex]\u2212\\sqrt{3}[\/latex]<\/td>\r\n<td style=\"width: 80px\">\u20131<\/td>\r\n<td style=\"width: 80px\">[latex]\u2212\\dfrac{\\sqrt{3}}{3}[\/latex]<\/td>\r\n<td style=\"width: 80px\">0<\/td>\r\n<td style=\"width: 80px\">[latex]\\dfrac{\\sqrt{3}}{3}[\/latex]<\/td>\r\n<td style=\"width: 80px\">1<\/td>\r\n<td style=\"width: 80px\">[latex]\\sqrt{3}[\/latex]<\/td>\r\n<td style=\"width: 80px\">undefined<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThese points will help us draw our graph, but we need to determine how the graph behaves where it is undefined. If we look more closely at values when [latex]\\frac{\\pi}{3}&lt;x&lt;\\frac{\\pi}{2}[\/latex], we can use a table to look for a trend. Because [latex]\\frac{\\pi}{3}\\approx 1.05[\/latex] and [latex]\\frac{\\pi}{2}\\approx 1.57[\/latex], we will evaluate x at radian measures 1.05 &lt; <em>x<\/em> &lt; 1.57 as shown in the table below.\r\n<table id=\"Table_06_02_01\" summary=\"Two rows and five columns. First row is labeled x and second row is labeled tangent of x. Th table has ordered pairs of these column values: (1.3, 3.6), (1.5, 14.1), (1.55, 48.1), (1.56, 92.6).\">\r\n<tbody>\r\n<tr>\r\n<td><em><strong> x <\/strong><\/em><\/td>\r\n<td>1.3<\/td>\r\n<td>1.5<\/td>\r\n<td>1.55<\/td>\r\n<td>1.56<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong> tan <em>x <\/em><\/strong><\/td>\r\n<td>3.6<\/td>\r\n<td>14.1<\/td>\r\n<td>48.1<\/td>\r\n<td>92.6<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAs <em>x<\/em> approaches [latex]\\frac{\\pi}{2}[\/latex], the outputs of the function get larger and larger. Because [latex]y=\\tan x[\/latex] is an odd function, we see the corresponding table of negative values in the table below.\r\n<table id=\"Table_06_02_02\" summary=\"Two rows and five columns. First row is labeled x and second row is labeled tangent of x. Th table has ordered pairs of these column values: (-1.3, -3.6), (-1.5, -14.1), (-1.55, -48.1), (-1.56, -92.6).\">\r\n<tbody>\r\n<tr>\r\n<td><em><strong> x <\/strong><\/em><\/td>\r\n<td>\u22121.3<\/td>\r\n<td>\u22121.5<\/td>\r\n<td>\u22121.55<\/td>\r\n<td>\u22121.56<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong> tan <em>x <\/em><\/strong><\/td>\r\n<td>\u22123.6<\/td>\r\n<td>\u221214.1<\/td>\r\n<td>\u221248.1<\/td>\r\n<td>\u221292.6<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can see that, as <em>x<\/em> approaches [latex]\u2212\\dfrac{\\pi}{2}[\/latex], the outputs get smaller and smaller. Remember that there are some values of <em>x<\/em> for which cos <em>x<\/em> = 0. For example, [latex]\\cos\\left(\\frac{\\pi}{2}\\right)=0[\/latex] and [latex]\\cos\\left(\\frac{3\\pi}{2}\\right)=0[\/latex]. At these values, the <strong>tangent function<\/strong> is undefined, so the graph of [latex]y=\\tan x[\/latex] has discontinuities at [latex]x=\\frac{\\pi}{2}[\/latex] and [latex]\\frac{3\\pi}{2}[\/latex]. At these values, the graph of the tangent has vertical asymptotes. Figure 1\u00a0represents the graph of [latex]y=\\tan x[\/latex]. The tangent is positive from 0 to [latex]\\frac{\\pi}{2}[\/latex] and from <em>\u03c0<\/em> to [latex]\\frac{3\\pi}{2}[\/latex], corresponding to quadrants I and III of the unit circle.\r\n<figure id=\"Figure_06_02_001\" class=\"small ui-has-child-figcaption\">\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\/3675\/2018\/09\/27163804\/CNX_Precalc_Figure_06_02_001.jpg\" alt=\"A graph of y=tangent of x. Asymptotes at -pi over 2 and pi over 2.\" width=\"487\" height=\"316\" \/> <b>Figure 26.<\/b> Graph of the tangent function[\/caption]<\/figure>\r\n<h2>Graphing Variations of <em>y<\/em> = tan <em>x<\/em><\/h2>\r\nAs with the sine and cosine functions, the <strong>tangent<\/strong> function can be described by a general equation.\r\n<div>\r\n<div style=\"text-align: center\">[latex]y=A\\tan(Bx)[\/latex]<\/div>\r\n<\/div>\r\nWe can identify horizontal and vertical stretches and compressions using values of A and B. The horizontal stretch can typically be determined from the period of the graph. With tangent graphs, it is often necessary to determine a vertical stretch using a point on the graph.\r\n\r\nBecause there are no maximum or minimum values of a tangent function, the term <em>amplitude<\/em> cannot be interpreted as it is for the sine and cosine functions. Instead, we will use the phrase <em>stretching\/compressing factor<\/em> when referring to the constant A.\r\n<div class=\"textbox\"><header>\r\n<h3>A General Note: Features of the Graph of <em>y<\/em> = <em>A<\/em>tan(<em>Bx<\/em>)<\/h3>\r\n<\/header>\r\n<ul>\r\n \t<li>The stretching factor is |<em>A<\/em>| .<\/li>\r\n \t<li>The period is [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\r\n \t<li>The domain is all real numbers <em>x<\/em>, where [latex]x\\ne \\frac{\\pi}{2|B|} + \\frac{\\pi}{|B|} k[\/latex] such that <em>k<\/em> is an integer.<\/li>\r\n \t<li>The range is [latex]\\left(-\\infty,\\infty\\right)[\/latex].<\/li>\r\n \t<li>The asymptotes occur at [latex]x=\\frac{\\pi}{2|B|} + \\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\r\n \t<li>[latex]y = A \\tan (Bx)[\/latex] is an odd function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Graphing One Period of a Stretched or Compressed Tangent Function<\/h2>\r\nWe can use what we know about the properties of the <strong>tangent function<\/strong> to quickly sketch a graph of any stretched and\/or compressed tangent function of the form [latex]f(x)=A\\tan(Bx)[\/latex]. We focus on a single <strong>period<\/strong> of the function including the origin, because the periodic property enables us to extend the graph to the rest of the function\u2019s domain if we wish. Our limited domain is then the interval [latex](\u2212\\frac{P}{2}, \\frac{P}{2})[\/latex] and the graph has vertical asymptotes at [latex]\\pm \\frac{P}{2}[\/latex] where [latex]P=\\frac{\\pi}{B}[\/latex]. On [latex](\u2212\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})[\/latex], the graph will come up from the left asymptote at [latex]x=\u2212\\dfrac{\\pi}{2}[\/latex], cross through the origin, and continue to increase as it approaches the right asymptote at [latex]x=\\frac{\\pi}{2}[\/latex]. To make the function approach the asymptotes at the correct rate, we also need to set the vertical scale by actually evaluating the function for at least one point that the graph will pass through. For example, we can use\r\n<div>\r\n<div style=\"text-align: center\">[latex]f\\left(\\frac{P}{4}\\right)=A \\tan\\left(B\\frac{P}{4}\\right)=A\\tan\\left(B\\frac{\\pi}{4B}\\right)=A[\/latex]<\/div>\r\n<\/div>\r\nbecause \u00a0[latex]\\tan\\left(\\frac{\\pi}{4}\\right)=1[\/latex].\r\n<div class=\"textbox\">\r\n<h3>How To: Given the function [latex]f(x)=A\\tan(Bx)[\/latex], graph one period.<\/h3>\r\n<ol>\r\n \t<li>Identify the stretching factor, |A|.<\/li>\r\n \t<li>Identify <em>B<\/em> and determine the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\r\n \t<li>Draw vertical asymptotes at \u00a0[latex]x=\u2212\\dfrac{P}{2}[\/latex] and [latex]x=\\frac{P}{2}[\/latex].<\/li>\r\n \t<li>For <em>A<\/em> &gt; 0 , the graph approaches the left asymptote at negative output values and the right asymptote at positive output values (reverse for <em>A<\/em> &lt; 0 ).<\/li>\r\n \t<li>Plot reference points at [latex]\\left(\\frac{P}{4},A\\right)[\/latex]\u00a0(0, 0), and ([latex]\u2212\\dfrac{P}{4}[\/latex],\u2212 A), and draw the graph through these points.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 14: Sketching a Compressed Tangent<\/h3>\r\nSketch a graph of one period of the function [latex]y=0.5\\tan\\left(\\frac{\\pi}{2}x\\right)[\/latex].\r\n\r\n[reveal-answer q=\"302986\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"302986\"]\r\n\r\nFirst, we identify <em>A<\/em> and B.\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\/3675\/2018\/09\/27163807\/CNX_Precalc_Figure_06_02_002.jpg\" alt=\"An illustration of equations showing that A is the coefficient of tangent and B is the coefficient of x, which is within the tangent function.\" width=\"487\" height=\"113\" \/> <b>Figure 27<\/b>[\/caption]\r\n\r\nBecause [latex]A=0.5[\/latex] and [latex]B=\\frac{\\pi}{2}[\/latex], we can find the <strong>stretching\/compressing factor<\/strong> and period. The period is [latex]\\frac{\\pi}{\\frac{\\pi}{2}}=2[\/latex], so the asymptotes are at [latex]x=\\pm 1[\/latex]. At a quarter period from the origin, we have\r\n<p style=\"text-align: center\">[latex]\\begin{align}f(0.5)&amp;=0.5\\tan\\left(\\frac{0.5\\pi}{2}\\right)\\\\ &amp;=0.5\\tan(\\frac{\\pi}{4})\\\\ &amp;=0.5 \\end{align}[\/latex]<\/p>\r\nThis means the curve must pass through the points(0.5,0.5),(0,0),and(\u22120.5,\u22120.5).The only inflection point is at the origin. Figure shows the graph of one period of the function.\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\/3675\/2018\/09\/27163810\/CNX_Precalc_Figure_06_02_003.jpg\" alt=\"A graph of one period of a modified tangent function, with asymptotes at x=-1 and x=1.\" width=\"487\" height=\"258\" \/> <b>Figure 28<\/b>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nSketch a graph of [latex]f(x)=3\\tan\\left(\\frac{\\pi}{6}x\\right)[\/latex].\r\n\r\n[reveal-answer q=\"547078\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"547078\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163917\/CNX_Precalc_Figure_06_02_004.jpg\" alt=\"A graph of two periods of a modified tangent function, with asymptotes at x=-3 and x=3.\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]174880[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Graphing One Period of a Shifted Tangent Function<\/h2>\r\nNow that we can graph a <strong>tangent function<\/strong> that is stretched or compressed, we will add a vertical and\/or horizontal (or phase) shift. In this case, we add <em>C<\/em> and <em>D<\/em> to the general form of the tangent function.\r\n<div>\r\n<div style=\"text-align: center\">[latex]f(x)=A\\tan(Bx\u2212C)+D[\/latex]<\/div>\r\n<\/div>\r\nThe graph of a transformed tangent function is different from the basic tangent function tan x in several ways:\r\n<div class=\"textbox\"><header>\r\n<h3>A General Note: Features of the Graph of [latex]y = A\\tan\\left(Bx\u2212C\\right)+D[\/latex]<\/h3>\r\n<\/header>\r\n<ul>\r\n \t<li>The stretching factor is |<em>A<\/em>|.<\/li>\r\n \t<li>The period is [latex]\\frac{\\pi}{|B|}[\/latex].<\/li>\r\n \t<li>The domain is [latex]x\\ne\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\r\n \t<li>The range is (\u2212\u221e,\u2212|<em>A<\/em>|] \u222a [|<em>A<\/em>|, \u221e).<\/li>\r\n \t<li>The vertical asymptotes occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\r\n \t<li>There is no amplitude.<\/li>\r\n \t<li>[latex]y=A\\tan(Bx)[\/latex] is an odd function because it is the quotient of odd and even functions (sine and cosine respectively).<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given the function [latex]y=A\\tan(Bx\u2212C)+D[\/latex], sketch the graph of one period.<\/h3>\r\n<ol>\r\n \t<li>Express the function given in the form [latex]y=A\\tan(Bx\u2212C)+D[\/latex].<\/li>\r\n \t<li>Identify the <strong>stretching\/compressing<\/strong> factor, |A|.<\/li>\r\n \t<li>Identify <em>B<\/em> and determine the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\r\n \t<li>Identify <em>C<\/em> and determine the phase shift, [latex]\\frac{C}{B}[\/latex].<\/li>\r\n \t<li>Draw the graph of [latex]y=A\\tan(Bx)[\/latex] shifted to the right by [latex]\\frac{C}{B}[\/latex] and up by <em>D<\/em>.<\/li>\r\n \t<li>Sketch the vertical asymptotes, which occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\r\n \t<li>Plot any three reference points and draw the graph through these points.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 15: Graphing One Period of a Shifted Tangent Function<\/h3>\r\nGraph one period of the function [latex]y=\u22122\\tan(\\pi x+\\pi)\u22121[\/latex].\r\n\r\n[reveal-answer q=\"385350\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"385350\"]\r\n\r\n<strong>Step 1.<\/strong> The function is already written in the form [latex]y=A\\tan(Bx\u2212C)+D[\/latex].\r\n\r\n<strong>Step 2.<\/strong>\u00a0[latex]A=\u22122[\/latex], so the stretching factor is [latex]|A|=2[\/latex].\r\n\r\n<strong>Step 3.<\/strong>\u00a0[latex]B=\\pi[\/latex], so the period is [latex]P=\\frac{\\pi}{|B|}=\\frac{\\pi}{\\pi}=1[\/latex].\r\n\r\n<strong>Step 4.<\/strong>\u00a0[latex]C=\u2212\\pi[\/latex], so the phase shift is [latex]\\dfrac{C}{B}=\\dfrac{\u2212\\pi}{\\pi}=\u22121[\/latex].\r\n\r\n<strong>Step 5\u20137.<\/strong> The asymptotes are at [latex]x=\u2212\\frac{3}{2}[\/latex] and [latex]x=\u2212\\frac{1}{2}[\/latex] and the three recommended reference points are (\u22121.25, 1), (\u22121,\u22121), and (\u22120.75, \u22123). The graph is shown in Figure 4.\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\/3675\/2018\/09\/27163812\/CNX_Precalc_Figure_06_02_005.jpg\" alt=\"A graph of one period of a shifted tangent function, with vertical asymptotes at x=-1.5 and x=-0.5.\" width=\"487\" height=\"193\" \/> <b>Figure 29<\/b>[\/caption]\r\n<h4>Analysis of the Solution<\/h4>\r\nNote that this is a decreasing function because <em>A<\/em> &lt; 0.\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nHow would the graph in Example 2\u00a0look different if we made <em>A<\/em> = 2 instead of \u22122?\r\n\r\n[reveal-answer q=\"560477\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"560477\"]\r\n\r\nIt would be reflected across the line [latex]y=\u22121[\/latex], becoming an increasing function.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given the graph of a tangent function, identify horizontal and vertical stretches.<\/h3>\r\n<ol>\r\n \t<li>Find the period <em>P<\/em> from the spacing between successive vertical asymptotes or <em>x<\/em>-intercepts.<\/li>\r\n \t<li>Write [latex]f(x)=A\\tan\\left(\\frac{\\pi}{P}x\\right)[\/latex].<\/li>\r\n \t<li>Determine a convenient point (<em>x<\/em>, <em>f<\/em>(<em>x<\/em>)) on the given graph and use it to determine <em>A<\/em>.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 16: Identifying the Graph of a Stretched Tangent<\/h3>\r\nFind a formula for the function graphed in Figure 5.\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\/3675\/2018\/09\/27163814\/CNX_Precalc_Figure_06_02_006.jpg\" alt=\"A graph of two periods of a modified tangent function, with asymptotes at x=-4 and x=4.\" width=\"487\" height=\"256\" \/> <b>Figure 30<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"606896\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"606896\"]\r\n\r\nThe graph has the shape of a tangent function.\r\n\r\n<strong>Step 1.<\/strong> One cycle extends from \u20134 to 4, so the period is [latex]P=8[\/latex]. Since [latex]P=\\frac{\\pi}{|B|}[\/latex], we have [latex]B=\\frac{\\pi}{P}=\\frac{\\pi}{8}[\/latex].\r\n\r\n<strong>Step 2.<\/strong> The equation must have the [latex]\\text{form}f(x)=A\\tan\\left(\\frac{\\pi}{8}x\\right)[\/latex].\r\n\r\n<strong>Step 3.<\/strong> To find the vertical stretch <em>A<\/em>, we can use the point (2,2).\r\n<p style=\"text-align: center\">[latex]2=A\\tan\\left(\\frac{\\pi}{8}\\times2\\right)=A\\tan\\left(\\frac{\\pi}{4}\\right)[\/latex]<\/p>\r\nBecause [latex]\\tan\\left(\\frac{\\pi}{4}\\right)=1[\/latex], <em>A<\/em> = 2.\r\n\r\nThis function would have a formula [latex]f(x)=2\\tan\\left(\\frac{\\pi}{8}x\\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nFind a formula for the function in Figure 6.\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\/3675\/2018\/09\/27163816\/CNX_Precalc_Figure_06_02_007.jpg\" alt=\"A graph of four periods of a modified tangent function, Vertical asymptotes at -3pi\/4, -pi\/4, pi\/4, and 3pi\/4.\" width=\"487\" height=\"315\" \/> <b>Figure 31<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"359527\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"359527\"]\r\n\r\n[latex]g(x)=4\\tan(2x)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]129731[\/ohm_question]\r\n\r\n<\/div>\r\n<div>\r\n<h2>Using the Graphs of Trigonometric Functions to Solve Real-World Problems<\/h2>\r\nMany real-world scenarios represent periodic functions and may be modeled by trigonometric functions. As an example, let\u2019s return to the scenario from the section opener. Have you ever observed the beam formed by the rotating light on a police car and wondered about the movement of the light beam itself across the wall? The periodic behavior of the distance the light shines as a function of time is obvious, but how do we determine the distance? We can use the tangent function .\r\n<div class=\"textbox shaded\">\r\n<h3>Example 17: Using Trigonometric Functions to Solve Real-World Scenarios<\/h3>\r\nSuppose the function [latex]y=5\\tan\\left(\\frac{\\pi}{4}t\\right)[\/latex] marks the distance in the movement of a light beam from the top of a police car across a wall where <em>t<\/em> is the time in seconds and <em>y<\/em> is the distance in feet from a point on the wall directly across from the police car.\r\n<ol>\r\n \t<li>Find and interpret the stretching factor and period.<\/li>\r\n \t<li>Graph on the interval [0, 5].<\/li>\r\n \t<li>Evaluate <em>f<\/em>(1) and discuss the function\u2019s value at that input.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"351813\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"351813\"]\r\n<ol>\r\n \t<li>We know from the general form of \u00a0[latex]y=A\\tan(Bt)\\\\[\/latex] \u00a0that |<em>A<\/em>| is the stretching factor and \u03c0 B is the period.\r\n<figure id=\"Image_06_02_022\" 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\/3675\/2018\/09\/27163819\/CNX_Precalc_Figure_06_02_022.jpg\" alt=\"A graph showing that variable A is the coefficient of the tangent function and variable B is the coefficient of x, which is within that tangent function.\" width=\"487\" height=\"107\" \/> <b>Figure 32<\/b>[\/caption]<\/figure>\r\nWe see that the stretching factor is 5. This means that the beam of light will have moved 5 ft after half the period.\r\n\r\nThe period is [latex]\\frac{\\pi}{\\frac{\\pi}{4}}=\\frac{\\pi}{1}\\times \\frac{4}{\\pi}=4[\/latex]. This means that every 4 seconds, the beam of light sweeps the wall. The distance from the spot across from the police car grows larger as the police car approaches.<\/li>\r\n \t<li>To graph the function, we draw an asymptote at [latex]t=2[\/latex] and use the stretching factor and period. See Figure 8.\r\n<figure id=\"Image_06_02_021\" 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\/3675\/2018\/09\/27163821\/CNX_Precalc_Figure_06_02_021n.jpg\" alt=\"A graph of one period of a modified tangent function, with a vertical asymptote at x=4.\" width=\"487\" height=\"319\" \/> <b>Figure 33<\/b>[\/caption]<\/figure>\r\n<\/li>\r\n \t<li>period: [latex]f(1)=5\\tan \\left(\\frac{\\pi}{4}\\left(1\\right)\\right)=5\\left(1\\right)=5[\/latex]; after 1 second, the beam of has moved 5 ft from the spot across from the police car.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nhttps:\/\/youtu.be\/ssjG9kE25OY\r\n<h2>Analyzing the Graphs of y = sec x and y = cscx and Their Variations<\/h2>\r\nThe <strong>secant<\/strong> was defined by the <strong>reciprocal identity<\/strong>\u00a0[latex]\\sec x=\\frac{1}{\\cos x}[\/latex]. Notice that the function is undefined when the cosine is 0, leading to vertical asymptotes at [latex]\\frac{\\pi}{2},\\frac{3\\pi}{2}\\text{, etc}[\/latex].\u00a0Because the cosine is never more than 1 in absolute value, the secant, being the reciprocal, will never be less than 1 in absolute value.\r\n\r\nWe can graph [latex]y=\\sec x[\/latex] by observing the graph of the cosine function because these two functions are reciprocals of one another. See Figure 9. The graph of the cosine is shown as a dashed orange wave so we can see the relationship. Where the graph of the cosine function decreases, the graph of the <strong>secant function<\/strong> increases. Where the graph of the cosine function increases, the graph of the secant function decreases. When the cosine function is zero, the secant is undefined.\r\n\r\nThe secant graph has vertical asymptotes at each value of <em>x<\/em> where the cosine graph crosses the <em>x<\/em>-axis; we show these in the graph below with dashed vertical lines, but will not show all the asymptotes explicitly on all later graphs involving the secant and cosecant.\r\n\r\nNote that, because cosine is an even function, secant is also an even function. That is, [latex]\\sec(\u2212x)=\\sec x[\/latex].\r\n<figure id=\"Figure_06_02_008\" class=\"small ui-has-child-figcaption\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163823\/CNX_Precalc_Figure_06_02_008.jpg\" alt=\"A graph of cosine of x and secant of x. Asymptotes for secant of x shown at -3pi\/2, -pi\/2, pi\/2, and 3pi\/2.\" \/>\r\n<div style=\"text-align: center\"><strong>Figure 34.\u00a0<\/strong>Graph of the secant function, [latex]f(x)=\\sec x=\\frac{1}{\\cos x}[\/latex]<\/div><\/figure>\r\nAs we did for the tangent function, we will again refer to the constant |<em>A<\/em>| as the stretching factor, not the amplitude.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Features of the Graph of <em>y<\/em> = <em>A<\/em>sec(<em>Bx<\/em>)<\/h3>\r\n<ul>\r\n \t<li>The stretching factor is |<em>A<\/em>|.<\/li>\r\n \t<li>The period is [latex]\\frac{2\\pi}{|B|}[\/latex].<\/li>\r\n \t<li>The domain is [latex]x\\ne \\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\r\n \t<li>The range is (\u2212\u221e, \u2212|<em>A<\/em>|] \u222a [|<em>A<\/em>|, \u221e).<\/li>\r\n \t<li>The vertical asymptotes occur at [latex]x=\\frac{\\pi}{2|B|}k [\/latex], where <em>k<\/em> is an odd integer.<\/li>\r\n \t<li>There is no amplitude.<\/li>\r\n \t<li>[latex]y=A\\sec(Bx)[\/latex] is an even function because cosine is an even function.<\/li>\r\n<\/ul>\r\n<\/div>\r\nSimilar to the secant, the <strong>cosecant<\/strong> is defined by the reciprocal identity [latex]\\csc x=1\\sin x[\/latex]. Notice that the function is undefined when the sine is 0, leading to a vertical asymptote in the graph at 0, \u03c0, etc. Since the sine is never more than 1 in absolute value, the cosecant, being the reciprocal, will never be less than 1 in absolute value.\r\n\r\nWe can graph [latex]y=\\csc x[\/latex] by observing the graph of the sine function because these two functions are reciprocals of one another. See Figure 10. The graph of sine is shown as a dashed orange wave so we can see the relationship. Where the graph of the sine function decreases, the graph of the <strong>cosecant function<\/strong> increases. Where the graph of the sine function increases, the graph of the cosecant function decreases.\r\n\r\nThe cosecant graph has vertical asymptotes at each value of <em>x<\/em> where the sine graph crosses the <em>x<\/em>-axis; we show these in the graph below with dashed vertical lines.\r\n\r\nNote that, since sine is an odd function, the cosecant function is also an odd function. That is, [latex]\\csc(\u2212x)=\u2212\\csc x[\/latex].\r\n\r\nThe graph of cosecant, which is shown in Figure 10, is similar to the graph of secant.\r\n<figure id=\"Figure_06_02_009\" class=\"small ui-has-child-figcaption\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163825\/CNX_Precalc_Figure_06_02_009.jpg\" alt=\"A graph of cosecant of x and sin of x. Five vertical asymptotes shown at multiples of pi.\" \/>\r\n<div style=\"text-align: center\"><strong>Figure 35.\u00a0<\/strong>The graph of the cosecant function, [latex]f(x)=\\csc x=\\frac{1}{\\sin x}\/latex]<\/div><\/figure>\r\n<div class=\"textbox\"><header>\r\n<h3>A General Note: Features of the Graph of [latex]y=A\\csc(Bx)<\/h3>\r\n<\/header>\r\n<ul>\r\n \t<li>The stretching factor is |<em>A<\/em>|.<\/li>\r\n \t<li>The period is [latex]\\frac{2\\pi}{|B|}[\/latex].<\/li>\r\n \t<li>The domain is [latex]x\\ne\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\r\n \t<li>The range is ( \u2212\u221e, \u2212|A|] \u222a [|A|, \u221e).<\/li>\r\n \t<li>The asymptotes occur at [latex]x=\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\r\n \t<li>[latex]y=A\\csc(Bx)[\/latex] is an odd function because sine is an odd function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Graphing Variations of <em>y<\/em> = sec <em>x<\/em> and <em>y\u00a0<\/em>= csc <em>x<\/em><\/h2>\r\nFor shifted, compressed, and\/or stretched versions of the secant and cosecant functions, we can follow similar methods to those we used for tangent and cotangent. That is, we locate the vertical asymptotes and also evaluate the functions for a few points (specifically the local extrema). If we want to graph only a single period, we can choose the interval for the period in more than one way. The procedure for secant is very similar, because the cofunction identity means that the secant graph is the same as the cosecant graph shifted half a period to the left. Vertical and phase shifts may be applied to the <strong>cosecant function<\/strong> in the same way as for the secant and other functions. The equations become the following.\r\n<div>\r\n<div style=\"text-align: center\">[latex]y=A\\sec(Bx\u2212C)+D[\/latex]<\/div>\r\n<\/div>\r\n<div style=\"text-align: center\">[latex]y=A\\csc(Bx\u2212C)+D[\/latex]<\/div>\r\n<div class=\"textbox\"><header>\r\n<h3>A General Note: Features of the Graph of [latex]y=A\\sec(Bx\u2212C)+D[\/latex]<\/h3>\r\n<\/header>\r\n<ul>\r\n \t<li>The stretching factor is |<em>A<\/em>|.<\/li>\r\n \t<li>The period is [latex]\\frac{2\\pi}{|B|}[\/latex].<\/li>\r\n \t<li>The domain is [latex]x\\ne \\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\r\n \t<li>The range is (\u2212\u221e, \u2212|<em>A<\/em>|] \u222a [|<em>A<\/em>|, \u221e).<\/li>\r\n \t<li>The vertical asymptotes occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\r\n \t<li>There is no amplitude.<\/li>\r\n \t<li>[latex]y=A\\sec(Bx)[\/latex] is an even function because cosine is an even function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\"><header>\r\n<h3>A General Note: Features of the Graph of [latex]y=A\\csc(Bx\u2212C)+D[\/latex]<\/h3>\r\n<\/header>\r\n<ul>\r\n \t<li>The stretching factor is |<em>A<\/em>|.<\/li>\r\n \t<li>The period is [latex]\\frac{2\\pi}{|B|}[\/latex].<\/li>\r\n \t<li>The domain is [latex]x\\ne\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\r\n \t<li>The range is (\u2212\u221e, \u2212|<em>A<\/em>|] \u222a [|<em>A<\/em>|, \u221e).<\/li>\r\n \t<li>The vertical asymptotes occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\r\n \t<li>There is no amplitude.<\/li>\r\n \t<li>[latex]y=A\\csc(Bx)[\/latex] is an odd function because sine is an odd function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a function of the form [latex]y=A\\sec(Bx)[\/latex], graph one period.<\/h3>\r\n<ol>\r\n \t<li>Express the function given in the form [latex]y=A\\sec(Bx)[\/latex].<\/li>\r\n \t<li>Identify the stretching\/compressing factor, |A|.<\/li>\r\n \t<li>Identify <em>B<\/em> and determine the period, [latex]P=\\frac{2\\pi}{|B|}[\/latex].<\/li>\r\n \t<li>Sketch the graph of [latex]y=A\\cos(Bx)[\/latex].<\/li>\r\n \t<li>Use the reciprocal relationship between [latex]y=\\cos x[\/latex] and [latex]y=\\sec x[\/latex] to draw the graph of [latex]y=A\\sec(Bx)[\/latex].<\/li>\r\n \t<li>Sketch the asymptotes.<\/li>\r\n \t<li>Plot any two reference points and draw the graph through these points.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 18: Graphing a Variation of the Secant Function<\/h3>\r\nGraph one period of [latex]f(x)=2.5\\sec(0.4x)[\/latex].\r\n\r\n[reveal-answer q=\"926159\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"926159\"]\r\n\r\n<strong>Step 1.<\/strong> The given function is already written in the general form, [latex]y=A\\sec(Bx)[\/latex].\r\n<strong>Step 2.<\/strong>\u00a0[latex]A=2.5[\/latex] so the stretching factor is 2.5.\r\n<strong>Step 3.<\/strong>\u00a0[latex]B=0.4[\/latex], so [latex]P=\\frac{2\\pi}{0.4}=5\\pi[\/latex]. The period is 5\u03c0 units.\r\n<strong>Step 4.<\/strong> Sketch the graph of the function [latex]g(x)=2.5\\cos(0.4x)[\/latex].\r\n<strong>Step 5.<\/strong> Use the reciprocal relationship of the cosine and secant functions to draw the cosecant function.\r\n<strong>Steps 6\u20137.<\/strong> Sketch two asymptotes at [latex]x=1.25\\pi[\/latex]\u00a0and [latex]x=3.75\\pi[\/latex]. We can use two reference points, the local minimum at (0, 2.5) and the local maximum at (2.5\u03c0, \u22122.5). Figure 11 shows the graph.\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\/3675\/2018\/09\/27163829\/CNX_Precalc_Figure_06_02_010.jpg\" alt=\"A graph of one period of a modified secant function, which looks like an upward facing prarbola and a downward facing parabola.\" width=\"487\" height=\"567\" \/> <b>Figure 36<\/b>[\/caption]\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nGraph one period of [latex]f(x)=\u22122.5\\sec(0.4x)[\/latex].\r\n\r\n[reveal-answer q=\"945046\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"945046\"]\r\n\r\nThis is a vertical reflection of the preceding graph because <em>A<\/em> is negative.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163919\/CNX_Precalc_Figure_06_02_011.jpg\" alt=\"A graph of one period of a modified secant function, which looks like an downward facing prarbola and a upward facing parabola.\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<h3>Do the vertical shift and stretch\/compression affect the secant\u2019s range?<\/h3>\r\n<em>Yes. The range of<\/em>\u00a0[latex]f(x) = A\\sec(Bx \u2212 C) + D[\/latex] is ( \u2212\u221e, \u2212|<em>A<\/em>| + <em>D<\/em>] \u222a [|<em>A<\/em>| + <em>D<\/em>, \u221e).\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a function of the form [latex]f(x)=A\\sec (Bx\u2212C)+D[\/latex], graph one period.<\/h3>\r\n<ol>\r\n \t<li>Express the function given in the form [latex]y=A\\sec(Bx\u2212C)+D[\/latex].<\/li>\r\n \t<li>Identify the stretching\/compressing factor, |<em>A<\/em>|.<\/li>\r\n \t<li>Identify <em>B<\/em> and determine the period, [latex]\\frac{2\\pi}{|B|}[\/latex].<\/li>\r\n \t<li>Identify <em>C<\/em> and determine the phase shift, [latex]\\frac{C}{B}[\/latex].<\/li>\r\n \t<li>Draw the graph of [latex]y=A\\sec(Bx)[\/latex]. but shift it to the right by [latex]\\frac{C}{B}[\/latex] and up by <em>D<\/em>.<\/li>\r\n \t<li>Sketch the vertical asymptotes, which occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 19: Graphing a Variation of the Secant Function<\/h3>\r\nGraph one period of [latex]y=4\\sec \\left(\\frac{\\pi}{3}x\u2212\\frac{\\pi}{2}\\right)+1[\/latex].\r\n\r\n[reveal-answer q=\"429424\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"429424\"]\r\n\r\n<strong>Step 1.<\/strong> Express the function given in the form [latex]y=4\\sec \\left(\\frac{\\pi}{3}x\u2212\\frac{\\pi}{2}\\right)+1[\/latex].\r\n\r\n<strong>Step 2.<\/strong> The stretching\/compressing factor is |<em>A<\/em>| = 4.\r\n\r\n<strong>Step 3.<\/strong> The period is\r\n<p style=\"text-align: center\">[latex]\\begin{align} \\frac{2\\pi}{|B|}&amp;=\\frac{2\\pi}{\\frac{\\pi}{3}}\\\\ &amp;=\\frac{2\\pi}{1}\\times\\frac{3}{\\pi}\\\\ &amp;=6 \\end{align}[\/latex]<\/p>\r\n<strong>Step 4.<\/strong> The phase shift is\r\n<p style=\"text-align: center\">[latex]\\begin{align}\\frac{C}{B}&amp;=\\frac{\\frac{\\pi}{2}}{\\frac{\\pi}{3}} \\\\ &amp;=\\frac{\\pi}{2} \\times \\frac{3}{\\pi} \\\\ &amp;=1.5 \\end{align}[\/latex]<\/p>\r\n<strong>Step 5.<\/strong> Draw the graph of [latex]y=A\\sec(Bx)[\/latex],but shift it to the right by [latex]\\frac{C}{B}=1.5[\/latex] and up by <em>D\u00a0<\/em>= 6.\r\n\r\n<strong>Step 6.<\/strong> Sketch the vertical asymptotes, which occur at <em>x\u00a0<\/em>= 0, <em>x<\/em> = 3, and <em>x<\/em> = 6. There is a local minimum at (1.5, 5) and a local maximum at (4.5, \u22123). Figure 12 shows the graph.\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\/3675\/2018\/09\/27163831\/CNX_Precalc_Figure_06_02_012-1.jpg\" alt=\"\" width=\"487\" height=\"318\" \/> <b>Figure 37<\/b>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nGraph one period of [latex]f(x)=\u22126\\sec(4x+2)\u22128[\/latex].\r\n\r\n[reveal-answer q=\"142167\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"142167\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163922\/CNX_Precalc_Figure_06_02_013.jpg\" alt=\"A graph of one period of a modified secant function. There are two vertical asymptotes, one at approximately x=-pi\/20 and one approximately at 3pi\/16.\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]174885[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<h4>The domain of [latex]\\csc x[\/latex] was given to be all <em>x<\/em> such that [latex]x\\ne k\\pi[\/latex] for any integer <em>k<\/em>. Would the domain of\u00a0[latex]y=A\\csc(Bx\u2212C)+D[\/latex] be [latex]x\\ne\\frac{C+k\\pi}{B}[\/latex]?<\/h4>\r\n<em>Yes. The excluded points of the domain follow the vertical asymptotes. Their locations show the horizontal shift and compression or expansion implied by the transformation to the original function\u2019s input.<\/em>\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a function of the form [latex]y=A\\csc(Bx)[\/latex], graph one period.<\/h3>\r\n<ol>\r\n \t<li>Express the function given in the form [latex]y=A\\csc(Bx)[\/latex].<\/li>\r\n \t<li>|<em>A<\/em>|.<\/li>\r\n \t<li>Identify <em>B<\/em> and determine the period, [latex]P=\\frac{2\\pi}{|B|}[\/latex].<\/li>\r\n \t<li>Draw the graph of [latex]y=A\\sin(Bx)[\/latex].<\/li>\r\n \t<li>Use the reciprocal relationship between [latex]y=\\sin x[\/latex] and [latex]y=\\csc x[\/latex] to draw the graph of [latex]y=A\\csc(Bx) [\/latex].<\/li>\r\n \t<li>Sketch the asymptotes.<\/li>\r\n \t<li>Plot any two reference points and draw the graph through these points.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 20: Graphing a Variation of the Cosecant Function<\/h3>\r\nGraph one period of [latex]f(x)=\u22123\\csc(4x)[\/latex].\r\n\r\n[reveal-answer q=\"194858\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"194858\"]\r\n\r\n<strong>Step 1.<\/strong> The given function is already written in the general form, [latex]y=A\\csc(Bx)[\/latex].\r\n\r\n<strong>Step 2. <\/strong>[latex]|A|=|\u22123|=3[\/latex], so the stretching factor is 3.\r\n\r\n<strong>Step 3.<\/strong> [latex]B=4\\text{, so}P=\\frac{2\\pi}{4}=\\frac{\\pi}{2}[\/latex].The period is [latex]\\frac{\\pi}{2}[\/latex] units.\r\n\r\n<strong>Step 4.<\/strong> Sketch the graph of the function [latex]g(x)=\u22123\\sin(4x)[\/latex].\r\n\r\n<strong>Step 5.<\/strong> Use the reciprocal relationship of the sine and cosecant functions to draw the cosecant function.\r\n\r\n<strong>Steps 6\u20137.<\/strong> Sketch three asymptotes at [latex]x=0\\text{, }x=\\frac{\\pi}{4}\\text{, and }x=\\frac{\\pi}{2}[\/latex].We can use two reference points, the local maximum at [latex]\\left(\\frac{\\pi}{8}\\text{, }\u22123\\right)[\/latex] and the local minimum at [latex]\\left(\\frac{3\\pi}{8}\\text{, }3\\right)[\/latex]. Figure 13 shows the graph.\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\/3675\/2018\/09\/27163833\/CNX_Precalc_Figure_06_02_023.jpg\" alt=\"A graph of one period of a cosecant function. There are vertical asymptotes at x=0, x=pi\/4, and x=pi\/2.\" width=\"487\" height=\"686\" \/> <b>Figure 38<\/b>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nGraph one period of [latex]f(x)=0.5\\csc(2x)[\/latex].\r\n\r\n[reveal-answer q=\"267711\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"267711\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163924\/CNX_Precalc_Figure_06_02_023b.jpg\" alt=\"A graph of one period of a modified secant function, which looks like an downward facing prarbola and a upward facing parabola.\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a function of the form [latex]f(x)=A\\csc(Bx\u2212C)+D[\/latex], graph one period.<\/h3>\r\n<ol>\r\n \t<li>Express the function given in the form [latex]y=A\\csc(Bx\u2212C)+D[\/latex].<\/li>\r\n \t<li>Identify the stretching\/compressing factor, |<em>A<\/em>|.<\/li>\r\n \t<li>Identify <em>B<\/em> and determine the period, [latex]\\frac{2\\pi}{|B|}[\/latex].<\/li>\r\n \t<li>Identify <em>C<\/em> and determine the phase shift, [latex]\\frac{C}{B}[\/latex].<\/li>\r\n \t<li>Draw the graph of [latex]y=A\\csc(Bx)[\/latex] but shift it to the right by and up by <em>D<\/em>.<\/li>\r\n \t<li>Sketch the vertical asymptotes, which occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 21: Graphing a Vertically Stretched, Horizontally Compressed, and Vertically Shifted Cosecant<\/h3>\r\nSketch a graph of [latex]y=2\\csc\\left(\\frac{\\pi}{2}x\\right)+1[\/latex]. What are the domain and range of this function?\r\n\r\n[reveal-answer q=\"993272\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"993272\"]\r\n\r\n<strong>Step 1.<\/strong> Express the function given in the form [latex]y=2\\csc\\left(\\frac{\\pi}{2}x\\right)+1[\/latex].\r\n\r\n<strong>Step 2.<\/strong> Identify the stretching\/compressing factor, [latex]|A|=2[\/latex].\r\n\r\n<strong>Step 3.<\/strong> The period is [latex]\\frac{2\\pi}{|B|}=\\frac{2\\pi}{\\frac{\\pi}{2}}=\\frac{2\\pi}{1}\\times \\frac{2}{\\pi}=4[\/latex].\r\n\r\n<strong>Step 4.<\/strong> The phase shift is [latex]\\frac{0}{\\frac{\\pi}{2}}=0[\/latex].\r\n\r\n<strong>Step 5.<\/strong> Draw the graph of [latex]y=A\\csc(Bx)[\/latex] but shift it up [latex]D=1[\/latex].\r\n\r\n<strong>Step 6.<\/strong> Sketch the vertical asymptotes, which occur at <em>x<\/em> = 0, <em>x<\/em> = 2, <em>x<\/em> = 4.\r\n\r\nThe graph for this function is shown in Figure 14.\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\/3675\/2018\/09\/27163835\/CNX_Precalc_Figure_06_02_014F.jpg\" alt=\"A graph of 3 periods of a modified cosecant function, with 3 vertical asymptotes, and a dotted sinusoidal function that has local maximums where the cosecant function has local minimums and local minimums where the cosecant function has local maximums.\" width=\"487\" height=\"377\" \/> <b>Figure 39<\/b>[\/caption]\r\n<h4>Analysis of the Solution<\/h4>\r\nThe vertical asymptotes shown on the graph mark off one period of the function, and the local extrema in this interval are shown by dots. Notice how the graph of the transformed cosecant relates to the graph of [latex]f(x)=2\\sin\\left(\\frac{\\pi}{2}x\\right)+1[\/latex], shown as the orange dashed wave.\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nGiven the graph of [latex]f(x)=2\\cos\\left(\\frac{\\pi}{2}x\\right)+1[\/latex] shown in Figure 15, sketch the graph of [latex]g(x)=2\\sec\\left(\\frac{\\pi}{2}x\\right)+1[\/latex] on the same axes.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163838\/CNX_Precalc_Figure_06_02_015.jpg\" alt=\"A graph of two periods of a modified cosine function. Range is [-1,3], graphed from x=-4 to x=4.\" width=\"488\" height=\"381\" \/> <b>Figure 40<\/b>[\/caption]\r\n[reveal-answer q=\"560894\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"560894\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163927\/CNX_Precalc_Figure_06_02_016.jpg\" alt=\"A graph of two periods of both a secant and consine function. Grpah shows that cosine function has local maximums where secant function has local minimums and vice versa.\" \/>[\/hidden-answer]<\/div>\r\n<h2>Analyzing the Graph of y = cot x and Its Variations<\/h2>\r\nThe last trigonometric function we need to explore is <strong>cotangent<\/strong>. The cotangent is defined by the <strong>reciprocal identity<\/strong> [latex]\\cot x=\\frac{1}{\\tan x}[\/latex]. Notice that the function is undefined when the tangent function is 0, leading to a vertical asymptote in the graph at 0, \u03c0, etc. Since the output of the tangent function is all real numbers, the output of the <strong>cotangent function<\/strong> is also all real numbers.\r\n\r\nWe can graph [latex]y=\\cot x[\/latex] by observing the graph of the tangent function because these two functions are reciprocals of one another. See Figure 16. Where the graph of the tangent function decreases, the graph of the cotangent function increases. Where the graph of the tangent function increases, the graph of the cotangent function decreases.\r\n\r\nThe cotangent graph has vertical asymptotes at each value of <em>x<\/em> where [latex]\\tan x=0[\/latex]; we show these in the graph below with dashed lines. Since the cotangent is the reciprocal of the tangent, [latex]\\cot x[\/latex] has vertical asymptotes at all values of <em>x<\/em> where [latex]\\tan x=0[\/latex] , and [latex]\\cot x=0[\/latex] at all values of x where tan x has its vertical asymptotes.\r\n<figure id=\"Figure_06_02_017\" class=\"small ui-has-child-figcaption\">\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\/3675\/2018\/09\/27163840\/CNX_Precalc_Figure_06_02_017.jpg\" alt=\"A graph of cotangent of x, with vertical asymptotes at multiples of pi.\" width=\"487\" height=\"439\" \/> <b>Figure 41.<\/b> The cotangent function[\/caption]\r\n\r\n<figcaption><\/figcaption><\/figure>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Features of the Graph of <em>y<\/em> = <em>A<\/em>cot(<em>Bx<\/em>)<\/h3>\r\n<ul>\r\n \t<li>The stretching factor is |<em>A<\/em>|.<\/li>\r\n \t<li>The period is [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\r\n \t<li>The domain is [latex]x\\ne\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\r\n \t<li>The range is (\u2212\u221e, \u221e).<\/li>\r\n \t<li>The asymptotes occur at [latex]x=\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\r\n \t<li>[latex]y=A\\cot(Bx)[\/latex] is an odd function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Graphing Variations of <em>y<\/em> = cot <em>x<\/em><\/h2>\r\nWe can transform the graph of the cotangent in much the same way as we did for the tangent. The equation becomes the following.\r\n<div>\r\n<div style=\"text-align: center\">[latex]y=A\\cot(Bx\u2212C)+D[\/latex]<\/div>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>A General Note: Properties of the Graph of <em>y<\/em> = <em>A<\/em>cot(<em>Bx<\/em>\u2212C)+<em>D<\/em><\/h3>\r\n<ul>\r\n \t<li>The stretching factor is |<em>A<\/em>|.<\/li>\r\n \t<li>The period is [latex]\\frac{\\pi}{|B|}[\/latex].<\/li>\r\n \t<li>The domain is [latex]x\\ne\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\r\n \t<li>The range is (\u2212\u221e, \u2212|<em>A<\/em>|] \u222a [|<em>A<\/em>|, \u221e).<\/li>\r\n \t<li>The vertical asymptotes occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\r\n \t<li>There is no amplitude.<\/li>\r\n \t<li>[latex]y=A\\cot(Bx)[\/latex] is an odd function because it is the quotient of even and odd functions (cosine and sine, respectively)<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a modified cotangent function of the form [latex]f(x)=A\\cot(Bx)[\/latex], graph one period.<\/h3>\r\n<ol>\r\n \t<li>Express the function in the form [latex]f(x)=A\\cot(Bx)[\/latex].<\/li>\r\n \t<li>Identify the stretching factor, |<em>A<\/em>|.<\/li>\r\n \t<li>Identify the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\r\n \t<li>Draw the graph of [latex]y=A\\tan(Bx)[\/latex].<\/li>\r\n \t<li>Plot any two reference points.<\/li>\r\n \t<li>Use the reciprocal relationship between tangent and cotangent to draw the graph of [latex]y=A\\cot(Bx)[\/latex].<\/li>\r\n \t<li>Sketch the asymptotes.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 22: Graphing Variations of the Cotangent Function<\/h3>\r\nDetermine the stretching factor, period, and phase shift of [latex]y=3\\cot(4x)[\/latex], and then sketch a graph.\r\n\r\n[reveal-answer q=\"32362\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"32362\"]\r\n\r\n<strong>Step 1.<\/strong> Expressing the function in the form [latex]f(x)=A\\cot(Bx)[\/latex] gives [latex]f(x)=3\\cot(4x)[\/latex].\r\n\r\n<strong>Step 2.<\/strong> The stretching factor is [latex]|A|=3[\/latex].\r\n\r\n<strong>Step 3.<\/strong> The period is [latex]P=\\frac{\\pi}{4}[\/latex].\r\n\r\n<strong>Step 4.<\/strong> Sketch the graph of [latex]y=3\\tan(4x)[\/latex].\r\n\r\n<strong>Step 5.<\/strong> Plot two reference points. Two such points are [latex]\\left(\\frac{\\pi}{16}\\text{, }3\\right)[\/latex] and [latex]\\left(\\frac{3\\pi}{16}\\text{, }\u22123\\right)[\/latex].\r\n\r\n<strong>Step 6.<\/strong> Use the reciprocal relationship to draw [latex]y=3\\cot(4x)[\/latex].\r\n\r\n<strong>Step 7.<\/strong> Sketch the asymptotes, [latex]x=0[\/latex], [latex]x=\\frac{\\pi}{4}[\/latex].\r\n\r\nThe orange graph in Figure 17 shows [latex]y=3\\tan(4x)[\/latex] and the blue graph shows [latex]y=3\\cot(4x)[\/latex].\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\/3675\/2018\/09\/27163843\/CNX_Precalc_Figure_06_02_019.jpg\" alt=\"A graph of two periods of a modified tangent function and a modified cotangent function. Vertical asymptotes at x=-pi\/4 and pi\/4.\" width=\"487\" height=\"592\" \/> <b>Figure 42<\/b>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a modified cotangent function of the form [latex]f(x)=A\\cot(Bx\u2212C)+D[\/latex], graph one period.<\/h3>\r\n<ol>\r\n \t<li>Express the function in the form [latex]f(x)=A\\cot(Bx\u2212C)+D[\/latex].<\/li>\r\n \t<li>Identify the stretching factor, |<em>A<\/em>|.<\/li>\r\n \t<li>Identify the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\r\n \t<li>Identify the phase shift, [latex]\\frac{C}{B}[\/latex].<\/li>\r\n \t<li>Draw the graph of [latex]y=A\\tan(Bx)[\/latex] shifted to the right by [latex]\\frac{C}{B}[\/latex] and up by <em>D<\/em>.<\/li>\r\n \t<li>Sketch the asymptotes [latex]x =\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\r\n \t<li>Plot any three reference points and draw the graph through these points.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Example 23: Graphing a Modified Cotangent<\/h3>\r\nSketch a graph of one period of the function [latex]f(x)=4\\cot\\left(\\frac{\\pi}{8}x\u2212\\frac{\\pi}{2}\\right)\u22122[\/latex].\r\n\r\n[reveal-answer q=\"706245\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"706245\"]\r\n\r\n<strong>Step 1.<\/strong> The function is already written in the general form [latex]f(x)=A\\cot(Bx\u2212C)+D[\/latex].\r\n\r\n<strong>Step 2.<\/strong>\u00a0[latex]A=4[\/latex], so the stretching factor is 4.\r\n\r\n<strong>Step 3.<\/strong>\u00a0[latex]B=\\frac{\\pi}{8}[\/latex], so the period is [latex]P=\\frac{\\pi}{|B|}=\\frac{\\pi}{\\frac{\\pi}{8}}=8[\/latex].\r\n\r\n<strong>Step 4.<\/strong>\u00a0[latex]C=\\frac{\\pi}{2}[\/latex], so the phase shift is [latex]\\frac{C}{B}=\\frac{\\frac{\\pi}{2}}{\\frac{\\pi}{8}}=4[\/latex].\r\n\r\n<strong>Step 5.<\/strong> We draw [latex]f(x)=4\\tan\\left(\\frac{\\pi}{8}x\u2212\\frac{\\pi}{2}\\right)\u22122[\/latex].\r\n\r\n<strong>Step 6-7.<\/strong> Three points we can use to guide the graph are (6,2), (8,\u22122), and (10,\u22126). We use the reciprocal relationship of tangent and cotangent to draw [latex]f(x)=4\\cot\\left(\\frac{\\pi}{8}x\u2212\\frac{\\pi}{2}\\right)\u22122[\/latex].\r\n\r\n<strong>Step 8.<\/strong> The vertical asymptotes are [latex]x=4[\/latex] and [latex]x=12[\/latex].\r\n\r\nThe graph is shown in Figure 18.\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\/3675\/2018\/09\/27163846\/CNX_Precalc_Figure_06_02_020.jpg\" alt=\"A graph of one period of a modified cotangent function. Vertical asymptotes at x=4 and x=12.\" width=\"487\" height=\"315\" \/> <b>Figure 43.<\/b> One period of a modified cotangent function.[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Shifted, compressed, and\/or stretched tangent function<\/td>\r\n<td>[latex]y=A\\tan(Bx\u2212C)+D[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Shifted, compressed, and\/or stretched secant function<\/td>\r\n<td>[latex]y=A\\sec(Bx\u2212C)+D[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Shifted, compressed, and\/or stretched cosecant<\/td>\r\n<td>[latex]y=A\\csc(Bx\u2212C)+D[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Shifted, compressed, and\/or stretched cotangent function<\/td>\r\n<td>[latex]y=A\\cot(Bx\u2212C)+D[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2><\/h2>\r\n<\/section><section id=\"fs-id1165137540392\" class=\"key-concepts\">\r\n<h1>Key Concepts<\/h1>\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 2\u03c0.<\/li>\r\n \t<li>The function sin <em>x<\/em> is odd, so its graph is symmetric about the origin. The function cos <em>x<\/em> 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]\\text{P}=\\frac{2\\pi}{|B|}[\/latex].<\/li>\r\n \t<li>In the general formula for a sinusoidal function, |<em>A<\/em>|represents amplitude. If |<em>A<\/em>| &gt; 1, the function is stretched, whereas if|<em>A<\/em>| &lt; 1, the function is compressed.<\/li>\r\n \t<li>The value [latex]\\frac{C}{B}[\/latex] in the general formula for a sinusoidal function indicates the phase shift.<\/li>\r\n \t<li>The value <em>D<\/em> in the general formula for a sinusoidal function indicates the vertical shift from the midline.<\/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 be graphed by identifying its amplitude and period.<\/li>\r\n \t<li>A function can also be graphed by identifying its amplitude, period, phase shift, and horizontal shift.<\/li>\r\n \t<li>Sinusoidal functions can be used to solve real-world problems.<\/li>\r\n \t<li>The tangent function has period \u03c0.<\/li>\r\n \t<li>[latex]f(x)=A\\tan(Bx\u2212C)+D[\/latex] is a tangent with vertical and\/or horizontal stretch\/compression and shift.<\/li>\r\n \t<li>The secant and cosecant are both periodic functions with a period of2\u03c0. [latex]f(x)=A\\sec(Bx\u2212C)+D[\/latex] gives a shifted, compressed, and\/or stretched secant function graph.<\/li>\r\n \t<li>[latex]f(x)=A\\csc(Bx\u2212C)+D[\/latex] gives a shifted, compressed, and\/or stretched cosecant function graph.<\/li>\r\n \t<li>The cotangent function has period \u03c0 and vertical asymptotes at 0, \u00b1\u03c0,\u00b12\u03c0,....<\/li>\r\n \t<li>The range of cotangent is (\u2212\u221e,\u221e),and the function is decreasing at each point in its range.<\/li>\r\n \t<li>The cotangent is zero at [latex]\\pm\\frac{\\pi}{2}\\text{, }\\pm\\frac{3\\pi}{2}[\/latex],....<\/li>\r\n \t<li>[latex]f(x)=A\\cot(Bx\u2212C)+D[\/latex] is a cotangent with vertical and\/or horizontal stretch\/compression and shift.<\/li>\r\n \t<li>Real-world scenarios can be solved using graphs of trigonometric functions.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165137414167\" class=\"definition\">\r\n \t<dt>amplitude<\/dt>\r\n \t<dd id=\"fs-id1165137463141\">the vertical height of a function; the constant <em>A<\/em> appearing in the definition of a sinusoidal function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137602032\" class=\"definition\">\r\n \t<dt>midline<\/dt>\r\n \t<dd id=\"fs-id1165137602037\">the horizontal line <em>y\u00a0<\/em>= <em>D<\/em>, where <em>D<\/em> appears in the general form of a sinusoidal function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137678058\" class=\"definition\">\r\n \t<dt>periodic function<\/dt>\r\n \t<dd id=\"fs-id1165137678063\">a function <em>f<\/em>(<em>x<\/em>) that satisfies [latex]f(x+P)=f(x)[\/latex] for a specific constant <em>P\u00a0<\/em>and any value of <em>x<\/em><\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137939683\" class=\"definition\">\r\n \t<dt>phase shift<\/dt>\r\n \t<dd id=\"fs-id1165137939688\">the horizontal displacement of the basic sine or cosine function; the constant [latex]\\frac{C}{B}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135160153\" class=\"definition\">\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(x)=A\\sin(Bx\u2212C)+D[\/latex] or [latex]f(x)=A\\cos(Bx\u2212C)+D[\/latex]<\/dd>\r\n<\/dl>\r\n<\/section>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Determine amplitude, period, phase shift, and vertical shift of a sine or cosine graph from its equation.<\/li>\n<li style=\"font-weight: 400\">Graph variations of\u2009y=cos x\u2009and y=sin x\u2009.<\/li>\n<li>Determine a function formula that would have a given sinusoidal graph.<\/li>\n<li style=\"font-weight: 400\">Determine functions that model circular and periodic motion.<\/li>\n<li style=\"font-weight: 400\">Analyze the graph of \u2009y=tan\u2009x and y=cot x.<\/li>\n<li style=\"font-weight: 400\">Graph variations of \u2009y=tan\u2009x and y=cot x.<\/li>\n<li>Determine a function formula from a tangent or cotangent graph.<\/li>\n<li style=\"font-weight: 400\">Analyze the graphs of \u2009y=sec\u2009x\u2009 and \u2009y=csc\u2009x.<\/li>\n<li style=\"font-weight: 400\">Graph variations of \u2009y=sec\u2009x\u2009 and \u2009y=csc\u2009x.<\/li>\n<li>Determine a function formula from a secant or cosecant graph.<\/li>\n<\/ul>\n<\/div>\n<h2>Graph variations of \u2009y=sin( x )\u2009 and \u2009y=cos( x )<\/h2>\n<p>Recall that the sine and cosine functions relate real number values to the <em>x<\/em>&#8211; and <em>y<\/em>-coordinates of a point on the unit circle. So what do they look like on a graph on a coordinate plane? Let\u2019s start with the <strong>sine function.<\/strong> We can create a table of values and use them to sketch a graph. The table below\u00a0lists some of the values for the sine function on a unit circle.<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>x<\/strong><\/td>\n<td>0<\/td>\n<td>[latex]\\frac{\\pi}{6}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi}{4}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi}{3}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi}{2}[\/latex]<\/td>\n<td>[latex]\\frac{2\\pi}{3}[\/latex]<\/td>\n<td>[latex]\\frac{3\\pi}{4}[\/latex]<\/td>\n<td>[latex]\\frac{5\\pi}{6}[\/latex]<\/td>\n<td>[latex]\\pi[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]\\sin(x)[\/latex]<\/strong><\/td>\n<td>0<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\n<td>1<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Plotting the points from the table and continuing along the <em>x<\/em>-axis gives the shape of the sine function. See Figure 2.<\/p>\n<figure id=\"Figure_06_01_002\" class=\"small ui-has-child-figcaption\">\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\/3675\/2018\/09\/27003914\/CNX_Precalc_Figure_06_01_002.jpg\" alt=\"A graph of sin(x). Local maximum at (pi\/2, 1). Local minimum at (3pi\/2, -1). Period of 2pi.\" width=\"487\" height=\"216\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2.<\/b> The sine function<\/p>\n<\/div>\n<\/figure>\n<p>Notice how the sine values are positive between 0 and \u03c0, 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 \u03c0 and 2\u03c0, which correspond to the values of the sine function in quadrants III and IV on the unit circle. See Figure 3.<\/p>\n<figure id=\"Figure_06_01_003\" class=\"small ui-has-child-figcaption\">\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\/3675\/2018\/09\/27003916\/CNX_Precalc_Figure_06_01_003.jpg\" alt=\"A side-by-side graph of a unit circle and a graph of sin(x). The two graphs show the equivalence of the coordinates.\" width=\"487\" height=\"219\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3.<\/b> Plotting values of the sine function<\/p>\n<\/div>\n<\/figure>\n<p>Now let\u2019s take a similar look at the <strong>cosine function<\/strong>. Again, we can create a table of values and use them to sketch a graph. The table below\u00a0lists some of the values for the cosine function on a unit circle.<\/p>\n<table>\n<tbody>\n<tr>\n<td><strong>x<\/strong><\/td>\n<td>0<\/td>\n<td>[latex]\\frac{\\pi}{6}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi}{4}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi}{3}[\/latex]<\/td>\n<td>[latex]\\frac{\\pi}{2}[\/latex]<\/td>\n<td>[latex]\\frac{2\\pi}{3}[\/latex]<\/td>\n<td>[latex]\\frac{3\\pi}{4}[\/latex]<\/td>\n<td>[latex]\\frac{5\\pi}{6}[\/latex]<\/td>\n<td>[latex]\\pi[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]\\cos(x)[\/latex]<\/td>\n<td>1<\/td>\n<td>[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>0<\/td>\n<td>[latex]-\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]-\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\n<td>[latex]-\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\n<td>\u22121<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>As with the sine function, we can plots points to create a graph of the cosine function as in\u00a0Figure 4.<\/p>\n<figure id=\"Figure_06_01_004\" class=\"medium ui-has-child-figcaption\">\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003918\/CNX_Precalc_Figure_06_01_004.jpg\" alt=\"A graph of cos(x). Local maxima at (0,1) and (2pi, 1). Local minimum at (pi, -1). Period of 2pi.\" width=\"731\" height=\"216\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 4.<\/b> The cosine function<\/p>\n<\/div>\n<\/figure>\n<p>Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval [\u22121,1].<\/p>\n<p>In both graphs, the shape of the graph repeats after 2\u03c0,which means the functions are periodic with a period of [latex]2\u03c0[\/latex]. A <strong>periodic function<\/strong> is a function for which a specific <strong>horizontal shift<\/strong>, <em>P<\/em>, results in a function equal to the original function: [latex]f (x + P) = f(x)[\/latex] for all values of <em>x<\/em> in the domain of <em>f<\/em>. When this occurs, we call the smallest such horizontal shift with [latex]P > 0[\/latex] the <strong>period<\/strong> of the function. Figure 5\u00a0shows several periods of the sine and cosine functions.<\/p>\n<figure id=\"Figure_06_01_005\" 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\/3675\/2018\/09\/27003920\/CNX_Precalc_Figure_06_01_005.jpg\" alt=\"Side-by-side graphs of sin(x) and cos(x). Graphs show period lengths for both functions, which is 2pi.\" width=\"487\" height=\"442\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 5<\/b><\/p>\n<\/div>\n<\/figure>\n<p>Looking again at the sine and cosine functions on a domain centered at the <em>y<\/em>-axis helps reveal symmetries. As we can see in Figure 6, the <strong>sine function<\/strong> is symmetric about the origin. Recall from <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/precalctwoxmaster\/chapter\/introduction-to-the-other-trigonometric-functions\/\" target=\"_blank\" rel=\"noopener\">The Other Trigonometric Functions<\/a> that we determined from the unit circle that the sine function is an odd function because [latex]\\sin(\u2212x)=\u2212\\sin x[\/latex]. Now we can clearly see this property from the graph.<\/p>\n<figure id=\"Figure_06_01_006\" class=\"small ui-has-child-figcaption\">\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\/3675\/2018\/09\/27003922\/CNX_Precalc_Figure_06_01_006.jpg\" alt=\"A graph of sin(x) that shows that sin(x) is an odd function due to the odd symmetry of the graph.\" width=\"487\" height=\"191\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 6.<\/b> Odd symmetry of the sine function<\/p>\n<\/div>\n<\/figure>\n<p>Figure 7\u00a0shows that the cosine function is symmetric about the <em>y<\/em>-axis. Again, we determined that the cosine function is an even function. Now we can see from the graph that [latex]\\cos(\u2212x)=\\cos x[\/latex].<\/p>\n<figure id=\"Figure_06_01_007\" class=\"small ui-has-child-figcaption\">\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\/3675\/2018\/09\/27003925\/CNX_Precalc_Figure_06_01_007.jpg\" alt=\"A graph of cos(x) that shows that cos(x) is an even function due to the even symmetry of the graph.\" width=\"487\" height=\"216\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 7.<\/b> Even symmetry of the cosine function<\/p>\n<\/div>\n<\/figure>\n<div class=\"textbox\">\n<h3>A General Note: Characteristics of Sine and Cosine Functions<\/h3>\n<p>The sine and cosine functions have several distinct characteristics:<\/p>\n<ul>\n<li>They are periodic functions with a period of 2\u03c0.<\/li>\n<li>The domain of each function is\u00a0[latex]\\left(-\\infty,\\infty\\right)[\/latex] and the range is [latex]\\left[\u22121,1\\right][\/latex].<\/li>\n<li>The graph of [latex]y=\\sin x[\/latex] is symmetric about the origin, because it is an odd function.<\/li>\n<li>The graph of [latex]y=\\cos x[\/latex] is symmetric about the <em>y<\/em>-axis, because it is an even function.<\/li>\n<\/ul>\n<\/div>\n<h2>Investigating Sinusoidal Functions<\/h2>\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 <strong>cosine function<\/strong> is known as a <strong>sinusoidal function<\/strong>. The general forms of sinusoidal functions are<\/p>\n<div style=\"text-align: center\">[latex]y = A\\sin (Bx\u2212C) + D[\/latex]<\/div>\n<p>and<\/p>\n<div style=\"text-align: center\">[latex]y = A\\cos (Bx\u2212C) + D[\/latex]<\/div>\n<h3>Determining the Period of Sinusoidal Functions<\/h3>\n<p>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>In the general formula, <em>B<\/em> is related to the period by [latex]P=\\frac{2\u03c0}{|B|}[\/latex]. If [latex]|B| > 1[\/latex], then the period is less than [latex]2\u03c0[\/latex] and the function undergoes a horizontal compression, whereas if [latex]|B| < 1[\/latex], then the period is greater than [latex]2\u03c0[\/latex] and the function undergoes a horizontal stretch. For example, [latex]f(x) = \\sin(x), B= 1[\/latex], so the period is [latex]2\u03c0[\/latex], which we knew. If [latex]f(x) =\\sin (2x)[\/latex], then [latex]B= 2[\/latex], so the period is [latex]\u03c0[\/latex] and the graph is compressed. If [latex]f(x) = \\sin\\left(\\frac{x}{2} \\right)[\/latex], then [latex]B=\\frac{1}{2}[\/latex], so the period is [latex]4\u03c0[\/latex] and the graph is stretched. Notice in Figure 8\u00a0how the period is indirectly related to [latex]|B|[\/latex].\n\n\n<figure id=\"Figure_06_01_008\" 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\/3675\/2018\/09\/27003927\/CNX_Precalc_Figure_06_01_008.jpg\" alt=\"A graph with three items. The x-axis ranges from 0 to 2pi. The y-axis ranges from -1 to 1. The first item is the graph of sin(x) for one full period. The second is the graph of sin(2x) over two periods. The third is the graph of sin(x\/2) for one half of a period.\" width=\"487\" height=\"274\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 8<\/b><\/p>\n<\/div>\n<\/figure>\n<div class=\"textbox\">\n<h3>A General Note: Period of Sinusoidal Functions<\/h3>\n<p>If we let <em>C<\/em> = 0 and <em>D<\/em> = 0 in the general form equations of the sine and cosine functions, we obtain the forms<\/p>\n<p style=\"text-align: center\"><span style=\"text-align: center;background-color: initial;font-size: 0.9em\">[latex]y=A\\sin\\left(Bx\\right)[\/latex]<\/span><\/p>\n<p style=\"text-align: center\"><span style=\"background-color: initial;font-size: 0.9em\">[latex]y=A\\cos\\left(Bx\\right)[\/latex]<\/span><\/p>\n<p>The period is [latex]\\frac{2\u03c0}{|B|}[\/latex].<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 1: Identifying the Period of a Sine or Cosine Function<\/h3>\n<p>Determine the period of the function [latex]f(x) = \\sin\\left(\\frac{\u03c0}{6}x\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q616023\">Show Solution<\/span><\/p>\n<div id=\"q616023\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let&#8217;s begin by comparing the equation to the general form [latex]y=A\\sin(Bx)[\/latex].<\/p>\n<p>In the given equation, [latex]B =\\frac{\u03c0}{6}[\/latex], so the period will be<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}P&=\\frac{\\frac{2}{\\pi}}{|B|} \\\\ &=\\frac{2\\pi}{\\frac{x}{6}} \\\\ &=2\\pi\\times \\frac{6}{\\pi} \\\\ &=12 \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Determine the period of the function [latex]g(x)=\\cos\\left(\\frac{x}{3}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q111765\">Show Solution<\/span><\/p>\n<div id=\"q111765\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]6 \\pi[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>Determining Amplitude<\/h3>\n<p>Returning to the general formula for a sinusoidal function, we have analyzed how the variable <em>B<\/em> relates to the period. Now let\u2019s turn to the variable <em>A<\/em> so we can analyze how it is related to the <strong>amplitude<\/strong>, or greatest distance from rest. <em>A<\/em> represents the vertical stretch factor, and its absolute value |<em>A<\/em>| is the amplitude. The local maxima will be a distance |<em>A<\/em>| above the vertical <strong>midline<\/strong> of the graph, which is the line <em>x\u00a0<\/em>= <em>D<\/em>; because <em>D<\/em> = 0 in this case, the midline is the <em>x<\/em>-axis. The local minima will be the same distance below the midline. If |<em>A<\/em>| &gt; 1, the function is stretched. For example, the amplitude of [latex]f(x)=4\\sin\\left(x\\right)[\/latex] is twice the amplitude of<\/p>\n<div style=\"text-align: center\">[latex]f(x)=2\\sin\\left(x\\right)[\/latex]<\/div>\n<p>If [latex]|<em>A<\/em>| < 1[\/latex], the function is compressed. Figure 9\u00a0compares several sine functions with different amplitudes.\n\n\n<figure id=\"Figure_06_01_009\">\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003929\/CNX_Precalc_Figure_06_01_009.jpg\" alt=\"A graph with four items. The x-axis ranges from -6pi to 6pi. The y-axis ranges from -4 to 4. The first item is the graph of sin(x), which has an amplitude of 1. The second is a graph of 2sin(x), which has amplitude of 2. The third is a graph of 3sin(x), which has an amplitude of 3. The fourth is a graph of 4 sin(x) with an amplitude of 4.\" width=\"975\" height=\"316\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 9<\/b><\/p>\n<\/div>\n<\/figure>\n<div class=\"textbox\">\n<h3>A General Note: Amplitude of Sinusoidal Functions<\/h3>\n<p>If we let <em>C\u00a0<\/em>= 0 and <em>D<\/em> = 0 in the general form equations of the sine and cosine functions, we obtain the forms<\/p>\n<p style=\"text-align: center\">[latex]y=A\\sin(Bx)[\/latex] and [latex]y=A\\cos(Bx)[\/latex]<\/p>\n<p>The <strong>amplitude<\/strong> is A, and the vertical height from the <strong>midline<\/strong> is |A|. In addition, notice in the example that<\/p>\n<p style=\"text-align: center\"><span style=\"text-align: center;background-color: initial;font-size: 0.9em\">[latex]|A|=\\text{amplitude}=\\frac{1}{2}|\\text{maximum}\u2212\\text{minimum}|[\/latex]<\/span><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 2: Identifying the Amplitude of a Sine or Cosine Function<\/h3>\n<p>What is the amplitude of the sinusoidal function\u00a0[latex]f(x)=\u22124\\sin(x)[\/latex]? Is the function stretched or compressed vertically?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q207317\">Show Solution<\/span><\/p>\n<div id=\"q207317\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let\u2019s begin by comparing the function to the simplified form [latex]y=A\\sin(Bx)[\/latex].<\/p>\n<p>In the given function, <em>A\u00a0<\/em>= \u22124, so the amplitude is |<em>A<\/em>|=|\u22124| = 4. The function is stretched.<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>The negative value of <em>A<\/em> results in a reflection across the <em>x<\/em>-axis of the <strong>sine function<\/strong>, as shown in Figure 10.<\/p>\n<figure id=\"Figure_06_01_010\" 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\/3675\/2018\/09\/27003932\/CNX_Precalc_Figure_06_01_010.jpg\" alt=\"A graph of -4sin(x). The function has an amplitude of 4. Local minima at (-3pi\/2, -4) and (pi\/2, -4). Local maxima at (-pi\/2, 4) and (3pi\/2, 4). Period of 2pi.\" width=\"487\" height=\"319\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 10<\/b><\/p>\n<\/div>\n<\/figure>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>What is the amplitude of the sinusoidal function [latex]f(x)=12\\sin (x)[\/latex]? Is the function stretched or compressed vertically?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q525586\">Show Solution<\/span><\/p>\n<div id=\"q525586\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{1}{2}[\/latex] compressed<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Analyzing Graphs of Variations of <em>y<\/em> = sin<em> x<\/em> and <em>y<\/em> = cos <em>x<\/em><\/h2>\n<p>Now that we understand how <em>A<\/em> and <em>B<\/em> relate to the general form equation for the sine and cosine functions, we will explore the variables <em>C\u00a0<\/em>and <em>D<\/em>. Recall the general form:<\/p>\n<div>\n<div style=\"text-align: center\">[latex]y = A \\sin(Bx\u2212C)+D[\/latex] and [latex]y=A\\cos(Bx\u2212C)+D[\/latex]<\/div>\n<div>or<\/div>\n<div style=\"text-align: center\">[latex]y=A\\sin(B(x\u2212\\frac{C}{B}))+D[\/latex] and [latex]y=A\\cos(B(x\u2212\\frac{C}{B}))+D[\/latex]<\/div>\n<\/div>\n<p>The value [latex]\\frac{C}{B}[\/latex] for a sinusoidal function is called the <strong>phase shift<\/strong>, or the horizontal displacement of the basic sine or <strong>cosine function<\/strong>. If C &gt; 0, the graph shifts to the right. If C &lt; 0,the graph shifts to the left. The greater the value of |<em>C<\/em>|, the more the graph is shifted. Figure 11\u00a0shows that the graph of [latex]f(x)=\\sin(x\u2212\u03c0)[\/latex] shifts to the right by \u03c0 units, which is more than we see in the graph of [latex]f(x)=\\sin(x\u2212\\frac{\u03c0}{4})[\/latex], which shifts to the right by [latex]\\frac{\u03c0}{4}[\/latex]units.<\/p>\n<figure id=\"Figure_06_01_011\" 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\/3675\/2018\/09\/27003934\/CNX_Precalc_Figure_06_01_011.jpg\" alt=\"A graph with three items. The first item is a graph of sin(x). The second item is a graph of sin(x-pi\/4), which is the same as sin(x) except shifted to the right by pi\/4. The third item is a graph of sin(x-pi), which is the same as sin(x) except shifted to the right by pi.\" width=\"487\" height=\"255\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 11<\/b><\/p>\n<\/div>\n<\/figure>\n<p>While <em>C<\/em> relates to the horizontal shift, <em>D<\/em> indicates the vertical shift from the midline in the general formula for a sinusoidal function. The function [latex]y=\\cos(x)+D[\/latex] has its midline at [latex]y=D[\/latex].<\/p>\n<figure id=\"Figure_06_01_012\" 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\/3675\/2018\/09\/27003936\/CNX_Precalc_Figure_06_01_012.jpg\" alt=\"A graph of y=Asin(x)+D. Graph shows the midline of the function at y=D.\" width=\"487\" height=\"255\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 12<\/b><\/p>\n<\/div>\n<\/figure>\n<p>Any value of <em>D<\/em> other than zero shifts the graph up or down. Figure 13\u00a0compares [latex]f(x)=\\sin x[\/latex] with [latex]f(x)=\\sin (x)+2[\/latex], which is shifted 2 units up on a graph.<\/p>\n<figure id=\"Figure_06_01_013\" 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\/3675\/2018\/09\/27003938\/CNX_Precalc_Figure_06_01_013.jpg\" alt=\"A graph with two items. The first item is a graph of sin(x). The second item is a graph of sin(x)+2, which is the same as sin(x) except shifted up by 2.\" width=\"487\" height=\"221\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 13<\/b><\/p>\n<\/div>\n<\/figure>\n<div class=\"textbox\">\n<h3>A General Note: Variations of Sine and Cosine Functions<\/h3>\n<p>Given an equation in the form [latex]f(x)=A\\sin(Bx\u2212C)+D[\/latex] or [latex]f(x)=A\\cos(Bx\u2212C)+D[\/latex], [latex]\\frac{C}{B}[\/latex]is the <strong>phase shift<\/strong> and <em>D<\/em> is the <strong>vertical shift<\/strong>.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 3: Identifying the Phase Shift of a Function<\/h3>\n<p>Determine the direction and magnitude of the phase shift for [latex]f(x)=\\sin(x+\\frac{\u03c0}{6})\u22122[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q673346\">Show Solution<\/span><\/p>\n<div id=\"q673346\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let\u2019s begin by comparing the equation to the general form [latex]y=A\\sin(Bx\u2212C)+D[\/latex].<\/p>\n<p>In the given equation, notice that <em>B<\/em> = 1 and [latex]C=\u2212\\frac{\u03c0}{6}[\/latex]. So the phase shift is<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}\\frac{C}{B}&=\u2212\\frac{\\frac{x}{6}}{1} \\\\ &=\u2212\\frac{\\pi}{6} \\end{align}[\/latex]<\/p>\n<p>or [latex]\\frac{\\pi}{6}[\/latex] units to the left.<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>We must pay attention to the sign in the equation for the general form of a sinusoidal function. The equation shows a minus sign before <em>C<\/em>. Therefore [latex]f(x)=\\sin(x+\\frac{\u03c0}{6})\u22122[\/latex] can be rewritten as [latex]f(x)=\\sin(x\u2212(\u2212\\frac{\u03c0}{6}))\u22122[\/latex]. If the value of <em>C<\/em> is negative, the shift is to the left.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Determine the direction and magnitude of the phase shift for [latex]f(x)=3\\cos(x\u2212\\frac{\\pi}{2})[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q739051\">Show Solution<\/span><\/p>\n<div id=\"q739051\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\frac{\u03c0}{2}[\/latex]; right<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 4: Identifying the Vertical Shift of a Function<\/h3>\n<p>Determine the direction and magnitude of the vertical shift for [latex]f(x)=\\cos(x)\u22123[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q920585\">Show Solution<\/span><\/p>\n<div id=\"q920585\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let&#8217;s begin by comparing the equation to the general form [latex]y=A\\cos(Bx\u2212C)+D[\/latex]. In the given equation, [latex]D=-3[\/latex], so the shift is 3 units downward.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Determine the direction and magnitude of the vertical shift for [latex]f(x)=3\\sin(x)+2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q558661\">Show Solution<\/span><\/p>\n<div id=\"q558661\" class=\"hidden-answer\" style=\"display: none\">\n<p>2 units up<\/p>\n<p><span style=\"font-size: 1rem;text-align: initial\"><\/div>\n<\/div>\n<p><\/span><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a sinusoidal function in the form [latex]f(x)=A\\sin(Bx\u2212C)+D[\/latex],\u00a0identify the midline, amplitude, period, and phase shift.<\/h3>\n<ol>\n<li>Determine the amplitude as |A|.<\/li>\n<li>Determine the period as [latex]P=\\frac{2\u03c0}{|B|}[\/latex].<\/li>\n<li>Determine the phase shift as [latex]\\frac{C}{B}[\/latex].<\/li>\n<li>Determine the midline as <em>y\u00a0<\/em>= D.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 5: Identifying the Variations of a Sinusoidal Function from an Equation<\/h3>\n<p>Determine the midline, amplitude, period, and phase shift of the function [latex]y=3\\sin(2x)+1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q622405\">Show Solution<\/span><\/p>\n<div id=\"q622405\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let\u2019s begin by comparing the equation to the general form [latex]y=A\\sin(Bx\u2212C)+D[\/latex].\u00a0<em>A<\/em> = 3, so the amplitude is |<em>A<\/em>| = 3.<\/p>\n<p>Next, <em>B<\/em> = 2, so the period is [latex]P=\\frac{2\u03c0}{|B|}=\\frac{2\u03c0}{2}=\u03c0[\/latex].<\/p>\n<p>There is no added constant inside the parentheses, so <em>C<\/em> = 0 and the phase shift is [latex]\\frac{C}{B}=\\frac{0}{2}=0[\/latex].<\/p>\n<p>Finally, <em>D<\/em> = 1, so the midline is <em>y<\/em> = 1.<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Inspecting the graph, we can determine that the period is \u03c0, the midline is <em>y<\/em> = 1,and the amplitude is 3. See Figure 14.<\/p>\n<figure id=\"Figure_06_01_014\" 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\/3675\/2018\/09\/27003941\/CNX_Precalc_Figure_06_01_014.jpg\" alt=\"A graph of y=3sin(2x)+1. The graph has an amplitude of 3. There is a midline at y=1. There is a period of pi. Local maximum at (pi\/4, 4) and local minimum at (3pi\/4, -2).\" width=\"487\" height=\"263\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 14<\/b><\/p>\n<\/div>\n<\/figure>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Determine the midline, amplitude, period, and phase shift of the function [latex]y=\\frac{1}{2}\\cos(\\frac{x}{3}\u2212\\frac{\u03c0}{3})[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q453453\">Show Solution<\/span><\/p>\n<div id=\"q453453\" class=\"hidden-answer\" style=\"display: none\">\n<p>midline: [latex]y=0[\/latex]; amplitude: |<em>A<\/em>|=[latex]\\frac{1}{2}[\/latex]; period: <em>P<\/em>=[latex]\\frac{2\u03c0}{|B|}=6\\pi[\/latex]; phase shift:[latex]\\frac{C}{B}=\\pi[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm105947\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=105947&theme=oea&iframe_resize_id=ohm105947\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 6: Identifying the Equation for a Sinusoidal Function from a Graph<\/h3>\n<p>Determine the formula for the cosine function in Figure 15.<\/p>\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\/3675\/2018\/09\/27003943\/CNX_Precalc_Figure_06_01_015.jpg\" alt=\"A graph of -0.5cos(x)+0.5. The graph has an amplitude of 0.5. The graph has a period of 2pi. The graph has a range of [0, 1]. The graph is also reflected about the x-axis from the parent function cos(x).\" width=\"487\" height=\"163\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 15<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q509662\">Show Solution<\/span><\/p>\n<div id=\"q509662\" 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<p style=\"text-align: center\">[latex]y=A\\sin\\left(Bx-C\\right)+D[\/latex]<span id=\"MathJax-Element-411-Frame\" class=\"MathJax\" style=\"font-style: normal;font-weight: normal;line-height: normal;font-size: 14px;text-indent: 0px;text-align: center;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;padding: 0px;margin: 0px\" role=\"presentation\"><\/span><\/p>\n<p style=\"text-align: center\">[latex]y=A\\cos\\left(Bx-C\\right)+D[\/latex]<\/p>\n<p id=\"fs-id1165137704661\">The graph could represent either a sine or a\u00a0<span class=\"no-emphasis\">cosine function<\/span>\u00a0that is shifted and\/or reflected. When [latex]x=0[\/latex], the graph has an extreme point, [latex](0,0)[\/latex]. Since the cosine function has an extreme point for [latex]x=0[\/latex], let us write our equation in terms of a cosine function.<\/p>\n<p id=\"fs-id1165135536557\">Let\u2019s start with the midline. We can see that the graph rises and falls an equal distance above and below [latex]y=0.5[\/latex]. This value, which is the midline, is\u00a0<em>D <\/em>in\u00a0the equation, so <em>D<\/em>=0.5.<\/p>\n<p id=\"fs-id1165137938642\">The greatest distance above and below the midline is the amplitude. The maxima are 0.5 units above the midline and the minima are 0.5 units below the midline. So |<em>A<\/em>|=0.5. Another way we could have determined the amplitude is by recognizing that the difference between the height of local maxima and minima is 1, so |<em>A<\/em>|=[latex]\\frac{1}{2}[\/latex]. Also, the graph is reflected about the\u00a0<em>x<\/em>-axis so that\u00a0<em>A<\/em>=0.5.<\/p>\n<p id=\"fs-id1165134204425\">The graph is not horizontally stretched or compressed, so\u00a0<em>B<\/em>=0 and the graph is not shifted horizontally, so\u00a0<em>C<\/em>=0.<\/p>\n<p id=\"fs-id1165135347312\">Putting this all together,<\/p>\n<p style=\"text-align: center\">[latex]g(x)=0.5\\cos\\left(x\\right)+0.5[\/latex]<\/p>\n<p style=\"text-align: left\"><span style=\"font-size: 1rem;text-align: initial\"><\/div>\n<\/div>\n<p><\/span><\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Determine the formula for the sine function in Figure 16.<\/p>\n<figure id=\"Figure_06_01_016\" 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\/3675\/2018\/09\/27003945\/CNX_Precalc_Figure_06_01_016.jpg\" alt=\"A graph of sin(x)+2. Period of 2pi, amplitude of 1, and range of [1, 3].\" width=\"487\" height=\"173\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 16<\/b><\/p>\n<\/div>\n<\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q448760\">Show Solution<\/span><\/p>\n<div id=\"q448760\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f(x)=\\sin(x)+2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm126732\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=126732&theme=oea&iframe_resize_id=ohm126732\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 7: Identifying the Equation for a Sinusoidal Function from a Graph<\/h3>\n<p>Determine the equation for the sinusoidal function in Figure 17.<\/p>\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003947\/CNX_Precalc_Figure_06_01_017.jpg\" alt=\"A graph of 3cos(pi\/3x-pi\/3)-2. Graph has amplitude of 3, period of 6, range of [-5,1].\" width=\"731\" height=\"565\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 17<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q680521\">Show Solution<\/span><\/p>\n<div id=\"q680521\" class=\"hidden-answer\" style=\"display: none\">With the highest value at 1 and the lowest value at\u22125, the midline will be halfway between at \u22122. So <em>D<\/em> = \u22122.The distance from the midline to the highest or lowest value gives an amplitude of |A|=3.The period of the graph is 6, which can be measured from the peak at <em>x\u00a0<\/em>= 1 to the next peak at <em>x<\/em> = 7,\u00a0or\u00a0from the distance between the lowest points. Therefore, [latex]\\text{P}=\\frac{2\\pi}{|B|}=6[\/latex]. Using the positive value for <em>B<\/em>, we find that<\/p>\n<p style=\"text-align: center\">[latex]B=\\frac{2\u03c0}{P}=\\frac{2\u03c0}{6}=\\frac{\u03c0}{3}[\/latex]<\/p>\n<p>So far, our equation is either [latex]y=3\\sin(\\frac{\\pi}{3}x\u2212C)\u22122[\/latex] or [latex]y=3\\cos(\\frac{\\pi}{3}x\u2212C)\u22122[\/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>\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>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<p style=\"text-align: center\">[latex]y=3\\cos(\\frac{\u03c0}{3}x\u2212\\frac{\u03c0}{3})\u22122[\/latex] or [latex]y=\u22123\\cos(\\frac{\u03c0}{3}x+\\frac{2\u03c0}{3})\u22122[\/latex]<\/p>\n<p>Again, these functions are equivalent, so both yield the same graph.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Write a formula for the function graphed in Figure 18.<\/p>\n<figure id=\"Figure_06_01_018\" class=\"medium\">\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003949\/CNX_Precalc_Figure_06_01_018n.jpg\" alt=\"A graph of 4sin((pi\/5)x-pi\/5)+4. Graph has period of 10, amplitude of 4, range of [0,8].\" width=\"731\" height=\"440\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 18<\/b><\/p>\n<\/div>\n<\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q993227\">Show Solution<\/span><\/p>\n<div id=\"q993227\" class=\"hidden-answer\" style=\"display: none\">\n<p>two possibilities are: [latex]y=4\\sin(\\frac{\u03c0}{5}x\u2212\\frac{\u03c0}{5})+4[\/latex] or [latex]y=\u22124sin(\\frac{\u03c0}{5}x+4\\frac{\u03c0}{5})+4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm126749\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=126749&theme=oea&iframe_resize_id=ohm126749\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Graphing Variations of <em>y<\/em> = sin <em>x<\/em> and <em>y<\/em> = cos <em>x<\/em><\/h2>\n<p>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>Instead of focusing on the general form equations<\/p>\n<div>\n<div style=\"text-align: center\">[latex]y=A\\sin(Bx\u2212C)+D[\/latex] and [latex]y=A\\cos(Bx\u2212C)+D[\/latex],<\/div>\n<\/div>\n<p>we will let <em>C<\/em> = 0 and <em>D<\/em> = 0 and work with a simplified form of the equations in the following examples.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given the function [latex]y=Asin(Bx)[\/latex], sketch its graph.<\/h3>\n<ol>\n<li>Identify the amplitude,|<em>A<\/em>|.<\/li>\n<li>Identify the period, [latex]P=\\frac{2\u03c0}{|B|}[\/latex].<\/li>\n<li>Start at the origin, with the function increasing to the right if <em>A<\/em> is positive or decreasing if <em>A<\/em> is negative.<\/li>\n<li>At [latex]x=\\frac{\u03c0}{2|B|}[\/latex] there is a local maximum for <em>A<\/em> &gt; 0 or a minimum for <em>A<\/em> &lt; 0, with <em>y<\/em> = <em>A<\/em>.<\/li>\n<li>The curve returns to the <em>x<\/em>-axis at [latex]x=\\frac{\u03c0}{|B|}[\/latex].<\/li>\n<li>There is a local minimum for <em>A<\/em> &gt; 0 (maximum for <em>A\u00a0<\/em>&lt; 0) at [latex]x=\\frac{3\u03c0}{2|B|}[\/latex] with <em>y\u00a0<\/em>= \u2013<em>A<\/em>.<\/li>\n<li>The curve returns again to the <em>x<\/em>-axis at [latex]x=\\frac{\u03c0}{2|B|}[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 8: Graphing a Function and Identifying the Amplitude and Period<\/h3>\n<p>Sketch a graph of [latex]f(x)=\u22122\\sin(\\frac{\u03c0x}{2})[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q699067\">Show Solution<\/span><\/p>\n<div id=\"q699067\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let\u2019s begin by comparing the equation to the form [latex]y=A\\sin(Bx)[\/latex].<\/p>\n<p><strong>Step 1.<\/strong> We can see from the equation that A=\u22122,so the amplitude is 2.<\/p>\n<p style=\"text-align: center\">|<em>A<\/em>| = 2<\/p>\n<p><strong>Step 2.<\/strong> The equation shows that [latex]B=\\frac{\u03c0}{2}[\/latex], so the period is<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}P&=\\frac{2\\pi}{\\frac{\\pi}{2}}\\\\&=2\\pi\\times\\frac{2}{\\pi}\\\\&=4 \\end{align}[\/latex]<\/p>\n<p><strong>Step 3.<\/strong> Because <em>A<\/em> is negative, the graph descends as we move to the right of the origin.<\/p>\n<p><strong>Step 4\u20137.<\/strong> The <em>x<\/em>-intercepts are at the beginning of one period, <em>x\u00a0<\/em>= 0, the horizontal midpoints are at <em>x\u00a0<\/em>= 2 and at the end of one period at <em>x<\/em> = 4.<\/p>\n<p>The quarter points include the minimum at <em>x<\/em> = 1 and the maximum at <em>x<\/em> = 3. A local minimum will occur 2 units below the midline, at <em>x<\/em> = 1, and a local maximum will occur at 2 units above the midline, at <em>x<\/em> = 3. Figure 19 shows the graph of the function.<\/p>\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\/3675\/2018\/09\/27003952\/CNX_Precalc_Figure_06_01_019.jpg\" alt=\"A graph of -2sin((pi\/2)x). Graph has range of &#091;-2,2&#093;, period of 4, and amplitude of 2.\" width=\"487\" height=\"252\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 19<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Sketch a graph of [latex]g(x)=\u22120.8\\cos(2x)[\/latex]. Determine the midline, amplitude, period, and phase shift.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q862743\">Show Solution<\/span><\/p>\n<div id=\"q862743\" class=\"hidden-answer\" style=\"display: none\">\n<p>midline: y=0; amplitude: |<em>A<\/em>|=0.8; period: P=[latex]\\frac{2\u03c0}{|B|}=\\pi[\/latex]; phase shift: [latex]\\frac{C}{B}=0[\/latex] or none<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004024\/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).\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a sinusoidal function with a phase shift and a vertical shift, sketch its graph.<\/h3>\n<ol>\n<li>Express the function in the general form [latex]y=A\\sin(Bx\u2212C)+D[\/latex] or [latex]y=A\\cos(Bx\u2212C)+D[\/latex].<\/li>\n<li>Identify the amplitude, |<em>A<\/em>|.<\/li>\n<li>Identify the period, [latex]P=2\u03c0|B|[\/latex].<\/li>\n<li>Identify the phase shift, [latex]\\frac{C}{B}[\/latex].<\/li>\n<li>Draw the graph of [latex]f(x)=A\\sin(Bx)[\/latex] shifted to the right or left by [latex]\\frac{C}{B}[\/latex] and up or down by <em>D<\/em>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 9: Graphing a Transformed Sinusoid<\/h3>\n<p>Sketch a graph of [latex]f(x)=3\\sin\\left(\\frac{\u03c0}{4}x\u2212\\frac{\u03c0}{4}\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q39590\">Show Solution<\/span><\/p>\n<div id=\"q39590\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Step 1.<\/strong> The function is already written in general form: [latex]f(x)=3\\sin\\left(\\frac{\u03c0}{4}x\u2212\\frac{\u03c0}{4}\\right)[\/latex]. This graph will have the shape of a <strong>sine function<\/strong>, starting at the midline and increasing to the right.<\/p>\n<p><strong>Step 2.<\/strong> |<em>A<\/em>|=|3|=3. The amplitude is 3.<\/p>\n<p><strong>Step 3.<\/strong> Since [latex]|B|=|\\frac{\u03c0}{4}|=\\frac{\u03c0}{4}[\/latex], we determine the period as follows.<\/p>\n<p style=\"text-align: center\">[latex]P=\\frac{2\u03c0}{|B|}=\\frac{2\u03c0}{\\frac{\u03c0}{4}}=2\u03c0\\times\\frac{4}{\u03c0}=8[\/latex]<\/p>\n<p>The period is 8.<\/p>\n<p><strong>Step 4.<\/strong> Since [latex]\\text{C}=\\frac{\u03c0}{4}[\/latex], the phase shift is<\/p>\n<p style=\"text-align: center\">[latex]\\frac{C}{B}=\\frac{\\frac{\\pi}{4}}{\\frac{\\pi}{4}}=1[\/latex].<\/p>\n<p>The phase shift is 1 unit.<\/p>\n<p><strong>Step 5.<\/strong> Figure 20 shows the graph of the function.<\/p>\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\/3675\/2018\/09\/27003954\/CNX_Precalc_Figure_06_01_021.jpg\" alt=\"A graph of 3sin(*(pi\/4)x-pi\/4). Graph has amplitude of 3, period of 8, and a phase shift of 1 to the right.\" width=\"487\" height=\"319\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 20.<\/b> A horizontally compressed, vertically stretched, and horizontally shifted sinusoid<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>\u00a0Try It<\/h3>\n<p>Draw a graph of [latex]g(x)=\u22122\\cos(\\frac{\\pi}{3}x+\\frac{\\pi}{6})[\/latex]. Determine the midline, amplitude, period, and phase shift.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q145991\">Show Solution<\/span><\/p>\n<div id=\"q145991\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\text{midline:}y=0;\\text{amplitude:}|A|=2;\\text{period:}\\text{P}=\\frac{2\\pi}{|B|}=6;\\text{phase shift:}\\frac{C}{B}=\u2212\\frac{1}{2}[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004026\/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.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm173422\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=173422&theme=oea&iframe_resize_id=ohm173422\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 10: Identifying the Properties of a Sinusoidal Function<\/h3>\n<p>Given [latex]y=\u22122\\cos\\left(\\frac{\\pi}{2}x+\\pi\\right)+3[\/latex], determine the amplitude, period, phase shift, and horizontal shift. Then graph the function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q603659\">Show Solution<\/span><\/p>\n<div id=\"q603659\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin by comparing the equation to the general form and use the steps outlined in Example 9.<\/p>\n<p>[latex]y=A\\cos(Bx\u2212C)+D[\/latex]<\/p>\n<p><strong>Step 1.<\/strong> The function is already written in general form.<\/p>\n<p><strong>Step 2.<\/strong> Since <em>A\u00a0<\/em>= \u22122, the amplitude is|<em>A<\/em>| = 2.<\/p>\n<p><strong>Step 3.<\/strong>\u00a0[latex]|B|=\\frac{\\pi}{2}[\/latex], so the period is [latex]P=\\frac{2\u03c0}{|B|}=\\frac{2\\pi}{\\frac{\\pi}{2}}\\times2\\pi=4[\/latex]. The period is 4.<\/p>\n<p><strong>Step 4.<\/strong>\u00a0[latex]C=\u2212\\pi[\/latex], so we calculate the phase shift as [latex]\\frac{C}{B}=\\frac{\u2212\\pi}{\\frac{\\pi}{2}}=\u2212\\pi\\times\\frac{2}{\\pi}=\u22122[\/latex]. The phase shift is \u22122.<\/p>\n<p><strong>Step 5.<\/strong> <em>D\u00a0<\/em>= 3, so the midline is <em>y\u00a0<\/em>= 3, and the vertical shift is up 3.<\/p>\n<p>Since <em>A<\/em> is negative, the graph of the cosine function has been reflected about the x-axis.<\/p>\n<p>Figure 21 shows one cycle of the graph of the function.<\/p>\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\/3675\/2018\/09\/27003956\/CNX_Precalc_Figure_06_01_028.jpg\" alt=\"A graph of -2cos((pi\/2)x+pi)+3. Graph shows an amplitude of 2, midline at y=3, and a period of 4.\" width=\"487\" height=\"317\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 21<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Animation:  Graphing the Sine Function Using The Unit Circle\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/QNQAkUUHNxo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"oembed-2\" title=\"Animation:  Graphing the Cosine Function Using the Unit Circle\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/tcjZOGaeoeo?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Using Transformations of Sine and Cosine Functions<\/h2>\n<p>We can use the transformations of sine and cosine functions in numerous applications. As mentioned at the beginning of the chapter, <strong>circular motion<\/strong> can be modeled using either the sine or <strong>cosine function<\/strong>.<\/p>\n<div class=\"textbox shaded\">\n<h3>Example 11: Finding the Vertical Component of Circular Motion<\/h3>\n<p>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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q140255\">Show Solution<\/span><\/p>\n<div id=\"q140255\" class=\"hidden-answer\" style=\"display: none\">\n<p>Recall that, for a point on a circle of radius r, the y-coordinate of the point is [latex]y=r\\sin(x)[\/latex], so in this case, we get the equation [latex]y(x)=3\\sin(x)[\/latex]. The constant 3 causes a vertical stretch of the y-values of the function by a factor of 3, which we can see in the graph in Figure 22.<\/p>\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\/3675\/2018\/09\/27003959\/CNX_Precalc_Figure_06_01_023.jpg\" alt=\"A graph of 3sin(x). Graph has period of 2pi, amplitude of 3, and range of &#091;-3,3&#093;.\" width=\"487\" height=\"319\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 22<\/b><\/p>\n<\/div>\n<h4>Analysis of the Solution<\/h4>\n<p>Notice that the period of the function is still 2\u03c0; as we travel around the circle, we return to the point (3,0) for [latex]x=2\\pi,4\\pi,6\\pi,\\dots[\/latex] Because the outputs of the graph will now oscillate between \u20133 and 3, the amplitude of the sine wave is 3.<\/p>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>What is the amplitude of the function [latex]f(x)=7\\cos(x)[\/latex]? Sketch a graph of this function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q317443\">Show Solution<\/span><\/p>\n<div id=\"q317443\" class=\"hidden-answer\" style=\"display: none\">\n<p>7<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004029\/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].\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 12: Finding the Vertical Component of Circular Motion<\/h3>\n<p>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 Figure 23. Sketch a graph of the height above the ground of the point <em>P<\/em> as the circle is rotated; then find a function that gives the height in terms of the angle of rotation.<\/p>\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\/3675\/2018\/09\/27004001\/CNX_Precalc_Figure_06_01_025.jpg\" alt=\"An illustration of a circle lifted 4 feet off the ground. Circle has radius of 3 ft. There is a point P labeled on the circle's circumference.\" width=\"487\" height=\"300\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 23<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q367979\">Show Solution<\/span><\/p>\n<div id=\"q367979\" class=\"hidden-answer\" style=\"display: none\">\n<p>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 Figure 24.<\/p>\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\/3675\/2018\/09\/27004004\/CNX_Precalc_Figure_06_01_026.jpg\" alt=\"A graph of -3cox(x)+4. Graph has midline at y=4, amplitude of 3, and period of 2pi.\" width=\"487\" height=\"521\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 24<\/b><\/p>\n<\/div>\n<p>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 <strong>sine function<\/strong> 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>Second, we see that the graph oscillates 3 above and below the center, while a basic cosine has an amplitude of 1, so this graph has been vertically stretched by 3, as in the last example.<\/p>\n<p>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<p style=\"text-align: center\">[latex]y=\u22123\\cos(x)+4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>A weight is attached to a spring that is then hung from a board, as shown in Figure 25. As the spring oscillates up and down, the position <em>y<\/em> of the weight relative to the board ranges from \u20131 in. (at time <em>x<\/em> = 0) to \u20137in. (at time <em>x<\/em> = \u03c0) below the board. Assume the position of <em>y<\/em> is given as a sinusoidal function of <em>x<\/em>. Sketch a graph of the function, and then find a cosine function that gives the position <em>y<\/em> in terms of x.<\/p>\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\/3675\/2018\/09\/27004006\/CNX_Precalc_Figure_06_01_029.jpg\" alt=\"An illustration of a spring with length y.\" width=\"487\" height=\"351\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 25<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q518116\">Show Solution<\/span><\/p>\n<div id=\"q518116\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]y=3\\cos(x)\u22124[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004032\/CNX_Precalc_Figure_06_01_027.jpg\" alt=\"A cosine graph with range &#091;-1,-7&#093;. Period is 2 pi. Local maximums at (0,-1), (2pi,-1), and (4pi, -1). Local minimums at (pi,-7) and (3pi, -7).\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 13: Determining 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=\"q304167\">Show Solution<\/span><\/p>\n<div id=\"q304167\" class=\"hidden-answer\" style=\"display: none\">\n<p>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>Passengers board 2 m above ground level, so the center of the wheel must be located 67.5 + 2 = 69.5 m above ground level. The midline of the oscillation will be at 69.5 m.<\/p>\n<p>The wheel takes 30 minutes to complete 1 revolution, so the height will oscillate with a period of 30 minutes.<\/p>\n<p>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>\n<li>Amplitude: 67.5, so <em>A\u00a0<\/em>= 67.5<\/li>\n<li>Midline: 69.5, so <em>D<\/em> = 69.5<\/li>\n<li>Period: 30, so [latex]B=\\frac{2\\pi}{30}=\\frac{\\pi}{15}[\/latex]<\/li>\n<li>Shape: \u2212cos(<em>t<\/em>)<\/li>\n<\/ul>\n<p>An equation for the rider\u2019s height would be<\/p>\n<p style=\"text-align: center\">[latex]y=\u221267.5\\cos\\left(\\frac{\\pi}{15}t\\right)+69.5[\/latex]<\/p>\n<p>where <em>t<\/em> is in minutes and <em>y<\/em> is measured in meters.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm127257\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=127257&theme=oea&iframe_resize_id=ohm127257\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<section id=\"fs-id1165137574576\" class=\"key-equations\">\n<h2>Analyzing the Graph of y = tan x and Its Variations<\/h2>\n<p>We will begin with the graph of the <strong>tangent<\/strong> function, plotting points as we did for the sine and cosine functions. Recall that<\/p>\n<div>\n<div style=\"text-align: center\">[latex]\\tan x=\\frac{\\sin x}{\\cos x}[\/latex]<\/div>\n<\/div>\n<p>The <strong>period<\/strong> of the tangent function is <em>\u03c0<\/em> because the graph repeats itself on intervals of <em>k\u03c0<\/em> where <em>k<\/em> is a constant. If we graph the tangent function on [latex]\u2212\\dfrac{\\pi}{2}\\text{ to }\\dfrac{\\pi}{2}[\/latex], we can see the behavior of the graph on one complete cycle. If we look at any larger interval, we will see that the characteristics of the graph repeat.<\/p>\n<p>We can determine whether tangent is an odd or even function by using the definition of tangent.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}\\tan(\u2212x)&=\\frac{\\sin(\u2212x)}{\\cos(\u2212x)} && \\text{Definition of tangent.} \\\\ &=\\frac{\u2212\\sin x}{\\cos x} && \\text{Sine is an odd function, cosine is even.} \\\\ &=\u2212\\frac{\\sin x}{\\cos x} && \\text{The quotient of an odd and an even function is odd.} \\\\ &=\u2212\\tan x && \\text{Definition of tangent.} \\end{align}[\/latex]<\/p>\n<p>Therefore, tangent is an odd function. We can further analyze the graphical behavior of the tangent function by looking at values for some of the special angles, as listed in the table below.<\/p>\n<table id=\"Table_06_02_00\" style=\"width: 1035px\" summary=\"Two rows and 10 columns. First row is labeled x and second row is labeled tangent of x. The table has ordered pairs of these column values: (-pi\/2,undefined), (-pi\/3, negative square root of 3), (-pi\/4, -1), (-pi\/6, negative square root of 3 over 3), (0, 0), (pi\/6, square root of 3 over 3), (pi\/4, 1), (pi\/3, square root of 3), (pi\/2, undefined).\">\n<tbody>\n<tr>\n<td style=\"width: 30px\"><em><strong> x <\/strong><\/em><\/td>\n<td style=\"width: 80px\">[latex]-\\frac{\\pi}{6}[\/latex]<\/td>\n<td style=\"width: 80px\">[latex]-\\frac{\\pi}{3}[\/latex]<\/td>\n<td style=\"width: 80px\">[latex]-\\frac{\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 80px\">[latex]-\\frac{\\pi}{6}[\/latex]<\/td>\n<td style=\"width: 80px\">0<\/td>\n<td style=\"width: 80px\">[latex]\\frac{\\pi}{6}[\/latex]<\/td>\n<td style=\"width: 80px\">[latex]\\frac{\\pi}{4}[\/latex]<\/td>\n<td style=\"width: 80px\">[latex]\\frac{\\pi}{3}[\/latex]<\/td>\n<td style=\"width: 80px\">[latex]\\frac{\\pi}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 30px\"><strong> tan (<em>x<\/em>) <\/strong><\/td>\n<td style=\"width: 80px\">undefined<\/td>\n<td style=\"width: 80px\">[latex]\u2212\\sqrt{3}[\/latex]<\/td>\n<td style=\"width: 80px\">\u20131<\/td>\n<td style=\"width: 80px\">[latex]\u2212\\dfrac{\\sqrt{3}}{3}[\/latex]<\/td>\n<td style=\"width: 80px\">0<\/td>\n<td style=\"width: 80px\">[latex]\\dfrac{\\sqrt{3}}{3}[\/latex]<\/td>\n<td style=\"width: 80px\">1<\/td>\n<td style=\"width: 80px\">[latex]\\sqrt{3}[\/latex]<\/td>\n<td style=\"width: 80px\">undefined<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>These points will help us draw our graph, but we need to determine how the graph behaves where it is undefined. If we look more closely at values when [latex]\\frac{\\pi}{3}<x<\\frac{\\pi}{2}[\/latex], we can use a table to look for a trend. Because [latex]\\frac{\\pi}{3}\\approx 1.05[\/latex] and [latex]\\frac{\\pi}{2}\\approx 1.57[\/latex], we will evaluate x at radian measures 1.05 &lt; <em>x<\/em> &lt; 1.57 as shown in the table below.<\/p>\n<table id=\"Table_06_02_01\" summary=\"Two rows and five columns. First row is labeled x and second row is labeled tangent of x. Th table has ordered pairs of these column values: (1.3, 3.6), (1.5, 14.1), (1.55, 48.1), (1.56, 92.6).\">\n<tbody>\n<tr>\n<td><em><strong> x <\/strong><\/em><\/td>\n<td>1.3<\/td>\n<td>1.5<\/td>\n<td>1.55<\/td>\n<td>1.56<\/td>\n<\/tr>\n<tr>\n<td><strong> tan <em>x <\/em><\/strong><\/td>\n<td>3.6<\/td>\n<td>14.1<\/td>\n<td>48.1<\/td>\n<td>92.6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>As <em>x<\/em> approaches [latex]\\frac{\\pi}{2}[\/latex], the outputs of the function get larger and larger. Because [latex]y=\\tan x[\/latex] is an odd function, we see the corresponding table of negative values in the table below.<\/p>\n<table id=\"Table_06_02_02\" summary=\"Two rows and five columns. First row is labeled x and second row is labeled tangent of x. Th table has ordered pairs of these column values: (-1.3, -3.6), (-1.5, -14.1), (-1.55, -48.1), (-1.56, -92.6).\">\n<tbody>\n<tr>\n<td><em><strong> x <\/strong><\/em><\/td>\n<td>\u22121.3<\/td>\n<td>\u22121.5<\/td>\n<td>\u22121.55<\/td>\n<td>\u22121.56<\/td>\n<\/tr>\n<tr>\n<td><strong> tan <em>x <\/em><\/strong><\/td>\n<td>\u22123.6<\/td>\n<td>\u221214.1<\/td>\n<td>\u221248.1<\/td>\n<td>\u221292.6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We can see that, as <em>x<\/em> approaches [latex]\u2212\\dfrac{\\pi}{2}[\/latex], the outputs get smaller and smaller. Remember that there are some values of <em>x<\/em> for which cos <em>x<\/em> = 0. For example, [latex]\\cos\\left(\\frac{\\pi}{2}\\right)=0[\/latex] and [latex]\\cos\\left(\\frac{3\\pi}{2}\\right)=0[\/latex]. At these values, the <strong>tangent function<\/strong> is undefined, so the graph of [latex]y=\\tan x[\/latex] has discontinuities at [latex]x=\\frac{\\pi}{2}[\/latex] and [latex]\\frac{3\\pi}{2}[\/latex]. At these values, the graph of the tangent has vertical asymptotes. Figure 1\u00a0represents the graph of [latex]y=\\tan x[\/latex]. The tangent is positive from 0 to [latex]\\frac{\\pi}{2}[\/latex] and from <em>\u03c0<\/em> to [latex]\\frac{3\\pi}{2}[\/latex], corresponding to quadrants I and III of the unit circle.<\/p>\n<figure id=\"Figure_06_02_001\" class=\"small ui-has-child-figcaption\">\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\/3675\/2018\/09\/27163804\/CNX_Precalc_Figure_06_02_001.jpg\" alt=\"A graph of y=tangent of x. Asymptotes at -pi over 2 and pi over 2.\" width=\"487\" height=\"316\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 26.<\/b> Graph of the tangent function<\/p>\n<\/div>\n<\/figure>\n<h2>Graphing Variations of <em>y<\/em> = tan <em>x<\/em><\/h2>\n<p>As with the sine and cosine functions, the <strong>tangent<\/strong> function can be described by a general equation.<\/p>\n<div>\n<div style=\"text-align: center\">[latex]y=A\\tan(Bx)[\/latex]<\/div>\n<\/div>\n<p>We can identify horizontal and vertical stretches and compressions using values of A and B. The horizontal stretch can typically be determined from the period of the graph. With tangent graphs, it is often necessary to determine a vertical stretch using a point on the graph.<\/p>\n<p>Because there are no maximum or minimum values of a tangent function, the term <em>amplitude<\/em> cannot be interpreted as it is for the sine and cosine functions. Instead, we will use the phrase <em>stretching\/compressing factor<\/em> when referring to the constant A.<\/p>\n<div class=\"textbox\">\n<header>\n<h3>A General Note: Features of the Graph of <em>y<\/em> = <em>A<\/em>tan(<em>Bx<\/em>)<\/h3>\n<\/header>\n<ul>\n<li>The stretching factor is |<em>A<\/em>| .<\/li>\n<li>The period is [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\n<li>The domain is all real numbers <em>x<\/em>, where [latex]x\\ne \\frac{\\pi}{2|B|} + \\frac{\\pi}{|B|} k[\/latex] such that <em>k<\/em> is an integer.<\/li>\n<li>The range is [latex]\\left(-\\infty,\\infty\\right)[\/latex].<\/li>\n<li>The asymptotes occur at [latex]x=\\frac{\\pi}{2|B|} + \\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n<li>[latex]y = A \\tan (Bx)[\/latex] is an odd function.<\/li>\n<\/ul>\n<\/div>\n<h2>Graphing One Period of a Stretched or Compressed Tangent Function<\/h2>\n<p>We can use what we know about the properties of the <strong>tangent function<\/strong> to quickly sketch a graph of any stretched and\/or compressed tangent function of the form [latex]f(x)=A\\tan(Bx)[\/latex]. We focus on a single <strong>period<\/strong> of the function including the origin, because the periodic property enables us to extend the graph to the rest of the function\u2019s domain if we wish. Our limited domain is then the interval [latex](\u2212\\frac{P}{2}, \\frac{P}{2})[\/latex] and the graph has vertical asymptotes at [latex]\\pm \\frac{P}{2}[\/latex] where [latex]P=\\frac{\\pi}{B}[\/latex]. On [latex](\u2212\\dfrac{\\pi}{2}, \\dfrac{\\pi}{2})[\/latex], the graph will come up from the left asymptote at [latex]x=\u2212\\dfrac{\\pi}{2}[\/latex], cross through the origin, and continue to increase as it approaches the right asymptote at [latex]x=\\frac{\\pi}{2}[\/latex]. To make the function approach the asymptotes at the correct rate, we also need to set the vertical scale by actually evaluating the function for at least one point that the graph will pass through. For example, we can use<\/p>\n<div>\n<div style=\"text-align: center\">[latex]f\\left(\\frac{P}{4}\\right)=A \\tan\\left(B\\frac{P}{4}\\right)=A\\tan\\left(B\\frac{\\pi}{4B}\\right)=A[\/latex]<\/div>\n<\/div>\n<p>because \u00a0[latex]\\tan\\left(\\frac{\\pi}{4}\\right)=1[\/latex].<\/p>\n<div class=\"textbox\">\n<h3>How To: Given the function [latex]f(x)=A\\tan(Bx)[\/latex], graph one period.<\/h3>\n<ol>\n<li>Identify the stretching factor, |A|.<\/li>\n<li>Identify <em>B<\/em> and determine the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\n<li>Draw vertical asymptotes at \u00a0[latex]x=\u2212\\dfrac{P}{2}[\/latex] and [latex]x=\\frac{P}{2}[\/latex].<\/li>\n<li>For <em>A<\/em> &gt; 0 , the graph approaches the left asymptote at negative output values and the right asymptote at positive output values (reverse for <em>A<\/em> &lt; 0 ).<\/li>\n<li>Plot reference points at [latex]\\left(\\frac{P}{4},A\\right)[\/latex]\u00a0(0, 0), and ([latex]\u2212\\dfrac{P}{4}[\/latex],\u2212 A), and draw the graph through these points.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 14: Sketching a Compressed Tangent<\/h3>\n<p>Sketch a graph of one period of the function [latex]y=0.5\\tan\\left(\\frac{\\pi}{2}x\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q302986\">Show Solution<\/span><\/p>\n<div id=\"q302986\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we identify <em>A<\/em> and B.<\/p>\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\/3675\/2018\/09\/27163807\/CNX_Precalc_Figure_06_02_002.jpg\" alt=\"An illustration of equations showing that A is the coefficient of tangent and B is the coefficient of x, which is within the tangent function.\" width=\"487\" height=\"113\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 27<\/b><\/p>\n<\/div>\n<p>Because [latex]A=0.5[\/latex] and [latex]B=\\frac{\\pi}{2}[\/latex], we can find the <strong>stretching\/compressing factor<\/strong> and period. The period is [latex]\\frac{\\pi}{\\frac{\\pi}{2}}=2[\/latex], so the asymptotes are at [latex]x=\\pm 1[\/latex]. At a quarter period from the origin, we have<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}f(0.5)&=0.5\\tan\\left(\\frac{0.5\\pi}{2}\\right)\\\\ &=0.5\\tan(\\frac{\\pi}{4})\\\\ &=0.5 \\end{align}[\/latex]<\/p>\n<p>This means the curve must pass through the points(0.5,0.5),(0,0),and(\u22120.5,\u22120.5).The only inflection point is at the origin. Figure shows the graph of one period of the function.<\/p>\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\/3675\/2018\/09\/27163810\/CNX_Precalc_Figure_06_02_003.jpg\" alt=\"A graph of one period of a modified tangent function, with asymptotes at x=-1 and x=1.\" width=\"487\" height=\"258\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 28<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Sketch a graph of [latex]f(x)=3\\tan\\left(\\frac{\\pi}{6}x\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q547078\">Show Solution<\/span><\/p>\n<div id=\"q547078\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163917\/CNX_Precalc_Figure_06_02_004.jpg\" alt=\"A graph of two periods of a modified tangent function, with asymptotes at x=-3 and x=3.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm174880\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174880&theme=oea&iframe_resize_id=ohm174880\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Graphing One Period of a Shifted Tangent Function<\/h2>\n<p>Now that we can graph a <strong>tangent function<\/strong> that is stretched or compressed, we will add a vertical and\/or horizontal (or phase) shift. In this case, we add <em>C<\/em> and <em>D<\/em> to the general form of the tangent function.<\/p>\n<div>\n<div style=\"text-align: center\">[latex]f(x)=A\\tan(Bx\u2212C)+D[\/latex]<\/div>\n<\/div>\n<p>The graph of a transformed tangent function is different from the basic tangent function tan x in several ways:<\/p>\n<div class=\"textbox\">\n<header>\n<h3>A General Note: Features of the Graph of [latex]y = A\\tan\\left(Bx\u2212C\\right)+D[\/latex]<\/h3>\n<\/header>\n<ul>\n<li>The stretching factor is |<em>A<\/em>|.<\/li>\n<li>The period is [latex]\\frac{\\pi}{|B|}[\/latex].<\/li>\n<li>The domain is [latex]x\\ne\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n<li>The range is (\u2212\u221e,\u2212|<em>A<\/em>|] \u222a [|<em>A<\/em>|, \u221e).<\/li>\n<li>The vertical asymptotes occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\n<li>There is no amplitude.<\/li>\n<li>[latex]y=A\\tan(Bx)[\/latex] is an odd function because it is the quotient of odd and even functions (sine and cosine respectively).<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the function [latex]y=A\\tan(Bx\u2212C)+D[\/latex], sketch the graph of one period.<\/h3>\n<ol>\n<li>Express the function given in the form [latex]y=A\\tan(Bx\u2212C)+D[\/latex].<\/li>\n<li>Identify the <strong>stretching\/compressing<\/strong> factor, |A|.<\/li>\n<li>Identify <em>B<\/em> and determine the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\n<li>Identify <em>C<\/em> and determine the phase shift, [latex]\\frac{C}{B}[\/latex].<\/li>\n<li>Draw the graph of [latex]y=A\\tan(Bx)[\/latex] shifted to the right by [latex]\\frac{C}{B}[\/latex] and up by <em>D<\/em>.<\/li>\n<li>Sketch the vertical asymptotes, which occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\n<li>Plot any three reference points and draw the graph through these points.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 15: Graphing One Period of a Shifted Tangent Function<\/h3>\n<p>Graph one period of the function [latex]y=\u22122\\tan(\\pi x+\\pi)\u22121[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q385350\">Show Solution<\/span><\/p>\n<div id=\"q385350\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Step 1.<\/strong> The function is already written in the form [latex]y=A\\tan(Bx\u2212C)+D[\/latex].<\/p>\n<p><strong>Step 2.<\/strong>\u00a0[latex]A=\u22122[\/latex], so the stretching factor is [latex]|A|=2[\/latex].<\/p>\n<p><strong>Step 3.<\/strong>\u00a0[latex]B=\\pi[\/latex], so the period is [latex]P=\\frac{\\pi}{|B|}=\\frac{\\pi}{\\pi}=1[\/latex].<\/p>\n<p><strong>Step 4.<\/strong>\u00a0[latex]C=\u2212\\pi[\/latex], so the phase shift is [latex]\\dfrac{C}{B}=\\dfrac{\u2212\\pi}{\\pi}=\u22121[\/latex].<\/p>\n<p><strong>Step 5\u20137.<\/strong> The asymptotes are at [latex]x=\u2212\\frac{3}{2}[\/latex] and [latex]x=\u2212\\frac{1}{2}[\/latex] and the three recommended reference points are (\u22121.25, 1), (\u22121,\u22121), and (\u22120.75, \u22123). The graph is shown in Figure 4.<\/p>\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\/3675\/2018\/09\/27163812\/CNX_Precalc_Figure_06_02_005.jpg\" alt=\"A graph of one period of a shifted tangent function, with vertical asymptotes at x=-1.5 and x=-0.5.\" width=\"487\" height=\"193\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 29<\/b><\/p>\n<\/div>\n<h4>Analysis of the Solution<\/h4>\n<p>Note that this is a decreasing function because <em>A<\/em> &lt; 0.<\/p>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>How would the graph in Example 2\u00a0look different if we made <em>A<\/em> = 2 instead of \u22122?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q560477\">Show Solution<\/span><\/p>\n<div id=\"q560477\" class=\"hidden-answer\" style=\"display: none\">\n<p>It would be reflected across the line [latex]y=\u22121[\/latex], becoming an increasing function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given the graph of a tangent function, identify horizontal and vertical stretches.<\/h3>\n<ol>\n<li>Find the period <em>P<\/em> from the spacing between successive vertical asymptotes or <em>x<\/em>-intercepts.<\/li>\n<li>Write [latex]f(x)=A\\tan\\left(\\frac{\\pi}{P}x\\right)[\/latex].<\/li>\n<li>Determine a convenient point (<em>x<\/em>, <em>f<\/em>(<em>x<\/em>)) on the given graph and use it to determine <em>A<\/em>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 16: Identifying the Graph of a Stretched Tangent<\/h3>\n<p>Find a formula for the function graphed in Figure 5.<\/p>\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\/3675\/2018\/09\/27163814\/CNX_Precalc_Figure_06_02_006.jpg\" alt=\"A graph of two periods of a modified tangent function, with asymptotes at x=-4 and x=4.\" width=\"487\" height=\"256\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 30<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q606896\">Show Solution<\/span><\/p>\n<div id=\"q606896\" class=\"hidden-answer\" style=\"display: none\">\n<p>The graph has the shape of a tangent function.<\/p>\n<p><strong>Step 1.<\/strong> One cycle extends from \u20134 to 4, so the period is [latex]P=8[\/latex]. Since [latex]P=\\frac{\\pi}{|B|}[\/latex], we have [latex]B=\\frac{\\pi}{P}=\\frac{\\pi}{8}[\/latex].<\/p>\n<p><strong>Step 2.<\/strong> The equation must have the [latex]\\text{form}f(x)=A\\tan\\left(\\frac{\\pi}{8}x\\right)[\/latex].<\/p>\n<p><strong>Step 3.<\/strong> To find the vertical stretch <em>A<\/em>, we can use the point (2,2).<\/p>\n<p style=\"text-align: center\">[latex]2=A\\tan\\left(\\frac{\\pi}{8}\\times2\\right)=A\\tan\\left(\\frac{\\pi}{4}\\right)[\/latex]<\/p>\n<p>Because [latex]\\tan\\left(\\frac{\\pi}{4}\\right)=1[\/latex], <em>A<\/em> = 2.<\/p>\n<p>This function would have a formula [latex]f(x)=2\\tan\\left(\\frac{\\pi}{8}x\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Find a formula for the function in Figure 6.<\/p>\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\/3675\/2018\/09\/27163816\/CNX_Precalc_Figure_06_02_007.jpg\" alt=\"A graph of four periods of a modified tangent function, Vertical asymptotes at -3pi\/4, -pi\/4, pi\/4, and 3pi\/4.\" width=\"487\" height=\"315\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 31<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q359527\">Show Solution<\/span><\/p>\n<div id=\"q359527\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]g(x)=4\\tan(2x)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm129731\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=129731&theme=oea&iframe_resize_id=ohm129731\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div>\n<h2>Using the Graphs of Trigonometric Functions to Solve Real-World Problems<\/h2>\n<p>Many real-world scenarios represent periodic functions and may be modeled by trigonometric functions. As an example, let\u2019s return to the scenario from the section opener. Have you ever observed the beam formed by the rotating light on a police car and wondered about the movement of the light beam itself across the wall? The periodic behavior of the distance the light shines as a function of time is obvious, but how do we determine the distance? We can use the tangent function .<\/p>\n<div class=\"textbox shaded\">\n<h3>Example 17: Using Trigonometric Functions to Solve Real-World Scenarios<\/h3>\n<p>Suppose the function [latex]y=5\\tan\\left(\\frac{\\pi}{4}t\\right)[\/latex] marks the distance in the movement of a light beam from the top of a police car across a wall where <em>t<\/em> is the time in seconds and <em>y<\/em> is the distance in feet from a point on the wall directly across from the police car.<\/p>\n<ol>\n<li>Find and interpret the stretching factor and period.<\/li>\n<li>Graph on the interval [0, 5].<\/li>\n<li>Evaluate <em>f<\/em>(1) and discuss the function\u2019s value at that input.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q351813\">Show Solution<\/span><\/p>\n<div id=\"q351813\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>We know from the general form of \u00a0[latex]y=A\\tan(Bt)\\\\[\/latex] \u00a0that |<em>A<\/em>| is the stretching factor and \u03c0 B is the period.<br \/>\n<figure id=\"Image_06_02_022\" 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\/3675\/2018\/09\/27163819\/CNX_Precalc_Figure_06_02_022.jpg\" alt=\"A graph showing that variable A is the coefficient of the tangent function and variable B is the coefficient of x, which is within that tangent function.\" width=\"487\" height=\"107\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 32<\/b><\/p>\n<\/div>\n<\/figure>\n<p>We see that the stretching factor is 5. This means that the beam of light will have moved 5 ft after half the period.<\/p>\n<p>The period is [latex]\\frac{\\pi}{\\frac{\\pi}{4}}=\\frac{\\pi}{1}\\times \\frac{4}{\\pi}=4[\/latex]. This means that every 4 seconds, the beam of light sweeps the wall. The distance from the spot across from the police car grows larger as the police car approaches.<\/li>\n<li>To graph the function, we draw an asymptote at [latex]t=2[\/latex] and use the stretching factor and period. See Figure 8.<br \/>\n<figure id=\"Image_06_02_021\" 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\/3675\/2018\/09\/27163821\/CNX_Precalc_Figure_06_02_021n.jpg\" alt=\"A graph of one period of a modified tangent function, with a vertical asymptote at x=4.\" width=\"487\" height=\"319\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 33<\/b><\/p>\n<\/div>\n<\/figure>\n<\/li>\n<li>period: [latex]f(1)=5\\tan \\left(\\frac{\\pi}{4}\\left(1\\right)\\right)=5\\left(1\\right)=5[\/latex]; after 1 second, the beam of has moved 5 ft from the spot across from the police car.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"oembed-3\" title=\"Animation:  Graphing the Tangent Function Using the Unit Circle\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/ssjG9kE25OY?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Analyzing the Graphs of y = sec x and y = cscx and Their Variations<\/h2>\n<p>The <strong>secant<\/strong> was defined by the <strong>reciprocal identity<\/strong>\u00a0[latex]\\sec x=\\frac{1}{\\cos x}[\/latex]. Notice that the function is undefined when the cosine is 0, leading to vertical asymptotes at [latex]\\frac{\\pi}{2},\\frac{3\\pi}{2}\\text{, etc}[\/latex].\u00a0Because the cosine is never more than 1 in absolute value, the secant, being the reciprocal, will never be less than 1 in absolute value.<\/p>\n<p>We can graph [latex]y=\\sec x[\/latex] by observing the graph of the cosine function because these two functions are reciprocals of one another. See Figure 9. The graph of the cosine is shown as a dashed orange wave so we can see the relationship. Where the graph of the cosine function decreases, the graph of the <strong>secant function<\/strong> increases. Where the graph of the cosine function increases, the graph of the secant function decreases. When the cosine function is zero, the secant is undefined.<\/p>\n<p>The secant graph has vertical asymptotes at each value of <em>x<\/em> where the cosine graph crosses the <em>x<\/em>-axis; we show these in the graph below with dashed vertical lines, but will not show all the asymptotes explicitly on all later graphs involving the secant and cosecant.<\/p>\n<p>Note that, because cosine is an even function, secant is also an even function. That is, [latex]\\sec(\u2212x)=\\sec x[\/latex].<\/p>\n<figure id=\"Figure_06_02_008\" class=\"small ui-has-child-figcaption\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163823\/CNX_Precalc_Figure_06_02_008.jpg\" alt=\"A graph of cosine of x and secant of x. Asymptotes for secant of x shown at -3pi\/2, -pi\/2, pi\/2, and 3pi\/2.\" \/><\/p>\n<div style=\"text-align: center\"><strong>Figure 34.\u00a0<\/strong>Graph of the secant function, [latex]f(x)=\\sec x=\\frac{1}{\\cos x}[\/latex]<\/div>\n<\/figure>\n<p>As we did for the tangent function, we will again refer to the constant |<em>A<\/em>| as the stretching factor, not the amplitude.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Features of the Graph of <em>y<\/em> = <em>A<\/em>sec(<em>Bx<\/em>)<\/h3>\n<ul>\n<li>The stretching factor is |<em>A<\/em>|.<\/li>\n<li>The period is [latex]\\frac{2\\pi}{|B|}[\/latex].<\/li>\n<li>The domain is [latex]x\\ne \\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\n<li>The range is (\u2212\u221e, \u2212|<em>A<\/em>|] \u222a [|<em>A<\/em>|, \u221e).<\/li>\n<li>The vertical asymptotes occur at [latex]x=\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\n<li>There is no amplitude.<\/li>\n<li>[latex]y=A\\sec(Bx)[\/latex] is an even function because cosine is an even function.<\/li>\n<\/ul>\n<\/div>\n<p>Similar to the secant, the <strong>cosecant<\/strong> is defined by the reciprocal identity [latex]\\csc x=1\\sin x[\/latex]. Notice that the function is undefined when the sine is 0, leading to a vertical asymptote in the graph at 0, \u03c0, etc. Since the sine is never more than 1 in absolute value, the cosecant, being the reciprocal, will never be less than 1 in absolute value.<\/p>\n<p>We can graph [latex]y=\\csc x[\/latex] by observing the graph of the sine function because these two functions are reciprocals of one another. See Figure 10. The graph of sine is shown as a dashed orange wave so we can see the relationship. Where the graph of the sine function decreases, the graph of the <strong>cosecant function<\/strong> increases. Where the graph of the sine function increases, the graph of the cosecant function decreases.<\/p>\n<p>The cosecant graph has vertical asymptotes at each value of <em>x<\/em> where the sine graph crosses the <em>x<\/em>-axis; we show these in the graph below with dashed vertical lines.<\/p>\n<p>Note that, since sine is an odd function, the cosecant function is also an odd function. That is, [latex]\\csc(\u2212x)=\u2212\\csc x[\/latex].<\/p>\n<p>The graph of cosecant, which is shown in Figure 10, is similar to the graph of secant.<\/p>\n<figure id=\"Figure_06_02_009\" class=\"small ui-has-child-figcaption\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163825\/CNX_Precalc_Figure_06_02_009.jpg\" alt=\"A graph of cosecant of x and sin of x. Five vertical asymptotes shown at multiples of pi.\" \/><\/p>\n<div style=\"text-align: center\"><strong>Figure 35.\u00a0<\/strong>The graph of the cosecant function, [latex]f(x)=\\csc x=\\frac{1}{\\sin x}\/latex]<\/div>\n<\/figure>\n<div class=\"textbox\">\n<header>\n<h3>A General Note: Features of the Graph of [latex]y=A\\csc(Bx)<\/h3>\n<\/header>\n<ul>\n<li>The stretching factor is |<em>A<\/em>|.<\/li>\n<li>The period is [latex]\\frac{2\\pi}{|B|}[\/latex].<\/li>\n<li>The domain is [latex]x\\ne\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n<li>The range is ( \u2212\u221e, \u2212|A|] \u222a [|A|, \u221e).<\/li>\n<li>The asymptotes occur at [latex]x=\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n<li>[latex]y=A\\csc(Bx)[\/latex] is an odd function because sine is an odd function.<\/li>\n<\/ul>\n<\/div>\n<h2>Graphing Variations of <em>y<\/em> = sec <em>x<\/em> and <em>y\u00a0<\/em>= csc <em>x<\/em><\/h2>\n<p>For shifted, compressed, and\/or stretched versions of the secant and cosecant functions, we can follow similar methods to those we used for tangent and cotangent. That is, we locate the vertical asymptotes and also evaluate the functions for a few points (specifically the local extrema). If we want to graph only a single period, we can choose the interval for the period in more than one way. The procedure for secant is very similar, because the cofunction identity means that the secant graph is the same as the cosecant graph shifted half a period to the left. Vertical and phase shifts may be applied to the <strong>cosecant function<\/strong> in the same way as for the secant and other functions. The equations become the following.<\/p>\n<div>\n<div style=\"text-align: center\">[latex]y=A\\sec(Bx\u2212C)+D[\/latex]<\/div>\n<\/div>\n<div style=\"text-align: center\">[latex]y=A\\csc(Bx\u2212C)+D[\/latex]<\/div>\n<div class=\"textbox\">\n<header>\n<h3>A General Note: Features of the Graph of [latex]y=A\\sec(Bx\u2212C)+D[\/latex]<\/h3>\n<\/header>\n<ul>\n<li>The stretching factor is |<em>A<\/em>|.<\/li>\n<li>The period is [latex]\\frac{2\\pi}{|B|}[\/latex].<\/li>\n<li>The domain is [latex]x\\ne \\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\n<li>The range is (\u2212\u221e, \u2212|<em>A<\/em>|] \u222a [|<em>A<\/em>|, \u221e).<\/li>\n<li>The vertical asymptotes occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\n<li>There is no amplitude.<\/li>\n<li>[latex]y=A\\sec(Bx)[\/latex] is an even function because cosine is an even function.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\">\n<header>\n<h3>A General Note: Features of the Graph of [latex]y=A\\csc(Bx\u2212C)+D[\/latex]<\/h3>\n<\/header>\n<ul>\n<li>The stretching factor is |<em>A<\/em>|.<\/li>\n<li>The period is [latex]\\frac{2\\pi}{|B|}[\/latex].<\/li>\n<li>The domain is [latex]x\\ne\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n<li>The range is (\u2212\u221e, \u2212|<em>A<\/em>|] \u222a [|<em>A<\/em>|, \u221e).<\/li>\n<li>The vertical asymptotes occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n<li>There is no amplitude.<\/li>\n<li>[latex]y=A\\csc(Bx)[\/latex] is an odd function because sine is an odd function.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a function of the form [latex]y=A\\sec(Bx)[\/latex], graph one period.<\/h3>\n<ol>\n<li>Express the function given in the form [latex]y=A\\sec(Bx)[\/latex].<\/li>\n<li>Identify the stretching\/compressing factor, |A|.<\/li>\n<li>Identify <em>B<\/em> and determine the period, [latex]P=\\frac{2\\pi}{|B|}[\/latex].<\/li>\n<li>Sketch the graph of [latex]y=A\\cos(Bx)[\/latex].<\/li>\n<li>Use the reciprocal relationship between [latex]y=\\cos x[\/latex] and [latex]y=\\sec x[\/latex] to draw the graph of [latex]y=A\\sec(Bx)[\/latex].<\/li>\n<li>Sketch the asymptotes.<\/li>\n<li>Plot any two reference points and draw the graph through these points.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 18: Graphing a Variation of the Secant Function<\/h3>\n<p>Graph one period of [latex]f(x)=2.5\\sec(0.4x)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q926159\">Show Solution<\/span><\/p>\n<div id=\"q926159\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Step 1.<\/strong> The given function is already written in the general form, [latex]y=A\\sec(Bx)[\/latex].<br \/>\n<strong>Step 2.<\/strong>\u00a0[latex]A=2.5[\/latex] so the stretching factor is 2.5.<br \/>\n<strong>Step 3.<\/strong>\u00a0[latex]B=0.4[\/latex], so [latex]P=\\frac{2\\pi}{0.4}=5\\pi[\/latex]. The period is 5\u03c0 units.<br \/>\n<strong>Step 4.<\/strong> Sketch the graph of the function [latex]g(x)=2.5\\cos(0.4x)[\/latex].<br \/>\n<strong>Step 5.<\/strong> Use the reciprocal relationship of the cosine and secant functions to draw the cosecant function.<br \/>\n<strong>Steps 6\u20137.<\/strong> Sketch two asymptotes at [latex]x=1.25\\pi[\/latex]\u00a0and [latex]x=3.75\\pi[\/latex]. We can use two reference points, the local minimum at (0, 2.5) and the local maximum at (2.5\u03c0, \u22122.5). Figure 11 shows the graph.<\/p>\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\/3675\/2018\/09\/27163829\/CNX_Precalc_Figure_06_02_010.jpg\" alt=\"A graph of one period of a modified secant function, which looks like an upward facing prarbola and a downward facing parabola.\" width=\"487\" height=\"567\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 36<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Graph one period of [latex]f(x)=\u22122.5\\sec(0.4x)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q945046\">Show Solution<\/span><\/p>\n<div id=\"q945046\" class=\"hidden-answer\" style=\"display: none\">\n<p>This is a vertical reflection of the preceding graph because <em>A<\/em> is negative.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163919\/CNX_Precalc_Figure_06_02_011.jpg\" alt=\"A graph of one period of a modified secant function, which looks like an downward facing prarbola and a upward facing parabola.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h3>Do the vertical shift and stretch\/compression affect the secant\u2019s range?<\/h3>\n<p><em>Yes. The range of<\/em>\u00a0[latex]f(x) = A\\sec(Bx \u2212 C) + D[\/latex] is ( \u2212\u221e, \u2212|<em>A<\/em>| + <em>D<\/em>] \u222a [|<em>A<\/em>| + <em>D<\/em>, \u221e).<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a function of the form [latex]f(x)=A\\sec (Bx\u2212C)+D[\/latex], graph one period.<\/h3>\n<ol>\n<li>Express the function given in the form [latex]y=A\\sec(Bx\u2212C)+D[\/latex].<\/li>\n<li>Identify the stretching\/compressing factor, |<em>A<\/em>|.<\/li>\n<li>Identify <em>B<\/em> and determine the period, [latex]\\frac{2\\pi}{|B|}[\/latex].<\/li>\n<li>Identify <em>C<\/em> and determine the phase shift, [latex]\\frac{C}{B}[\/latex].<\/li>\n<li>Draw the graph of [latex]y=A\\sec(Bx)[\/latex]. but shift it to the right by [latex]\\frac{C}{B}[\/latex] and up by <em>D<\/em>.<\/li>\n<li>Sketch the vertical asymptotes, which occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{2|B|}k[\/latex], where <em>k<\/em> is an odd integer.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 19: Graphing a Variation of the Secant Function<\/h3>\n<p>Graph one period of [latex]y=4\\sec \\left(\\frac{\\pi}{3}x\u2212\\frac{\\pi}{2}\\right)+1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q429424\">Show Solution<\/span><\/p>\n<div id=\"q429424\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Step 1.<\/strong> Express the function given in the form [latex]y=4\\sec \\left(\\frac{\\pi}{3}x\u2212\\frac{\\pi}{2}\\right)+1[\/latex].<\/p>\n<p><strong>Step 2.<\/strong> The stretching\/compressing factor is |<em>A<\/em>| = 4.<\/p>\n<p><strong>Step 3.<\/strong> The period is<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align} \\frac{2\\pi}{|B|}&=\\frac{2\\pi}{\\frac{\\pi}{3}}\\\\ &=\\frac{2\\pi}{1}\\times\\frac{3}{\\pi}\\\\ &=6 \\end{align}[\/latex]<\/p>\n<p><strong>Step 4.<\/strong> The phase shift is<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}\\frac{C}{B}&=\\frac{\\frac{\\pi}{2}}{\\frac{\\pi}{3}} \\\\ &=\\frac{\\pi}{2} \\times \\frac{3}{\\pi} \\\\ &=1.5 \\end{align}[\/latex]<\/p>\n<p><strong>Step 5.<\/strong> Draw the graph of [latex]y=A\\sec(Bx)[\/latex],but shift it to the right by [latex]\\frac{C}{B}=1.5[\/latex] and up by <em>D\u00a0<\/em>= 6.<\/p>\n<p><strong>Step 6.<\/strong> Sketch the vertical asymptotes, which occur at <em>x\u00a0<\/em>= 0, <em>x<\/em> = 3, and <em>x<\/em> = 6. There is a local minimum at (1.5, 5) and a local maximum at (4.5, \u22123). Figure 12 shows the graph.<\/p>\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\/3675\/2018\/09\/27163831\/CNX_Precalc_Figure_06_02_012-1.jpg\" alt=\"\" width=\"487\" height=\"318\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 37<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Graph one period of [latex]f(x)=\u22126\\sec(4x+2)\u22128[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q142167\">Show Solution<\/span><\/p>\n<div id=\"q142167\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163922\/CNX_Precalc_Figure_06_02_013.jpg\" alt=\"A graph of one period of a modified secant function. There are two vertical asymptotes, one at approximately x=-pi\/20 and one approximately at 3pi\/16.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm174885\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=174885&theme=oea&iframe_resize_id=ohm174885\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<h4>The domain of [latex]\\csc x[\/latex] was given to be all <em>x<\/em> such that [latex]x\\ne k\\pi[\/latex] for any integer <em>k<\/em>. Would the domain of\u00a0[latex]y=A\\csc(Bx\u2212C)+D[\/latex] be [latex]x\\ne\\frac{C+k\\pi}{B}[\/latex]?<\/h4>\n<p><em>Yes. The excluded points of the domain follow the vertical asymptotes. Their locations show the horizontal shift and compression or expansion implied by the transformation to the original function\u2019s input.<\/em><\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a function of the form [latex]y=A\\csc(Bx)[\/latex], graph one period.<\/h3>\n<ol>\n<li>Express the function given in the form [latex]y=A\\csc(Bx)[\/latex].<\/li>\n<li>|<em>A<\/em>|.<\/li>\n<li>Identify <em>B<\/em> and determine the period, [latex]P=\\frac{2\\pi}{|B|}[\/latex].<\/li>\n<li>Draw the graph of [latex]y=A\\sin(Bx)[\/latex].<\/li>\n<li>Use the reciprocal relationship between [latex]y=\\sin x[\/latex] and [latex]y=\\csc x[\/latex] to draw the graph of [latex]y=A\\csc(Bx)[\/latex].<\/li>\n<li>Sketch the asymptotes.<\/li>\n<li>Plot any two reference points and draw the graph through these points.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 20: Graphing a Variation of the Cosecant Function<\/h3>\n<p>Graph one period of [latex]f(x)=\u22123\\csc(4x)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q194858\">Show Solution<\/span><\/p>\n<div id=\"q194858\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Step 1.<\/strong> The given function is already written in the general form, [latex]y=A\\csc(Bx)[\/latex].<\/p>\n<p><strong>Step 2. <\/strong>[latex]|A|=|\u22123|=3[\/latex], so the stretching factor is 3.<\/p>\n<p><strong>Step 3.<\/strong> [latex]B=4\\text{, so}P=\\frac{2\\pi}{4}=\\frac{\\pi}{2}[\/latex].The period is [latex]\\frac{\\pi}{2}[\/latex] units.<\/p>\n<p><strong>Step 4.<\/strong> Sketch the graph of the function [latex]g(x)=\u22123\\sin(4x)[\/latex].<\/p>\n<p><strong>Step 5.<\/strong> Use the reciprocal relationship of the sine and cosecant functions to draw the cosecant function.<\/p>\n<p><strong>Steps 6\u20137.<\/strong> Sketch three asymptotes at [latex]x=0\\text{, }x=\\frac{\\pi}{4}\\text{, and }x=\\frac{\\pi}{2}[\/latex].We can use two reference points, the local maximum at [latex]\\left(\\frac{\\pi}{8}\\text{, }\u22123\\right)[\/latex] and the local minimum at [latex]\\left(\\frac{3\\pi}{8}\\text{, }3\\right)[\/latex]. Figure 13 shows the graph.<\/p>\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\/3675\/2018\/09\/27163833\/CNX_Precalc_Figure_06_02_023.jpg\" alt=\"A graph of one period of a cosecant function. There are vertical asymptotes at x=0, x=pi\/4, and x=pi\/2.\" width=\"487\" height=\"686\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 38<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Graph one period of [latex]f(x)=0.5\\csc(2x)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q267711\">Show Solution<\/span><\/p>\n<div id=\"q267711\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163924\/CNX_Precalc_Figure_06_02_023b.jpg\" alt=\"A graph of one period of a modified secant function, which looks like an downward facing prarbola and a upward facing parabola.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a function of the form [latex]f(x)=A\\csc(Bx\u2212C)+D[\/latex], graph one period.<\/h3>\n<ol>\n<li>Express the function given in the form [latex]y=A\\csc(Bx\u2212C)+D[\/latex].<\/li>\n<li>Identify the stretching\/compressing factor, |<em>A<\/em>|.<\/li>\n<li>Identify <em>B<\/em> and determine the period, [latex]\\frac{2\\pi}{|B|}[\/latex].<\/li>\n<li>Identify <em>C<\/em> and determine the phase shift, [latex]\\frac{C}{B}[\/latex].<\/li>\n<li>Draw the graph of [latex]y=A\\csc(Bx)[\/latex] but shift it to the right by and up by <em>D<\/em>.<\/li>\n<li>Sketch the vertical asymptotes, which occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 21: Graphing a Vertically Stretched, Horizontally Compressed, and Vertically Shifted Cosecant<\/h3>\n<p>Sketch a graph of [latex]y=2\\csc\\left(\\frac{\\pi}{2}x\\right)+1[\/latex]. What are the domain and range of this function?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q993272\">Show Solution<\/span><\/p>\n<div id=\"q993272\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Step 1.<\/strong> Express the function given in the form [latex]y=2\\csc\\left(\\frac{\\pi}{2}x\\right)+1[\/latex].<\/p>\n<p><strong>Step 2.<\/strong> Identify the stretching\/compressing factor, [latex]|A|=2[\/latex].<\/p>\n<p><strong>Step 3.<\/strong> The period is [latex]\\frac{2\\pi}{|B|}=\\frac{2\\pi}{\\frac{\\pi}{2}}=\\frac{2\\pi}{1}\\times \\frac{2}{\\pi}=4[\/latex].<\/p>\n<p><strong>Step 4.<\/strong> The phase shift is [latex]\\frac{0}{\\frac{\\pi}{2}}=0[\/latex].<\/p>\n<p><strong>Step 5.<\/strong> Draw the graph of [latex]y=A\\csc(Bx)[\/latex] but shift it up [latex]D=1[\/latex].<\/p>\n<p><strong>Step 6.<\/strong> Sketch the vertical asymptotes, which occur at <em>x<\/em> = 0, <em>x<\/em> = 2, <em>x<\/em> = 4.<\/p>\n<p>The graph for this function is shown in Figure 14.<\/p>\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\/3675\/2018\/09\/27163835\/CNX_Precalc_Figure_06_02_014F.jpg\" alt=\"A graph of 3 periods of a modified cosecant function, with 3 vertical asymptotes, and a dotted sinusoidal function that has local maximums where the cosecant function has local minimums and local minimums where the cosecant function has local maximums.\" width=\"487\" height=\"377\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 39<\/b><\/p>\n<\/div>\n<h4>Analysis of the Solution<\/h4>\n<p>The vertical asymptotes shown on the graph mark off one period of the function, and the local extrema in this interval are shown by dots. Notice how the graph of the transformed cosecant relates to the graph of [latex]f(x)=2\\sin\\left(\\frac{\\pi}{2}x\\right)+1[\/latex], shown as the orange dashed wave.<\/p>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Given the graph of [latex]f(x)=2\\cos\\left(\\frac{\\pi}{2}x\\right)+1[\/latex] shown in Figure 15, sketch the graph of [latex]g(x)=2\\sec\\left(\\frac{\\pi}{2}x\\right)+1[\/latex] on the same axes.<\/p>\n<p>&nbsp;<\/p>\n<div style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163838\/CNX_Precalc_Figure_06_02_015.jpg\" alt=\"A graph of two periods of a modified cosine function. Range is [-1,3], graphed from x=-4 to x=4.\" width=\"488\" height=\"381\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 40<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q560894\">Show Solution<\/span><\/p>\n<div id=\"q560894\" class=\"hidden-answer\" style=\"display: none\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27163927\/CNX_Precalc_Figure_06_02_016.jpg\" alt=\"A graph of two periods of both a secant and consine function. Grpah shows that cosine function has local maximums where secant function has local minimums and vice versa.\" \/><\/div>\n<\/div>\n<\/div>\n<h2>Analyzing the Graph of y = cot x and Its Variations<\/h2>\n<p>The last trigonometric function we need to explore is <strong>cotangent<\/strong>. The cotangent is defined by the <strong>reciprocal identity<\/strong> [latex]\\cot x=\\frac{1}{\\tan x}[\/latex]. Notice that the function is undefined when the tangent function is 0, leading to a vertical asymptote in the graph at 0, \u03c0, etc. Since the output of the tangent function is all real numbers, the output of the <strong>cotangent function<\/strong> is also all real numbers.<\/p>\n<p>We can graph [latex]y=\\cot x[\/latex] by observing the graph of the tangent function because these two functions are reciprocals of one another. See Figure 16. Where the graph of the tangent function decreases, the graph of the cotangent function increases. Where the graph of the tangent function increases, the graph of the cotangent function decreases.<\/p>\n<p>The cotangent graph has vertical asymptotes at each value of <em>x<\/em> where [latex]\\tan x=0[\/latex]; we show these in the graph below with dashed lines. Since the cotangent is the reciprocal of the tangent, [latex]\\cot x[\/latex] has vertical asymptotes at all values of <em>x<\/em> where [latex]\\tan x=0[\/latex] , and [latex]\\cot x=0[\/latex] at all values of x where tan x has its vertical asymptotes.<\/p>\n<figure id=\"Figure_06_02_017\" class=\"small ui-has-child-figcaption\">\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\/3675\/2018\/09\/27163840\/CNX_Precalc_Figure_06_02_017.jpg\" alt=\"A graph of cotangent of x, with vertical asymptotes at multiples of pi.\" width=\"487\" height=\"439\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 41.<\/b> The cotangent function<\/p>\n<\/div><figcaption><\/figcaption><\/figure>\n<div class=\"textbox\">\n<h3>A General Note: Features of the Graph of <em>y<\/em> = <em>A<\/em>cot(<em>Bx<\/em>)<\/h3>\n<ul>\n<li>The stretching factor is |<em>A<\/em>|.<\/li>\n<li>The period is [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\n<li>The domain is [latex]x\\ne\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n<li>The range is (\u2212\u221e, \u221e).<\/li>\n<li>The asymptotes occur at [latex]x=\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n<li>[latex]y=A\\cot(Bx)[\/latex] is an odd function.<\/li>\n<\/ul>\n<\/div>\n<h2>Graphing Variations of <em>y<\/em> = cot <em>x<\/em><\/h2>\n<p>We can transform the graph of the cotangent in much the same way as we did for the tangent. The equation becomes the following.<\/p>\n<div>\n<div style=\"text-align: center\">[latex]y=A\\cot(Bx\u2212C)+D[\/latex]<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Properties of the Graph of <em>y<\/em> = <em>A<\/em>cot(<em>Bx<\/em>\u2212C)+<em>D<\/em><\/h3>\n<ul>\n<li>The stretching factor is |<em>A<\/em>|.<\/li>\n<li>The period is [latex]\\frac{\\pi}{|B|}[\/latex].<\/li>\n<li>The domain is [latex]x\\ne\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n<li>The range is (\u2212\u221e, \u2212|<em>A<\/em>|] \u222a [|<em>A<\/em>|, \u221e).<\/li>\n<li>The vertical asymptotes occur at [latex]x=\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n<li>There is no amplitude.<\/li>\n<li>[latex]y=A\\cot(Bx)[\/latex] is an odd function because it is the quotient of even and odd functions (cosine and sine, respectively)<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a modified cotangent function of the form [latex]f(x)=A\\cot(Bx)[\/latex], graph one period.<\/h3>\n<ol>\n<li>Express the function in the form [latex]f(x)=A\\cot(Bx)[\/latex].<\/li>\n<li>Identify the stretching factor, |<em>A<\/em>|.<\/li>\n<li>Identify the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\n<li>Draw the graph of [latex]y=A\\tan(Bx)[\/latex].<\/li>\n<li>Plot any two reference points.<\/li>\n<li>Use the reciprocal relationship between tangent and cotangent to draw the graph of [latex]y=A\\cot(Bx)[\/latex].<\/li>\n<li>Sketch the asymptotes.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 22: Graphing Variations of the Cotangent Function<\/h3>\n<p>Determine the stretching factor, period, and phase shift of [latex]y=3\\cot(4x)[\/latex], and then sketch a graph.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q32362\">Show Solution<\/span><\/p>\n<div id=\"q32362\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Step 1.<\/strong> Expressing the function in the form [latex]f(x)=A\\cot(Bx)[\/latex] gives [latex]f(x)=3\\cot(4x)[\/latex].<\/p>\n<p><strong>Step 2.<\/strong> The stretching factor is [latex]|A|=3[\/latex].<\/p>\n<p><strong>Step 3.<\/strong> The period is [latex]P=\\frac{\\pi}{4}[\/latex].<\/p>\n<p><strong>Step 4.<\/strong> Sketch the graph of [latex]y=3\\tan(4x)[\/latex].<\/p>\n<p><strong>Step 5.<\/strong> Plot two reference points. Two such points are [latex]\\left(\\frac{\\pi}{16}\\text{, }3\\right)[\/latex] and [latex]\\left(\\frac{3\\pi}{16}\\text{, }\u22123\\right)[\/latex].<\/p>\n<p><strong>Step 6.<\/strong> Use the reciprocal relationship to draw [latex]y=3\\cot(4x)[\/latex].<\/p>\n<p><strong>Step 7.<\/strong> Sketch the asymptotes, [latex]x=0[\/latex], [latex]x=\\frac{\\pi}{4}[\/latex].<\/p>\n<p>The orange graph in Figure 17 shows [latex]y=3\\tan(4x)[\/latex] and the blue graph shows [latex]y=3\\cot(4x)[\/latex].<\/p>\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\/3675\/2018\/09\/27163843\/CNX_Precalc_Figure_06_02_019.jpg\" alt=\"A graph of two periods of a modified tangent function and a modified cotangent function. Vertical asymptotes at x=-pi\/4 and pi\/4.\" width=\"487\" height=\"592\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 42<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a modified cotangent function of the form [latex]f(x)=A\\cot(Bx\u2212C)+D[\/latex], graph one period.<\/h3>\n<ol>\n<li>Express the function in the form [latex]f(x)=A\\cot(Bx\u2212C)+D[\/latex].<\/li>\n<li>Identify the stretching factor, |<em>A<\/em>|.<\/li>\n<li>Identify the period, [latex]P=\\frac{\\pi}{|B|}[\/latex].<\/li>\n<li>Identify the phase shift, [latex]\\frac{C}{B}[\/latex].<\/li>\n<li>Draw the graph of [latex]y=A\\tan(Bx)[\/latex] shifted to the right by [latex]\\frac{C}{B}[\/latex] and up by <em>D<\/em>.<\/li>\n<li>Sketch the asymptotes [latex]x =\\frac{C}{B}+\\frac{\\pi}{|B|}k[\/latex], where <em>k<\/em> is an integer.<\/li>\n<li>Plot any three reference points and draw the graph through these points.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Example 23: Graphing a Modified Cotangent<\/h3>\n<p>Sketch a graph of one period of the function [latex]f(x)=4\\cot\\left(\\frac{\\pi}{8}x\u2212\\frac{\\pi}{2}\\right)\u22122[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q706245\">Show Solution<\/span><\/p>\n<div id=\"q706245\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>Step 1.<\/strong> The function is already written in the general form [latex]f(x)=A\\cot(Bx\u2212C)+D[\/latex].<\/p>\n<p><strong>Step 2.<\/strong>\u00a0[latex]A=4[\/latex], so the stretching factor is 4.<\/p>\n<p><strong>Step 3.<\/strong>\u00a0[latex]B=\\frac{\\pi}{8}[\/latex], so the period is [latex]P=\\frac{\\pi}{|B|}=\\frac{\\pi}{\\frac{\\pi}{8}}=8[\/latex].<\/p>\n<p><strong>Step 4.<\/strong>\u00a0[latex]C=\\frac{\\pi}{2}[\/latex], so the phase shift is [latex]\\frac{C}{B}=\\frac{\\frac{\\pi}{2}}{\\frac{\\pi}{8}}=4[\/latex].<\/p>\n<p><strong>Step 5.<\/strong> We draw [latex]f(x)=4\\tan\\left(\\frac{\\pi}{8}x\u2212\\frac{\\pi}{2}\\right)\u22122[\/latex].<\/p>\n<p><strong>Step 6-7.<\/strong> Three points we can use to guide the graph are (6,2), (8,\u22122), and (10,\u22126). We use the reciprocal relationship of tangent and cotangent to draw [latex]f(x)=4\\cot\\left(\\frac{\\pi}{8}x\u2212\\frac{\\pi}{2}\\right)\u22122[\/latex].<\/p>\n<p><strong>Step 8.<\/strong> The vertical asymptotes are [latex]x=4[\/latex] and [latex]x=12[\/latex].<\/p>\n<p>The graph is shown in Figure 18.<\/p>\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\/3675\/2018\/09\/27163846\/CNX_Precalc_Figure_06_02_020.jpg\" alt=\"A graph of one period of a modified cotangent function. Vertical asymptotes at x=4 and x=12.\" width=\"487\" height=\"315\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 43.<\/b> One period of a modified cotangent function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2>Key Equations<\/h2>\n<table>\n<tbody>\n<tr>\n<td>Shifted, compressed, and\/or stretched tangent function<\/td>\n<td>[latex]y=A\\tan(Bx\u2212C)+D[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Shifted, compressed, and\/or stretched secant function<\/td>\n<td>[latex]y=A\\sec(Bx\u2212C)+D[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Shifted, compressed, and\/or stretched cosecant<\/td>\n<td>[latex]y=A\\csc(Bx\u2212C)+D[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Shifted, compressed, and\/or stretched cotangent function<\/td>\n<td>[latex]y=A\\cot(Bx\u2212C)+D[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2><\/h2>\n<\/section>\n<section id=\"fs-id1165137540392\" class=\"key-concepts\">\n<h1>Key Concepts<\/h1>\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 2\u03c0.<\/li>\n<li>The function sin <em>x<\/em> is odd, so its graph is symmetric about the origin. The function cos <em>x<\/em> 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]\\text{P}=\\frac{2\\pi}{|B|}[\/latex].<\/li>\n<li>In the general formula for a sinusoidal function, |<em>A<\/em>|represents amplitude. If |<em>A<\/em>| &gt; 1, the function is stretched, whereas if|<em>A<\/em>| &lt; 1, the function is compressed.<\/li>\n<li>The value [latex]\\frac{C}{B}[\/latex] in the general formula for a sinusoidal function indicates the phase shift.<\/li>\n<li>The value <em>D<\/em> in the general formula for a sinusoidal function indicates the vertical shift from the midline.<\/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 be graphed by identifying its amplitude and period.<\/li>\n<li>A function can also be graphed by identifying its amplitude, period, phase shift, and horizontal shift.<\/li>\n<li>Sinusoidal functions can be used to solve real-world problems.<\/li>\n<li>The tangent function has period \u03c0.<\/li>\n<li>[latex]f(x)=A\\tan(Bx\u2212C)+D[\/latex] is a tangent with vertical and\/or horizontal stretch\/compression and shift.<\/li>\n<li>The secant and cosecant are both periodic functions with a period of2\u03c0. [latex]f(x)=A\\sec(Bx\u2212C)+D[\/latex] gives a shifted, compressed, and\/or stretched secant function graph.<\/li>\n<li>[latex]f(x)=A\\csc(Bx\u2212C)+D[\/latex] gives a shifted, compressed, and\/or stretched cosecant function graph.<\/li>\n<li>The cotangent function has period \u03c0 and vertical asymptotes at 0, \u00b1\u03c0,\u00b12\u03c0,....<\/li>\n<li>The range of cotangent is (\u2212\u221e,\u221e),and the function is decreasing at each point in its range.<\/li>\n<li>The cotangent is zero at [latex]\\pm\\frac{\\pi}{2}\\text{, }\\pm\\frac{3\\pi}{2}[\/latex],....<\/li>\n<li>[latex]f(x)=A\\cot(Bx\u2212C)+D[\/latex] is a cotangent with vertical and\/or horizontal stretch\/compression and shift.<\/li>\n<li>Real-world scenarios can be solved using graphs of trigonometric functions.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165137414167\" class=\"definition\">\n<dt>amplitude<\/dt>\n<dd id=\"fs-id1165137463141\">the vertical height of a function; the constant <em>A<\/em> appearing in the definition of a sinusoidal function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137602032\" class=\"definition\">\n<dt>midline<\/dt>\n<dd id=\"fs-id1165137602037\">the horizontal line <em>y\u00a0<\/em>= <em>D<\/em>, where <em>D<\/em> appears in the general form of a sinusoidal function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137678058\" class=\"definition\">\n<dt>periodic function<\/dt>\n<dd id=\"fs-id1165137678063\">a function <em>f<\/em>(<em>x<\/em>) that satisfies [latex]f(x+P)=f(x)[\/latex] for a specific constant <em>P\u00a0<\/em>and any value of <em>x<\/em><\/dd>\n<\/dl>\n<dl id=\"fs-id1165137939683\" class=\"definition\">\n<dt>phase shift<\/dt>\n<dd id=\"fs-id1165137939688\">the horizontal displacement of the basic sine or cosine function; the constant [latex]\\frac{C}{B}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135160153\" class=\"definition\">\n<dt>sinusoidal function<\/dt>\n<dd id=\"fs-id1165137737500\">any function that can be expressed in the form [latex]f(x)=A\\sin(Bx\u2212C)+D[\/latex] or [latex]f(x)=A\\cos(Bx\u2212C)+D[\/latex]<\/dd>\n<\/dl>\n<\/section>\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-14093\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <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 class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Animation: Graphing the Sine Function Using The Unit Circle . <strong>Authored by<\/strong>: Mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/QNQAkUUHNxo\">https:\/\/youtu.be\/QNQAkUUHNxo<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><li>Animation: Graphing the Cosine Function Using the Unit Circle. <strong>Authored by<\/strong>: Mathispower4u. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/tcjZOGaeoeo\">https:\/\/youtu.be\/tcjZOGaeoeo<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/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":17533,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Animation: Graphing the Sine Function Using The Unit Circle \",\"author\":\"Mathispower4u\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/QNQAkUUHNxo\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Animation: Graphing the Cosine Function Using the Unit Circle\",\"author\":\"Mathispower4u\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/tcjZOGaeoeo\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube 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