{"id":245,"date":"2017-08-23T10:31:21","date_gmt":"2017-08-23T10:31:21","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=245"},"modified":"2017-08-31T05:09:29","modified_gmt":"2017-08-31T05:09:29","slug":"amplitude-frequency-period-and-phase-shift","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/amplitude-frequency-period-and-phase-shift\/","title":{"rendered":"Amplitude, Frequency, Period, and Phase Shift"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/pythagorean-trigonometric-identities\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/trigonometry\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/graphing-trigonometric-functions-and-inverse-trigonometric-functions\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Amplitude, Frequency, Period, and Phase Shift<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will learn the definitions of amplitude, frequency, period, and phase shift. You will also learn\u00a0how to locate amplitude, frequency, period, and phase shift on a graph.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>Two important trigonometric ratios used to help solve for\u00a0the lengths of the sides of triangles are sine and cosine. Their\u00a0formulas are:<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p1_clip_image002.gif\" width=\"127\" height=\"44\" name=\"graphics2\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p1_clip_image004.gif\" width=\"129\" height=\"44\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/><\/li>\n<\/ul>\n<section>\n<h3>Amplitude, frequency, period, and phase shift on a graph<\/h3>\n<p>The common form when graphing amplitude, frequency, period, and\u00a0phase shift is<\/p>\n<p><em><span style=\"text-decoration: none;\">f<\/span><\/em><span style=\"text-decoration: none;\">(<\/span><em><span style=\"text-decoration: none;\">t<\/span><\/em><span style=\"text-decoration: none;\">)\u00a0<\/span><em><span style=\"text-decoration: none;\">= A <\/span><\/em><span style=\"text-decoration: none;\">sin (<\/span><em><span style=\"text-decoration: none;\">Bt\u00a0+ C<\/span><\/em><span style=\"text-decoration: none;\">) or <\/span><em><span style=\"text-decoration: none;\">f<\/span><\/em><span style=\"text-decoration: none;\">(<\/span><em><span style=\"text-decoration: none;\">t<\/span><\/em><span style=\"text-decoration: none;\">)\u00a0<\/span><em><span style=\"text-decoration: none;\">= A <\/span><\/em><span style=\"text-decoration: none;\">cos (<\/span><em><span style=\"text-decoration: none;\">Bt\u00a0+ C<\/span><\/em><span style=\"text-decoration: none;\">), where <\/span><em><span style=\"text-decoration: none;\">A\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is the amplitude,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p2_clip_image002.gif\" width=\"16\" height=\"17\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is the frequency,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p2_clip_image004.gif\" width=\"27\" height=\"41\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is the period, and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p2_clip_image006.gif\" width=\"31\" height=\"41\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is the phase shift. You are probably wondering where these\u00a0variable formulas came from and what the amplitude, frequency,\u00a0period, and phase shift look like on a graph. We will look at\u00a0these formulas in more detail in this module. <\/span><\/p>\n<h4>What is the amplitude?<\/h4>\n<p>The <abbr title=\" the height of a peak in a wave pattern, measured from the mid-line between the peaks and troughs \">amplitude<\/abbr> is the height of each peak, or highest point, in a wave pattern\u00a0measured from the middle of the wave. Equivalently, it is half the\u00a0vertical height from lowest point, or trough to the peak.<\/p>\n<p><span style=\"text-decoration: none;\">We know that the amplitude\u00a0is <\/span><em><span style=\"text-decoration: none;\">A\u00a0<\/span><\/em><span style=\"text-decoration: none;\">in the formulas <\/span><em><span style=\"text-decoration: none;\">f<\/span><\/em><span style=\"text-decoration: none;\">(<\/span><em><span style=\"text-decoration: none;\">t<\/span><\/em><span style=\"text-decoration: none;\">)\u00a0<\/span><em><span style=\"text-decoration: none;\">= A <\/span><\/em><span style=\"text-decoration: none;\">sin (<\/span><em><span style=\"text-decoration: none;\">Bt\u00a0+ C<\/span><\/em><span style=\"text-decoration: none;\">) and <\/span><em><span style=\"text-decoration: none;\">f<\/span><\/em><span style=\"text-decoration: none;\">(<\/span><em><span style=\"text-decoration: none;\">t<\/span><\/em><span style=\"text-decoration: none;\">)\u00a0<\/span><em><span style=\"text-decoration: none;\">= A <\/span><\/em><span style=\"text-decoration: none;\">cos\u00a0(<\/span><em><span style=\"text-decoration: none;\">Bt+ C<\/span><\/em><span style=\"text-decoration: none;\">),\u00a0but how can we identify the amplitude on a graph? <\/span><\/p>\n<p><span style=\"text-decoration: none;\">Start with the function cos\u00a0(<\/span><em><span style=\"text-decoration: none;\">Bt+C<\/span><\/em><span style=\"text-decoration: none;\">)\u00a0in the equation, since cosine is simply a function and, by\u00a0graphing, we can easily see what the function looks like. Below is\u00a0a graph of the cosine function. You can see that the range in a\u00a0cosine function is always from \u20131 to 1. <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/Math%20Mod%206.3%20Art%20001.GIF\" alt=\"Wave amplitude\" width=\"487\" height=\"126\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>So you know that<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p2_clip_image008.gif\" width=\"135\" height=\"21\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p><span style=\"text-decoration: none;\">But we know that the\u00a0amplitude is <\/span><em><span style=\"text-decoration: none;\">A,\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">A\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is not used here. So, what can you do to get <\/span><em><span style=\"text-decoration: none;\">A\u00a0<\/span><\/em><span style=\"text-decoration: none;\">in the equation? Multiply both sides by <\/span><em><span style=\"text-decoration: none;\">A<\/span><\/em><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p2_clip_image010.gif\" width=\"159\" height=\"21\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>I<span style=\"text-decoration: none;\">f <\/span><em><span style=\"text-decoration: none;\">A\u00a0<\/span><\/em><span style=\"text-decoration: none;\">were negative, then <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p2_clip_image012.gif\" width=\"156\" height=\"21\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p style=\"text-decoration: none;\">So the range on the function\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p2_clip_image014.gif\" width=\"92\" height=\"21\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p2_clip_image016.gif\" width=\"60\" height=\"27\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0where A is the amplitude. So using this information, you can see\u00a0what the amplitude is on the graph of the function\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p2_clip_image018.gif\" width=\"89\" height=\"21\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Using the formula\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p2_clip_image019.gif\" width=\"92\" height=\"21\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0you know that the amplitude is 3. But as you can see, the graph\u00a0itself tells you the amplitude.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/Math%20Mod%206.3%20Art%20002.JPG\" alt=\"Graph with amplitude 3\" width=\"431\" height=\"237\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The same method can be applied to\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p2_clip_image021.gif\" width=\"89\" height=\"21\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the amplitude on the graph below?<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/Math%20Mod%206.3%20Art%20003.JPG\" alt=\"Graph with amplitude 2\" width=\"454\" height=\"156\" name=\"graphics16\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<ol>\n<li>1<\/li>\n<li>2<\/li>\n<li>3<\/li>\n<li>1.5<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is B. As you can see, the range is from\u00a0[\u20132, 2], so the amplitude is 2.<\/p>\n<p>T<span style=\"text-decoration: none;\">he fact that the wave\u00a0discussed here is \u201cbalanced\u201d on the line <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">= 0, with a range that is symmetric about the point 0, is purely\u00a0coincidental. A wave with a range of [4, 10] would have an\u00a0amplitude of half the height from 4 to 10. Such a wave would have\u00a0an amplitude of 3.<\/span><\/p>\n<\/div>\n<\/section>\n<h3>What are the period and the frequency?<\/h3>\n<p>The <abbr title=\"the number of wave patterns within a distance from 0 toa \">frequency<\/abbr> is the number of wave patterns within a distance from 0 to\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p3_clip_image002.gif\" width=\"24\" height=\"19\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p><span style=\"text-decoration: none;\">The <\/span><abbr title=\"in a wave pattern, it is the horizontal distance (on the x-axis) from a point on the x-axis to the next equal (in terms of the y-axis) point on the x-axis, where the wave pattern begins to repeat itself\"><span style=\"text-decoration: none;\">period<\/span><\/abbr><span style=\"text-decoration: none;\"> is equal to the horizontal distance on the <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-axis\u00a0between corresponding portions of the wave\u2013from peak to\u00a0peak, or trough to trough. In other words, the period shows the\u00a0distance from where a wave pattern starts to where the wave\u00a0pattern begins to repeat itself. <\/span><\/p>\n<p><span style=\"text-decoration: none;\">T<\/span>he period and\u00a0frequency on a graph are always the reciprocal of each other. So,\u00a0try and prove that the period is equal to\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p3_clip_image004.gif\" width=\"27\" height=\"41\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p><em><strong>Hint:<\/strong> We need to look at the\u00a0period in terms of cycles and a complete cycle is from 0 to\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p3_clip_image005.gif\" width=\"24\" height=\"19\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0or in degrees from 0\u00b0 to 360\u00b0. <\/em><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p3_clip_image007.gif\" width=\"92\" height=\"21\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><span style=\"text-decoration: none;\">We know that we need to use\u00a0the cycle information given above, and since we know that <\/span><em><span style=\"text-decoration: none;\">Bt\u00a0<\/span><\/em><span style=\"text-decoration: none;\">+ <\/span><em><span style=\"text-decoration: none;\">C\u00a0<\/span><\/em><span style=\"text-decoration: none;\">can be written in terms of degrees, we know that <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p3_clip_image009.gif\" width=\"107\" height=\"19\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<p><span style=\"text-decoration: none;\">Subtract <\/span><em><span style=\"text-decoration: none;\">C\u00a0<\/span><\/em><span style=\"text-decoration: none;\">from both sides. <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p3_clip_image011.gif\" width=\"123\" height=\"19\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Divide both<span style=\"text-decoration: none;\"> sides by <\/span><em><span style=\"text-decoration: none;\">B.<\/span><\/em><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p3_clip_image013.gif\" width=\"117\" height=\"41\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><span style=\"text-decoration: none;\">Add\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/s6_p3_html_m1dead725.gif\" width=\"19\" height=\"41\" name=\"graphics10\" align=\"absmiddle\" border=\"0\" \/>to\u00a0the right side so that the manipulated equation will appear as\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p3_clip_image017.gif\" width=\"120\" height=\"41\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0So the period is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p3_clip_image019.gif\" width=\"27\" height=\"41\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and you can see that the variable <\/span><em><span style=\"text-decoration: none;\">C\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is not taken into account in the period. <\/span><\/p>\n<p>Using the graph below, you can see that the period is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p3_clip_image021.gif\" width=\"15\" height=\"15\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0and you can see that the frequency is 2, because the wave repeats\u00a0twice between 0 and\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p3_clip_image023.gif\" width=\"24\" height=\"19\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/Math%20Mod%206.3%20Art%20004.JPG\" alt=\" Wave's period\" width=\"476\" height=\"267\" name=\"graphics15\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the period on the graph below?<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/Math%20Mod%206.3%20Art%20005.JPG\" alt=\"Graph with period pi over 2\" width=\"393\" height=\"163\" name=\"graphics16\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p3_clip_image025.gif\" width=\"17\" height=\"41\" name=\"graphics17\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li>0.5<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p3_clip_image027.gif\" width=\"15\" height=\"15\" name=\"graphics18\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li>4<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is A. You can see a maximum point is at 0,\u00a0and the next maximum point is between 1 and 2. The only answer\u00a0choice between 1 and 2 is A.<\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>Using the equation\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p3_clip_image029.gif\" width=\"87\" height=\"21\" name=\"graphics19\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0what is the frequency?<\/p>\n<ol>\n<li>6<\/li>\n<li>3<\/li>\n<li>2<\/li>\n<li>4<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p><span style=\"text-decoration: none;\">The correct choice is C.\u00a0From the equation\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p3_clip_image031.gif\" width=\"116\" height=\"21\" name=\"graphics20\" align=\"ABSMIDDLE\" border=\"0\" \/>, we know that <\/span><em><span style=\"text-decoration: none;\">B\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is the frequency. Therefore, 2 is the frequency in the equation\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p3_clip_image032.gif\" width=\"87\" height=\"21\" name=\"graphics21\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/span><\/p>\n<\/div>\n<\/section>\n<h3>What is the phase shift?<\/h3>\n<p>A <abbr title=\"a horizontal shift of a line representing a wave pattern to the left or right on a graph\">phase shift<\/abbr> is a horizontal shift of a line to the left or right on a graph.\u00a0The amplitude, period, and frequency do not change with the phase\u00a0shift.<\/p>\n<p><span style=\"text-decoration: none;\">You may not have realized\u00a0it, but you have already proved that the phase shift is equal to <\/span><em><span style=\"text-decoration: none;\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image002.gif\" width=\"31\" height=\"41\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0<\/span><\/em><span style=\"text-decoration: none;\">using the equations <\/span><em><span style=\"text-decoration: none;\">f(t)\u00a0= A sin (Bt + C)\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">f(t) = A cos\u00a0(Bt +C)<\/span><\/em><span style=\"text-decoration: none;\"> in the\u00a0proof of the period. <\/span><\/p>\n<p><span style=\"text-decoration: none;\">At<\/span> the end of the\u00a0proof, you showed that<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image004.gif\" width=\"117\" height=\"41\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>This is equal to<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image006.gif\" width=\"120\" height=\"41\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>This shows that there is a phase shift equal to\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image008.gif\" width=\"31\" height=\"41\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0If\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image010.gif\" width=\"31\" height=\"41\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is negative, there is a shift to the left, and if<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image011.gif\" width=\"31\" height=\"41\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is positive, there is a shift to the right.<\/p>\n<p>In the graph below, you can see the phase shift. The function\u00a0in black is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image013.gif\" width=\"64\" height=\"21\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0and the function in red is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image015.gif\" width=\"87\" height=\"21\" name=\"graphics10\" align=\"TEXTTOP\" border=\"0\" \/>.<br \/>\nUsing the formula\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image017.gif\" width=\"31\" height=\"41\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0you get\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image019.gif\" width=\"61\" height=\"41\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Since\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image020.gif\" width=\"31\" height=\"41\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is negative, there is a shift to the left by 2. This shift is\u00a0represented by the blue arrow.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/Math%20Mod%206.3%20Art%20006.JPG\" alt=\"phase shift\" width=\"550\" height=\"250\" name=\"graphics14\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the phase shift of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image022.gif\" width=\"109\" height=\"45\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/p>\n<ol>\n<li>1<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image024.gif\" width=\"29\" height=\"41\" name=\"graphics16\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li>4<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image026.gif\" width=\"17\" height=\"41\" name=\"graphics17\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is D. Using the formula\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image028.gif\" width=\"115\" height=\"23\" name=\"graphics18\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0we know that the phase shift equals\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image029.gif\" width=\"31\" height=\"41\" name=\"graphics19\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0So using substitution, you get <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image031.gif\" width=\"75\" height=\"60\" name=\"graphics20\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<p>Now that you know what the amplitude, frequency, period, and\u00a0phase shift are, can you prove that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image033.gif\" width=\"136\" height=\"21\" name=\"graphics21\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0can be written in the form\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image035.gif\" width=\"83\" height=\"21\" name=\"graphics22\" align=\"ABSMIDDLE\" border=\"0\" \/>?\u00a0Try this proof yourself. You should be able to do this using what\u00a0you have learned in this section.<\/p>\n<p>An efficient way of doing this proof is by using an addition\u00a0formula that we will go over in a future section. So if you are\u00a0stuck, try and prove this using the formula\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image037.gif\" width=\"273\" height=\"23\" name=\"graphics23\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image039.gif\" width=\"163\" height=\"21\" name=\"graphics24\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"LEFT\">Let\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image041.gif\" width=\"76\" height=\"21\" name=\"graphics25\" align=\"TEXTTOP\" border=\"0\" \/>\u00a0and let\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image043.gif\" width=\"79\" height=\"21\" name=\"graphics26\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image045.gif\" width=\"261\" height=\"21\" name=\"graphics27\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image047.gif\" width=\"238\" height=\"21\" name=\"graphics28\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Using the addition formula given above, you get<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image049.gif\" width=\"117\" height=\"23\" name=\"graphics29\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>So we know that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p4_clip_image051.gif\" width=\"231\" height=\"21\" name=\"graphics30\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<h3>Review of New Vocabulary and Concepts<\/h3>\n<ul>\n<li>The <em><strong>amplitude\u00a0<\/strong><\/em>is the height of each peak in a wave pattern, measured from the\u00a0midline of the wave. Equivalently, it is half the vertical\u00a0distance from the trough of the wave to the peak of the wave.<\/li>\n<li>The <em><strong>frequency\u00a0<\/strong><\/em>is the number of wave patterns within a horizontal interval from\u00a00 to\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/6\/images\/s3_p5_clip_image002.gif\" width=\"24\" height=\"19\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<li>The <em><strong>period\u00a0<\/strong><\/em>is<span style=\"text-decoration: none;\"> defined as the horizontal\u00a0distance (on the <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-axis)\u00a0from a point on the <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-axis\u00a0to the next equal (in terms of the <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">-axis)\u00a0point on the <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">-axis\u00a0where the wave pattern begins to repeat itself. <\/span><\/li>\n<li>A <em><strong>phase shift<\/strong><\/em> is a horizontal\u00a0shift of a line representing a wave pattern to the left or right\u00a0on the graph.<\/li>\n<\/ul>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/pythagorean-trigonometric-identities\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/trigonometry\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/graphing-trigonometric-functions-and-inverse-trigonometric-functions\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Amplitude, Frequency, Period, and Phase Shift Objective In this lesson, you will learn the definitions of amplitude, frequency, period, and phase shift. You will also learn\u00a0how to locate amplitude, frequency, period, and phase shift on a graph. Previously Covered: Two important trigonometric ratios used to help solve for\u00a0the lengths [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-245","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/245","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=245"}],"version-history":[{"count":9,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/245\/revisions"}],"predecessor-version":[{"id":602,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/245\/revisions\/602"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=245"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}