{"id":981,"date":"2025-06-20T17:26:43","date_gmt":"2025-06-20T17:26:43","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/?post_type=chapter&#038;p=981"},"modified":"2025-09-10T17:42:52","modified_gmt":"2025-09-10T17:42:52","slug":"fundamentals-of-parametric-equations-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/fundamentals-of-parametric-equations-learn-it-1\/","title":{"raw":"Fundamentals of Parametric Equations: Learn It 1","rendered":"Fundamentals of Parametric Equations: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Create and sketch graphs of curves given their parametric equations<\/li>\r\n \t<li>Convert parametric equations into a regular y = f(x) form by eliminating the parameter<\/li>\r\n \t<li>Recognize and describe the curve called a cycloid<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Parametric Equations and Their Graphs<\/h2>\r\nTraditional functions like [latex]y = f(x)[\/latex] work well for many curves, but some curves can't be described this way. What if a curve loops back on itself or is vertical in places? This is where <strong>parametric equations<\/strong> become essential.\r\n\r\nIn parametric equations, both [latex]x[\/latex] and [latex]y[\/latex] are defined as functions of a third variable called a <strong>parameter<\/strong>. This parameter, often denoted as [latex]t[\/latex], acts as an independent variable that controls both coordinates simultaneously.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>parametric equations<\/h3>\r\n<p id=\"fs-id1169293185419\">If [latex]x[\/latex] and [latex]y[\/latex] are continuous functions of [latex]t[\/latex] on an interval [latex]I[\/latex], then the equations<\/p>\r\n\r\n<div id=\"fs-id1169295419559\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]x=x\\left(t\\right)\\text{and}y=y\\left(t\\right)[\/latex]<\/div>\r\n<p id=\"fs-id1169295804821\">are called<strong> parametric equations<\/strong> and [latex]t[\/latex] is called the <strong>parameter<\/strong>.\r\n[latex]\\\\[\/latex]\r\nThe set of points [latex](x,y)[\/latex] obtained as [latex]t[\/latex] varies over the interval [latex]I[\/latex] is called the graph of the parametric equations. The graph of parametric equations is called a <strong>parametric curve<\/strong> or <em data-effect=\"italics\">plane curve<\/em>, and is denoted by [latex]C[\/latex].<\/p>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">Consider Earth's orbit around the Sun. Earth's position changes continuously throughout the year, tracing an elliptical path. We can use the day of the year as our parameter [latex]t[\/latex].<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Day 1 = January 1<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Day 31 = January 31<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Day 59 = February 28<\/li>\r\n \t<li class=\"whitespace-normal break-words\">And so on...<\/li>\r\n<\/ul>\r\n[caption id=\"\" align=\"aligncenter\" width=\"458\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234603\/CNX_Calc_Figure_11_01_001.jpg\" alt=\"An ellipse with January 1 (t = 1) at the top, April 2 (t = 92) on the left, July 1 (t = 182) on the bottom, and October 1 (t = 274) on the right. The focal points of the ellipse have F2 on the left and the Sun on the right.\" width=\"458\" height=\"346\" data-media-type=\"image\/jpeg\" \/> Figure 1. Earth\u2019s orbit around the Sun during one year. The point labeled [latex]{F}_{2}[\/latex] is one of the foci of the ellipse; the other focus is occupied by the Sun.[\/caption]\r\n<p class=\"whitespace-normal break-words\">As [latex]t[\/latex] (the day) increases from 1 to 365, Earth's position [latex](x(t), y(t))[\/latex] traces out its complete orbital path. After 365 days, we return to the starting position and begin a new cycle.<\/p>\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">Think of the parameter [latex]t[\/latex] as time on a stopwatch. As time progresses, both the [latex]x[\/latex] and [latex]y[\/latex] coordinates change according to their respective functions, creating a path through the coordinate plane.<\/section>\r\n<p class=\"whitespace-normal break-words\">It's crucial to understand that [latex]x[\/latex] and [latex]y[\/latex] serve dual roles in parametric equations:<\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\"><strong>As functions<\/strong>: [latex]x(t)[\/latex] and [latex]y(t)[\/latex] are functions of the parameter [latex]t[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\"><strong>As variables<\/strong>: [latex]x[\/latex] and [latex]y[\/latex] represent the coordinates of points on the curve<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-normal break-words\">When [latex]t[\/latex] varies over an interval, the functions [latex]x(t)[\/latex] and [latex]y(t)[\/latex] generate ordered pairs [latex](x,y)[\/latex] that form the parametric curve.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<div id=\"fs-id1169295520035\" data-type=\"problem\">\r\n<p id=\"fs-id1169295605391\">Sketch the curves described by the following parametric equations:<\/p>\r\n\r\n<ol id=\"fs-id1169295858133\" type=\"a\">\r\n \t<li>[latex]x\\left(t\\right)=t - 1,y\\left(t\\right)=2t+4,-3\\le t\\le 2[\/latex]<\/li>\r\n \t<li>[latex]x\\left(t\\right)={t}^{2}-3,y\\left(t\\right)=2t+1,-2\\le t\\le 3[\/latex]<\/li>\r\n \t<li>[latex]x\\left(t\\right)=4\\cos{t},y\\left(t\\right)=4\\sin{t},0\\le t\\le 2\\pi [\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n[reveal-answer q=\"44558899\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558899\"]\r\n<div id=\"fs-id1169295754533\" data-type=\"solution\">\r\n<ol id=\"fs-id1169295439556\" type=\"a\">\r\n \t<li>To create a graph of this curve, first set up a table of values. Since the independent variable in both [latex]x\\left(t\\right)[\/latex] and [latex]y\\left(t\\right)[\/latex] is <em data-effect=\"italics\">t<\/em>, let <em data-effect=\"italics\">t<\/em> appear in the first column. Then [latex]x\\left(t\\right)[\/latex] and [latex]y\\left(t\\right)[\/latex] will appear in the second and third columns of the table.<span data-type=\"newline\">\r\n<\/span>\r\n<table id=\"fs-id1169295361650\" class=\"unnumbered\" summary=\"This table has three columns and seven rows. The first row is a header row, and it reads from left to right t, x(t), and y(t). Below the header row, in the first column, the values read \u22123, \u22122, \u22121, 0, 1, and 2. In the second column, the values read \u22124, \u22123, \u22122, \u22121, 0, and 1. In the third column, the values read \u22122, 0, 2, 4, 6, and 8.\" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th data-valign=\"top\" data-align=\"center\"><em data-effect=\"italics\">t<\/em><\/th>\r\n<th data-valign=\"top\" data-align=\"center\">[latex]x\\left(t\\right)[\/latex]<\/th>\r\n<th data-valign=\"top\" data-align=\"center\">[latex]y\\left(t\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">\u22123<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">\u22124<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">\u22122<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">\u22122<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">\u22123<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">\u22121<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">\u22122<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">2<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">\u22121<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">4<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">1<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">6<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">2<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">1<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<span data-type=\"newline\">\r\n<\/span>\r\nThe second and third columns in this table provide a set of points to be plotted. The graph of these points appears in Figure 3. The arrows on the graph indicate the <span data-type=\"term\">orientation<\/span> of the graph, that is, the direction that a point moves on the graph as <em data-effect=\"italics\">t<\/em> varies from \u22123 to 2.<span data-type=\"newline\">\r\n<\/span>\r\n<figure id=\"CNX_Calc_Figure_11_01_003\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234610\/CNX_Calc_Figure_11_01_003.jpg\" alt=\"A straight line going from (\u22124, \u22122) through (\u22123, 0), (\u22122, 2), and (0, 6) to (1, 8) with arrow pointed up and to the right. The point (\u22124, \u22122) is marked t = \u22123, the point (\u22122, 2) is marked t = \u22121, and the point (1, 8) is marked t = 2. On the graph there are also written three equations: x(t) = t \u22121, y(t) = 2t + 4, and \u22123 \u2264 t \u2264 2.\" width=\"488\" height=\"497\" data-media-type=\"image\/jpeg\" \/> Figure 3. Graph of the plane curve described by the parametric equations in part a.[\/caption]<\/figure>\r\n<\/li>\r\n \t<li>To create a graph of this curve, again set up a table of values.<span data-type=\"newline\">\r\n<\/span>\r\n<table id=\"fs-id1169295730684\" class=\"unnumbered\" summary=\"This table has three columns and seven rows. The first row is a header row, and it reads from left to right t, x(t), and y(t). Below the header row, in the first column, the values read \u22122, \u22121, 0, 1, 2, and 3. In the second column, the values read 1, \u22122, \u22122, \u22122, 1, and 6. In the third column, the values read \u22123, \u22121, 1, 3, 5, and 7.\" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th data-valign=\"top\" data-align=\"center\"><em data-effect=\"italics\">t<\/em><\/th>\r\n<th data-valign=\"top\" data-align=\"center\">[latex]x\\left(t\\right)[\/latex]<\/th>\r\n<th data-valign=\"top\" data-align=\"center\">[latex]y\\left(t\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">\u22122<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">1<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">\u22123<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">\u22121<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">\u22122<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">\u22121<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">\u22123<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">1<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">1<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">\u22122<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">3<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">2<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">1<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">5<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">3<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">6<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">7<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<span data-type=\"newline\">\r\n<\/span>\r\nThe second and third columns in this table give a set of points to be plotted (Figure 4). The first point on the graph (corresponding to [latex]t=-2[\/latex]) has coordinates [latex]\\left(1,-3\\right)[\/latex], and the last point (corresponding to [latex]t=3[\/latex]) has coordinates [latex]\\left(6,7\\right)[\/latex]. As [latex]t[\/latex] progresses from \u22122 to 3, the point on the curve travels along a parabola. The direction the point moves is again called the orientation and is indicated on the graph.<span data-type=\"newline\">\r\n<\/span>\r\n<figure id=\"CNX_Calc_Figure_11_01_004\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234613\/CNX_Calc_Figure_11_01_004.jpg\" alt=\"A curved line going from (1, \u22123) through (\u22123, 1) to (6, 7) with arrow pointing in that order. The point (1, \u22123) is marked t = \u22122, the point (\u22123, 1) is marked t = 0, and the point (6, 7) is marked t = 3. On the graph there are also written three equations: x(t) = t2 \u2212 3, y(t) = 2t + 1, and \u22122 \u2264 t \u2264 3.\" width=\"488\" height=\"497\" data-media-type=\"image\/jpeg\" \/> Figure 4. Graph of the plane curve described by the parametric equations in part b.[\/caption]<\/figure>\r\n<\/li>\r\n \t<li>In this case, use multiples of [latex]\\frac{\\pi}{6}[\/latex] for <em data-effect=\"italics\">t<\/em> and create another table of values:<span data-type=\"newline\">\r\n<\/span>\r\n<table id=\"fs-id1169295591562\" class=\"unnumbered\" summary=\"This table has three columns and 14 rows. The first row is a header row, and it reads from left to right t, x(t), and y(t). Below the header row, in the first column, the values read 0, \u03c0\/6, \u03c0\/3, \u03c0\/2, 2\u03c0\/3, 5\u03c0\/6, \u03c0, 7\u03c0\/6, 4\u03c0\/3, 3\u03c0\/2, 5\u03c0\/3, 11\u03c0\/6, and 2\u03c0. In the second column, the values read 2 times the square root of 3, which is approximately equal to 3.5, 2, 0, \u22122, \u22122 times the square root of 3, which is approximately equal to \u22123.5, \u22124, \u22122 times the square root of 3, which is approximately equal to \u22123.5, \u22122, 0, 2, 2 times the square root of 3, which is approximately equal to 3.5, and 5. In the third column, the values read 0, 2, 2 times the square root of 3, which is approximately equal to 3.5, 4, 2 times the square root of 3, which is approximately equal to 3.5, 2, 0, 2, \u22122 times the square root of 3, which is approximately equal to \u22123.5, \u22124, \u22122 times the square root of 3, which is approximately equal to \u22123.5, 2, and 0.\" data-label=\"\">\r\n<thead>\r\n<tr valign=\"top\">\r\n<th data-valign=\"top\" data-align=\"center\"><em data-effect=\"italics\">t<\/em><\/th>\r\n<th data-valign=\"top\" data-align=\"center\">[latex]x\\left(t\\right)[\/latex]<\/th>\r\n<th data-valign=\"top\" data-align=\"center\">[latex]y\\left(t\\right)[\/latex]<\/th>\r\n<th data-valign=\"top\" data-align=\"center\"><\/th>\r\n<th data-valign=\"top\" data-align=\"center\"><em data-effect=\"italics\">t<\/em><\/th>\r\n<th data-valign=\"top\" data-align=\"center\">[latex]x\\left(t\\right)[\/latex]<\/th>\r\n<th data-valign=\"top\" data-align=\"center\">[latex]y\\left(t\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">4<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\r\n<td rowspan=\"7\" data-valign=\"top\" data-align=\"center\"><\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{7\\pi }{6}[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]-2\\sqrt{3}\\approx -3.5[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">2<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{\\pi }{6}[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]2\\sqrt{3}\\approx 3.5[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]2[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{4\\pi }{3}[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">\u22122<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]-2\\sqrt{3}\\approx -3.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{\\pi }{3}[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]2[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]2\\sqrt{3}\\approx 3.5[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{3\\pi }{2}[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">\u22124<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{\\pi }{2}[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">4<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{5\\pi }{3}[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">2<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]-2\\sqrt{3}\\approx -3.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{2\\pi }{3}[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">\u22122<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]2\\sqrt{3}\\approx 3.5[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{11\\pi }{6}[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]2\\sqrt{3}\\approx 3.5[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">2<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{5\\pi }{6}[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]-2\\sqrt{3}\\approx -3.5[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">2<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]2\\pi [\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">4<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">[latex]\\pi [\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">\u22124<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<td data-valign=\"top\" data-align=\"left\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<span data-type=\"newline\">\r\n<\/span>\r\nThe graph of this plane curve appears in the following graph.<span data-type=\"newline\">\r\n<\/span>\r\n<figure id=\"CNX_Calc_Figure_11_01_005\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"417\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234617\/CNX_Calc_Figure_11_01_005.jpg\" alt=\"A circle with radius 4 centered at the origin is graphed with arrow going counterclockwise. The point (4, 0) is marked t = 0, the point (0, 4) is marked t = \u03c0\/2, the point (\u22124, 0) is marked t = \u03c0, and the point (0, \u22124) is marked t = 3\u03c0\/2. On the graph there are also written three equations: x(t) = 4 cos(t), y(t) = 4 sin(t), and 0 \u2264 t \u2264 2\u03c0.\" width=\"417\" height=\"423\" data-media-type=\"image\/jpeg\" \/> Figure 5. Graph of the plane curve described by the parametric equations in part c.[\/caption]<\/figure>\r\nThis is the graph of a circle with radius 4 centered at the origin, with a counterclockwise orientation. The starting point and ending points of the curve both have coordinates [latex]\\left(4,0\\right)[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the example above.<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ZEIm-ZV-8Lw?controls=0&amp;start=44&amp;end=485&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/center>\r\n<p class=\"p1\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\nYou can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/7.1ParametricEquations44to485_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"7.1 Parametric Equations\" here (opens in new window)<\/a>.\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]310277[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Create and sketch graphs of curves given their parametric equations<\/li>\n<li>Convert parametric equations into a regular y = f(x) form by eliminating the parameter<\/li>\n<li>Recognize and describe the curve called a cycloid<\/li>\n<\/ul>\n<\/section>\n<h2>Parametric Equations and Their Graphs<\/h2>\n<p>Traditional functions like [latex]y = f(x)[\/latex] work well for many curves, but some curves can&#8217;t be described this way. What if a curve loops back on itself or is vertical in places? This is where <strong>parametric equations<\/strong> become essential.<\/p>\n<p>In parametric equations, both [latex]x[\/latex] and [latex]y[\/latex] are defined as functions of a third variable called a <strong>parameter<\/strong>. This parameter, often denoted as [latex]t[\/latex], acts as an independent variable that controls both coordinates simultaneously.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>parametric equations<\/h3>\n<p id=\"fs-id1169293185419\">If [latex]x[\/latex] and [latex]y[\/latex] are continuous functions of [latex]t[\/latex] on an interval [latex]I[\/latex], then the equations<\/p>\n<div id=\"fs-id1169295419559\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]x=x\\left(t\\right)\\text{and}y=y\\left(t\\right)[\/latex]<\/div>\n<p id=\"fs-id1169295804821\">are called<strong> parametric equations<\/strong> and [latex]t[\/latex] is called the <strong>parameter<\/strong>.<br \/>\n[latex]\\\\[\/latex]<br \/>\nThe set of points [latex](x,y)[\/latex] obtained as [latex]t[\/latex] varies over the interval [latex]I[\/latex] is called the graph of the parametric equations. The graph of parametric equations is called a <strong>parametric curve<\/strong> or <em data-effect=\"italics\">plane curve<\/em>, and is denoted by [latex]C[\/latex].<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">Consider Earth&#8217;s orbit around the Sun. Earth&#8217;s position changes continuously throughout the year, tracing an elliptical path. We can use the day of the year as our parameter [latex]t[\/latex].<\/p>\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Day 1 = January 1<\/li>\n<li class=\"whitespace-normal break-words\">Day 31 = January 31<\/li>\n<li class=\"whitespace-normal break-words\">Day 59 = February 28<\/li>\n<li class=\"whitespace-normal break-words\">And so on&#8230;<\/li>\n<\/ul>\n<figure style=\"width: 458px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234603\/CNX_Calc_Figure_11_01_001.jpg\" alt=\"An ellipse with January 1 (t = 1) at the top, April 2 (t = 92) on the left, July 1 (t = 182) on the bottom, and October 1 (t = 274) on the right. The focal points of the ellipse have F2 on the left and the Sun on the right.\" width=\"458\" height=\"346\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 1. Earth\u2019s orbit around the Sun during one year. The point labeled [latex]{F}_{2}[\/latex] is one of the foci of the ellipse; the other focus is occupied by the Sun.<\/figcaption><\/figure>\n<p class=\"whitespace-normal break-words\">As [latex]t[\/latex] (the day) increases from 1 to 365, Earth&#8217;s position [latex](x(t), y(t))[\/latex] traces out its complete orbital path. After 365 days, we return to the starting position and begin a new cycle.<\/p>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">Think of the parameter [latex]t[\/latex] as time on a stopwatch. As time progresses, both the [latex]x[\/latex] and [latex]y[\/latex] coordinates change according to their respective functions, creating a path through the coordinate plane.<\/section>\n<p class=\"whitespace-normal break-words\">It&#8217;s crucial to understand that [latex]x[\/latex] and [latex]y[\/latex] serve dual roles in parametric equations:<\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\"><strong>As functions<\/strong>: [latex]x(t)[\/latex] and [latex]y(t)[\/latex] are functions of the parameter [latex]t[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\"><strong>As variables<\/strong>: [latex]x[\/latex] and [latex]y[\/latex] represent the coordinates of points on the curve<\/li>\n<\/ol>\n<p class=\"whitespace-normal break-words\">When [latex]t[\/latex] varies over an interval, the functions [latex]x(t)[\/latex] and [latex]y(t)[\/latex] generate ordered pairs [latex](x,y)[\/latex] that form the parametric curve.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<div id=\"fs-id1169295520035\" data-type=\"problem\">\n<p id=\"fs-id1169295605391\">Sketch the curves described by the following parametric equations:<\/p>\n<ol id=\"fs-id1169295858133\" type=\"a\">\n<li>[latex]x\\left(t\\right)=t - 1,y\\left(t\\right)=2t+4,-3\\le t\\le 2[\/latex]<\/li>\n<li>[latex]x\\left(t\\right)={t}^{2}-3,y\\left(t\\right)=2t+1,-2\\le t\\le 3[\/latex]<\/li>\n<li>[latex]x\\left(t\\right)=4\\cos{t},y\\left(t\\right)=4\\sin{t},0\\le t\\le 2\\pi[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558899\">Show Solution<\/button><\/p>\n<div id=\"q44558899\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1169295754533\" data-type=\"solution\">\n<ol id=\"fs-id1169295439556\" type=\"a\">\n<li>To create a graph of this curve, first set up a table of values. Since the independent variable in both [latex]x\\left(t\\right)[\/latex] and [latex]y\\left(t\\right)[\/latex] is <em data-effect=\"italics\">t<\/em>, let <em data-effect=\"italics\">t<\/em> appear in the first column. Then [latex]x\\left(t\\right)[\/latex] and [latex]y\\left(t\\right)[\/latex] will appear in the second and third columns of the table.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<table id=\"fs-id1169295361650\" class=\"unnumbered\" summary=\"This table has three columns and seven rows. The first row is a header row, and it reads from left to right t, x(t), and y(t). Below the header row, in the first column, the values read \u22123, \u22122, \u22121, 0, 1, and 2. In the second column, the values read \u22124, \u22123, \u22122, \u22121, 0, and 1. In the third column, the values read \u22122, 0, 2, 4, 6, and 8.\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"center\"><em data-effect=\"italics\">t<\/em><\/th>\n<th data-valign=\"top\" data-align=\"center\">[latex]x\\left(t\\right)[\/latex]<\/th>\n<th data-valign=\"top\" data-align=\"center\">[latex]y\\left(t\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">\u22123<\/td>\n<td data-valign=\"top\" data-align=\"center\">\u22124<\/td>\n<td data-valign=\"top\" data-align=\"center\">\u22122<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">\u22122<\/td>\n<td data-valign=\"top\" data-align=\"center\">\u22123<\/td>\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">\u22121<\/td>\n<td data-valign=\"top\" data-align=\"center\">\u22122<\/td>\n<td data-valign=\"top\" data-align=\"center\">2<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\n<td data-valign=\"top\" data-align=\"center\">\u22121<\/td>\n<td data-valign=\"top\" data-align=\"center\">4<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">1<\/td>\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\n<td data-valign=\"top\" data-align=\"center\">6<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">2<\/td>\n<td data-valign=\"top\" data-align=\"center\">1<\/td>\n<td data-valign=\"top\" data-align=\"center\">8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span data-type=\"newline\"><br \/>\n<\/span><br \/>\nThe second and third columns in this table provide a set of points to be plotted. The graph of these points appears in Figure 3. The arrows on the graph indicate the <span data-type=\"term\">orientation<\/span> of the graph, that is, the direction that a point moves on the graph as <em data-effect=\"italics\">t<\/em> varies from \u22123 to 2.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<figure id=\"CNX_Calc_Figure_11_01_003\">\n<figure style=\"width: 488px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234610\/CNX_Calc_Figure_11_01_003.jpg\" alt=\"A straight line going from (\u22124, \u22122) through (\u22123, 0), (\u22122, 2), and (0, 6) to (1, 8) with arrow pointed up and to the right. The point (\u22124, \u22122) is marked t = \u22123, the point (\u22122, 2) is marked t = \u22121, and the point (1, 8) is marked t = 2. On the graph there are also written three equations: x(t) = t \u22121, y(t) = 2t + 4, and \u22123 \u2264 t \u2264 2.\" width=\"488\" height=\"497\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 3. Graph of the plane curve described by the parametric equations in part a.<\/figcaption><\/figure>\n<\/figure>\n<\/li>\n<li>To create a graph of this curve, again set up a table of values.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<table id=\"fs-id1169295730684\" class=\"unnumbered\" summary=\"This table has three columns and seven rows. The first row is a header row, and it reads from left to right t, x(t), and y(t). Below the header row, in the first column, the values read \u22122, \u22121, 0, 1, 2, and 3. In the second column, the values read 1, \u22122, \u22122, \u22122, 1, and 6. In the third column, the values read \u22123, \u22121, 1, 3, 5, and 7.\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"center\"><em data-effect=\"italics\">t<\/em><\/th>\n<th data-valign=\"top\" data-align=\"center\">[latex]x\\left(t\\right)[\/latex]<\/th>\n<th data-valign=\"top\" data-align=\"center\">[latex]y\\left(t\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">\u22122<\/td>\n<td data-valign=\"top\" data-align=\"center\">1<\/td>\n<td data-valign=\"top\" data-align=\"center\">\u22123<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">\u22121<\/td>\n<td data-valign=\"top\" data-align=\"center\">\u22122<\/td>\n<td data-valign=\"top\" data-align=\"center\">\u22121<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\n<td data-valign=\"top\" data-align=\"center\">\u22123<\/td>\n<td data-valign=\"top\" data-align=\"center\">1<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">1<\/td>\n<td data-valign=\"top\" data-align=\"center\">\u22122<\/td>\n<td data-valign=\"top\" data-align=\"center\">3<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">2<\/td>\n<td data-valign=\"top\" data-align=\"center\">1<\/td>\n<td data-valign=\"top\" data-align=\"center\">5<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">3<\/td>\n<td data-valign=\"top\" data-align=\"center\">6<\/td>\n<td data-valign=\"top\" data-align=\"center\">7<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span data-type=\"newline\"><br \/>\n<\/span><br \/>\nThe second and third columns in this table give a set of points to be plotted (Figure 4). The first point on the graph (corresponding to [latex]t=-2[\/latex]) has coordinates [latex]\\left(1,-3\\right)[\/latex], and the last point (corresponding to [latex]t=3[\/latex]) has coordinates [latex]\\left(6,7\\right)[\/latex]. As [latex]t[\/latex] progresses from \u22122 to 3, the point on the curve travels along a parabola. The direction the point moves is again called the orientation and is indicated on the graph.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<figure id=\"CNX_Calc_Figure_11_01_004\">\n<figure style=\"width: 488px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234613\/CNX_Calc_Figure_11_01_004.jpg\" alt=\"A curved line going from (1, \u22123) through (\u22123, 1) to (6, 7) with arrow pointing in that order. The point (1, \u22123) is marked t = \u22122, the point (\u22123, 1) is marked t = 0, and the point (6, 7) is marked t = 3. On the graph there are also written three equations: x(t) = t2 \u2212 3, y(t) = 2t + 1, and \u22122 \u2264 t \u2264 3.\" width=\"488\" height=\"497\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 4. Graph of the plane curve described by the parametric equations in part b.<\/figcaption><\/figure>\n<\/figure>\n<\/li>\n<li>In this case, use multiples of [latex]\\frac{\\pi}{6}[\/latex] for <em data-effect=\"italics\">t<\/em> and create another table of values:<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<table id=\"fs-id1169295591562\" class=\"unnumbered\" summary=\"This table has three columns and 14 rows. The first row is a header row, and it reads from left to right t, x(t), and y(t). Below the header row, in the first column, the values read 0, \u03c0\/6, \u03c0\/3, \u03c0\/2, 2\u03c0\/3, 5\u03c0\/6, \u03c0, 7\u03c0\/6, 4\u03c0\/3, 3\u03c0\/2, 5\u03c0\/3, 11\u03c0\/6, and 2\u03c0. In the second column, the values read 2 times the square root of 3, which is approximately equal to 3.5, 2, 0, \u22122, \u22122 times the square root of 3, which is approximately equal to \u22123.5, \u22124, \u22122 times the square root of 3, which is approximately equal to \u22123.5, \u22122, 0, 2, 2 times the square root of 3, which is approximately equal to 3.5, and 5. In the third column, the values read 0, 2, 2 times the square root of 3, which is approximately equal to 3.5, 4, 2 times the square root of 3, which is approximately equal to 3.5, 2, 0, 2, \u22122 times the square root of 3, which is approximately equal to \u22123.5, \u22124, \u22122 times the square root of 3, which is approximately equal to \u22123.5, 2, and 0.\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"top\" data-align=\"center\"><em data-effect=\"italics\">t<\/em><\/th>\n<th data-valign=\"top\" data-align=\"center\">[latex]x\\left(t\\right)[\/latex]<\/th>\n<th data-valign=\"top\" data-align=\"center\">[latex]y\\left(t\\right)[\/latex]<\/th>\n<th data-valign=\"top\" data-align=\"center\"><\/th>\n<th data-valign=\"top\" data-align=\"center\"><em data-effect=\"italics\">t<\/em><\/th>\n<th data-valign=\"top\" data-align=\"center\">[latex]x\\left(t\\right)[\/latex]<\/th>\n<th data-valign=\"top\" data-align=\"center\">[latex]y\\left(t\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\n<td data-valign=\"top\" data-align=\"center\">4<\/td>\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\n<td rowspan=\"7\" data-valign=\"top\" data-align=\"center\"><\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{7\\pi }{6}[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]-2\\sqrt{3}\\approx -3.5[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">2<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{\\pi }{6}[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]2\\sqrt{3}\\approx 3.5[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]2[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{4\\pi }{3}[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">\u22122<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]-2\\sqrt{3}\\approx -3.5[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{\\pi }{3}[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]2[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]2\\sqrt{3}\\approx 3.5[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{3\\pi }{2}[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\n<td data-valign=\"top\" data-align=\"center\">\u22124<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{\\pi }{2}[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\n<td data-valign=\"top\" data-align=\"center\">4<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{5\\pi }{3}[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">2<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]-2\\sqrt{3}\\approx -3.5[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{2\\pi }{3}[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">\u22122<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]2\\sqrt{3}\\approx 3.5[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{11\\pi }{6}[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]2\\sqrt{3}\\approx 3.5[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">2<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{5\\pi }{6}[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]-2\\sqrt{3}\\approx -3.5[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">2<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]2\\pi[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">4<\/td>\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">[latex]\\pi[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">\u22124<\/td>\n<td data-valign=\"top\" data-align=\"center\">0<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><span data-type=\"newline\"><br \/>\n<\/span><br \/>\nThe graph of this plane curve appears in the following graph.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<figure id=\"CNX_Calc_Figure_11_01_005\">\n<figure style=\"width: 417px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234617\/CNX_Calc_Figure_11_01_005.jpg\" alt=\"A circle with radius 4 centered at the origin is graphed with arrow going counterclockwise. The point (4, 0) is marked t = 0, the point (0, 4) is marked t = \u03c0\/2, the point (\u22124, 0) is marked t = \u03c0, and the point (0, \u22124) is marked t = 3\u03c0\/2. On the graph there are also written three equations: x(t) = 4 cos(t), y(t) = 4 sin(t), and 0 \u2264 t \u2264 2\u03c0.\" width=\"417\" height=\"423\" data-media-type=\"image\/jpeg\" \/><figcaption class=\"wp-caption-text\">Figure 5. Graph of the plane curve described by the parametric equations in part c.<\/figcaption><\/figure>\n<\/figure>\n<p>This is the graph of a circle with radius 4 centered at the origin, with a counterclockwise orientation. The starting point and ending points of the curve both have coordinates [latex]\\left(4,0\\right)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">Watch the following video to see the worked solution to the example above.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ZEIm-ZV-8Lw?controls=0&amp;start=44&amp;end=485&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/div>\n<p class=\"p1\">For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus+II\/Transcripts\/7.1ParametricEquations44to485_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;7.1 Parametric Equations&#8221; here (opens in new window)<\/a>.<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm310277\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=310277&theme=lumen&iframe_resize_id=ohm310277&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":5,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":675,"module-header":"- Select Header -","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/981"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/981\/revisions"}],"predecessor-version":[{"id":2298,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/981\/revisions\/2298"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/675"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/981\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=981"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=981"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=981"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=981"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}