{"id":1517,"date":"2025-07-25T02:31:07","date_gmt":"2025-07-25T02:31:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1517"},"modified":"2026-03-12T05:44:42","modified_gmt":"2026-03-12T05:44:42","slug":"graphs-of-the-sine-and-cosine-function-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-the-sine-and-cosine-function-fresh-take\/","title":{"raw":"Graphs of the Sine and Cosine Function: Fresh Take","rendered":"Graphs of the Sine and Cosine Function: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\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>Graph transformations of y=cos x and y=sin x .<\/li>\r\n \t<li>Determine a function formula that would have a given sinusoidal graph.<\/li>\r\n \t<li>Determine functions that model circular and periodic motion.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Sine and Cosine Graph Features<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"225\" data-end=\"378\">The graphs of sine and cosine can be stretched, compressed, shifted, or moved up and down by changing parts of their equations. In general, the form is<\/p>\r\n<p style=\"padding-left: 280px;\" data-start=\"380\" data-end=\"461\">[latex]y = a \\sin(bx)[\/latex]<\/p>\r\n<p style=\"padding-left: 280px;\" data-start=\"380\" data-end=\"461\">[latex]y = a \\cos(bx)[\/latex].<\/p>\r\n<p data-start=\"463\" data-end=\"513\">Each letter tells you something about the graph:<\/p>\r\n\r\n<ul>\r\n \t<li data-start=\"467\" data-end=\"525\">\r\n<p data-start=\"469\" data-end=\"525\"><strong data-start=\"469\" data-end=\"474\">a<\/strong> controls the <em data-start=\"488\" data-end=\"499\">amplitude<\/em> (how tall the wave is).<\/p>\r\n<\/li>\r\n \t<li data-start=\"526\" data-end=\"600\">\r\n<p data-start=\"528\" data-end=\"600\"><strong data-start=\"528\" data-end=\"533\">b<\/strong> affects the <em data-start=\"546\" data-end=\"554\">period<\/em> (how long it takes for the wave to repeat).<\/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: Reading Transformations from the Equation<\/strong>\r\n<ol>\r\n \t<li data-start=\"831\" data-end=\"846\"><strong data-start=\"831\" data-end=\"844\">Amplitude<\/strong>\r\n<ul data-start=\"850\" data-end=\"962\">\r\n \t<li data-start=\"850\" data-end=\"901\">\r\n<p data-start=\"852\" data-end=\"901\">Formula: [latex]\\text{Amplitude} = |a|[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"905\" data-end=\"962\">\r\n<p data-start=\"907\" data-end=\"962\">Example: [latex]y = 3\\sin(x)[\/latex] \u2192 amplitude = 3.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"967\" data-end=\"979\"><strong data-start=\"967\" data-end=\"977\">Period<\/strong>\r\n<ul data-start=\"983\" data-end=\"1136\">\r\n \t<li data-start=\"983\" data-end=\"1043\">\r\n<p data-start=\"985\" data-end=\"1043\">Formula: [latex]\\text{Period} = \\dfrac{2\\pi}{b}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1047\" data-end=\"1136\">\r\n<p data-start=\"1049\" data-end=\"1136\">Example: [latex]y = \\cos(2x)[\/latex] \u2192 period = [latex]\\dfrac{2\\pi}{2} = \\pi[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<ul>\r\n \t<li style=\"list-style-type: none;\"><\/li>\r\n<\/ul>\r\n<\/div>\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-aafhbhfd-QNQAkUUHNxo\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/QNQAkUUHNxo?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-aafhbhfd-QNQAkUUHNxo\" 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=14660582&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-aafhbhfd-QNQAkUUHNxo&vembed=0&video_id=QNQAkUUHNxo&video_target=tpm-plugin-aafhbhfd-QNQAkUUHNxo'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Animation+-+Graphing+the+Sine+Function+Using+The+Unit+Circle_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cAnimation: Graphing the Sine Function Using The Unit Circle\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Sine and Cosine Graph Transformations<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"225\" data-end=\"378\">The graphs of [latex]y=\\sin x[\/latex] and [latex]y=\\cos x[\/latex] form smooth, repeating waves. Transformations let us reshape these waves by stretching them taller or shorter, squeezing them to fit more cycles, sliding them left or right, or lifting and lowering the whole curve. These changes don\u2019t alter the wave\u2019s basic pattern\u2014it still oscillates smoothly\u2014but they make the graph flexible enough to model real-world cycles like sound waves, tides, or seasonal patterns. Recognizing transformations helps us quickly sketch graphs and understand how the wave has been shifted from its standard position.<\/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: Transforming Sine and Cosine Graphs<\/strong>\r\n<ol>\r\n \t<li data-start=\"864\" data-end=\"981\">\r\n<p data-start=\"867\" data-end=\"981\"><strong data-start=\"867\" data-end=\"880\">Amplitude<\/strong>: [latex]|a|[\/latex] is the max distance from the midline. Bigger [latex]|a|[\/latex] = taller wave.<\/p>\r\n<\/li>\r\n \t<li data-start=\"982\" data-end=\"1141\">\r\n<p data-start=\"985\" data-end=\"1141\"><strong data-start=\"985\" data-end=\"995\">Period<\/strong>: [latex]\\dfrac{2\\pi}{b}[\/latex] is the length of one cycle. Larger [latex]b[\/latex] compresses the wave, smaller [latex]b[\/latex] stretches it.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1142\" data-end=\"1269\">\r\n<p data-start=\"1145\" data-end=\"1269\"><strong data-start=\"1145\" data-end=\"1160\">Phase Shift<\/strong>: [latex]\\dfrac{c}{b}[\/latex]. If [latex]c &gt; 0[\/latex] \u2192 shift right, if [latex]c &lt; 0[\/latex] \u2192 shift left.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1270\" data-end=\"1344\">\r\n<p data-start=\"1273\" data-end=\"1344\"><strong data-start=\"1273\" data-end=\"1291\">Vertical Shift<\/strong>: [latex]d[\/latex]. Positive \u2192 up, Negative \u2192 down.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1345\" data-end=\"1521\">\r\n<p data-start=\"1348\" data-end=\"1367\"><strong data-start=\"1348\" data-end=\"1364\">Start Points<\/strong>:<\/p>\r\n\r\n<ul data-start=\"1371\" data-end=\"1521\">\r\n \t<li data-start=\"1371\" data-end=\"1451\">\r\n<p data-start=\"1373\" data-end=\"1451\">For sine: normally starts at (0,0); apply shifts to move the starting point.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1455\" data-end=\"1521\">\r\n<p data-start=\"1457\" data-end=\"1521\">For cosine: normally starts at (0,1); apply shifts to move it.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1522\" data-end=\"1614\">\r\n<p data-start=\"1525\" data-end=\"1614\"><strong data-start=\"1525\" data-end=\"1543\">Graph in Steps<\/strong>: Midline first, mark amplitude, then adjust cycle length and shifts.<\/p>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Given [latex]f(x) = 3\\sin\\left(2x - \\frac{\\pi}{2}\\right) + 1[\/latex], identify the amplitude, period, phase shift, and vertical shift.[reveal-answer q=\"sine-trans-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"sine-trans-001\"]\r\nThe function is in the form [latex]f(x) = A\\sin(B(x - C)) + D[\/latex].Compare: [latex]f(x) = 3\\sin\\left(2\\left(x - \\frac{\\pi}{4}\\right)\\right) + 1[\/latex]Amplitude: [latex]|A| = |3| = 3[\/latex]Period: [latex]\\frac{2\\pi}{|B|} = \\frac{2\\pi}{2} = \\pi[\/latex]Phase shift: [latex]C = \\frac{\\pi}{4}[\/latex] (right)Vertical shift: [latex]D = 1[\/latex] (up)\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-befgfgca-POVfta32Ghc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/POVfta32Ghc?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-befgfgca-POVfta32Ghc\" 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=14660583&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-befgfgca-POVfta32Ghc&vembed=0&video_id=POVfta32Ghc&video_target=tpm-plugin-befgfgca-POVfta32Ghc'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Exploring+Transformations+of+Sine+and+Cosine+-+y%3DAsin(Bx-C)%2BD+with+Desmos_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExploring Transformations of Sine and Cosine: y=Asin(Bx-C)+D with Desmos\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<section><section class=\"textbox example\">Sketch the graph of [latex]f(x) = 2\\cos\\left(x + \\frac{\\pi}{3}\\right) - 1[\/latex].[reveal-answer q=\"sine-trans-002\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"sine-trans-002\"]\r\nFrom [latex]f(x) = 2\\cos\\left(x + \\frac{\\pi}{3}\\right) - 1[\/latex]:Amplitude: 2\r\nPeriod: [latex]2\\pi[\/latex]\r\nPhase shift: [latex]\\frac{\\pi}{3}[\/latex] left\r\nVertical shift: 1 down\r\nMidline: [latex]y = -1[\/latex]\r\nMaximum: [latex]-1 + 2 = 1[\/latex]\r\nMinimum: [latex]-1 - 2 = -3[\/latex]Start with key point [latex]\\left(-\\frac{\\pi}{3}, 1\\right)[\/latex] and plot one complete cycle over [latex]2\\pi[\/latex] (each critical [latex]x[\/latex] point will be [latex]\\frac{2\\pi}{4}=\\frac{\\pi}{2}[\/latex] more than the previous).\r\n<table style=\"border-collapse: collapse; width: 100%; height: 198px;\">\r\n<tbody>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 50%; height: 22px;\">x<\/td>\r\n<td style=\"width: 50%; height: 22px;\">f(x)<\/td>\r\n<\/tr>\r\n<tr style=\"height: 44px;\">\r\n<td style=\"width: 50%; height: 44px;\">[latex]-\\frac{\\pi}{3}[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 44px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 50%; height: 22px;\">[latex]\\frac{\\pi}{6}[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 22px;\">[latex]-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 44px;\">\r\n<td style=\"width: 50%; height: 44px;\">[latex]\\frac{2\\pi}{3}[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 44px;\">[latex]-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 44px;\">\r\n<td style=\"width: 50%; height: 44px;\">[latex]\\frac{7\\pi}{6}[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 44px;\">[latex]-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 50%; height: 22px;\">[latex]\\frac{5\\pi}{3}[\/latex]<\/td>\r\n<td style=\"width: 50%; height: 22px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\"alignnone wp-image-5546\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11164415\/Screenshot-2026-02-11-at-9.43.00%E2%80%AFAM.png\" alt=\"The graph is a cosine function with amplitude 2 and midline y equals -1. The graph passes through the points: (-pi over 3, 1), (0, 0), (pi over 3, -2), (2pi over 3, -3), (pi, -2), (4pi over 3, 0), (5pi over 3, 1) The maximum value is 1 and the minimum value is -3. The period is 2pi.\" width=\"579\" height=\"372\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><\/section>\r\n<h2>Formulas of Sinusoidal Graphs<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"239\" data-end=\"663\">A sinusoidal graph is a shifted, stretched, and possibly reflected version of [latex]y=\\sin x[\/latex] or [latex]y=\\cos x[\/latex]. To build its formula, read the graph\u2019s <strong data-start=\"408\" data-end=\"419\">midline<\/strong> (vertical shift), <strong data-start=\"438\" data-end=\"451\">amplitude<\/strong> (height of peaks above the midline), <strong data-start=\"489\" data-end=\"499\">period<\/strong> (cycle length), and <strong data-start=\"520\" data-end=\"535\">phase shift<\/strong> (horizontal shift). Then choose sine or cosine to match a convenient key point (like a peak or a midline crossing) and write:<\/p>\r\n<p style=\"padding-left: 200px;\" data-start=\"665\" data-end=\"764\">[latex]y = a\\sin\\big(b(x-h)\\big)+d \\quad \\text{or} \\quad y = a\\cos\\big(b(x-h)\\big)+d[\/latex].<\/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: Transforming Sine and Cosine Graphs<\/strong>\r\n<ol>\r\n \t<li data-start=\"789\" data-end=\"930\">\r\n<p data-start=\"792\" data-end=\"822\"><strong data-start=\"792\" data-end=\"820\">Midline &amp; Vertical Shift<\/strong><\/p>\r\n\r\n<ul data-start=\"826\" data-end=\"930\">\r\n \t<li data-start=\"826\" data-end=\"870\">\r\n<p data-start=\"828\" data-end=\"870\">Midline = average of max and min values.<\/p>\r\n<\/li>\r\n \t<li data-start=\"874\" data-end=\"930\">\r\n<p data-start=\"876\" data-end=\"930\">[latex]d = \\dfrac{\\text{max}+\\text{min}}{2}[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"932\" data-end=\"1219\">\r\n<p data-start=\"935\" data-end=\"950\"><strong data-start=\"935\" data-end=\"948\">Amplitude<\/strong><\/p>\r\n\r\n<ul data-start=\"954\" data-end=\"1219\">\r\n \t<li data-start=\"954\" data-end=\"1000\">\r\n<p data-start=\"956\" data-end=\"1000\">Distance from midline to a peak or valley.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1004\" data-end=\"1062\">\r\n<p data-start=\"1006\" data-end=\"1062\">[latex]|a| = \\dfrac{\\text{max}-\\text{min}}{2}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1066\" data-end=\"1219\">\r\n<p data-start=\"1068\" data-end=\"1219\">If the graph starts by going down from a midline crossing or has peaks where cosine would normally have troughs, let [latex]a&lt;0[\/latex] (reflection).<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1221\" data-end=\"1431\">\r\n<p data-start=\"1224\" data-end=\"1255\"><strong data-start=\"1224\" data-end=\"1253\">Period \u2192 [latex]b[\/latex]<\/strong><\/p>\r\n\r\n<ul data-start=\"1259\" data-end=\"1431\">\r\n \t<li data-start=\"1259\" data-end=\"1388\">\r\n<p data-start=\"1261\" data-end=\"1388\">Period [latex]P[\/latex] = horizontal distance of one full cycle (peak-to-peak, trough-to-trough, or midline-up to next same).<\/p>\r\n<\/li>\r\n \t<li data-start=\"1392\" data-end=\"1431\">\r\n<p data-start=\"1394\" data-end=\"1431\">[latex]b = \\dfrac{2\\pi}{P}[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1433\" data-end=\"1639\">\r\n<p data-start=\"1436\" data-end=\"1470\"><strong data-start=\"1436\" data-end=\"1468\">Phase Shift [latex]h[\/latex]<\/strong><\/p>\r\n\r\n<ul data-start=\"1474\" data-end=\"1639\">\r\n \t<li data-start=\"1474\" data-end=\"1639\">\r\n<p data-start=\"1476\" data-end=\"1499\">Pick an anchor point:<\/p>\r\n\r\n<ul data-start=\"1505\" data-end=\"1639\">\r\n \t<li data-start=\"1505\" data-end=\"1560\">\r\n<p data-start=\"1507\" data-end=\"1560\">Cosine form \u2192 use a <strong data-start=\"1527\" data-end=\"1535\">peak<\/strong> at [latex]x=h[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1566\" data-end=\"1639\">\r\n<p data-start=\"1568\" data-end=\"1639\">Sine form \u2192 use an <strong data-start=\"1587\" data-end=\"1614\">upward midline crossing<\/strong> at [latex]x=h[\/latex].<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1641\" data-end=\"1852\">\r\n<p data-start=\"1644\" data-end=\"1672\"><strong data-start=\"1644\" data-end=\"1670\">Choose Sine vs. Cosine<\/strong><\/p>\r\n\r\n<ul data-start=\"1676\" data-end=\"1852\">\r\n \t<li data-start=\"1676\" data-end=\"1721\">\r\n<p data-start=\"1678\" data-end=\"1721\">Starts at a <strong data-start=\"1690\" data-end=\"1698\">peak<\/strong> \u2192 cosine is natural.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1725\" data-end=\"1780\">\r\n<p data-start=\"1727\" data-end=\"1780\">Starts at a <strong data-start=\"1739\" data-end=\"1759\">midline going up<\/strong> \u2192 sine is natural.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1784\" data-end=\"1852\">\r\n<p data-start=\"1786\" data-end=\"1852\">Both work with the right [latex]h[\/latex]; pick the simpler one.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1854\" data-end=\"2034\">\r\n<p data-start=\"1857\" data-end=\"1876\"><strong data-start=\"1857\" data-end=\"1874\">Final Formula<\/strong><\/p>\r\n\r\n<ul data-start=\"1880\" data-end=\"2034\">\r\n \t<li data-start=\"1880\" data-end=\"1933\">\r\n<p data-start=\"1882\" data-end=\"1933\">Keep values exact in terms of [latex]\\pi[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1937\" data-end=\"2034\">\r\n<p data-start=\"1939\" data-end=\"2034\">[latex]y = a\\sin\\big(b(x-h)\\big)+d \\quad \\text{or} \\quad y = a\\cos\\big(b(x-h)\\big)+d[\/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 sine function for a graph with amplitude 4, period [latex]\\pi[\/latex], phase shift [latex]\\frac{\\pi}{4}[\/latex] right, and vertical shift 2 up.[reveal-answer q=\"sine-trans-003\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"sine-trans-003\"]\r\nUse [latex]f(x) = A\\sin(B(x - C)) + D[\/latex]Amplitude: [latex]A = 4[\/latex]Period: [latex]\\frac{2\\pi}{B} = \\pi[\/latex], so [latex]B = 2[\/latex]Phase shift: [latex]C = \\frac{\\pi}{4}[\/latex]Vertical shift: [latex]D = 2[\/latex]The function is: [latex]f(x) = 4\\sin\\left(2\\left(x - \\frac{\\pi}{4}\\right)\\right) + 2[\/latex]Or simplified: [latex]f(x) = 4\\sin\\left(2x - \\frac{\\pi}{2}\\right) + 2[\/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-fccgfdba-Vp6uOaOxUS0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Vp6uOaOxUS0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fccgfdba-Vp6uOaOxUS0\" 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=14660584&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-fccgfdba-Vp6uOaOxUS0&vembed=0&video_id=Vp6uOaOxUS0&video_target=tpm-plugin-fccgfdba-Vp6uOaOxUS0'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Determining+the+Equation+of+a+Sine+and+Cosine+Graph_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermining the Equation of a Sine and Cosine Graph\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><\/div>\r\n<h2>Modeling Circular and Periodic Motion<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<p data-start=\"201\" data-end=\"488\">Circular and periodic motion can be modeled using sine and cosine because these functions naturally repeat in cycles. When an object moves around a circle or follows a repeating up-and-down pattern, its position over time can be described with sinusoidal functions. The general form is<\/p>\r\n<p style=\"padding-left: 200px;\" data-start=\"490\" data-end=\"573\">[latex]y = a\\sin(bx - c) + d \\quad \\text{or} \\quad y = a\\cos(bx - c) + d[\/latex].<\/p>\r\n<p data-start=\"575\" data-end=\"997\">Here, [latex]a[\/latex] represents the maximum displacement (amplitude), [latex]b[\/latex] sets how quickly the cycle repeats (related to period or frequency), [latex]c[\/latex] shifts the motion left or right in time (phase shift), and [latex]d[\/latex] raises or lowers the entire path (vertical shift). These formulas allow us to model phenomena such as a Ferris wheel, a pendulum, tides, or seasonal temperature changes.<\/p>\r\n<p data-start=\"239\" data-end=\"663\"><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Transforming Sine and Cosine Graphs<\/strong><\/p>\r\n\r\n<ol>\r\n \t<li data-start=\"1061\" data-end=\"1192\">\r\n<p data-start=\"1064\" data-end=\"1096\"><strong data-start=\"1064\" data-end=\"1094\">Amplitude [latex]a[\/latex]<\/strong><\/p>\r\n\r\n<ul data-start=\"1100\" data-end=\"1192\">\r\n \t<li data-start=\"1100\" data-end=\"1134\">\r\n<p data-start=\"1102\" data-end=\"1134\">Distance from midline to peak.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1138\" data-end=\"1192\">\r\n<p data-start=\"1140\" data-end=\"1192\">Represents the maximum displacement of the motion.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1194\" data-end=\"1348\">\r\n<p data-start=\"1197\" data-end=\"1230\"><strong data-start=\"1197\" data-end=\"1228\">Period and [latex]b[\/latex]<\/strong><\/p>\r\n\r\n<ul data-start=\"1234\" data-end=\"1348\">\r\n \t<li data-start=\"1234\" data-end=\"1280\">\r\n<p data-start=\"1236\" data-end=\"1280\">Period [latex]P = \\dfrac{2\\pi}{b}[\/latex].<\/p>\r\n<\/li>\r\n \t<li data-start=\"1284\" data-end=\"1348\">\r\n<p data-start=\"1286\" data-end=\"1348\">Shorter period = faster cycle; longer period = slower cycle.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1350\" data-end=\"1529\">\r\n<p data-start=\"1353\" data-end=\"1398\"><strong data-start=\"1353\" data-end=\"1396\">Phase Shift [latex]\\dfrac{c}{b}[\/latex]<\/strong><\/p>\r\n\r\n<ul data-start=\"1402\" data-end=\"1529\">\r\n \t<li data-start=\"1402\" data-end=\"1441\">\r\n<p data-start=\"1404\" data-end=\"1441\">Determines where the motion starts.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1445\" data-end=\"1529\">\r\n<p data-start=\"1447\" data-end=\"1529\">Positive [latex]c[\/latex] \u2192 shift right; Negative [latex]c[\/latex] \u2192 shift left.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1531\" data-end=\"1726\">\r\n<p data-start=\"1534\" data-end=\"1571\"><strong data-start=\"1534\" data-end=\"1569\">Vertical Shift [latex]d[\/latex]<\/strong><\/p>\r\n\r\n<ul data-start=\"1575\" data-end=\"1726\">\r\n \t<li data-start=\"1575\" data-end=\"1622\">\r\n<p data-start=\"1577\" data-end=\"1622\">Moves the midline of the motion up or down.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1626\" data-end=\"1726\">\r\n<p data-start=\"1628\" data-end=\"1726\">Useful for modeling motions above the ground (e.g., a Ferris wheel that never dips below 10 ft).<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1728\" data-end=\"1893\">\r\n<p data-start=\"1731\" data-end=\"1759\"><strong data-start=\"1731\" data-end=\"1757\">Choose Sine vs. Cosine<\/strong><\/p>\r\n\r\n<ul data-start=\"1763\" data-end=\"1893\">\r\n \t<li data-start=\"1763\" data-end=\"1823\">\r\n<p data-start=\"1765\" data-end=\"1823\">Use cosine if the motion begins at a maximum or minimum.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1827\" data-end=\"1893\">\r\n<p data-start=\"1829\" data-end=\"1893\">Use sine if the motion begins at the midline going up or down.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li data-start=\"1895\" data-end=\"2085\">\r\n<p data-start=\"1898\" data-end=\"1925\"><strong data-start=\"1898\" data-end=\"1923\">Real-World Connection<\/strong><\/p>\r\n\r\n<ul data-start=\"1929\" data-end=\"2085\">\r\n \t<li data-start=\"1929\" data-end=\"1976\">\r\n<p data-start=\"1931\" data-end=\"1976\">Ferris wheel: height as a function of time.<\/p>\r\n<\/li>\r\n \t<li data-start=\"1980\" data-end=\"2031\">\r\n<p data-start=\"1982\" data-end=\"2031\">Tides: water level rising and falling each day.<\/p>\r\n<\/li>\r\n \t<li data-start=\"2035\" data-end=\"2085\">\r\n<p data-start=\"2037\" data-end=\"2085\">Springs\/Pendulums: back-and-forth oscillation.<\/p>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dehdacfc-R9dKBH2vunc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/R9dKBH2vunc?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-dehdacfc-R9dKBH2vunc\" 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=14660585&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-dehdacfc-R9dKBH2vunc&vembed=0&video_id=R9dKBH2vunc&video_target=tpm-plugin-dehdacfc-R9dKBH2vunc'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Sine+%26+Cosine+Graphs+Word+Problems+(Writing+the+Equation)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSine &amp; Cosine Graphs Word Problems (Writing the Equation)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Determine amplitude, period, phase shift, and vertical shift of a sine or cosine graph from its equation.<\/li>\n<li>Graph transformations of y=cos x and y=sin x .<\/li>\n<li>Determine a function formula that would have a given sinusoidal graph.<\/li>\n<li>Determine functions that model circular and periodic motion.<\/li>\n<\/ul>\n<\/section>\n<h2>Sine and Cosine Graph Features<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"225\" data-end=\"378\">The graphs of sine and cosine can be stretched, compressed, shifted, or moved up and down by changing parts of their equations. In general, the form is<\/p>\n<p style=\"padding-left: 280px;\" data-start=\"380\" data-end=\"461\">[latex]y = a \\sin(bx)[\/latex]<\/p>\n<p style=\"padding-left: 280px;\" data-start=\"380\" data-end=\"461\">[latex]y = a \\cos(bx)[\/latex].<\/p>\n<p data-start=\"463\" data-end=\"513\">Each letter tells you something about the graph:<\/p>\n<ul>\n<li data-start=\"467\" data-end=\"525\">\n<p data-start=\"469\" data-end=\"525\"><strong data-start=\"469\" data-end=\"474\">a<\/strong> controls the <em data-start=\"488\" data-end=\"499\">amplitude<\/em> (how tall the wave is).<\/p>\n<\/li>\n<li data-start=\"526\" data-end=\"600\">\n<p data-start=\"528\" data-end=\"600\"><strong data-start=\"528\" data-end=\"533\">b<\/strong> affects the <em data-start=\"546\" data-end=\"554\">period<\/em> (how long it takes for the wave to repeat).<\/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: Reading Transformations from the Equation<\/strong><\/p>\n<ol>\n<li data-start=\"831\" data-end=\"846\"><strong data-start=\"831\" data-end=\"844\">Amplitude<\/strong>\n<ul data-start=\"850\" data-end=\"962\">\n<li data-start=\"850\" data-end=\"901\">\n<p data-start=\"852\" data-end=\"901\">Formula: [latex]\\text{Amplitude} = |a|[\/latex].<\/p>\n<\/li>\n<li data-start=\"905\" data-end=\"962\">\n<p data-start=\"907\" data-end=\"962\">Example: [latex]y = 3\\sin(x)[\/latex] \u2192 amplitude = 3.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"967\" data-end=\"979\"><strong data-start=\"967\" data-end=\"977\">Period<\/strong>\n<ul data-start=\"983\" data-end=\"1136\">\n<li data-start=\"983\" data-end=\"1043\">\n<p data-start=\"985\" data-end=\"1043\">Formula: [latex]\\text{Period} = \\dfrac{2\\pi}{b}[\/latex].<\/p>\n<\/li>\n<li data-start=\"1047\" data-end=\"1136\">\n<p data-start=\"1049\" data-end=\"1136\">Example: [latex]y = \\cos(2x)[\/latex] \u2192 period = [latex]\\dfrac{2\\pi}{2} = \\pi[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<ul>\n<li style=\"list-style-type: none;\"><\/li>\n<\/ul>\n<\/div>\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-aafhbhfd-QNQAkUUHNxo\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/QNQAkUUHNxo?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-aafhbhfd-QNQAkUUHNxo\" 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=14660582&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-aafhbhfd-QNQAkUUHNxo&#38;vembed=0&#38;video_id=QNQAkUUHNxo&#38;video_target=tpm-plugin-aafhbhfd-QNQAkUUHNxo\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Animation+-+Graphing+the+Sine+Function+Using+The+Unit+Circle_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cAnimation: Graphing the Sine Function Using The Unit Circle\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Sine and Cosine Graph Transformations<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"225\" data-end=\"378\">The graphs of [latex]y=\\sin x[\/latex] and [latex]y=\\cos x[\/latex] form smooth, repeating waves. Transformations let us reshape these waves by stretching them taller or shorter, squeezing them to fit more cycles, sliding them left or right, or lifting and lowering the whole curve. These changes don\u2019t alter the wave\u2019s basic pattern\u2014it still oscillates smoothly\u2014but they make the graph flexible enough to model real-world cycles like sound waves, tides, or seasonal patterns. Recognizing transformations helps us quickly sketch graphs and understand how the wave has been shifted from its standard position.<\/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: Transforming Sine and Cosine Graphs<\/strong><\/p>\n<ol>\n<li data-start=\"864\" data-end=\"981\">\n<p data-start=\"867\" data-end=\"981\"><strong data-start=\"867\" data-end=\"880\">Amplitude<\/strong>: [latex]|a|[\/latex] is the max distance from the midline. Bigger [latex]|a|[\/latex] = taller wave.<\/p>\n<\/li>\n<li data-start=\"982\" data-end=\"1141\">\n<p data-start=\"985\" data-end=\"1141\"><strong data-start=\"985\" data-end=\"995\">Period<\/strong>: [latex]\\dfrac{2\\pi}{b}[\/latex] is the length of one cycle. Larger [latex]b[\/latex] compresses the wave, smaller [latex]b[\/latex] stretches it.<\/p>\n<\/li>\n<li data-start=\"1142\" data-end=\"1269\">\n<p data-start=\"1145\" data-end=\"1269\"><strong data-start=\"1145\" data-end=\"1160\">Phase Shift<\/strong>: [latex]\\dfrac{c}{b}[\/latex]. If [latex]c > 0[\/latex] \u2192 shift right, if [latex]c < 0[\/latex] \u2192 shift left.<\/p>\n<\/li>\n<li data-start=\"1270\" data-end=\"1344\">\n<p data-start=\"1273\" data-end=\"1344\"><strong data-start=\"1273\" data-end=\"1291\">Vertical Shift<\/strong>: [latex]d[\/latex]. Positive \u2192 up, Negative \u2192 down.<\/p>\n<\/li>\n<li data-start=\"1345\" data-end=\"1521\">\n<p data-start=\"1348\" data-end=\"1367\"><strong data-start=\"1348\" data-end=\"1364\">Start Points<\/strong>:<\/p>\n<ul data-start=\"1371\" data-end=\"1521\">\n<li data-start=\"1371\" data-end=\"1451\">\n<p data-start=\"1373\" data-end=\"1451\">For sine: normally starts at (0,0); apply shifts to move the starting point.<\/p>\n<\/li>\n<li data-start=\"1455\" data-end=\"1521\">\n<p data-start=\"1457\" data-end=\"1521\">For cosine: normally starts at (0,1); apply shifts to move it.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1522\" data-end=\"1614\">\n<p data-start=\"1525\" data-end=\"1614\"><strong data-start=\"1525\" data-end=\"1543\">Graph in Steps<\/strong>: Midline first, mark amplitude, then adjust cycle length and shifts.<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Given [latex]f(x) = 3\\sin\\left(2x - \\frac{\\pi}{2}\\right) + 1[\/latex], identify the amplitude, period, phase shift, and vertical shift.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qsine-trans-001\">Show Solution<\/button><\/p>\n<div id=\"qsine-trans-001\" class=\"hidden-answer\" style=\"display: none\">\nThe function is in the form [latex]f(x) = A\\sin(B(x - C)) + D[\/latex].Compare: [latex]f(x) = 3\\sin\\left(2\\left(x - \\frac{\\pi}{4}\\right)\\right) + 1[\/latex]Amplitude: [latex]|A| = |3| = 3[\/latex]Period: [latex]\\frac{2\\pi}{|B|} = \\frac{2\\pi}{2} = \\pi[\/latex]Phase shift: [latex]C = \\frac{\\pi}{4}[\/latex] (right)Vertical shift: [latex]D = 1[\/latex] (up)\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-befgfgca-POVfta32Ghc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/POVfta32Ghc?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-befgfgca-POVfta32Ghc\" 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=14660583&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-befgfgca-POVfta32Ghc&#38;vembed=0&#38;video_id=POVfta32Ghc&#38;video_target=tpm-plugin-befgfgca-POVfta32Ghc\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Exploring+Transformations+of+Sine+and+Cosine+-+y%3DAsin(Bx-C)%2BD+with+Desmos_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cExploring Transformations of Sine and Cosine: y=Asin(Bx-C)+D with Desmos\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<section>\n<section class=\"textbox example\">Sketch the graph of [latex]f(x) = 2\\cos\\left(x + \\frac{\\pi}{3}\\right) - 1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qsine-trans-002\">Show Solution<\/button><\/p>\n<div id=\"qsine-trans-002\" class=\"hidden-answer\" style=\"display: none\">\nFrom [latex]f(x) = 2\\cos\\left(x + \\frac{\\pi}{3}\\right) - 1[\/latex]:Amplitude: 2<br \/>\nPeriod: [latex]2\\pi[\/latex]<br \/>\nPhase shift: [latex]\\frac{\\pi}{3}[\/latex] left<br \/>\nVertical shift: 1 down<br \/>\nMidline: [latex]y = -1[\/latex]<br \/>\nMaximum: [latex]-1 + 2 = 1[\/latex]<br \/>\nMinimum: [latex]-1 - 2 = -3[\/latex]Start with key point [latex]\\left(-\\frac{\\pi}{3}, 1\\right)[\/latex] and plot one complete cycle over [latex]2\\pi[\/latex] (each critical [latex]x[\/latex] point will be [latex]\\frac{2\\pi}{4}=\\frac{\\pi}{2}[\/latex] more than the previous).<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 198px;\">\n<tbody>\n<tr style=\"height: 22px;\">\n<td style=\"width: 50%; height: 22px;\">x<\/td>\n<td style=\"width: 50%; height: 22px;\">f(x)<\/td>\n<\/tr>\n<tr style=\"height: 44px;\">\n<td style=\"width: 50%; height: 44px;\">[latex]-\\frac{\\pi}{3}[\/latex]<\/td>\n<td style=\"width: 50%; height: 44px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 50%; height: 22px;\">[latex]\\frac{\\pi}{6}[\/latex]<\/td>\n<td style=\"width: 50%; height: 22px;\">[latex]-1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 44px;\">\n<td style=\"width: 50%; height: 44px;\">[latex]\\frac{2\\pi}{3}[\/latex]<\/td>\n<td style=\"width: 50%; height: 44px;\">[latex]-3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 44px;\">\n<td style=\"width: 50%; height: 44px;\">[latex]\\frac{7\\pi}{6}[\/latex]<\/td>\n<td style=\"width: 50%; height: 44px;\">[latex]-1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 50%; height: 22px;\">[latex]\\frac{5\\pi}{3}[\/latex]<\/td>\n<td style=\"width: 50%; height: 22px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-5546\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11164415\/Screenshot-2026-02-11-at-9.43.00%E2%80%AFAM.png\" alt=\"The graph is a cosine function with amplitude 2 and midline y equals -1. The graph passes through the points: (-pi over 3, 1), (0, 0), (pi over 3, -2), (2pi over 3, -3), (pi, -2), (4pi over 3, 0), (5pi over 3, 1) The maximum value is 1 and the minimum value is -3. The period is 2pi.\" width=\"579\" height=\"372\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11164415\/Screenshot-2026-02-11-at-9.43.00%E2%80%AFAM.png 1282w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11164415\/Screenshot-2026-02-11-at-9.43.00%E2%80%AFAM-300x193.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11164415\/Screenshot-2026-02-11-at-9.43.00%E2%80%AFAM-1024x658.png 1024w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11164415\/Screenshot-2026-02-11-at-9.43.00%E2%80%AFAM-768x494.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11164415\/Screenshot-2026-02-11-at-9.43.00%E2%80%AFAM-65x42.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11164415\/Screenshot-2026-02-11-at-9.43.00%E2%80%AFAM-225x145.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/61\/2025\/07\/11164415\/Screenshot-2026-02-11-at-9.43.00%E2%80%AFAM-350x225.png 350w\" sizes=\"(max-width: 579px) 100vw, 579px\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<h2>Formulas of Sinusoidal Graphs<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"239\" data-end=\"663\">A sinusoidal graph is a shifted, stretched, and possibly reflected version of [latex]y=\\sin x[\/latex] or [latex]y=\\cos x[\/latex]. To build its formula, read the graph\u2019s <strong data-start=\"408\" data-end=\"419\">midline<\/strong> (vertical shift), <strong data-start=\"438\" data-end=\"451\">amplitude<\/strong> (height of peaks above the midline), <strong data-start=\"489\" data-end=\"499\">period<\/strong> (cycle length), and <strong data-start=\"520\" data-end=\"535\">phase shift<\/strong> (horizontal shift). Then choose sine or cosine to match a convenient key point (like a peak or a midline crossing) and write:<\/p>\n<p style=\"padding-left: 200px;\" data-start=\"665\" data-end=\"764\">[latex]y = a\\sin\\big(b(x-h)\\big)+d \\quad \\text{or} \\quad y = a\\cos\\big(b(x-h)\\big)+d[\/latex].<\/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: Transforming Sine and Cosine Graphs<\/strong><\/p>\n<ol>\n<li data-start=\"789\" data-end=\"930\">\n<p data-start=\"792\" data-end=\"822\"><strong data-start=\"792\" data-end=\"820\">Midline &amp; Vertical Shift<\/strong><\/p>\n<ul data-start=\"826\" data-end=\"930\">\n<li data-start=\"826\" data-end=\"870\">\n<p data-start=\"828\" data-end=\"870\">Midline = average of max and min values.<\/p>\n<\/li>\n<li data-start=\"874\" data-end=\"930\">\n<p data-start=\"876\" data-end=\"930\">[latex]d = \\dfrac{\\text{max}+\\text{min}}{2}[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"932\" data-end=\"1219\">\n<p data-start=\"935\" data-end=\"950\"><strong data-start=\"935\" data-end=\"948\">Amplitude<\/strong><\/p>\n<ul data-start=\"954\" data-end=\"1219\">\n<li data-start=\"954\" data-end=\"1000\">\n<p data-start=\"956\" data-end=\"1000\">Distance from midline to a peak or valley.<\/p>\n<\/li>\n<li data-start=\"1004\" data-end=\"1062\">\n<p data-start=\"1006\" data-end=\"1062\">[latex]|a| = \\dfrac{\\text{max}-\\text{min}}{2}[\/latex].<\/p>\n<\/li>\n<li data-start=\"1066\" data-end=\"1219\">\n<p data-start=\"1068\" data-end=\"1219\">If the graph starts by going down from a midline crossing or has peaks where cosine would normally have troughs, let [latex]a<0[\/latex] (reflection).<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1221\" data-end=\"1431\">\n<p data-start=\"1224\" data-end=\"1255\"><strong data-start=\"1224\" data-end=\"1253\">Period \u2192 [latex]b[\/latex]<\/strong><\/p>\n<ul data-start=\"1259\" data-end=\"1431\">\n<li data-start=\"1259\" data-end=\"1388\">\n<p data-start=\"1261\" data-end=\"1388\">Period [latex]P[\/latex] = horizontal distance of one full cycle (peak-to-peak, trough-to-trough, or midline-up to next same).<\/p>\n<\/li>\n<li data-start=\"1392\" data-end=\"1431\">\n<p data-start=\"1394\" data-end=\"1431\">[latex]b = \\dfrac{2\\pi}{P}[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1433\" data-end=\"1639\">\n<p data-start=\"1436\" data-end=\"1470\"><strong data-start=\"1436\" data-end=\"1468\">Phase Shift [latex]h[\/latex]<\/strong><\/p>\n<ul data-start=\"1474\" data-end=\"1639\">\n<li data-start=\"1474\" data-end=\"1639\">\n<p data-start=\"1476\" data-end=\"1499\">Pick an anchor point:<\/p>\n<ul data-start=\"1505\" data-end=\"1639\">\n<li data-start=\"1505\" data-end=\"1560\">\n<p data-start=\"1507\" data-end=\"1560\">Cosine form \u2192 use a <strong data-start=\"1527\" data-end=\"1535\">peak<\/strong> at [latex]x=h[\/latex].<\/p>\n<\/li>\n<li data-start=\"1566\" data-end=\"1639\">\n<p data-start=\"1568\" data-end=\"1639\">Sine form \u2192 use an <strong data-start=\"1587\" data-end=\"1614\">upward midline crossing<\/strong> at [latex]x=h[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1641\" data-end=\"1852\">\n<p data-start=\"1644\" data-end=\"1672\"><strong data-start=\"1644\" data-end=\"1670\">Choose Sine vs. Cosine<\/strong><\/p>\n<ul data-start=\"1676\" data-end=\"1852\">\n<li data-start=\"1676\" data-end=\"1721\">\n<p data-start=\"1678\" data-end=\"1721\">Starts at a <strong data-start=\"1690\" data-end=\"1698\">peak<\/strong> \u2192 cosine is natural.<\/p>\n<\/li>\n<li data-start=\"1725\" data-end=\"1780\">\n<p data-start=\"1727\" data-end=\"1780\">Starts at a <strong data-start=\"1739\" data-end=\"1759\">midline going up<\/strong> \u2192 sine is natural.<\/p>\n<\/li>\n<li data-start=\"1784\" data-end=\"1852\">\n<p data-start=\"1786\" data-end=\"1852\">Both work with the right [latex]h[\/latex]; pick the simpler one.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1854\" data-end=\"2034\">\n<p data-start=\"1857\" data-end=\"1876\"><strong data-start=\"1857\" data-end=\"1874\">Final Formula<\/strong><\/p>\n<ul data-start=\"1880\" data-end=\"2034\">\n<li data-start=\"1880\" data-end=\"1933\">\n<p data-start=\"1882\" data-end=\"1933\">Keep values exact in terms of [latex]\\pi[\/latex].<\/p>\n<\/li>\n<li data-start=\"1937\" data-end=\"2034\">\n<p data-start=\"1939\" data-end=\"2034\">[latex]y = a\\sin\\big(b(x-h)\\big)+d \\quad \\text{or} \\quad y = a\\cos\\big(b(x-h)\\big)+d[\/latex].<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Write a sine function for a graph with amplitude 4, period [latex]\\pi[\/latex], phase shift [latex]\\frac{\\pi}{4}[\/latex] right, and vertical shift 2 up.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qsine-trans-003\">Show Solution<\/button><\/p>\n<div id=\"qsine-trans-003\" class=\"hidden-answer\" style=\"display: none\">\nUse [latex]f(x) = A\\sin(B(x - C)) + D[\/latex]Amplitude: [latex]A = 4[\/latex]Period: [latex]\\frac{2\\pi}{B} = \\pi[\/latex], so [latex]B = 2[\/latex]Phase shift: [latex]C = \\frac{\\pi}{4}[\/latex]Vertical shift: [latex]D = 2[\/latex]The function is: [latex]f(x) = 4\\sin\\left(2\\left(x - \\frac{\\pi}{4}\\right)\\right) + 2[\/latex]Or simplified: [latex]f(x) = 4\\sin\\left(2x - \\frac{\\pi}{2}\\right) + 2[\/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-fccgfdba-Vp6uOaOxUS0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Vp6uOaOxUS0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fccgfdba-Vp6uOaOxUS0\" 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=14660584&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-fccgfdba-Vp6uOaOxUS0&#38;vembed=0&#38;video_id=Vp6uOaOxUS0&#38;video_target=tpm-plugin-fccgfdba-Vp6uOaOxUS0\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Determining+the+Equation+of+a+Sine+and+Cosine+Graph_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermining the Equation of a Sine and Cosine Graph\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<\/div>\n<h2>Modeling Circular and Periodic Motion<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<p data-start=\"201\" data-end=\"488\">Circular and periodic motion can be modeled using sine and cosine because these functions naturally repeat in cycles. When an object moves around a circle or follows a repeating up-and-down pattern, its position over time can be described with sinusoidal functions. The general form is<\/p>\n<p style=\"padding-left: 200px;\" data-start=\"490\" data-end=\"573\">[latex]y = a\\sin(bx - c) + d \\quad \\text{or} \\quad y = a\\cos(bx - c) + d[\/latex].<\/p>\n<p data-start=\"575\" data-end=\"997\">Here, [latex]a[\/latex] represents the maximum displacement (amplitude), [latex]b[\/latex] sets how quickly the cycle repeats (related to period or frequency), [latex]c[\/latex] shifts the motion left or right in time (phase shift), and [latex]d[\/latex] raises or lowers the entire path (vertical shift). These formulas allow us to model phenomena such as a Ferris wheel, a pendulum, tides, or seasonal temperature changes.<\/p>\n<p data-start=\"239\" data-end=\"663\"><strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">Quick Tips: Transforming Sine and Cosine Graphs<\/strong><\/p>\n<ol>\n<li data-start=\"1061\" data-end=\"1192\">\n<p data-start=\"1064\" data-end=\"1096\"><strong data-start=\"1064\" data-end=\"1094\">Amplitude [latex]a[\/latex]<\/strong><\/p>\n<ul data-start=\"1100\" data-end=\"1192\">\n<li data-start=\"1100\" data-end=\"1134\">\n<p data-start=\"1102\" data-end=\"1134\">Distance from midline to peak.<\/p>\n<\/li>\n<li data-start=\"1138\" data-end=\"1192\">\n<p data-start=\"1140\" data-end=\"1192\">Represents the maximum displacement of the motion.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1194\" data-end=\"1348\">\n<p data-start=\"1197\" data-end=\"1230\"><strong data-start=\"1197\" data-end=\"1228\">Period and [latex]b[\/latex]<\/strong><\/p>\n<ul data-start=\"1234\" data-end=\"1348\">\n<li data-start=\"1234\" data-end=\"1280\">\n<p data-start=\"1236\" data-end=\"1280\">Period [latex]P = \\dfrac{2\\pi}{b}[\/latex].<\/p>\n<\/li>\n<li data-start=\"1284\" data-end=\"1348\">\n<p data-start=\"1286\" data-end=\"1348\">Shorter period = faster cycle; longer period = slower cycle.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1350\" data-end=\"1529\">\n<p data-start=\"1353\" data-end=\"1398\"><strong data-start=\"1353\" data-end=\"1396\">Phase Shift [latex]\\dfrac{c}{b}[\/latex]<\/strong><\/p>\n<ul data-start=\"1402\" data-end=\"1529\">\n<li data-start=\"1402\" data-end=\"1441\">\n<p data-start=\"1404\" data-end=\"1441\">Determines where the motion starts.<\/p>\n<\/li>\n<li data-start=\"1445\" data-end=\"1529\">\n<p data-start=\"1447\" data-end=\"1529\">Positive [latex]c[\/latex] \u2192 shift right; Negative [latex]c[\/latex] \u2192 shift left.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1531\" data-end=\"1726\">\n<p data-start=\"1534\" data-end=\"1571\"><strong data-start=\"1534\" data-end=\"1569\">Vertical Shift [latex]d[\/latex]<\/strong><\/p>\n<ul data-start=\"1575\" data-end=\"1726\">\n<li data-start=\"1575\" data-end=\"1622\">\n<p data-start=\"1577\" data-end=\"1622\">Moves the midline of the motion up or down.<\/p>\n<\/li>\n<li data-start=\"1626\" data-end=\"1726\">\n<p data-start=\"1628\" data-end=\"1726\">Useful for modeling motions above the ground (e.g., a Ferris wheel that never dips below 10 ft).<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1728\" data-end=\"1893\">\n<p data-start=\"1731\" data-end=\"1759\"><strong data-start=\"1731\" data-end=\"1757\">Choose Sine vs. Cosine<\/strong><\/p>\n<ul data-start=\"1763\" data-end=\"1893\">\n<li data-start=\"1763\" data-end=\"1823\">\n<p data-start=\"1765\" data-end=\"1823\">Use cosine if the motion begins at a maximum or minimum.<\/p>\n<\/li>\n<li data-start=\"1827\" data-end=\"1893\">\n<p data-start=\"1829\" data-end=\"1893\">Use sine if the motion begins at the midline going up or down.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"1895\" data-end=\"2085\">\n<p data-start=\"1898\" data-end=\"1925\"><strong data-start=\"1898\" data-end=\"1923\">Real-World Connection<\/strong><\/p>\n<ul data-start=\"1929\" data-end=\"2085\">\n<li data-start=\"1929\" data-end=\"1976\">\n<p data-start=\"1931\" data-end=\"1976\">Ferris wheel: height as a function of time.<\/p>\n<\/li>\n<li data-start=\"1980\" data-end=\"2031\">\n<p data-start=\"1982\" data-end=\"2031\">Tides: water level rising and falling each day.<\/p>\n<\/li>\n<li data-start=\"2035\" data-end=\"2085\">\n<p data-start=\"2037\" data-end=\"2085\">Springs\/Pendulums: back-and-forth oscillation.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-dehdacfc-R9dKBH2vunc\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/R9dKBH2vunc?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-dehdacfc-R9dKBH2vunc\" 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=14660585&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-dehdacfc-R9dKBH2vunc&#38;vembed=0&#38;video_id=R9dKBH2vunc&#38;video_target=tpm-plugin-dehdacfc-R9dKBH2vunc\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Sine+%26+Cosine+Graphs+Word+Problems+(Writing+the+Equation)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSine &amp; Cosine Graphs Word Problems (Writing the Equation)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Animation: Graphing the Sine Function Using The Unit Circle\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/QNQAkUUHNxo\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Exploring Transformations of Sine and Cosine: y=Asin(Bx-C)+D with 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