{"id":1520,"date":"2025-07-25T02:31:51","date_gmt":"2025-07-25T02:31:51","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1520"},"modified":"2026-03-12T05:50:37","modified_gmt":"2026-03-12T05:50:37","slug":"graphs-of-the-other-trigonometric-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-the-other-trigonometric-functions-fresh-take\/","title":{"raw":"Graphs of the Other Trigonometric Functions: Fresh Take","rendered":"Graphs of the Other Trigonometric Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Graph variations of y=tan x and y=cot x.<\/li>\r\n \t<li>Determine a function formula from a tangent or cotangent graph.<\/li>\r\n \t<li>Graph variations of y=sec x and y=csc x.<\/li>\r\n \t<li>Determine a function formula from a secant or cosecant graph.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Graph Variations of Tangent and Cotangent<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"360\" data-end=\"664\">The graphs of [latex]y=\\tan x[\/latex] and [latex]y=\\cot x[\/latex] look different from sine and cosine because they have <strong data-start=\"480\" data-end=\"503\">vertical asymptotes<\/strong> and repeat every [latex]\\pi[\/latex] units (instead of [latex]2\\pi[\/latex]). Variations of these graphs come from changing the parameters in the general forms:<\/p>\r\n\r\n<ul data-start=\"666\" data-end=\"735\">\r\n \t<li data-start=\"666\" data-end=\"700\">\r\n<p data-start=\"668\" data-end=\"700\">[latex]y=a\\tan(bx-c)+d[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"701\" data-end=\"735\">\r\n<p data-start=\"703\" data-end=\"735\">[latex]y=a\\cot(bx-c)+d[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<p data-start=\"737\" data-end=\"792\">Each parameter changes the graph in predictable ways:<\/p>\r\n\r\n<ul data-start=\"793\" data-end=\"1058\">\r\n \t<li data-start=\"793\" data-end=\"857\">\r\n<p data-start=\"795\" data-end=\"857\">[latex]a[\/latex] stretches or reflects the graph vertically.<\/p>\r\n<\/li>\r\n \t<li data-start=\"858\" data-end=\"924\">\r\n<p data-start=\"860\" data-end=\"924\">[latex]b[\/latex] changes the period (the length of one cycle).<\/p>\r\n<\/li>\r\n \t<li data-start=\"925\" data-end=\"991\">\r\n<p data-start=\"927\" data-end=\"991\">[latex]c[\/latex] shifts the graph left or right (phase shift).<\/p>\r\n<\/li>\r\n \t<li data-start=\"992\" data-end=\"1058\">\r\n<p data-start=\"994\" data-end=\"1058\">[latex]d[\/latex] shifts the graph up or down (vertical shift).<\/p>\r\n<\/li>\r\n<\/ul>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Graphing Variations of Tangent and Cotangent<\/strong>\r\n<ol>\r\n \t<li data-start=\"1129\" data-end=\"1364\">\r\n<p data-start=\"1132\" data-end=\"1157\"><strong data-start=\"1132\" data-end=\"1155\">Base Graph Features<\/strong><\/p>\r\n\r\n<ul data-start=\"1161\" data-end=\"1364\">\r\n \t<li data-start=\"1161\" data-end=\"1268\">\r\n<p data-start=\"1163\" data-end=\"1268\">[latex]y=\\tan x[\/latex]: period [latex]\\pi[\/latex], asymptotes at [latex]x=\\dfrac{\\pi}{2}+k\\pi[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1272\" data-end=\"1364\">\r\n<p data-start=\"1274\" data-end=\"1364\">[latex]y=\\cot x[\/latex]: period [latex]\\pi[\/latex], asymptotes at [latex]x=k\\pi[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1366\" data-end=\"1533\">\r\n<p data-start=\"1369\" data-end=\"1389\"><strong data-start=\"1369\" data-end=\"1387\">Period Changes<\/strong><\/p>\r\n\r\n<ul data-start=\"1393\" data-end=\"1533\">\r\n \t<li data-start=\"1393\" data-end=\"1450\">\r\n<p data-start=\"1395\" data-end=\"1450\">Formula: [latex]\\text{Period}=\\dfrac{\\pi}{b}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1454\" data-end=\"1533\">\r\n<p data-start=\"1456\" data-end=\"1533\">Larger [latex]b[\/latex] = compressed, smaller [latex]b[\/latex] = stretched.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1535\" data-end=\"1658\">\r\n<p data-start=\"1538\" data-end=\"1583\"><strong data-start=\"1538\" data-end=\"1581\">Phase Shift [latex]\\dfrac{c}{b}[\/latex]<\/strong><\/p>\r\n\r\n<ul data-start=\"1587\" data-end=\"1658\">\r\n \t<li data-start=\"1587\" data-end=\"1621\">\r\n<p data-start=\"1589\" data-end=\"1621\">Moves the graph left or right.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1625\" data-end=\"1658\">\r\n<p data-start=\"1627\" data-end=\"1658\">Asymptotes shift accordingly.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1660\" data-end=\"1813\">\r\n<p data-start=\"1663\" data-end=\"1696\"><strong data-start=\"1663\" data-end=\"1694\">Vertical Stretch\/Reflection<\/strong><\/p>\r\n\r\n<ul data-start=\"1700\" data-end=\"1813\">\r\n \t<li data-start=\"1700\" data-end=\"1752\">\r\n<p data-start=\"1702\" data-end=\"1752\">[latex]a[\/latex] changes steepness of the graph.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1756\" data-end=\"1813\">\r\n<p data-start=\"1758\" data-end=\"1813\">Negative [latex]a[\/latex] reflects across the x-axis.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1815\" data-end=\"1927\">\r\n<p data-start=\"1818\" data-end=\"1855\"><strong data-start=\"1818\" data-end=\"1853\">Vertical Shift [latex]d[\/latex]<\/strong><\/p>\r\n\r\n<ul data-start=\"1859\" data-end=\"1927\">\r\n \t<li data-start=\"1859\" data-end=\"1892\">\r\n<p data-start=\"1861\" data-end=\"1892\">Moves the midline up or down.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1896\" data-end=\"1927\">\r\n<p data-start=\"1898\" data-end=\"1927\">Asymptotes remain vertical.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1929\" data-end=\"2263\">\r\n<p data-start=\"1932\" data-end=\"1955\"><strong data-start=\"1932\" data-end=\"1953\">Graphing Strategy<\/strong><\/p>\r\n\r\n<ul data-start=\"1959\" data-end=\"2263\">\r\n \t<li data-start=\"1959\" data-end=\"2023\">\r\n<p data-start=\"1961\" data-end=\"2023\">Step 1: Find the period using [latex]\\dfrac{\\pi}{b}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"2027\" data-end=\"2061\">\r\n<p data-start=\"2029\" data-end=\"2061\">Step 2: Locate the asymptotes.<\/p>\r\n<\/li>\r\n \t<li data-start=\"2065\" data-end=\"2109\">\r\n<p data-start=\"2067\" data-end=\"2109\">Step 3: Apply phase and vertical shifts.<\/p>\r\n<\/li>\r\n \t<li data-start=\"2113\" data-end=\"2211\">\r\n<p data-start=\"2115\" data-end=\"2211\">Step 4: Plot key points ([latex]\\pm 1[\/latex] for tangent; reciprocal behavior for cotangent).<\/p>\r\n<\/li>\r\n \t<li data-start=\"2215\" data-end=\"2263\">\r\n<p data-start=\"2217\" data-end=\"2263\">Step 5: Sketch the curve between asymptotes.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Sketch the graph of [latex]f(x) = 2\\tan\\left(\\pi x\\right)[\/latex]. Identify the period and vertical asymptotes.[reveal-answer q=\"tan-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"tan-001\"]From [latex]f(x) = 2\\tan(\\pi x)[\/latex]Stretching factor: [latex]A = 2[\/latex]Period: [latex]\\frac{\\pi}{|B|} = \\frac{\\pi}{\\pi} = 1[\/latex]Vertical asymptotes: occur at [latex]x = -\\frac{1}{2}, \\frac{1}{2}, \\frac{3}{2}, ...[\/latex]The graph is stretched vertically by a factor of 2 and completes one cycle over an interval of length 1.<img class=\"alignnone wp-image-5554\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11173858\/Screenshot-2026-02-11-at-10.38.24%E2%80%AFAM.png\" alt=\"The graph shows a tangent function with repeating vertical asymptotes. Vertical dashed lines appear at x equals negative 0.5, 0.5, 1.5, 2.5, and so on, spaced one unit apart. Between each pair of asymptotes, the graph is an increasing curve that goes from negative infinity to positive infinity. Each branch crosses the x-axis at the midpoint between consecutive asymptotes.\" width=\"375\" height=\"370\" \/>\r\n[\/hidden-answer]<\/section>\r\n<div><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ahafafhe-4FF-zSGYnaM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/4FF-zSGYnaM?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ahafafhe-4FF-zSGYnaM\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660596&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ahafafhe-4FF-zSGYnaM&vembed=0&video_id=4FF-zSGYnaM&video_target=tpm-plugin-ahafafhe-4FF-zSGYnaM'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Trigonometry+-+The+graphs+of+tan+and+cot_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cTrigonometry - The graphs of tan and cot\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Function Formulas from Tangent and Cotangent Graphs<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"80\" data-end=\"412\">To write a function formula from a tangent or cotangent graph, we analyze its key features: the <strong data-start=\"176\" data-end=\"186\">period<\/strong>, the <strong data-start=\"192\" data-end=\"207\">phase shift<\/strong> (location of asymptotes or intercepts), any <strong data-start=\"252\" data-end=\"270\">vertical shift<\/strong>, and the <strong data-start=\"280\" data-end=\"293\">steepness<\/strong> of the curve. Tangent and cotangent graphs both repeat every [latex]\\pi[\/latex], but parameters in the general forms<\/p>\r\n\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul data-start=\"414\" data-end=\"483\">\r\n \t<li data-start=\"414\" data-end=\"448\">\r\n<p data-start=\"416\" data-end=\"448\">[latex]y=a\\tan(bx-c)+d[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"449\" data-end=\"483\">\r\n<p data-start=\"451\" data-end=\"483\">[latex]y=a\\cot(bx-c)+d[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">control how the graph is stretched, shifted, or reflected. By identifying these features from the graph, we can reconstruct the exact equation.<\/span>\r\n\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Building Tangent or Cotangent Formulas<\/strong>\r\n<ol data-start=\"695\" data-end=\"1981\">\r\n \t<li data-start=\"695\" data-end=\"910\">\r\n<p data-start=\"698\" data-end=\"724\"><strong data-start=\"698\" data-end=\"722\">Determine the Period<\/strong><\/p>\r\n\r\n<ul data-start=\"728\" data-end=\"910\">\r\n \t<li data-start=\"728\" data-end=\"803\">\r\n<p data-start=\"730\" data-end=\"803\">For tangent and cotangent, [latex]\\text{Period}=\\dfrac{\\pi}{b}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"807\" data-end=\"910\">\r\n<p data-start=\"809\" data-end=\"910\">Measure the distance between consecutive asymptotes (or repeating points) to find [latex]b[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"912\" data-end=\"1205\">\r\n<p data-start=\"915\" data-end=\"939\"><strong data-start=\"915\" data-end=\"937\">Locate Phase Shift<\/strong><\/p>\r\n\r\n<ul data-start=\"943\" data-end=\"1205\">\r\n \t<li data-start=\"943\" data-end=\"1029\">\r\n<p data-start=\"945\" data-end=\"1029\">Tangent asymptotes: [latex]x=\\dfrac{c}{b}+\\dfrac{\\pi}{2b}+k\\dfrac{\\pi}{b}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1033\" data-end=\"1105\">\r\n<p data-start=\"1035\" data-end=\"1105\">Cotangent asymptotes: [latex]x=\\dfrac{c}{b}+k\\dfrac{\\pi}{b}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1109\" data-end=\"1205\">\r\n<p data-start=\"1111\" data-end=\"1205\">Identify where the central asymptote (for tangent) or intercept (for cotangent) has shifted.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1207\" data-end=\"1356\">\r\n<p data-start=\"1210\" data-end=\"1252\"><strong data-start=\"1210\" data-end=\"1250\">Find Vertical Shift [latex]d[\/latex]<\/strong><\/p>\r\n\r\n<ul data-start=\"1256\" data-end=\"1356\">\r\n \t<li data-start=\"1256\" data-end=\"1303\">\r\n<p data-start=\"1258\" data-end=\"1303\">Midline of the graph is [latex]y=d[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1307\" data-end=\"1356\">\r\n<p data-start=\"1309\" data-end=\"1356\">Check if the curve has been moved up or down.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1358\" data-end=\"1530\">\r\n<p data-start=\"1361\" data-end=\"1414\"><strong data-start=\"1361\" data-end=\"1412\">Determine [latex]a[\/latex] (Stretch\/Reflection)<\/strong><\/p>\r\n\r\n<ul data-start=\"1418\" data-end=\"1530\">\r\n \t<li data-start=\"1418\" data-end=\"1461\">\r\n<p data-start=\"1420\" data-end=\"1461\">[latex]a[\/latex] changes the steepness.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1465\" data-end=\"1530\">\r\n<p data-start=\"1467\" data-end=\"1530\">Negative [latex]a[\/latex] flips the graph across the midline.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1532\" data-end=\"1743\">\r\n<p data-start=\"1535\" data-end=\"1573\"><strong data-start=\"1535\" data-end=\"1571\">Choose Tangent or Cotangent Form<\/strong><\/p>\r\n\r\n<ul data-start=\"1577\" data-end=\"1743\">\r\n \t<li data-start=\"1577\" data-end=\"1659\">\r\n<p data-start=\"1579\" data-end=\"1659\">Tangent passes through the origin (before shifts) and increases left to right.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1663\" data-end=\"1743\">\r\n<p data-start=\"1665\" data-end=\"1743\">Cotangent decreases left to right, starting with an asymptote at the origin.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1745\" data-end=\"1981\">\r\n<p data-start=\"1748\" data-end=\"1772\"><strong data-start=\"1748\" data-end=\"1770\">Write the Equation<\/strong><\/p>\r\n\r\n<ul data-start=\"1776\" data-end=\"1981\">\r\n \t<li data-start=\"1776\" data-end=\"1981\">\r\n<p data-start=\"1778\" data-end=\"1981\">Plug amplitude [latex]a[\/latex], period factor [latex]b[\/latex], phase shift [latex]c[\/latex], and vertical shift [latex]d[\/latex] into [latex]y=a\\tan(bx-c)+d[\/latex] or [latex]y=a\\cot(bx-c)+d[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">A tangent graph has period [latex]2\\pi[\/latex] and passes through the point [latex]\\left(\\frac{\\pi}{2}, 3\\right)[\/latex]. Write a possible equation.[reveal-answer q=\"tan-002\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"tan-002\"]\r\nUse [latex]f(x) = A\\tan(Bx)[\/latex]Period:\r\n[latex]\\frac{\\pi}{B} = 2\\pi[\/latex], so [latex]B = \\frac{1}{2}[\/latex]Find A:\r\nSubstitute [latex]\\left(\\frac{\\pi}{2}, 3\\right)[\/latex]:\r\n[latex]\\begin{align*}\r\n3 &amp;= A\\tan\\left(\\frac{1}{2} \\cdot \\frac{\\pi}{2}\\right) \\\\\r\n3 &amp;= A\\tan\\left(\\frac{\\pi}{4}\\right) \\\\\r\n3 &amp;= A(1) \\\\\r\nA &amp;= 3\r\n\\end{align*}[\/latex]The function is:[latex]f(x) = 3\\tan\\left(\\frac{x}{2}\\right)[\/latex]\r\n[\/hidden-answer]<\/section>\r\n<div><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dcedddaf-x_yn02gwnPA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/x_yn02gwnPA?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dcedddaf-x_yn02gwnPA\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660597&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dcedddaf-x_yn02gwnPA&vembed=0&video_id=x_yn02gwnPA&video_target=tpm-plugin-dcedddaf-x_yn02gwnPA'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Writing+Equations+for+Tangent+Graphs_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWriting Equations for Tangent Graphs\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Graph Variations of Secant and Cosecant<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"68\" data-end=\"480\">The graphs of [latex]y=\\sec x[\/latex] and [latex]y=\\csc x[\/latex] are built from cosine and sine, since [latex]\\sec x=\\dfrac{1}{\\cos x}[\/latex] and [latex]\\csc x=\\dfrac{1}{\\sin x}[\/latex]. They feature repeating <strong data-start=\"280\" data-end=\"292\">U-shaped<\/strong> and <strong data-start=\"297\" data-end=\"318\">inverted U-shaped<\/strong> branches with <strong data-start=\"333\" data-end=\"356\">vertical asymptotes<\/strong> where sine or cosine equals zero. Variations of these graphs are created by changing the parameters in the general forms:<\/p>\r\n\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul data-start=\"482\" data-end=\"551\">\r\n \t<li data-start=\"482\" data-end=\"516\">\r\n<p data-start=\"484\" data-end=\"516\">[latex]y=a\\sec(bx-c)+d[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"517\" data-end=\"551\">\r\n<p data-start=\"519\" data-end=\"551\">[latex]y=a\\csc(bx-c)+d[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p data-start=\"553\" data-end=\"629\">Each parameter controls how the graph is stretched, shifted, or reflected.<\/p>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Graphing Variations of Secant and Cosecant<\/strong>\r\n<ol>\r\n \t<li data-start=\"698\" data-end=\"935\">\r\n<p data-start=\"701\" data-end=\"726\"><strong data-start=\"701\" data-end=\"724\">Base Graph Features<\/strong><\/p>\r\n\r\n<ul data-start=\"730\" data-end=\"935\">\r\n \t<li data-start=\"730\" data-end=\"838\">\r\n<p data-start=\"732\" data-end=\"838\">[latex]y=\\sec x[\/latex]: period [latex]2\\pi[\/latex], asymptotes at [latex]x=\\dfrac{\\pi}{2}+k\\pi[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"842\" data-end=\"935\">\r\n<p data-start=\"844\" data-end=\"935\">[latex]y=\\csc x[\/latex]: period [latex]2\\pi[\/latex], asymptotes at [latex]x=k\\pi[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"937\" data-end=\"1071\">\r\n<p data-start=\"940\" data-end=\"960\"><strong data-start=\"940\" data-end=\"958\">Period Changes<\/strong><\/p>\r\n\r\n<ul data-start=\"964\" data-end=\"1071\">\r\n \t<li data-start=\"964\" data-end=\"1022\">\r\n<p data-start=\"966\" data-end=\"1022\">Formula: [latex]\\text{Period}=\\dfrac{2\\pi}{b}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1026\" data-end=\"1071\">\r\n<p data-start=\"1028\" data-end=\"1071\">Adjusts how wide each repeating cycle is.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1073\" data-end=\"1209\">\r\n<p data-start=\"1076\" data-end=\"1121\"><strong data-start=\"1076\" data-end=\"1119\">Phase Shift [latex]\\dfrac{c}{b}[\/latex]<\/strong><\/p>\r\n\r\n<ul data-start=\"1125\" data-end=\"1209\">\r\n \t<li data-start=\"1125\" data-end=\"1159\">\r\n<p data-start=\"1127\" data-end=\"1159\">Moves the graph left or right.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1163\" data-end=\"1209\">\r\n<p data-start=\"1165\" data-end=\"1209\">Asymptotes and branches shift accordingly.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1211\" data-end=\"1447\">\r\n<p data-start=\"1214\" data-end=\"1264\"><strong data-start=\"1214\" data-end=\"1262\">Vertical Stretch\/Reflection [latex]a[\/latex]<\/strong><\/p>\r\n\r\n<ul data-start=\"1268\" data-end=\"1447\">\r\n \t<li data-start=\"1268\" data-end=\"1372\">\r\n<p data-start=\"1270\" data-end=\"1372\">[latex]a[\/latex] changes the distance from the midline to the minimum\/maximum points of each branch.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1376\" data-end=\"1447\">\r\n<p data-start=\"1378\" data-end=\"1447\">Negative [latex]a[\/latex] reflects the branches across the midline.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1449\" data-end=\"1615\">\r\n<p data-start=\"1452\" data-end=\"1489\"><strong data-start=\"1452\" data-end=\"1487\">Vertical Shift [latex]d[\/latex]<\/strong><\/p>\r\n\r\n<ul data-start=\"1493\" data-end=\"1615\">\r\n \t<li data-start=\"1493\" data-end=\"1553\">\r\n<p data-start=\"1495\" data-end=\"1553\">Moves the midline up or down, shifting the entire graph.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1557\" data-end=\"1615\">\r\n<p data-start=\"1559\" data-end=\"1615\">Asymptotes stay vertical, but branch positions adjust.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1617\" data-end=\"1917\">\r\n<p data-start=\"1620\" data-end=\"1643\"><strong data-start=\"1620\" data-end=\"1641\">Graphing Strategy<\/strong><\/p>\r\n\r\n<ul data-start=\"1647\" data-end=\"1917\">\r\n \t<li data-start=\"1647\" data-end=\"1696\">\r\n<p data-start=\"1649\" data-end=\"1696\">Step 1: Start with sine or cosine as a guide.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1700\" data-end=\"1757\">\r\n<p data-start=\"1702\" data-end=\"1757\">Step 2: Identify asymptotes where sine or cosine = 0.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1761\" data-end=\"1835\">\r\n<p data-start=\"1763\" data-end=\"1835\">Step 3: Plot key points at maximum\/minimum distances from the midline.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1839\" data-end=\"1917\">\r\n<p data-start=\"1841\" data-end=\"1917\">Step 4: Sketch U-shaped and inverted U-shaped branches between asymptotes.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Sketch the graph of [latex]g(x) = 3\\csc(2x)[\/latex]. Identify the period and vertical asymptotes.[reveal-answer q=\"sec-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"sec-001\"]\r\nFrom [latex]g(x) = 3\\csc(2x)[\/latex]Stretching factor: [latex]A = 3[\/latex]Period:\r\n[latex]\\frac{2\\pi}{|B|} = \\frac{2\\pi}{2} = \\pi[\/latex]Vertical asymptotes:\r\noccur where [latex]\\sin(2x) = 0[\/latex], at [latex]x = 0, \\frac{\\pi}{2}, \\pi, \\frac{3\\pi}{2}, ...[\/latex]Graph [latex]y = 3\\sin(2x)[\/latex] first, then draw the cosecant as the reciprocal with asymptotes where sine equals zero.<img class=\"alignnone wp-image-5555\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11175204\/Screenshot-2026-02-11-at-10.44.49%E2%80%AFAM.png\" alt=\"The dashed curve is a sine function with amplitude 3 and period pi. It oscillates between y equals 3 and y equals negative 3. It passes through the points: (0, 0), (pi\/4, 3), (pi\/2, 0), (3pi\/4, -3), (pi, 0) The solid black curves are branches of the cosecant function with a vertical stretch factor of 3 and period pi. There are vertical asymptotes at: x equals 0, pi\/2, and pi. Between 0 and pi\/2, the cosecant branch forms a U-shape with a minimum at (pi\/4, 3). Between pi\/2 and pi, the branch forms an upside-down U-shape with a maximum at (3pi\/4, -3). The cosecant graph touches the sine graph at the sine function\u2019s maximum and minimum points.\" width=\"373\" height=\"268\" \/>\r\n[\/hidden-answer]<\/section>\r\n<div><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ceggaece-2o_dxyKAXLQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/2o_dxyKAXLQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ceggaece-2o_dxyKAXLQ\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660598&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ceggaece-2o_dxyKAXLQ&vembed=0&video_id=2o_dxyKAXLQ&video_target=tpm-plugin-ceggaece-2o_dxyKAXLQ'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+to+Graph+Secant+and+Cosecant_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Graph Secant and Cosecant\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Function Formula from Secant and Cosecant Graphs<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"78\" data-end=\"485\">To determine a function formula from a secant or cosecant graph, we use the fact that these graphs are built as reciprocals of cosine and sine. The graph\u2019s <strong data-start=\"234\" data-end=\"245\">midline<\/strong>, <strong data-start=\"247\" data-end=\"265\">vertical shift<\/strong>, <strong data-start=\"267\" data-end=\"277\">period<\/strong>, <strong data-start=\"279\" data-end=\"294\">phase shift<\/strong>, and <strong data-start=\"300\" data-end=\"322\">stretch\/reflection<\/strong> can all be read directly from its repeating U-shaped or inverted U-shaped branches. Once these features are identified, they are plugged into the general forms:<\/p>\r\n\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul>\r\n \t<li style=\"list-style-type: none;\">\r\n<ul data-start=\"487\" data-end=\"556\">\r\n \t<li data-start=\"487\" data-end=\"521\">\r\n<p data-start=\"489\" data-end=\"521\">[latex]y=a\\sec(bx-c)+d[\/latex]<\/p>\r\n<\/li>\r\n \t<li data-start=\"522\" data-end=\"556\">\r\n<p data-start=\"524\" data-end=\"556\">[latex]y=a\\csc(bx-c)+d[\/latex]<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p data-start=\"558\" data-end=\"631\">Recognizing the asymptotes and midline first helps anchor the equation.<\/p>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Building Secant or Cosecant Formulas<\/strong>\r\n<ol>\r\n \t<li data-start=\"694\" data-end=\"855\">\r\n<p data-start=\"697\" data-end=\"723\"><strong data-start=\"697\" data-end=\"721\">Determine the Period<\/strong><\/p>\r\n\r\n<ul data-start=\"727\" data-end=\"855\">\r\n \t<li data-start=\"727\" data-end=\"785\">\r\n<p data-start=\"729\" data-end=\"785\">Formula: [latex]\\text{Period}=\\dfrac{2\\pi}{b}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"789\" data-end=\"855\">\r\n<p data-start=\"791\" data-end=\"855\">Measure the distance between repeating branches or asymptotes.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"857\" data-end=\"1144\">\r\n<p data-start=\"860\" data-end=\"888\"><strong data-start=\"860\" data-end=\"886\">Locate the Phase Shift<\/strong><\/p>\r\n\r\n<ul data-start=\"892\" data-end=\"1144\">\r\n \t<li data-start=\"892\" data-end=\"986\">\r\n<p data-start=\"894\" data-end=\"986\">Secant asymptotes: align with where cosine = 0, i.e. [latex]x=\\dfrac{\\pi}{2}+k\\pi[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"990\" data-end=\"1069\">\r\n<p data-start=\"992\" data-end=\"1069\">Cosecant asymptotes: align with where sine = 0, i.e. [latex]x=k\\pi[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1073\" data-end=\"1144\">\r\n<p data-start=\"1075\" data-end=\"1144\">Compare the shifted asymptotes to find [latex]\\dfrac{c}{b}[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1146\" data-end=\"1304\">\r\n<p data-start=\"1149\" data-end=\"1195\"><strong data-start=\"1149\" data-end=\"1193\">Find the Vertical Shift [latex]d[\/latex]<\/strong><\/p>\r\n\r\n<ul data-start=\"1199\" data-end=\"1304\">\r\n \t<li data-start=\"1199\" data-end=\"1250\">\r\n<p data-start=\"1201\" data-end=\"1250\">The midline of the graph is [latex]y=d[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1254\" data-end=\"1304\">\r\n<p data-start=\"1256\" data-end=\"1304\">This is halfway between a maximum and minimum.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1306\" data-end=\"1553\">\r\n<p data-start=\"1309\" data-end=\"1363\"><strong data-start=\"1309\" data-end=\"1361\">Identify the Stretch\/Reflection [latex]a[\/latex]<\/strong><\/p>\r\n\r\n<ul data-start=\"1367\" data-end=\"1553\">\r\n \t<li data-start=\"1367\" data-end=\"1455\">\r\n<p data-start=\"1369\" data-end=\"1455\">Distance from the midline to the \u201ctop\u201d or \u201cbottom\u201d of a branch = [latex]|a|[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1459\" data-end=\"1553\">\r\n<p data-start=\"1461\" data-end=\"1553\">If the branch opens downward where it normally opens upward, [latex]a[\/latex] is negative.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1555\" data-end=\"1767\">\r\n<p data-start=\"1558\" data-end=\"1590\"><strong data-start=\"1558\" data-end=\"1588\">Choose Secant vs. Cosecant<\/strong><\/p>\r\n\r\n<ul data-start=\"1594\" data-end=\"1767\">\r\n \t<li data-start=\"1594\" data-end=\"1678\">\r\n<p data-start=\"1596\" data-end=\"1678\">If the branches line up with cosine (max\/min at [latex]x=0[\/latex]), use secant.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1682\" data-end=\"1767\">\r\n<p data-start=\"1684\" data-end=\"1767\">If the branches line up with sine (crossing at [latex]x=0[\/latex]), use cosecant.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1769\" data-end=\"2156\">\r\n<p data-start=\"1772\" data-end=\"1796\"><strong data-start=\"1772\" data-end=\"1794\">Write the Equation<\/strong><\/p>\r\n\r\n<ul data-start=\"1800\" data-end=\"2156\">\r\n \t<li data-start=\"1800\" data-end=\"1917\">\r\n<p data-start=\"1802\" data-end=\"1917\">Substitute values of [latex]a[\/latex], [latex]b[\/latex], [latex]c[\/latex], and [latex]d[\/latex] into the formula.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1921\" data-end=\"2156\">\r\n<p data-start=\"1923\" data-end=\"2156\">Example: A secant graph with midline [latex]y=1[\/latex], amplitude [latex]2[\/latex], period [latex]\\pi[\/latex], and shift right [latex]\\dfrac{\\pi}{4}[\/latex] would be<br data-start=\"2089\" data-end=\"2092\" \/>[latex]y=2\\sec!\\left(2x-\\dfrac{\\pi}{2}\\right)+1[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Write a secant equation for the graph shown.\r\n<img class=\"alignnone wp-image-5556\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11180436\/Screenshot-2026-02-11-at-10.53.08%E2%80%AFAM.png\" alt=\"The graph shows a reciprocal function with a midline at y = 1. Vertical asymptotes occur at x equals pi, 3pi, and 5pi. The curve approaches very large positive or negative values near each asymptote and does not touch the asymptote lines. Key points shown on the curve: (0, 4), (2pi, 2), (4pi, 4), (6pi, 2) The pattern repeats every 4pi.\" width=\"346\" height=\"186\" \/>\r\n[reveal-answer q=\"sec-002\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"sec-002\"]\r\nBecause the initial curve has a minimum at (0,4) we can use secant, the reciprocal of cosine.Use [latex]f(x) = A\\sec(Bx) + D[\/latex]Period: The pattern repeats every [latex]4\\pi[\/latex].[latex]\\frac{2\\pi}{B} = 4\\pi[\/latex], so [latex]B = \\frac{1}{2}[\/latex]The minimum value of the upward facing curves is [latex]2[\/latex] and the maximum value of the downward facing curves is [latex]2[\/latex].To find the stretch factor: [latex]\\frac{4-2}{2}=1[\/latex] so [latex]A=1[\/latex].To find the midline: [latex]\\frac{4+2}{2}=3[\/latex] so [latex]D=3[\/latex]The function is: [latex]f(x) = \\sec\\left(\\frac{x}{2}\\right) + 3[\/latex]\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<div><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gagafgbd-3JpiWHFFRzs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/3JpiWHFFRzs?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gagafgbd-3JpiWHFFRzs\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660599&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-gagafgbd-3JpiWHFFRzs&vembed=0&video_id=3JpiWHFFRzs&video_target=tpm-plugin-gagafgbd-3JpiWHFFRzs'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Ex+-+Find+the+Equation+of+a+Transformed+Secant+Function+From+The+Graph_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find the Equation of a Transformed Secant Function From The Graph\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Graph variations of y=tan x and y=cot x.<\/li>\n<li>Determine a function formula from a tangent or cotangent graph.<\/li>\n<li>Graph variations of y=sec x and y=csc x.<\/li>\n<li>Determine a function formula from a secant or cosecant graph.<\/li>\n<\/ul>\n<\/section>\n<h2>Graph Variations of Tangent and Cotangent<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"360\" data-end=\"664\">The graphs of [latex]y=\\tan x[\/latex] and [latex]y=\\cot x[\/latex] look different from sine and cosine because they have <strong data-start=\"480\" data-end=\"503\">vertical asymptotes<\/strong> and repeat every [latex]\\pi[\/latex] units (instead of [latex]2\\pi[\/latex]). Variations of these graphs come from changing the parameters in the general forms:<\/p>\n<ul data-start=\"666\" data-end=\"735\">\n<li data-start=\"666\" data-end=\"700\">\n<p data-start=\"668\" data-end=\"700\">[latex]y=a\\tan(bx-c)+d[\/latex]<\/p>\n<\/li>\n<li data-start=\"701\" data-end=\"735\">\n<p data-start=\"703\" data-end=\"735\">[latex]y=a\\cot(bx-c)+d[\/latex]<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"737\" data-end=\"792\">Each parameter changes the graph in predictable ways:<\/p>\n<ul data-start=\"793\" data-end=\"1058\">\n<li data-start=\"793\" data-end=\"857\">\n<p data-start=\"795\" data-end=\"857\">[latex]a[\/latex] stretches or reflects the graph vertically.<\/p>\n<\/li>\n<li data-start=\"858\" data-end=\"924\">\n<p data-start=\"860\" data-end=\"924\">[latex]b[\/latex] changes the period (the length of one cycle).<\/p>\n<\/li>\n<li data-start=\"925\" data-end=\"991\">\n<p data-start=\"927\" data-end=\"991\">[latex]c[\/latex] shifts the graph left or right (phase shift).<\/p>\n<\/li>\n<li data-start=\"992\" data-end=\"1058\">\n<p data-start=\"994\" data-end=\"1058\">[latex]d[\/latex] shifts the graph up or down (vertical shift).<\/p>\n<\/li>\n<\/ul>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Graphing Variations of Tangent and Cotangent<\/strong><\/p>\n<ol>\n<li data-start=\"1129\" data-end=\"1364\">\n<p data-start=\"1132\" data-end=\"1157\"><strong data-start=\"1132\" data-end=\"1155\">Base Graph Features<\/strong><\/p>\n<ul data-start=\"1161\" data-end=\"1364\">\n<li data-start=\"1161\" data-end=\"1268\">\n<p data-start=\"1163\" data-end=\"1268\">[latex]y=\\tan x[\/latex]: period [latex]\\pi[\/latex], asymptotes at [latex]x=\\dfrac{\\pi}{2}+k\\pi[\/latex].<\/p>\n<\/li>\n<li data-start=\"1272\" data-end=\"1364\">\n<p data-start=\"1274\" data-end=\"1364\">[latex]y=\\cot x[\/latex]: period [latex]\\pi[\/latex], asymptotes at [latex]x=k\\pi[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1366\" data-end=\"1533\">\n<p data-start=\"1369\" data-end=\"1389\"><strong data-start=\"1369\" data-end=\"1387\">Period Changes<\/strong><\/p>\n<ul data-start=\"1393\" data-end=\"1533\">\n<li data-start=\"1393\" data-end=\"1450\">\n<p data-start=\"1395\" data-end=\"1450\">Formula: [latex]\\text{Period}=\\dfrac{\\pi}{b}[\/latex].<\/p>\n<\/li>\n<li data-start=\"1454\" data-end=\"1533\">\n<p data-start=\"1456\" data-end=\"1533\">Larger [latex]b[\/latex] = compressed, smaller [latex]b[\/latex] = stretched.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1535\" data-end=\"1658\">\n<p data-start=\"1538\" data-end=\"1583\"><strong data-start=\"1538\" data-end=\"1581\">Phase Shift [latex]\\dfrac{c}{b}[\/latex]<\/strong><\/p>\n<ul data-start=\"1587\" data-end=\"1658\">\n<li data-start=\"1587\" data-end=\"1621\">\n<p data-start=\"1589\" data-end=\"1621\">Moves the graph left or right.<\/p>\n<\/li>\n<li data-start=\"1625\" data-end=\"1658\">\n<p data-start=\"1627\" data-end=\"1658\">Asymptotes shift accordingly.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1660\" data-end=\"1813\">\n<p data-start=\"1663\" data-end=\"1696\"><strong data-start=\"1663\" data-end=\"1694\">Vertical Stretch\/Reflection<\/strong><\/p>\n<ul data-start=\"1700\" data-end=\"1813\">\n<li data-start=\"1700\" data-end=\"1752\">\n<p data-start=\"1702\" data-end=\"1752\">[latex]a[\/latex] changes steepness of the graph.<\/p>\n<\/li>\n<li data-start=\"1756\" data-end=\"1813\">\n<p data-start=\"1758\" data-end=\"1813\">Negative [latex]a[\/latex] reflects across the x-axis.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1815\" data-end=\"1927\">\n<p data-start=\"1818\" data-end=\"1855\"><strong data-start=\"1818\" data-end=\"1853\">Vertical Shift [latex]d[\/latex]<\/strong><\/p>\n<ul data-start=\"1859\" data-end=\"1927\">\n<li data-start=\"1859\" data-end=\"1892\">\n<p data-start=\"1861\" data-end=\"1892\">Moves the midline up or down.<\/p>\n<\/li>\n<li data-start=\"1896\" data-end=\"1927\">\n<p data-start=\"1898\" data-end=\"1927\">Asymptotes remain vertical.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1929\" data-end=\"2263\">\n<p data-start=\"1932\" data-end=\"1955\"><strong data-start=\"1932\" data-end=\"1953\">Graphing Strategy<\/strong><\/p>\n<ul data-start=\"1959\" data-end=\"2263\">\n<li data-start=\"1959\" data-end=\"2023\">\n<p data-start=\"1961\" data-end=\"2023\">Step 1: Find the period using [latex]\\dfrac{\\pi}{b}[\/latex].<\/p>\n<\/li>\n<li data-start=\"2027\" data-end=\"2061\">\n<p data-start=\"2029\" data-end=\"2061\">Step 2: Locate the asymptotes.<\/p>\n<\/li>\n<li data-start=\"2065\" data-end=\"2109\">\n<p data-start=\"2067\" data-end=\"2109\">Step 3: Apply phase and vertical shifts.<\/p>\n<\/li>\n<li data-start=\"2113\" data-end=\"2211\">\n<p data-start=\"2115\" data-end=\"2211\">Step 4: Plot key points ([latex]\\pm 1[\/latex] for tangent; reciprocal behavior for cotangent).<\/p>\n<\/li>\n<li data-start=\"2215\" data-end=\"2263\">\n<p data-start=\"2217\" data-end=\"2263\">Step 5: Sketch the curve between asymptotes.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Sketch the graph of [latex]f(x) = 2\\tan\\left(\\pi x\\right)[\/latex]. Identify the period and vertical asymptotes.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qtan-001\">Show Solution<\/button><\/p>\n<div id=\"qtan-001\" class=\"hidden-answer\" style=\"display: none\">From [latex]f(x) = 2\\tan(\\pi x)[\/latex]Stretching factor: [latex]A = 2[\/latex]Period: [latex]\\frac{\\pi}{|B|} = \\frac{\\pi}{\\pi} = 1[\/latex]Vertical asymptotes: occur at [latex]x = -\\frac{1}{2}, \\frac{1}{2}, \\frac{3}{2}, ...[\/latex]The graph is stretched vertically by a factor of 2 and completes one cycle over an interval of length 1.<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-5554\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11173858\/Screenshot-2026-02-11-at-10.38.24%E2%80%AFAM.png\" alt=\"The graph shows a tangent function with repeating vertical asymptotes. Vertical dashed lines appear at x equals negative 0.5, 0.5, 1.5, 2.5, and so on, spaced one unit apart. Between each pair of asymptotes, the graph is an increasing curve that goes from negative infinity to positive infinity. Each branch crosses the x-axis at the midpoint between consecutive asymptotes.\" width=\"375\" height=\"370\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11173858\/Screenshot-2026-02-11-at-10.38.24%E2%80%AFAM.png 920w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11173858\/Screenshot-2026-02-11-at-10.38.24%E2%80%AFAM-300x296.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11173858\/Screenshot-2026-02-11-at-10.38.24%E2%80%AFAM-768x758.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11173858\/Screenshot-2026-02-11-at-10.38.24%E2%80%AFAM-65x64.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11173858\/Screenshot-2026-02-11-at-10.38.24%E2%80%AFAM-225x222.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11173858\/Screenshot-2026-02-11-at-10.38.24%E2%80%AFAM-350x345.png 350w\" sizes=\"(max-width: 375px) 100vw, 375px\" \/>\n<\/div>\n<\/div>\n<\/section>\n<div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ahafafhe-4FF-zSGYnaM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/4FF-zSGYnaM?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ahafafhe-4FF-zSGYnaM\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14660596&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ahafafhe-4FF-zSGYnaM&#38;vembed=0&#38;video_id=4FF-zSGYnaM&#38;video_target=tpm-plugin-ahafafhe-4FF-zSGYnaM\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Trigonometry+-+The+graphs+of+tan+and+cot_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cTrigonometry &#8211; The graphs of tan and cot\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Function Formulas from Tangent and Cotangent Graphs<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"80\" data-end=\"412\">To write a function formula from a tangent or cotangent graph, we analyze its key features: the <strong data-start=\"176\" data-end=\"186\">period<\/strong>, the <strong data-start=\"192\" data-end=\"207\">phase shift<\/strong> (location of asymptotes or intercepts), any <strong data-start=\"252\" data-end=\"270\">vertical shift<\/strong>, and the <strong data-start=\"280\" data-end=\"293\">steepness<\/strong> of the curve. Tangent and cotangent graphs both repeat every [latex]\\pi[\/latex], but parameters in the general forms<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li style=\"list-style-type: none;\">\n<ul data-start=\"414\" data-end=\"483\">\n<li data-start=\"414\" data-end=\"448\">\n<p data-start=\"416\" data-end=\"448\">[latex]y=a\\tan(bx-c)+d[\/latex]<\/p>\n<\/li>\n<li data-start=\"449\" data-end=\"483\">\n<p data-start=\"451\" data-end=\"483\">[latex]y=a\\cot(bx-c)+d[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">control how the graph is stretched, shifted, or reflected. By identifying these features from the graph, we can reconstruct the exact equation.<\/span><\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Building Tangent or Cotangent Formulas<\/strong><\/p>\n<ol data-start=\"695\" data-end=\"1981\">\n<li data-start=\"695\" data-end=\"910\">\n<p data-start=\"698\" data-end=\"724\"><strong data-start=\"698\" data-end=\"722\">Determine the Period<\/strong><\/p>\n<ul data-start=\"728\" data-end=\"910\">\n<li data-start=\"728\" data-end=\"803\">\n<p data-start=\"730\" data-end=\"803\">For tangent and cotangent, [latex]\\text{Period}=\\dfrac{\\pi}{b}[\/latex].<\/p>\n<\/li>\n<li data-start=\"807\" data-end=\"910\">\n<p data-start=\"809\" data-end=\"910\">Measure the distance between consecutive asymptotes (or repeating points) to find [latex]b[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"912\" data-end=\"1205\">\n<p data-start=\"915\" data-end=\"939\"><strong data-start=\"915\" data-end=\"937\">Locate Phase Shift<\/strong><\/p>\n<ul data-start=\"943\" data-end=\"1205\">\n<li data-start=\"943\" data-end=\"1029\">\n<p data-start=\"945\" data-end=\"1029\">Tangent asymptotes: [latex]x=\\dfrac{c}{b}+\\dfrac{\\pi}{2b}+k\\dfrac{\\pi}{b}[\/latex].<\/p>\n<\/li>\n<li data-start=\"1033\" data-end=\"1105\">\n<p data-start=\"1035\" data-end=\"1105\">Cotangent asymptotes: [latex]x=\\dfrac{c}{b}+k\\dfrac{\\pi}{b}[\/latex].<\/p>\n<\/li>\n<li data-start=\"1109\" data-end=\"1205\">\n<p data-start=\"1111\" data-end=\"1205\">Identify where the central asymptote (for tangent) or intercept (for cotangent) has shifted.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1207\" data-end=\"1356\">\n<p data-start=\"1210\" data-end=\"1252\"><strong data-start=\"1210\" data-end=\"1250\">Find Vertical Shift [latex]d[\/latex]<\/strong><\/p>\n<ul data-start=\"1256\" data-end=\"1356\">\n<li data-start=\"1256\" data-end=\"1303\">\n<p data-start=\"1258\" data-end=\"1303\">Midline of the graph is [latex]y=d[\/latex].<\/p>\n<\/li>\n<li data-start=\"1307\" data-end=\"1356\">\n<p data-start=\"1309\" data-end=\"1356\">Check if the curve has been moved up or down.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1358\" data-end=\"1530\">\n<p data-start=\"1361\" data-end=\"1414\"><strong data-start=\"1361\" data-end=\"1412\">Determine [latex]a[\/latex] (Stretch\/Reflection)<\/strong><\/p>\n<ul data-start=\"1418\" data-end=\"1530\">\n<li data-start=\"1418\" data-end=\"1461\">\n<p data-start=\"1420\" data-end=\"1461\">[latex]a[\/latex] changes the steepness.<\/p>\n<\/li>\n<li data-start=\"1465\" data-end=\"1530\">\n<p data-start=\"1467\" data-end=\"1530\">Negative [latex]a[\/latex] flips the graph across the midline.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1532\" data-end=\"1743\">\n<p data-start=\"1535\" data-end=\"1573\"><strong data-start=\"1535\" data-end=\"1571\">Choose Tangent or Cotangent Form<\/strong><\/p>\n<ul data-start=\"1577\" data-end=\"1743\">\n<li data-start=\"1577\" data-end=\"1659\">\n<p data-start=\"1579\" data-end=\"1659\">Tangent passes through the origin (before shifts) and increases left to right.<\/p>\n<\/li>\n<li data-start=\"1663\" data-end=\"1743\">\n<p data-start=\"1665\" data-end=\"1743\">Cotangent decreases left to right, starting with an asymptote at the origin.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1745\" data-end=\"1981\">\n<p data-start=\"1748\" data-end=\"1772\"><strong data-start=\"1748\" data-end=\"1770\">Write the Equation<\/strong><\/p>\n<ul data-start=\"1776\" data-end=\"1981\">\n<li data-start=\"1776\" data-end=\"1981\">\n<p data-start=\"1778\" data-end=\"1981\">Plug amplitude [latex]a[\/latex], period factor [latex]b[\/latex], phase shift [latex]c[\/latex], and vertical shift [latex]d[\/latex] into [latex]y=a\\tan(bx-c)+d[\/latex] or [latex]y=a\\cot(bx-c)+d[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">A tangent graph has period [latex]2\\pi[\/latex] and passes through the point [latex]\\left(\\frac{\\pi}{2}, 3\\right)[\/latex]. Write a possible equation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qtan-002\">Show Solution<\/button><\/p>\n<div id=\"qtan-002\" class=\"hidden-answer\" style=\"display: none\">\nUse [latex]f(x) = A\\tan(Bx)[\/latex]Period:<br \/>\n[latex]\\frac{\\pi}{B} = 2\\pi[\/latex], so [latex]B = \\frac{1}{2}[\/latex]Find A:<br \/>\nSubstitute [latex]\\left(\\frac{\\pi}{2}, 3\\right)[\/latex]:<br \/>\n[latex]\\begin{align*}  3 &= A\\tan\\left(\\frac{1}{2} \\cdot \\frac{\\pi}{2}\\right) \\\\  3 &= A\\tan\\left(\\frac{\\pi}{4}\\right) \\\\  3 &= A(1) \\\\  A &= 3  \\end{align*}[\/latex]The function is:[latex]f(x) = 3\\tan\\left(\\frac{x}{2}\\right)[\/latex]\n<\/div>\n<\/div>\n<\/section>\n<div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dcedddaf-x_yn02gwnPA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/x_yn02gwnPA?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dcedddaf-x_yn02gwnPA\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14660597&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-dcedddaf-x_yn02gwnPA&#38;vembed=0&#38;video_id=x_yn02gwnPA&#38;video_target=tpm-plugin-dcedddaf-x_yn02gwnPA\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Writing+Equations+for+Tangent+Graphs_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cWriting Equations for Tangent Graphs\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Graph Variations of Secant and Cosecant<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"68\" data-end=\"480\">The graphs of [latex]y=\\sec x[\/latex] and [latex]y=\\csc x[\/latex] are built from cosine and sine, since [latex]\\sec x=\\dfrac{1}{\\cos x}[\/latex] and [latex]\\csc x=\\dfrac{1}{\\sin x}[\/latex]. They feature repeating <strong data-start=\"280\" data-end=\"292\">U-shaped<\/strong> and <strong data-start=\"297\" data-end=\"318\">inverted U-shaped<\/strong> branches with <strong data-start=\"333\" data-end=\"356\">vertical asymptotes<\/strong> where sine or cosine equals zero. Variations of these graphs are created by changing the parameters in the general forms:<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li style=\"list-style-type: none;\">\n<ul data-start=\"482\" data-end=\"551\">\n<li data-start=\"482\" data-end=\"516\">\n<p data-start=\"484\" data-end=\"516\">[latex]y=a\\sec(bx-c)+d[\/latex]<\/p>\n<\/li>\n<li data-start=\"517\" data-end=\"551\">\n<p data-start=\"519\" data-end=\"551\">[latex]y=a\\csc(bx-c)+d[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p data-start=\"553\" data-end=\"629\">Each parameter controls how the graph is stretched, shifted, or reflected.<\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Graphing Variations of Secant and Cosecant<\/strong><\/p>\n<ol>\n<li data-start=\"698\" data-end=\"935\">\n<p data-start=\"701\" data-end=\"726\"><strong data-start=\"701\" data-end=\"724\">Base Graph Features<\/strong><\/p>\n<ul data-start=\"730\" data-end=\"935\">\n<li data-start=\"730\" data-end=\"838\">\n<p data-start=\"732\" data-end=\"838\">[latex]y=\\sec x[\/latex]: period [latex]2\\pi[\/latex], asymptotes at [latex]x=\\dfrac{\\pi}{2}+k\\pi[\/latex].<\/p>\n<\/li>\n<li data-start=\"842\" data-end=\"935\">\n<p data-start=\"844\" data-end=\"935\">[latex]y=\\csc x[\/latex]: period [latex]2\\pi[\/latex], asymptotes at [latex]x=k\\pi[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"937\" data-end=\"1071\">\n<p data-start=\"940\" data-end=\"960\"><strong data-start=\"940\" data-end=\"958\">Period Changes<\/strong><\/p>\n<ul data-start=\"964\" data-end=\"1071\">\n<li data-start=\"964\" data-end=\"1022\">\n<p data-start=\"966\" data-end=\"1022\">Formula: [latex]\\text{Period}=\\dfrac{2\\pi}{b}[\/latex].<\/p>\n<\/li>\n<li data-start=\"1026\" data-end=\"1071\">\n<p data-start=\"1028\" data-end=\"1071\">Adjusts how wide each repeating cycle is.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1073\" data-end=\"1209\">\n<p data-start=\"1076\" data-end=\"1121\"><strong data-start=\"1076\" data-end=\"1119\">Phase Shift [latex]\\dfrac{c}{b}[\/latex]<\/strong><\/p>\n<ul data-start=\"1125\" data-end=\"1209\">\n<li data-start=\"1125\" data-end=\"1159\">\n<p data-start=\"1127\" data-end=\"1159\">Moves the graph left or right.<\/p>\n<\/li>\n<li data-start=\"1163\" data-end=\"1209\">\n<p data-start=\"1165\" data-end=\"1209\">Asymptotes and branches shift accordingly.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1211\" data-end=\"1447\">\n<p data-start=\"1214\" data-end=\"1264\"><strong data-start=\"1214\" data-end=\"1262\">Vertical Stretch\/Reflection [latex]a[\/latex]<\/strong><\/p>\n<ul data-start=\"1268\" data-end=\"1447\">\n<li data-start=\"1268\" data-end=\"1372\">\n<p data-start=\"1270\" data-end=\"1372\">[latex]a[\/latex] changes the distance from the midline to the minimum\/maximum points of each branch.<\/p>\n<\/li>\n<li data-start=\"1376\" data-end=\"1447\">\n<p data-start=\"1378\" data-end=\"1447\">Negative [latex]a[\/latex] reflects the branches across the midline.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1449\" data-end=\"1615\">\n<p data-start=\"1452\" data-end=\"1489\"><strong data-start=\"1452\" data-end=\"1487\">Vertical Shift [latex]d[\/latex]<\/strong><\/p>\n<ul data-start=\"1493\" data-end=\"1615\">\n<li data-start=\"1493\" data-end=\"1553\">\n<p data-start=\"1495\" data-end=\"1553\">Moves the midline up or down, shifting the entire graph.<\/p>\n<\/li>\n<li data-start=\"1557\" data-end=\"1615\">\n<p data-start=\"1559\" data-end=\"1615\">Asymptotes stay vertical, but branch positions adjust.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1617\" data-end=\"1917\">\n<p data-start=\"1620\" data-end=\"1643\"><strong data-start=\"1620\" data-end=\"1641\">Graphing Strategy<\/strong><\/p>\n<ul data-start=\"1647\" data-end=\"1917\">\n<li data-start=\"1647\" data-end=\"1696\">\n<p data-start=\"1649\" data-end=\"1696\">Step 1: Start with sine or cosine as a guide.<\/p>\n<\/li>\n<li data-start=\"1700\" data-end=\"1757\">\n<p data-start=\"1702\" data-end=\"1757\">Step 2: Identify asymptotes where sine or cosine = 0.<\/p>\n<\/li>\n<li data-start=\"1761\" data-end=\"1835\">\n<p data-start=\"1763\" data-end=\"1835\">Step 3: Plot key points at maximum\/minimum distances from the midline.<\/p>\n<\/li>\n<li data-start=\"1839\" data-end=\"1917\">\n<p data-start=\"1841\" data-end=\"1917\">Step 4: Sketch U-shaped and inverted U-shaped branches between asymptotes.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Sketch the graph of [latex]g(x) = 3\\csc(2x)[\/latex]. Identify the period and vertical asymptotes.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qsec-001\">Show Solution<\/button><\/p>\n<div id=\"qsec-001\" class=\"hidden-answer\" style=\"display: none\">\nFrom [latex]g(x) = 3\\csc(2x)[\/latex]Stretching factor: [latex]A = 3[\/latex]Period:<br \/>\n[latex]\\frac{2\\pi}{|B|} = \\frac{2\\pi}{2} = \\pi[\/latex]Vertical asymptotes:<br \/>\noccur where [latex]\\sin(2x) = 0[\/latex], at [latex]x = 0, \\frac{\\pi}{2}, \\pi, \\frac{3\\pi}{2}, ...[\/latex]Graph [latex]y = 3\\sin(2x)[\/latex] first, then draw the cosecant as the reciprocal with asymptotes where sine equals zero.<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-5555\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11175204\/Screenshot-2026-02-11-at-10.44.49%E2%80%AFAM.png\" alt=\"The dashed curve is a sine function with amplitude 3 and period pi. It oscillates between y equals 3 and y equals negative 3. It passes through the points: (0, 0), (pi\/4, 3), (pi\/2, 0), (3pi\/4, -3), (pi, 0) The solid black curves are branches of the cosecant function with a vertical stretch factor of 3 and period pi. There are vertical asymptotes at: x equals 0, pi\/2, and pi. Between 0 and pi\/2, the cosecant branch forms a U-shape with a minimum at (pi\/4, 3). Between pi\/2 and pi, the branch forms an upside-down U-shape with a maximum at (3pi\/4, -3). The cosecant graph touches the sine graph at the sine function\u2019s maximum and minimum points.\" width=\"373\" height=\"268\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11175204\/Screenshot-2026-02-11-at-10.44.49%E2%80%AFAM.png 1546w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11175204\/Screenshot-2026-02-11-at-10.44.49%E2%80%AFAM-300x215.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11175204\/Screenshot-2026-02-11-at-10.44.49%E2%80%AFAM-1024x734.png 1024w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11175204\/Screenshot-2026-02-11-at-10.44.49%E2%80%AFAM-768x550.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11175204\/Screenshot-2026-02-11-at-10.44.49%E2%80%AFAM-1536x1101.png 1536w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11175204\/Screenshot-2026-02-11-at-10.44.49%E2%80%AFAM-65x47.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11175204\/Screenshot-2026-02-11-at-10.44.49%E2%80%AFAM-225x161.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11175204\/Screenshot-2026-02-11-at-10.44.49%E2%80%AFAM-350x251.png 350w\" sizes=\"(max-width: 373px) 100vw, 373px\" \/>\n<\/div>\n<\/div>\n<\/section>\n<div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ceggaece-2o_dxyKAXLQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/2o_dxyKAXLQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ceggaece-2o_dxyKAXLQ\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14660598&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ceggaece-2o_dxyKAXLQ&#38;vembed=0&#38;video_id=2o_dxyKAXLQ&#38;video_target=tpm-plugin-ceggaece-2o_dxyKAXLQ\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/How+to+Graph+Secant+and+Cosecant_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Graph Secant and Cosecant\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Function Formula from Secant and Cosecant Graphs<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"78\" data-end=\"485\">To determine a function formula from a secant or cosecant graph, we use the fact that these graphs are built as reciprocals of cosine and sine. The graph\u2019s <strong data-start=\"234\" data-end=\"245\">midline<\/strong>, <strong data-start=\"247\" data-end=\"265\">vertical shift<\/strong>, <strong data-start=\"267\" data-end=\"277\">period<\/strong>, <strong data-start=\"279\" data-end=\"294\">phase shift<\/strong>, and <strong data-start=\"300\" data-end=\"322\">stretch\/reflection<\/strong> can all be read directly from its repeating U-shaped or inverted U-shaped branches. Once these features are identified, they are plugged into the general forms:<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li style=\"list-style-type: none;\">\n<ul data-start=\"487\" data-end=\"556\">\n<li data-start=\"487\" data-end=\"521\">\n<p data-start=\"489\" data-end=\"521\">[latex]y=a\\sec(bx-c)+d[\/latex]<\/p>\n<\/li>\n<li data-start=\"522\" data-end=\"556\">\n<p data-start=\"524\" data-end=\"556\">[latex]y=a\\csc(bx-c)+d[\/latex]<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p data-start=\"558\" data-end=\"631\">Recognizing the asymptotes and midline first helps anchor the equation.<\/p>\n<p><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Building Secant or Cosecant Formulas<\/strong><\/p>\n<ol>\n<li data-start=\"694\" data-end=\"855\">\n<p data-start=\"697\" data-end=\"723\"><strong data-start=\"697\" data-end=\"721\">Determine the Period<\/strong><\/p>\n<ul data-start=\"727\" data-end=\"855\">\n<li data-start=\"727\" data-end=\"785\">\n<p data-start=\"729\" data-end=\"785\">Formula: [latex]\\text{Period}=\\dfrac{2\\pi}{b}[\/latex].<\/p>\n<\/li>\n<li data-start=\"789\" data-end=\"855\">\n<p data-start=\"791\" data-end=\"855\">Measure the distance between repeating branches or asymptotes.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"857\" data-end=\"1144\">\n<p data-start=\"860\" data-end=\"888\"><strong data-start=\"860\" data-end=\"886\">Locate the Phase Shift<\/strong><\/p>\n<ul data-start=\"892\" data-end=\"1144\">\n<li data-start=\"892\" data-end=\"986\">\n<p data-start=\"894\" data-end=\"986\">Secant asymptotes: align with where cosine = 0, i.e. [latex]x=\\dfrac{\\pi}{2}+k\\pi[\/latex].<\/p>\n<\/li>\n<li data-start=\"990\" data-end=\"1069\">\n<p data-start=\"992\" data-end=\"1069\">Cosecant asymptotes: align with where sine = 0, i.e. [latex]x=k\\pi[\/latex].<\/p>\n<\/li>\n<li data-start=\"1073\" data-end=\"1144\">\n<p data-start=\"1075\" data-end=\"1144\">Compare the shifted asymptotes to find [latex]\\dfrac{c}{b}[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1146\" data-end=\"1304\">\n<p data-start=\"1149\" data-end=\"1195\"><strong data-start=\"1149\" data-end=\"1193\">Find the Vertical Shift [latex]d[\/latex]<\/strong><\/p>\n<ul data-start=\"1199\" data-end=\"1304\">\n<li data-start=\"1199\" data-end=\"1250\">\n<p data-start=\"1201\" data-end=\"1250\">The midline of the graph is [latex]y=d[\/latex].<\/p>\n<\/li>\n<li data-start=\"1254\" data-end=\"1304\">\n<p data-start=\"1256\" data-end=\"1304\">This is halfway between a maximum and minimum.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1306\" data-end=\"1553\">\n<p data-start=\"1309\" data-end=\"1363\"><strong data-start=\"1309\" data-end=\"1361\">Identify the Stretch\/Reflection [latex]a[\/latex]<\/strong><\/p>\n<ul data-start=\"1367\" data-end=\"1553\">\n<li data-start=\"1367\" data-end=\"1455\">\n<p data-start=\"1369\" data-end=\"1455\">Distance from the midline to the \u201ctop\u201d or \u201cbottom\u201d of a branch = [latex]|a|[\/latex].<\/p>\n<\/li>\n<li data-start=\"1459\" data-end=\"1553\">\n<p data-start=\"1461\" data-end=\"1553\">If the branch opens downward where it normally opens upward, [latex]a[\/latex] is negative.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1555\" data-end=\"1767\">\n<p data-start=\"1558\" data-end=\"1590\"><strong data-start=\"1558\" data-end=\"1588\">Choose Secant vs. Cosecant<\/strong><\/p>\n<ul data-start=\"1594\" data-end=\"1767\">\n<li data-start=\"1594\" data-end=\"1678\">\n<p data-start=\"1596\" data-end=\"1678\">If the branches line up with cosine (max\/min at [latex]x=0[\/latex]), use secant.<\/p>\n<\/li>\n<li data-start=\"1682\" data-end=\"1767\">\n<p data-start=\"1684\" data-end=\"1767\">If the branches line up with sine (crossing at [latex]x=0[\/latex]), use cosecant.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1769\" data-end=\"2156\">\n<p data-start=\"1772\" data-end=\"1796\"><strong data-start=\"1772\" data-end=\"1794\">Write the Equation<\/strong><\/p>\n<ul data-start=\"1800\" data-end=\"2156\">\n<li data-start=\"1800\" data-end=\"1917\">\n<p data-start=\"1802\" data-end=\"1917\">Substitute values of [latex]a[\/latex], [latex]b[\/latex], [latex]c[\/latex], and [latex]d[\/latex] into the formula.<\/p>\n<\/li>\n<li data-start=\"1921\" data-end=\"2156\">\n<p data-start=\"1923\" data-end=\"2156\">Example: A secant graph with midline [latex]y=1[\/latex], amplitude [latex]2[\/latex], period [latex]\\pi[\/latex], and shift right [latex]\\dfrac{\\pi}{4}[\/latex] would be<br data-start=\"2089\" data-end=\"2092\" \/>[latex]y=2\\sec!\\left(2x-\\dfrac{\\pi}{2}\\right)+1[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Write a secant equation for the graph shown.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-5556\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11180436\/Screenshot-2026-02-11-at-10.53.08%E2%80%AFAM.png\" alt=\"The graph shows a reciprocal function with a midline at y = 1. Vertical asymptotes occur at x equals pi, 3pi, and 5pi. The curve approaches very large positive or negative values near each asymptote and does not touch the asymptote lines. Key points shown on the curve: (0, 4), (2pi, 2), (4pi, 4), (6pi, 2) The pattern repeats every 4pi.\" width=\"346\" height=\"186\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11180436\/Screenshot-2026-02-11-at-10.53.08%E2%80%AFAM.png 1902w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11180436\/Screenshot-2026-02-11-at-10.53.08%E2%80%AFAM-300x161.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11180436\/Screenshot-2026-02-11-at-10.53.08%E2%80%AFAM-1024x548.png 1024w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11180436\/Screenshot-2026-02-11-at-10.53.08%E2%80%AFAM-768x411.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11180436\/Screenshot-2026-02-11-at-10.53.08%E2%80%AFAM-1536x822.png 1536w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11180436\/Screenshot-2026-02-11-at-10.53.08%E2%80%AFAM-65x35.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11180436\/Screenshot-2026-02-11-at-10.53.08%E2%80%AFAM-225x120.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11180436\/Screenshot-2026-02-11-at-10.53.08%E2%80%AFAM-350x187.png 350w\" sizes=\"(max-width: 346px) 100vw, 346px\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qsec-002\">Show Solution<\/button><\/p>\n<div id=\"qsec-002\" class=\"hidden-answer\" style=\"display: none\">\nBecause the initial curve has a minimum at (0,4) we can use secant, the reciprocal of cosine.Use [latex]f(x) = A\\sec(Bx) + D[\/latex]Period: The pattern repeats every [latex]4\\pi[\/latex].[latex]\\frac{2\\pi}{B} = 4\\pi[\/latex], so [latex]B = \\frac{1}{2}[\/latex]The minimum value of the upward facing curves is [latex]2[\/latex] and the maximum value of the downward facing curves is [latex]2[\/latex].To find the stretch factor: [latex]\\frac{4-2}{2}=1[\/latex] so [latex]A=1[\/latex].To find the midline: [latex]\\frac{4+2}{2}=3[\/latex] so [latex]D=3[\/latex]The function is: [latex]f(x) = \\sec\\left(\\frac{x}{2}\\right) + 3[\/latex]\n<\/div>\n<\/div>\n<\/section>\n<div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gagafgbd-3JpiWHFFRzs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/3JpiWHFFRzs?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gagafgbd-3JpiWHFFRzs\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14660599&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-gagafgbd-3JpiWHFFRzs&#38;vembed=0&#38;video_id=3JpiWHFFRzs&#38;video_target=tpm-plugin-gagafgbd-3JpiWHFFRzs\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Ex+-+Find+the+Equation+of+a+Transformed+Secant+Function+From+The+Graph_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find the Equation of a Transformed Secant Function From The Graph\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n","protected":false},"author":67,"menu_order":18,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Trigonometry - The graphs of tan and 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