{"id":1153,"date":"2021-11-11T17:37:24","date_gmt":"2021-11-11T17:37:24","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/graphing-and-representing-parametric-equations\/"},"modified":"2022-10-20T23:15:57","modified_gmt":"2022-10-20T23:15:57","slug":"graphing-and-representing-parametric-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/graphing-and-representing-parametric-equations\/","title":{"raw":"Graphing and Representing Parametric Equations","rendered":"Graphing and Representing Parametric Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\" data-type=\"abstract\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Plot a curve described by parametric equations<\/li>\r\n \t<li>Convert the parametric equations of a curve into the form [latex]y=f\\left(x\\right)[\/latex]<\/li>\r\n \t<li>Recognize the parametric equations of basic curves, such as a line and a circle<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Parametric Equations and Their Graphs<\/h2>\r\n<section id=\"fs-id1169295516903\" data-depth=\"1\">\r\n<p id=\"fs-id1169295553685\">Consider the orbit of Earth around the Sun. Our year lasts approximately [latex]365.25[\/latex] days, but for this discussion we will use [latex]365[\/latex] days. On January 1 of each year, the physical location of Earth with respect to the Sun is nearly the same, except for leap years, when the lag introduced by the extra [latex]\\frac{1}{4}[\/latex] day of orbiting time is built into the calendar. We call January 1 \"day [latex]1[\/latex]\" of the year. Then, for example, day [latex]31[\/latex] is January 31, day 59 is February 28, and so on.<\/p>\r\n<p id=\"fs-id1169293250149\">The number of the day in a year can be considered a variable that determines Earth\u2019s position in its orbit. As Earth revolves around the Sun, its physical location changes relative to the Sun. After one full year, we are back where we started, and a new year begins. According to Kepler\u2019s laws of planetary motion, the shape of the orbit is elliptical, with the Sun at one focus of the ellipse. We study this idea in more detail in Conic Sections.<\/p>\r\n\r\n<figure id=\"CNX_Calc_Figure_11_01_001\"><figcaption><\/figcaption>\r\n\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 in one year.[\/caption]<\/figure>\r\n<p id=\"fs-id1169293338092\">Figure 1 depicts <span class=\"no-emphasis\" data-type=\"term\">Earth\u2019s orbit<\/span> 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. If we superimpose coordinate axes over this graph, then we can assign ordered pairs to each point on the ellipse (Figure 2). Then each\u00a0[latex]x[\/latex] value on the graph is a value of position as a function of time, and each [latex]y[\/latex] value is also a value of position as a function of time. Therefore, each point on the graph corresponds to a value of Earth\u2019s position as a function of time.<\/p>\r\n\r\n<figure id=\"CNX_Calc_Figure_11_01_002\"><figcaption><\/figcaption>[caption id=\"\" align=\"aligncenter\" width=\"492\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234606\/CNX_Calc_Figure_11_01_002.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. There is a line going from [latex]t[\/latex] = 1 to [latex]t[\/latex] = 182. There is also a line going from [latex]t[\/latex] = 92 to [latex]t[\/latex] = 274 that passes through F2 and the Sun. On the upper left side, there is a point marked (x(t), y(t)) with a tangent line pointing down and to the left.\" width=\"492\" height=\"422\" data-media-type=\"image\/jpeg\" \/> Figure 2. Coordinate axes superimposed on the orbit of Earth.[\/caption]<\/figure>\r\n<p id=\"fs-id1169293137188\">We can determine the functions for [latex]x\\left(t\\right)[\/latex] and [latex]y\\left(t\\right)[\/latex], thereby parameterizing the orbit of Earth around the Sun. The variable [latex]t[\/latex] is called an independent parameter and, in this context, represents time relative to the beginning of each year.<\/p>\r\n<p id=\"fs-id1169295637245\">A curve in the [latex]\\left(x,y\\right)[\/latex] plane can be represented parametrically. The equations that are used to define the curve are called <span data-type=\"term\">parametric equations<\/span>.<\/p>\r\n\r\n<div id=\"fs-id1169295820852\" data-type=\"note\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\r\n\r\n<hr \/>\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&nbsp;\r\n<p id=\"fs-id1169295804821\">are called parametric equations and [latex]t[\/latex] is called the <strong>parameter<\/strong>. The set of points [latex]\\left(x,y\\right)[\/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<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1169295465331\">Notice in this definition that [latex]x[\/latex]\u00a0and [latex]y[\/latex] are used in two ways. The first is as functions of the independent variable\u00a0[latex]t[\/latex]. As [latex]t[\/latex] varies over the interval\u00a0[latex]I[\/latex], the functions [latex]x\\left(t\\right)[\/latex] and [latex]y\\left(t\\right)[\/latex] generate a set of ordered pairs [latex]\\left(x,y\\right)[\/latex]. This set of ordered pairs generates the graph of the parametric equations. In this second usage, to designate the ordered pairs, [latex]x[\/latex] and [latex]y[\/latex] are variables. It is important to distinguish the variables [latex]x[\/latex] and [latex]y[\/latex] from the functions [latex]x\\left(t\\right)[\/latex] and [latex]y\\left(t\\right)[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1169295362672\" data-type=\"example\">\r\n<div id=\"fs-id1169295870973\" data-type=\"exercise\">\r\n<div id=\"fs-id1169295520035\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing a Parametrically Defined Curve<\/h3>\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 [latex]t[\/latex], let [latex]t[\/latex] 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\" style=\"height: 84px;\" 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 style=\"height: 12px;\" valign=\"top\">\r\n<th style=\"height: 12px; width: 105.994px;\" data-valign=\"top\" data-align=\"center\">[latex]t[\/latex]<\/th>\r\n<th style=\"height: 12px; width: 192.131px;\" data-valign=\"top\" data-align=\"center\">[latex]x\\left(t\\right)[\/latex]<\/th>\r\n<th style=\"height: 12px; width: 192.145px;\" 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 style=\"height: 12px;\" valign=\"top\">\r\n<td style=\"height: 12px; width: 105.994px;\" data-valign=\"top\" data-align=\"center\">[latex]-3[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 192.131px;\" data-valign=\"top\" data-align=\"center\">[latex]-4[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 192.145px;\" data-valign=\"top\" data-align=\"center\">[latex]-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\" valign=\"top\">\r\n<td style=\"height: 12px; width: 105.994px;\" data-valign=\"top\" data-align=\"center\">[latex]-2[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 192.131px;\" data-valign=\"top\" data-align=\"center\">[latex]-3[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 192.145px;\" data-valign=\"top\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\" valign=\"top\">\r\n<td style=\"height: 12px; width: 105.994px;\" data-valign=\"top\" data-align=\"center\">[latex]-1[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 192.131px;\" data-valign=\"top\" data-align=\"center\">[latex]-2[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 192.145px;\" data-valign=\"top\" data-align=\"center\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\" valign=\"top\">\r\n<td style=\"height: 12px; width: 105.994px;\" data-valign=\"top\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 192.131px;\" data-valign=\"top\" data-align=\"center\">[latex]-1[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 192.145px;\" data-valign=\"top\" data-align=\"center\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\" valign=\"top\">\r\n<td style=\"height: 12px; width: 105.994px;\" data-valign=\"top\" data-align=\"center\">[latex]1[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 192.131px;\" data-valign=\"top\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 192.145px;\" data-valign=\"top\" data-align=\"center\">[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\" valign=\"top\">\r\n<td style=\"height: 12px; width: 105.994px;\" data-valign=\"top\" data-align=\"center\">[latex]2[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 192.131px;\" data-valign=\"top\" data-align=\"center\">[latex]1[\/latex]<\/td>\r\n<td style=\"height: 12px; width: 192.145px;\" data-valign=\"top\" data-align=\"center\">[latex]8[\/latex]<\/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 [latex]t[\/latex] varies from\u00a0[latex]-3[\/latex] to [latex]2[\/latex] .<span data-type=\"newline\">\r\n<\/span>\r\n<figure id=\"CNX_Calc_Figure_11_01_003\">[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 [latex]t[\/latex] = \u22123, the point (\u22122, 2) is marked [latex]t[\/latex] = \u22121, and the point (1, 8) is marked [latex]t[\/latex] = 2. On the graph there are also written three equations: x(t) = [latex]t[\/latex] \u22121, y(t) = 2t + 4, and \u22123 \u2264 [latex]t[\/latex] \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\">[latex]t[\/latex]<\/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\">[latex]-2[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]1[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]-3[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">[latex]-1[\/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]-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">[latex]0[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]-3[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">[latex]1[\/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]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">[latex]2[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]1[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td data-valign=\"top\" data-align=\"center\">[latex]3[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]6[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]7[\/latex]<\/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\u00a0[latex]-2[\/latex] to[latex]3[\/latex], 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\">[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 [latex]t[\/latex] = \u22122, the point (\u22123, 1) is marked [latex]t[\/latex] = 0, and the point (6, 7) is marked [latex]t[\/latex] = 3. On the graph there are also written three equations: x(t) = t2 \u2212 3, y(t) = 2t + 1, and \u22122 \u2264 [latex]t[\/latex] \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 [latex]t[\/latex] 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\">[latex]t[\/latex]<\/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\">[latex]t[\/latex]<\/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\">[latex]0[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]4[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]0[\/latex]<\/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\">[latex]2[\/latex]<\/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\">[latex]-2[\/latex]<\/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\">[latex]0[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]-4[\/latex]<\/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\">[latex]0[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]4[\/latex]<\/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\">[latex]2[\/latex]<\/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\">[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{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\">[latex]2[\/latex]<\/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\">[latex]2[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]2\\pi [\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]4[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]0[\/latex]<\/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\">[latex]-4[\/latex]<\/td>\r\n<td data-valign=\"top\" data-align=\"center\">[latex]0[\/latex]<\/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\">[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 [latex]t[\/latex] = 0, the point (0, 4) is marked [latex]t[\/latex] = \u03c0\/2, the point (\u22124, 0) is marked [latex]t[\/latex] = \u03c0, and the point (0, \u22124) is marked [latex]t[\/latex] = 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 [latex]t[\/latex] \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\u00a0[latex]4[\/latex] 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<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Graphing a Parametrically Defined Curve.\r\n\r\n<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\"><\/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<div id=\"fs-id1169295454074\" class=\"checkpoint\" data-type=\"note\">\r\n<div id=\"fs-id1169293200600\" data-type=\"exercise\">\r\n<div id=\"fs-id1169295589408\" data-type=\"problem\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-id1169295589408\" data-type=\"problem\">\r\n<p id=\"fs-id1169295759487\">Sketch the curve described by the parametric equations<\/p>\r\n\r\n<div id=\"fs-id1169295431091\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]x\\left(t\\right)=3t+2,y\\left(t\\right)={t}^{2}-1,-3\\le t\\le 2[\/latex].<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558898\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558898\"]\r\n<div id=\"fs-id1169295557869\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1169295652237\">Make a table of values for [latex]x\\left(t\\right)[\/latex] and [latex]y\\left(t\\right)[\/latex] using [latex]t[\/latex] values from\u00a0[latex]-3[\/latex] to[latex]2[\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558897\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558897\"]\r\n<div id=\"fs-id1169295453647\" data-type=\"solution\">[caption id=\"\" align=\"aligncenter\" width=\"642\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234620\/CNX_Calc_Figure_11_01_006.jpg\" alt=\"A curved line going from (\u22127, 8) through (\u22121, 0) and (2, \u22121) to (8, 3) with arrow going in that order. The point (\u22127, 8) is marked [latex]t[\/latex] = \u22123, the point (2, \u22121) is marked [latex]t[\/latex] = 0, and the point (8, 3) is marked [latex]t[\/latex] = 2. On the graph there are also written three equations: x(t) = 3t + 2, y(t) = t2 \u2212 1, and \u22123 \u2264 [latex]t[\/latex] \u2264 2.\" width=\"642\" height=\"423\" data-media-type=\"image\/jpeg\" \/> Figure 6.[\/caption]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]173890[\/ohm_question]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1169295807788\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Eliminating the Parameter<\/h2>\r\n<p id=\"fs-id1169293341434\">To better understand the graph of a curve represented parametrically, it is useful to rewrite the two equations as a single equation relating the variables [latex]x[\/latex] and <em data-effect=\"italics\">y.<\/em> Then we can apply any previous knowledge of equations of curves in the plane to identify the curve. For example, the equations describing the plane curve in part (b) of the previous example are<\/p>\r\n\r\n<div id=\"fs-id1169295565020\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]x\\left(t\\right)={t}^{2}-3,y\\left(t\\right)=2t+1,-2\\le t\\le 3[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169295882380\">Solving the second equation for [latex]t[\/latex] gives<\/p>\r\n\r\n<div id=\"fs-id1169293137841\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\large{t=\\frac{y - 1}{2}}[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169293144942\">This can be substituted into the first equation:<\/p>\r\n\r\n<div id=\"fs-id1169293130361\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]x={\\left(\\frac{y - 1}{2}\\right)}^{2}-3=\\frac{{y}^{2}-2y+1}{4}-3=\\frac{{y}^{2}-2y - 11}{4}[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169295454203\">This equation describes [latex]x[\/latex] as a function of <em data-effect=\"italics\">y.<\/em> These steps give an example of <em data-effect=\"italics\">eliminating the parameter<\/em>. The graph of this function is a parabola opening to the right. Recall that the plane curve started at [latex]\\left(1,-3\\right)[\/latex] and ended at [latex]\\left(6,7\\right)[\/latex]. These terminations were due to the restriction on the parameter [latex]t[\/latex]<em data-effect=\"italics\">.<\/em><\/p>\r\nBefore working through an example on how to eliminate the parameter, it is useful to recall the Pythagorean Identity as well as the equations of circles.\r\n<div id=\"fs-id1169295694719\" data-type=\"example\">\r\n<div id=\"fs-id1169293393055\" data-type=\"exercise\">\r\n<div id=\"fs-id1169295362943\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox examples\">\r\n<h3>Recall: Pythagorean identity and equation of circles<\/h3>\r\n<ul>\r\n \t<li>For any angle [latex] t, \\sin^2 t + \\cos^2 t = 1 [\/latex]<\/li>\r\n \t<li>A circle of radius [latex] a [\/latex] centered at the origin is given by [latex] x^2 + y^2 = a^2 [\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Eliminating the Parameter<\/h3>\r\n<div id=\"fs-id1169295362943\" data-type=\"problem\">\r\n<p id=\"fs-id1169295422772\">Eliminate the parameter for each of the plane curves described by the following parametric equations and describe the resulting graph.<\/p>\r\n\r\n<ol id=\"fs-id1169293340647\" type=\"a\">\r\n \t<li>[latex]x\\left(t\\right)=\\sqrt{2t+4},y\\left(t\\right)=2t+1,-2\\le t\\le 6[\/latex]<\/li>\r\n \t<li>[latex]x\\left(t\\right)=4\\cos{t},y\\left(t\\right)=3\\sin{t},0\\le t\\le 2\\pi [\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n[reveal-answer q=\"44558895\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558895\"]\r\n<div id=\"fs-id1169293275836\" data-type=\"solution\">\r\n<ol id=\"fs-id1169293199708\" type=\"a\">\r\n \t<li>To eliminate the parameter, we can solve either of the equations for <em data-effect=\"italics\">t.<\/em> For example, solving the first equation for [latex]t[\/latex] gives<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169293390065\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{ccc}\\hfill x&amp; =\\hfill &amp; \\sqrt{2t+4}\\hfill \\\\ \\hfill {x}^{2}&amp; =\\hfill &amp; 2t+4\\hfill \\\\ \\hfill {x}^{2}-4&amp; =\\hfill &amp; 2t\\hfill \\\\ \\hfill t&amp; =\\hfill &amp; \\frac{{x}^{2}-4}{2}.\\hfill \\end{array}[\/latex]<\/div>\r\n<span data-type=\"newline\">\r\n<\/span>\r\nNote that when we square both sides it is important to observe that [latex]x\\ge 0[\/latex]. Substituting [latex]t=\\frac{{x}^{2}-4}{2}[\/latex] this into [latex]y\\left(t\\right)[\/latex] yields<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169293387312\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{ccc}\\hfill y\\left(t\\right)&amp; =\\hfill &amp; 2t+1\\hfill \\\\ \\hfill y&amp; =\\hfill &amp; 2\\left(\\frac{{x}^{2}-4}{2}\\right)+1\\hfill \\\\ \\hfill y&amp; =\\hfill &amp; {x}^{2}-4+1\\hfill \\\\ \\hfill y&amp; =\\hfill &amp; {x}^{2}-3.\\hfill \\end{array}[\/latex]<\/div>\r\n<span data-type=\"newline\">\r\n<\/span>\r\nThis is the equation of a parabola opening upward. There is, however, a domain restriction because of the limits on the parameter [latex]t[\/latex]. When [latex]t=-2[\/latex], [latex]x=\\sqrt{2\\left(-2\\right)+4}=0[\/latex], and when [latex]t=6[\/latex], [latex]x=\\sqrt{2\\left(6\\right)+4}=4[\/latex]. The graph of this plane curve follows.<span data-type=\"newline\">\r\n<\/span>\r\n<figure id=\"CNX_Calc_Figure_11_01_007\">[caption id=\"\" align=\"aligncenter\" width=\"267\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234623\/CNX_Calc_Figure_11_01_007.jpg\" alt=\"A curved line going from (\u22123, 0) through (2, 1) to (4, 13) with arrow going in that order. The point (\u22123, 0) is marked [latex]t[\/latex] = \u22122, the point (2, 1) is marked [latex]t[\/latex] = 0, the point (2 times the square root of 2, 5) is marked [latex]t[\/latex] = 2, the point (3 times the square root of 2, 9) is marked [latex]t[\/latex] = 4, and the point (4, 13) is marked [latex]t[\/latex] = 6. On the graph there are also written three equations: x(t) = square root of the quantity (2t + 4), y(t) = 2t + 1, and \u22122 \u2264 [latex]t[\/latex] \u2264 6.\" width=\"267\" height=\"385\" data-media-type=\"image\/jpeg\" \/> Figure 7. Graph of the plane curve described by the parametric equations in part a.[\/caption]<\/figure>\r\n<\/li>\r\n \t<li>Sometimes it is necessary to be a bit creative in eliminating the parameter. The parametric equations for this example are<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169295773361\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]x\\left(t\\right)=4\\cos{t}\\text{ and }y\\left(t\\right)=3\\sin{t}[\/latex].<\/div>\r\n<span data-type=\"newline\">\r\n<\/span>\r\nSolving either equation for [latex]t[\/latex] directly is not advisable because sine and cosine are not one-to-one functions. However, dividing the first equation by 4 and the second equation by 3 (and suppressing the [latex]t[\/latex]) gives us<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169295447894\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\cos{t}=\\frac{x}{4}\\text{ and }\\sin{t}=\\frac{y}{3}[\/latex].<\/div>\r\n<span data-type=\"newline\">\r\n<\/span>\r\nNow use the Pythagorean identity [latex]{\\cos}^{2}t+{\\sin}^{2}t=1[\/latex] and replace the expressions for [latex]\\sin{t}[\/latex] and [latex]\\cos{t}[\/latex] with the equivalent expressions in terms of [latex]x[\/latex] and [latex]y[\/latex]. This gives<span data-type=\"newline\">\r\n<\/span>\r\n<div id=\"fs-id1169295591977\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{ccc}\\hfill {\\left(\\frac{x}{4}\\right)}^{2}+{\\left(\\frac{y}{3}\\right)}^{2}&amp; =\\hfill &amp; 1\\hfill \\\\ \\hfill \\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{9}&amp; =\\hfill &amp; 1.\\hfill \\end{array}[\/latex]<\/div>\r\n<span data-type=\"newline\">\r\n<\/span>\r\nThis is the equation of a horizontal ellipse centered at the origin, with semimajor axis [latex]4[\/latex] and semiminor axis [latex]3[\/latex] as shown in the following graph.<span data-type=\"newline\">\r\n<\/span>\r\n<figure id=\"CNX_Calc_Figure_11_01_008\">[caption id=\"\" align=\"aligncenter\" width=\"417\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234626\/CNX_Calc_Figure_11_01_008.jpg\" alt=\"An ellipse with major axis horizontal and of length 8 and with minor radius vertical and of length 6 that is centered at the origin with arrow going counterclockwise. The point (4, 0) is marked [latex]t[\/latex] = 0, the point (0, 3) is marked [latex]t[\/latex] = \u03c0\/2, the point (\u22124, 0) is marked [latex]t[\/latex] = \u03c0, and the point (0, \u22123) is marked [latex]t[\/latex] = 3\u03c0\/2. On the graph there are also written three equations: x(t) = 4 cos(t), y(t) = 3 sin(t), and 0 \u2264 [latex]t[\/latex] \u2264 2\u03c0.\" width=\"417\" height=\"423\" data-media-type=\"image\/jpeg\" \/> Figure 8. Graph of the plane curve described by the parametric equations in part b.[\/caption]<\/figure>\r\nAs [latex]t[\/latex] progresses from [latex]0[\/latex] to [latex]2\\pi [\/latex], a point on the curve traverses the ellipse once, in a counterclockwise direction. Recall from the section opener that the orbit of Earth around the Sun is also elliptical. This is a perfect example of using parameterized curves to model a real-world phenomenon.<\/li>\r\n<\/ol>\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Eliminating the Parameter.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ZEIm-ZV-8Lw?controls=0&amp;start=495&amp;end=846&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/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\/7.1ParametricEquations495to846_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"7.1 Parametric Equations\" here (opens in new window)<\/a>.\r\n<div id=\"fs-id1169295362509\" class=\"checkpoint\" data-type=\"note\">\r\n<div id=\"fs-id1169295362512\" data-type=\"exercise\">\r\n<div id=\"fs-id1169295362514\" data-type=\"problem\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-id1169295362514\" data-type=\"problem\">\r\n<p id=\"fs-id1169295555110\">Eliminate the parameter for the plane curve defined by the following parametric equations and describe the resulting graph.<\/p>\r\n\r\n<div id=\"fs-id1169295555115\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]x\\left(t\\right)=2+\\frac{3}{t},y\\left(t\\right)=t - 1,2\\le t\\le 6[\/latex]<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558893\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558893\"]\r\n<div id=\"fs-id1169293185389\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1169293313963\">Solve one of the equations for [latex]t[\/latex] and substitute into the other equation.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558894\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558894\"]\r\n<div id=\"fs-id1169293397158\" data-type=\"solution\">\r\n<p id=\"fs-id1169293397160\">[latex]x=2+\\frac{3}{y+1}[\/latex], or [latex]y=-1+\\frac{3}{x - 2}[\/latex]. This equation describes a portion of a rectangular hyperbola centered at [latex]\\left(2,-1\\right)[\/latex]. <span data-type=\"newline\">\r\n<\/span><\/p>\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"330\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234629\/CNX_Calc_Figure_11_01_009.jpg\" alt=\"A curved line going from (3.5, 1) to (2.5, 5) with arrow going in that order. The point (3.5, 1) is marked [latex]t[\/latex] = 2 and the point (2.5, 5) is marked [latex]t[\/latex] = 6. On the graph there are also written three equations: x(t) = 2 + 3\/t, y(t) = [latex]t[\/latex] \u2212 1, and 2 \u2264 [latex]t[\/latex] \u2264 6.\" width=\"330\" height=\"310\" data-media-type=\"image\/jpeg\" \/> Figure 9.[\/caption]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]4733[\/ohm_question]\r\n\r\n<\/div>\r\n<p id=\"fs-id1169295460497\">So far we have seen the method of eliminating the parameter, assuming we know a set of parametric equations that describe a plane curve. What if we would like to start with the equation of a curve and determine a pair of parametric equations for that curve? This is certainly possible, and in fact it is possible to do so in many different ways for a given curve. The process is known as <span data-type=\"term\">parameterization of a curve<\/span>.<\/p>\r\n\r\n<div id=\"fs-id1169295460500\" data-type=\"example\">\r\n<div id=\"fs-id1169295398174\" data-type=\"exercise\">\r\n<div id=\"fs-id1169295398176\" data-type=\"problem\">\r\n<div data-type=\"title\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Parameterizing a Curve<\/h3>\r\n<div id=\"fs-id1169295398176\" data-type=\"problem\">\r\n<p id=\"fs-id1169293341360\">Find two different pairs of parametric equations to represent the graph of [latex]y=2{x}^{2}-3[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558892\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558892\"]\r\n<div id=\"fs-id1169295712606\" data-type=\"solution\">\r\n<p id=\"fs-id1169295712608\">First, it is always possible to parameterize a curve by defining [latex]x\\left(t\\right)=t[\/latex], then replacing [latex]x[\/latex] with [latex]t[\/latex] in the equation for [latex]y\\left(t\\right)[\/latex]. This gives the parameterization<\/p>\r\n\r\n<div id=\"fs-id1169295409716\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]x\\left(t\\right)=t,y\\left(t\\right)=2{t}^{2}-3[\/latex].<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169295554644\">Since there is no restriction on the domain in the original graph, there is no restriction on the values of <em data-effect=\"italics\">t.<\/em><\/p>\r\n<p id=\"fs-id1169295353796\">We have complete freedom in the choice for the second parameterization. For example, we can choose [latex]x\\left(t\\right)=3t - 2[\/latex]. The only thing we need to check is that there are no restrictions imposed on [latex]x[\/latex]; that is, the range of [latex]x\\left(t\\right)[\/latex] is all real numbers. This is the case for [latex]x\\left(t\\right)=3t - 2[\/latex]. Now since [latex]y=2{x}^{2}-3[\/latex], we can substitute [latex]x\\left(t\\right)=3t - 2[\/latex] for\u00a0[latex]x[\/latex]. This gives<\/p>\r\n\r\n<div id=\"fs-id1169295558068\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cc}\\hfill y\\left(t\\right)&amp; =2{\\left(3t - 2\\right)}^{2}-2\\hfill \\\\ &amp; =2\\left(9{t}^{2}-12t+4\\right)-2\\hfill \\\\ &amp; =18{t}^{2}-24t+8 - 2\\hfill \\\\ &amp; =18{t}^{2}-24t+6.\\hfill \\end{array}[\/latex]<\/div>\r\n&nbsp;\r\n<p id=\"fs-id1169293401741\">Therefore, a second parameterization of the curve can be written as<\/p>\r\n\r\n<div id=\"fs-id1169293401744\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]x\\left(t\\right)=3t - 2\\text{ and }y\\left(t\\right)=18{t}^{2}-24t+6[\/latex].<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nWatch the following video to see the worked solution to Example: Parameterizing a Curve.\r\n\r\n<center><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/ZEIm-ZV-8Lw?controls=0&amp;start=848&amp;end=944&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/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\/7.1ParametricEquations848to944_transcript.html\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"7.1 Parametric Equations\" here (opens in new window)<\/a>.\r\n<div id=\"fs-id1169295461539\" class=\"checkpoint\" data-type=\"note\">\r\n<div id=\"fs-id1169295461542\" data-type=\"exercise\">\r\n<div id=\"fs-id1169295461544\" data-type=\"problem\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n<div id=\"fs-id1169295461544\" data-type=\"problem\">\r\n<p id=\"fs-id1169293398615\">Find two different sets of parametric equations to represent the graph of [latex]y={x}^{2}+2x[\/latex].<\/p>\r\n\r\n<\/div>\r\n[reveal-answer q=\"44558890\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"44558890\"]\r\n<div id=\"fs-id1169293392950\" data-type=\"commentary\" data-element-type=\"hint\">\r\n<p id=\"fs-id1169293253923\">Follow the steps from the example. Remember we have freedom in choosing the parameterization for [latex]x\\left(t\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n[reveal-answer q=\"44558891\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"44558891\"]\r\n<div id=\"fs-id1169293390102\" data-type=\"solution\">\r\n<p id=\"fs-id1169293390104\">One possibility is [latex]x\\left(t\\right)=t,y\\left(t\\right)={t}^{2}+2t[\/latex]. Another possibility is [latex]x\\left(t\\right)=2t - 3,y\\left(t\\right)={\\left(2t - 3\\right)}^{2}+2\\left(2t - 3\\right)=4{t}^{2}-8t+3[\/latex].<\/p>\r\n<p id=\"fs-id1169295712634\">There are, in fact, an infinite number of possibilities.<\/p>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1169295349662\" data-depth=\"1\"><\/section>","rendered":"<div class=\"textbox learning-objectives\" data-type=\"abstract\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Plot a curve described by parametric equations<\/li>\n<li>Convert the parametric equations of a curve into the form [latex]y=f\\left(x\\right)[\/latex]<\/li>\n<li>Recognize the parametric equations of basic curves, such as a line and a circle<\/li>\n<\/ul>\n<\/div>\n<h2>Parametric Equations and Their Graphs<\/h2>\n<section id=\"fs-id1169295516903\" data-depth=\"1\">\n<p id=\"fs-id1169295553685\">Consider the orbit of Earth around the Sun. Our year lasts approximately [latex]365.25[\/latex] days, but for this discussion we will use [latex]365[\/latex] days. On January 1 of each year, the physical location of Earth with respect to the Sun is nearly the same, except for leap years, when the lag introduced by the extra [latex]\\frac{1}{4}[\/latex] day of orbiting time is built into the calendar. We call January 1 &#8220;day [latex]1[\/latex]&#8221; of the year. Then, for example, day [latex]31[\/latex] is January 31, day 59 is February 28, and so on.<\/p>\n<p id=\"fs-id1169293250149\">The number of the day in a year can be considered a variable that determines Earth\u2019s position in its orbit. As Earth revolves around the Sun, its physical location changes relative to the Sun. After one full year, we are back where we started, and a new year begins. According to Kepler\u2019s laws of planetary motion, the shape of the orbit is elliptical, with the Sun at one focus of the ellipse. We study this idea in more detail in Conic Sections.<\/p>\n<figure id=\"CNX_Calc_Figure_11_01_001\"><figcaption><\/figcaption><div style=\"width: 468px\" 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\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1. Earth\u2019s orbit around the Sun in one year.<\/p>\n<\/div>\n<\/figure>\n<p id=\"fs-id1169293338092\">Figure 1 depicts <span class=\"no-emphasis\" data-type=\"term\">Earth\u2019s orbit<\/span> 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. If we superimpose coordinate axes over this graph, then we can assign ordered pairs to each point on the ellipse (Figure 2). Then each\u00a0[latex]x[\/latex] value on the graph is a value of position as a function of time, and each [latex]y[\/latex] value is also a value of position as a function of time. Therefore, each point on the graph corresponds to a value of Earth\u2019s position as a function of time.<\/p>\n<figure id=\"CNX_Calc_Figure_11_01_002\"><figcaption><\/figcaption><div style=\"width: 502px\" 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\/11234606\/CNX_Calc_Figure_11_01_002.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. There is a line going from [latex]t[\/latex] = 1 to [latex]t[\/latex] = 182. There is also a line going from [latex]t[\/latex] = 92 to [latex]t[\/latex] = 274 that passes through F2 and the Sun. On the upper left side, there is a point marked (x(t), y(t)) with a tangent line pointing down and to the left.\" width=\"492\" height=\"422\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 2. Coordinate axes superimposed on the orbit of Earth.<\/p>\n<\/div>\n<\/figure>\n<p id=\"fs-id1169293137188\">We can determine the functions for [latex]x\\left(t\\right)[\/latex] and [latex]y\\left(t\\right)[\/latex], thereby parameterizing the orbit of Earth around the Sun. The variable [latex]t[\/latex] is called an independent parameter and, in this context, represents time relative to the beginning of each year.<\/p>\n<p id=\"fs-id1169295637245\">A curve in the [latex]\\left(x,y\\right)[\/latex] plane can be represented parametrically. The equations that are used to define the curve are called <span data-type=\"term\">parametric equations<\/span>.<\/p>\n<div id=\"fs-id1169295820852\" data-type=\"note\">\n<div data-type=\"title\">\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\" data-type=\"title\">Definition<\/h3>\n<hr \/>\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>&nbsp;<\/p>\n<p id=\"fs-id1169295804821\">are called parametric equations and [latex]t[\/latex] is called the <strong>parameter<\/strong>. The set of points [latex]\\left(x,y\\right)[\/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<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1169295465331\">Notice in this definition that [latex]x[\/latex]\u00a0and [latex]y[\/latex] are used in two ways. The first is as functions of the independent variable\u00a0[latex]t[\/latex]. As [latex]t[\/latex] varies over the interval\u00a0[latex]I[\/latex], the functions [latex]x\\left(t\\right)[\/latex] and [latex]y\\left(t\\right)[\/latex] generate a set of ordered pairs [latex]\\left(x,y\\right)[\/latex]. This set of ordered pairs generates the graph of the parametric equations. In this second usage, to designate the ordered pairs, [latex]x[\/latex] and [latex]y[\/latex] are variables. It is important to distinguish the variables [latex]x[\/latex] and [latex]y[\/latex] from the functions [latex]x\\left(t\\right)[\/latex] and [latex]y\\left(t\\right)[\/latex].<\/p>\n<div id=\"fs-id1169295362672\" data-type=\"example\">\n<div id=\"fs-id1169295870973\" data-type=\"exercise\">\n<div id=\"fs-id1169295520035\" data-type=\"problem\">\n<div data-type=\"title\">\n<div class=\"textbox exercises\">\n<h3>Example: Graphing a Parametrically Defined Curve<\/h3>\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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558899\">Show Solution<\/span><\/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 [latex]t[\/latex], let [latex]t[\/latex] 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\" style=\"height: 84px;\" 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 style=\"height: 12px;\" valign=\"top\">\n<th style=\"height: 12px; width: 105.994px;\" data-valign=\"top\" data-align=\"center\">[latex]t[\/latex]<\/th>\n<th style=\"height: 12px; width: 192.131px;\" data-valign=\"top\" data-align=\"center\">[latex]x\\left(t\\right)[\/latex]<\/th>\n<th style=\"height: 12px; width: 192.145px;\" data-valign=\"top\" data-align=\"center\">[latex]y\\left(t\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 12px;\" valign=\"top\">\n<td style=\"height: 12px; width: 105.994px;\" data-valign=\"top\" data-align=\"center\">[latex]-3[\/latex]<\/td>\n<td style=\"height: 12px; width: 192.131px;\" data-valign=\"top\" data-align=\"center\">[latex]-4[\/latex]<\/td>\n<td style=\"height: 12px; width: 192.145px;\" data-valign=\"top\" data-align=\"center\">[latex]-2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\" valign=\"top\">\n<td style=\"height: 12px; width: 105.994px;\" data-valign=\"top\" data-align=\"center\">[latex]-2[\/latex]<\/td>\n<td style=\"height: 12px; width: 192.131px;\" data-valign=\"top\" data-align=\"center\">[latex]-3[\/latex]<\/td>\n<td style=\"height: 12px; width: 192.145px;\" data-valign=\"top\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\" valign=\"top\">\n<td style=\"height: 12px; width: 105.994px;\" data-valign=\"top\" data-align=\"center\">[latex]-1[\/latex]<\/td>\n<td style=\"height: 12px; width: 192.131px;\" data-valign=\"top\" data-align=\"center\">[latex]-2[\/latex]<\/td>\n<td style=\"height: 12px; width: 192.145px;\" data-valign=\"top\" data-align=\"center\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\" valign=\"top\">\n<td style=\"height: 12px; width: 105.994px;\" data-valign=\"top\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"height: 12px; width: 192.131px;\" data-valign=\"top\" data-align=\"center\">[latex]-1[\/latex]<\/td>\n<td style=\"height: 12px; width: 192.145px;\" data-valign=\"top\" data-align=\"center\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\" valign=\"top\">\n<td style=\"height: 12px; width: 105.994px;\" data-valign=\"top\" data-align=\"center\">[latex]1[\/latex]<\/td>\n<td style=\"height: 12px; width: 192.131px;\" data-valign=\"top\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td style=\"height: 12px; width: 192.145px;\" data-valign=\"top\" data-align=\"center\">[latex]6[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\" valign=\"top\">\n<td style=\"height: 12px; width: 105.994px;\" data-valign=\"top\" data-align=\"center\">[latex]2[\/latex]<\/td>\n<td style=\"height: 12px; width: 192.131px;\" data-valign=\"top\" data-align=\"center\">[latex]1[\/latex]<\/td>\n<td style=\"height: 12px; width: 192.145px;\" data-valign=\"top\" data-align=\"center\">[latex]8[\/latex]<\/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 [latex]t[\/latex] varies from\u00a0[latex]-3[\/latex] to [latex]2[\/latex] .<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<figure id=\"CNX_Calc_Figure_11_01_003\">\n<div style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/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 &#091;latex&#093;t&#091;\/latex&#093; = \u22123, the point (\u22122, 2) is marked &#091;latex&#093;t&#091;\/latex&#093; = \u22121, and the point (1, 8) is marked &#091;latex&#093;t&#091;\/latex&#093; = 2. On the graph there are also written three equations: x(t) = &#091;latex&#093;t&#091;\/latex&#093; \u22121, y(t) = 2t + 4, and \u22123 \u2264 &#091;latex&#093;t&#091;\/latex&#093; \u2264 2.\" width=\"488\" height=\"497\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 3. Graph of the plane curve described by the parametric equations in part a.<\/p>\n<\/div>\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\">[latex]t[\/latex]<\/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\">[latex]-2[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]1[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]-3[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">[latex]-1[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]-2[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]-1[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">[latex]0[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]-3[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">[latex]1[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]-2[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">[latex]2[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]1[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]5[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\">[latex]3[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]6[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]7[\/latex]<\/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\u00a0[latex]-2[\/latex] to[latex]3[\/latex], 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<div style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/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 &#091;latex&#093;t&#091;\/latex&#093; = \u22122, the point (\u22123, 1) is marked &#091;latex&#093;t&#091;\/latex&#093; = 0, and the point (6, 7) is marked &#091;latex&#093;t&#091;\/latex&#093; = 3. On the graph there are also written three equations: x(t) = t2 \u2212 3, y(t) = 2t + 1, and \u22122 \u2264 &#091;latex&#093;t&#091;\/latex&#093; \u2264 3.\" width=\"488\" height=\"497\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 4. Graph of the plane curve described by the parametric equations in part b.<\/p>\n<\/div>\n<\/figure>\n<\/li>\n<li>In this case, use multiples of [latex]\\frac{\\pi}{6}[\/latex] for [latex]t[\/latex] 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\">[latex]t[\/latex]<\/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\">[latex]t[\/latex]<\/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\">[latex]0[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]4[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]0[\/latex]<\/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\">[latex]2[\/latex]<\/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\">[latex]-2[\/latex]<\/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\">[latex]0[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]-4[\/latex]<\/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\">[latex]0[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]4[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]\\frac{5\\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<\/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\">[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{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\">[latex]2[\/latex]<\/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\">[latex]2[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]2\\pi[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]4[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]0[\/latex]<\/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\">[latex]-4[\/latex]<\/td>\n<td data-valign=\"top\" data-align=\"center\">[latex]0[\/latex]<\/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<div style=\"width: 427px\" 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 &#091;latex&#093;t&#091;\/latex&#093; = 0, the point (0, 4) is marked &#091;latex&#093;t&#091;\/latex&#093; = \u03c0\/2, the point (\u22124, 0) is marked &#091;latex&#093;t&#091;\/latex&#093; = \u03c0, and the point (0, \u22124) is marked &#091;latex&#093;t&#091;\/latex&#093; = 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 &#091;latex&#093;t&#091;\/latex&#093; \u2264 2\u03c0.\" width=\"417\" height=\"423\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 5. Graph of the plane curve described by the parametric equations in part c.<\/p>\n<\/div>\n<\/figure>\n<p>This is the graph of a circle with radius\u00a0[latex]4[\/latex] 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<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Graphing a Parametrically Defined Curve.<\/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\"><\/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<div id=\"fs-id1169295454074\" class=\"checkpoint\" data-type=\"note\">\n<div id=\"fs-id1169293200600\" data-type=\"exercise\">\n<div id=\"fs-id1169295589408\" data-type=\"problem\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-id1169295589408\" data-type=\"problem\">\n<p id=\"fs-id1169295759487\">Sketch the curve described by the parametric equations<\/p>\n<div id=\"fs-id1169295431091\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]x\\left(t\\right)=3t+2,y\\left(t\\right)={t}^{2}-1,-3\\le t\\le 2[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558898\">Hint<\/span><\/p>\n<div id=\"q44558898\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1169295557869\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1169295652237\">Make a table of values for [latex]x\\left(t\\right)[\/latex] and [latex]y\\left(t\\right)[\/latex] using [latex]t[\/latex] values from\u00a0[latex]-3[\/latex] to[latex]2[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558897\">Show Solution<\/span><\/p>\n<div id=\"q44558897\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1169295453647\" data-type=\"solution\">\n<div style=\"width: 652px\" 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\/11234620\/CNX_Calc_Figure_11_01_006.jpg\" alt=\"A curved line going from (\u22127, 8) through (\u22121, 0) and (2, \u22121) to (8, 3) with arrow going in that order. The point (\u22127, 8) is marked &#091;latex&#093;t&#091;\/latex&#093; = \u22123, the point (2, \u22121) is marked &#091;latex&#093;t&#091;\/latex&#093; = 0, and the point (8, 3) is marked &#091;latex&#093;t&#091;\/latex&#093; = 2. On the graph there are also written three equations: x(t) = 3t + 2, y(t) = t2 \u2212 1, and \u22123 \u2264 &#091;latex&#093;t&#091;\/latex&#093; \u2264 2.\" width=\"642\" height=\"423\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 6.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm173890\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=173890&theme=oea&iframe_resize_id=ohm173890&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1169295807788\" data-depth=\"1\">\n<h2 data-type=\"title\">Eliminating the Parameter<\/h2>\n<p id=\"fs-id1169293341434\">To better understand the graph of a curve represented parametrically, it is useful to rewrite the two equations as a single equation relating the variables [latex]x[\/latex] and <em data-effect=\"italics\">y.<\/em> Then we can apply any previous knowledge of equations of curves in the plane to identify the curve. For example, the equations describing the plane curve in part (b) of the previous example are<\/p>\n<div id=\"fs-id1169295565020\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]x\\left(t\\right)={t}^{2}-3,y\\left(t\\right)=2t+1,-2\\le t\\le 3[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169295882380\">Solving the second equation for [latex]t[\/latex] gives<\/p>\n<div id=\"fs-id1169293137841\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\large{t=\\frac{y - 1}{2}}[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169293144942\">This can be substituted into the first equation:<\/p>\n<div id=\"fs-id1169293130361\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]x={\\left(\\frac{y - 1}{2}\\right)}^{2}-3=\\frac{{y}^{2}-2y+1}{4}-3=\\frac{{y}^{2}-2y - 11}{4}[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169295454203\">This equation describes [latex]x[\/latex] as a function of <em data-effect=\"italics\">y.<\/em> These steps give an example of <em data-effect=\"italics\">eliminating the parameter<\/em>. The graph of this function is a parabola opening to the right. Recall that the plane curve started at [latex]\\left(1,-3\\right)[\/latex] and ended at [latex]\\left(6,7\\right)[\/latex]. These terminations were due to the restriction on the parameter [latex]t[\/latex]<em data-effect=\"italics\">.<\/em><\/p>\n<p>Before working through an example on how to eliminate the parameter, it is useful to recall the Pythagorean Identity as well as the equations of circles.<\/p>\n<div id=\"fs-id1169295694719\" data-type=\"example\">\n<div id=\"fs-id1169293393055\" data-type=\"exercise\">\n<div id=\"fs-id1169295362943\" data-type=\"problem\">\n<div data-type=\"title\">\n<div class=\"textbox examples\">\n<h3>Recall: Pythagorean identity and equation of circles<\/h3>\n<ul>\n<li>For any angle [latex]t, \\sin^2 t + \\cos^2 t = 1[\/latex]<\/li>\n<li>A circle of radius [latex]a[\/latex] centered at the origin is given by [latex]x^2 + y^2 = a^2[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Eliminating the Parameter<\/h3>\n<div id=\"fs-id1169295362943\" data-type=\"problem\">\n<p id=\"fs-id1169295422772\">Eliminate the parameter for each of the plane curves described by the following parametric equations and describe the resulting graph.<\/p>\n<ol id=\"fs-id1169293340647\" type=\"a\">\n<li>[latex]x\\left(t\\right)=\\sqrt{2t+4},y\\left(t\\right)=2t+1,-2\\le t\\le 6[\/latex]<\/li>\n<li>[latex]x\\left(t\\right)=4\\cos{t},y\\left(t\\right)=3\\sin{t},0\\le t\\le 2\\pi[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558895\">Show Solution<\/span><\/p>\n<div id=\"q44558895\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1169293275836\" data-type=\"solution\">\n<ol id=\"fs-id1169293199708\" type=\"a\">\n<li>To eliminate the parameter, we can solve either of the equations for <em data-effect=\"italics\">t.<\/em> For example, solving the first equation for [latex]t[\/latex] gives<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169293390065\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{ccc}\\hfill x& =\\hfill & \\sqrt{2t+4}\\hfill \\\\ \\hfill {x}^{2}& =\\hfill & 2t+4\\hfill \\\\ \\hfill {x}^{2}-4& =\\hfill & 2t\\hfill \\\\ \\hfill t& =\\hfill & \\frac{{x}^{2}-4}{2}.\\hfill \\end{array}[\/latex]<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span><br \/>\nNote that when we square both sides it is important to observe that [latex]x\\ge 0[\/latex]. Substituting [latex]t=\\frac{{x}^{2}-4}{2}[\/latex] this into [latex]y\\left(t\\right)[\/latex] yields<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169293387312\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{ccc}\\hfill y\\left(t\\right)& =\\hfill & 2t+1\\hfill \\\\ \\hfill y& =\\hfill & 2\\left(\\frac{{x}^{2}-4}{2}\\right)+1\\hfill \\\\ \\hfill y& =\\hfill & {x}^{2}-4+1\\hfill \\\\ \\hfill y& =\\hfill & {x}^{2}-3.\\hfill \\end{array}[\/latex]<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span><br \/>\nThis is the equation of a parabola opening upward. There is, however, a domain restriction because of the limits on the parameter [latex]t[\/latex]. When [latex]t=-2[\/latex], [latex]x=\\sqrt{2\\left(-2\\right)+4}=0[\/latex], and when [latex]t=6[\/latex], [latex]x=\\sqrt{2\\left(6\\right)+4}=4[\/latex]. The graph of this plane curve follows.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<figure id=\"CNX_Calc_Figure_11_01_007\">\n<div style=\"width: 277px\" 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\/11234623\/CNX_Calc_Figure_11_01_007.jpg\" alt=\"A curved line going from (\u22123, 0) through (2, 1) to (4, 13) with arrow going in that order. The point (\u22123, 0) is marked &#091;latex&#093;t&#091;\/latex&#093; = \u22122, the point (2, 1) is marked &#091;latex&#093;t&#091;\/latex&#093; = 0, the point (2 times the square root of 2, 5) is marked &#091;latex&#093;t&#091;\/latex&#093; = 2, the point (3 times the square root of 2, 9) is marked &#091;latex&#093;t&#091;\/latex&#093; = 4, and the point (4, 13) is marked &#091;latex&#093;t&#091;\/latex&#093; = 6. On the graph there are also written three equations: x(t) = square root of the quantity (2t + 4), y(t) = 2t + 1, and \u22122 \u2264 &#091;latex&#093;t&#091;\/latex&#093; \u2264 6.\" width=\"267\" height=\"385\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 7. Graph of the plane curve described by the parametric equations in part a.<\/p>\n<\/div>\n<\/figure>\n<\/li>\n<li>Sometimes it is necessary to be a bit creative in eliminating the parameter. The parametric equations for this example are<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169295773361\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]x\\left(t\\right)=4\\cos{t}\\text{ and }y\\left(t\\right)=3\\sin{t}[\/latex].<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span><br \/>\nSolving either equation for [latex]t[\/latex] directly is not advisable because sine and cosine are not one-to-one functions. However, dividing the first equation by 4 and the second equation by 3 (and suppressing the [latex]t[\/latex]) gives us<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169295447894\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\cos{t}=\\frac{x}{4}\\text{ and }\\sin{t}=\\frac{y}{3}[\/latex].<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span><br \/>\nNow use the Pythagorean identity [latex]{\\cos}^{2}t+{\\sin}^{2}t=1[\/latex] and replace the expressions for [latex]\\sin{t}[\/latex] and [latex]\\cos{t}[\/latex] with the equivalent expressions in terms of [latex]x[\/latex] and [latex]y[\/latex]. This gives<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"fs-id1169295591977\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{ccc}\\hfill {\\left(\\frac{x}{4}\\right)}^{2}+{\\left(\\frac{y}{3}\\right)}^{2}& =\\hfill & 1\\hfill \\\\ \\hfill \\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{9}& =\\hfill & 1.\\hfill \\end{array}[\/latex]<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span><br \/>\nThis is the equation of a horizontal ellipse centered at the origin, with semimajor axis [latex]4[\/latex] and semiminor axis [latex]3[\/latex] as shown in the following graph.<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<figure id=\"CNX_Calc_Figure_11_01_008\">\n<div style=\"width: 427px\" 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\/11234626\/CNX_Calc_Figure_11_01_008.jpg\" alt=\"An ellipse with major axis horizontal and of length 8 and with minor radius vertical and of length 6 that is centered at the origin with arrow going counterclockwise. The point (4, 0) is marked &#091;latex&#093;t&#091;\/latex&#093; = 0, the point (0, 3) is marked &#091;latex&#093;t&#091;\/latex&#093; = \u03c0\/2, the point (\u22124, 0) is marked &#091;latex&#093;t&#091;\/latex&#093; = \u03c0, and the point (0, \u22123) is marked &#091;latex&#093;t&#091;\/latex&#093; = 3\u03c0\/2. On the graph there are also written three equations: x(t) = 4 cos(t), y(t) = 3 sin(t), and 0 \u2264 &#091;latex&#093;t&#091;\/latex&#093; \u2264 2\u03c0.\" width=\"417\" height=\"423\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 8. Graph of the plane curve described by the parametric equations in part b.<\/p>\n<\/div>\n<\/figure>\n<p>As [latex]t[\/latex] progresses from [latex]0[\/latex] to [latex]2\\pi[\/latex], a point on the curve traverses the ellipse once, in a counterclockwise direction. Recall from the section opener that the orbit of Earth around the Sun is also elliptical. This is a perfect example of using parameterized curves to model a real-world phenomenon.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Eliminating the Parameter.<\/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=495&amp;end=846&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/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\/7.1ParametricEquations495to846_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<div id=\"fs-id1169295362509\" class=\"checkpoint\" data-type=\"note\">\n<div id=\"fs-id1169295362512\" data-type=\"exercise\">\n<div id=\"fs-id1169295362514\" data-type=\"problem\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-id1169295362514\" data-type=\"problem\">\n<p id=\"fs-id1169295555110\">Eliminate the parameter for the plane curve defined by the following parametric equations and describe the resulting graph.<\/p>\n<div id=\"fs-id1169295555115\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]x\\left(t\\right)=2+\\frac{3}{t},y\\left(t\\right)=t - 1,2\\le t\\le 6[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558893\">Hint<\/span><\/p>\n<div id=\"q44558893\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1169293185389\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1169293313963\">Solve one of the equations for [latex]t[\/latex] and substitute into the other equation.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558894\">Show Solution<\/span><\/p>\n<div id=\"q44558894\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1169293397158\" data-type=\"solution\">\n<p id=\"fs-id1169293397160\">[latex]x=2+\\frac{3}{y+1}[\/latex], or [latex]y=-1+\\frac{3}{x - 2}[\/latex]. This equation describes a portion of a rectangular hyperbola centered at [latex]\\left(2,-1\\right)[\/latex]. <span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div style=\"width: 340px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/4175\/2019\/04\/11234629\/CNX_Calc_Figure_11_01_009.jpg\" alt=\"A curved line going from (3.5, 1) to (2.5, 5) with arrow going in that order. The point (3.5, 1) is marked &#091;latex&#093;t&#091;\/latex&#093; = 2 and the point (2.5, 5) is marked &#091;latex&#093;t&#091;\/latex&#093; = 6. On the graph there are also written three equations: x(t) = 2 + 3\/t, y(t) = &#091;latex&#093;t&#091;\/latex&#093; \u2212 1, and 2 \u2264 &#091;latex&#093;t&#091;\/latex&#093; \u2264 6.\" width=\"330\" height=\"310\" data-media-type=\"image\/jpeg\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 9.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm4733\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=4733&theme=oea&iframe_resize_id=ohm4733&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p id=\"fs-id1169295460497\">So far we have seen the method of eliminating the parameter, assuming we know a set of parametric equations that describe a plane curve. What if we would like to start with the equation of a curve and determine a pair of parametric equations for that curve? This is certainly possible, and in fact it is possible to do so in many different ways for a given curve. The process is known as <span data-type=\"term\">parameterization of a curve<\/span>.<\/p>\n<div id=\"fs-id1169295460500\" data-type=\"example\">\n<div id=\"fs-id1169295398174\" data-type=\"exercise\">\n<div id=\"fs-id1169295398176\" data-type=\"problem\">\n<div data-type=\"title\">\n<div class=\"textbox exercises\">\n<h3>Example: Parameterizing a Curve<\/h3>\n<div id=\"fs-id1169295398176\" data-type=\"problem\">\n<p id=\"fs-id1169293341360\">Find two different pairs of parametric equations to represent the graph of [latex]y=2{x}^{2}-3[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558892\">Show Solution<\/span><\/p>\n<div id=\"q44558892\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1169295712606\" data-type=\"solution\">\n<p id=\"fs-id1169295712608\">First, it is always possible to parameterize a curve by defining [latex]x\\left(t\\right)=t[\/latex], then replacing [latex]x[\/latex] with [latex]t[\/latex] in the equation for [latex]y\\left(t\\right)[\/latex]. This gives the parameterization<\/p>\n<div id=\"fs-id1169295409716\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]x\\left(t\\right)=t,y\\left(t\\right)=2{t}^{2}-3[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169295554644\">Since there is no restriction on the domain in the original graph, there is no restriction on the values of <em data-effect=\"italics\">t.<\/em><\/p>\n<p id=\"fs-id1169295353796\">We have complete freedom in the choice for the second parameterization. For example, we can choose [latex]x\\left(t\\right)=3t - 2[\/latex]. The only thing we need to check is that there are no restrictions imposed on [latex]x[\/latex]; that is, the range of [latex]x\\left(t\\right)[\/latex] is all real numbers. This is the case for [latex]x\\left(t\\right)=3t - 2[\/latex]. Now since [latex]y=2{x}^{2}-3[\/latex], we can substitute [latex]x\\left(t\\right)=3t - 2[\/latex] for\u00a0[latex]x[\/latex]. This gives<\/p>\n<div id=\"fs-id1169295558068\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]\\begin{array}{cc}\\hfill y\\left(t\\right)& =2{\\left(3t - 2\\right)}^{2}-2\\hfill \\\\ & =2\\left(9{t}^{2}-12t+4\\right)-2\\hfill \\\\ & =18{t}^{2}-24t+8 - 2\\hfill \\\\ & =18{t}^{2}-24t+6.\\hfill \\end{array}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<p id=\"fs-id1169293401741\">Therefore, a second parameterization of the curve can be written as<\/p>\n<div id=\"fs-id1169293401744\" class=\"unnumbered\" style=\"text-align: center;\" data-type=\"equation\" data-label=\"\">[latex]x\\left(t\\right)=3t - 2\\text{ and }y\\left(t\\right)=18{t}^{2}-24t+6[\/latex].<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to Example: Parameterizing a Curve.<\/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=848&amp;end=944&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/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\/7.1ParametricEquations848to944_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<div id=\"fs-id1169295461539\" class=\"checkpoint\" data-type=\"note\">\n<div id=\"fs-id1169295461542\" data-type=\"exercise\">\n<div id=\"fs-id1169295461544\" data-type=\"problem\">\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<div id=\"fs-id1169295461544\" data-type=\"problem\">\n<p id=\"fs-id1169293398615\">Find two different sets of parametric equations to represent the graph of [latex]y={x}^{2}+2x[\/latex].<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558890\">Hint<\/span><\/p>\n<div id=\"q44558890\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1169293392950\" data-type=\"commentary\" data-element-type=\"hint\">\n<p id=\"fs-id1169293253923\">Follow the steps from the example. Remember we have freedom in choosing the parameterization for [latex]x\\left(t\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q44558891\">Show Solution<\/span><\/p>\n<div id=\"q44558891\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1169293390102\" data-type=\"solution\">\n<p id=\"fs-id1169293390104\">One possibility is [latex]x\\left(t\\right)=t,y\\left(t\\right)={t}^{2}+2t[\/latex]. Another possibility is [latex]x\\left(t\\right)=2t - 3,y\\left(t\\right)={\\left(2t - 3\\right)}^{2}+2\\left(2t - 3\\right)=4{t}^{2}-8t+3[\/latex].<\/p>\n<p id=\"fs-id1169295712634\">There are, in fact, an infinite number of possibilities.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1169295349662\" data-depth=\"1\"><\/section>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1153\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>7.1 Parametric Equations. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 3. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":349141,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"7.1 Parametric Equations\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1153","chapter","type-chapter","status-publish","hentry"],"part":1150,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1153","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/349141"}],"version-history":[{"count":7,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1153\/revisions"}],"predecessor-version":[{"id":4695,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1153\/revisions\/4695"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/1150"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1153\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=1153"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=1153"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=1153"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=1153"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}