{"id":227,"date":"2025-02-13T22:45:17","date_gmt":"2025-02-13T22:45:17","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/parametric-equations\/"},"modified":"2025-10-21T17:23:06","modified_gmt":"2025-10-21T17:23:06","slug":"parametric-equations","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/parametric-equations\/","title":{"raw":"Parametric Equations: Learn It 1","rendered":"Parametric Equations: Learn It 1"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find a rectangular equation for a curve defined parametrically.<\/li>\r\n \t<li>Find parametric equations for curves defined by rectangular equations.<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\nConsider sets of equations given by [latex]x\\left(t\\right)[\/latex] and [latex]y\\left(t\\right)[\/latex] where [latex]t[\/latex] is the independent variable of time. We can use these parametric equations in a number of applications when we are looking for not only a particular position but also the direction of the movement. As we trace out successive values of [latex]t[\/latex], the orientation of the curve becomes clear. This is one of the primary advantages of using parametric equations: we are able to trace the movement of an object along a path according to time.\r\n<h2>Parameterizing a Curve<\/h2>\r\nWhen an object moves along a curve\u2014or <strong>curvilinear path<\/strong>\u2014in a given direction and in a given amount of time, the position of the object in the plane is given by the <em>x-<\/em>coordinate and the <em>y-<\/em>coordinate. However, both [latex]x[\/latex] and [latex]y[\/latex] vary over time and so are functions of time. For this reason, we add another variable, the <strong>parameter<\/strong>, upon which both [latex]x[\/latex] and [latex]y[\/latex] are dependent functions. Parametric equations primarily describe motion and direction.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<div>\r\n<h3>parameter<\/h3>\r\nThe variable that [latex]x[\/latex] and [latex]y[\/latex] are both dependent on.\r\n\r\n<\/div>\r\n<\/section>When we parameterize a curve, we are translating a single equation in two variables, such as [latex]x[\/latex] and [latex]y [\/latex], into an equivalent pair of equations in three variables, [latex]x,y[\/latex], and [latex]t[\/latex]. One of the reasons we parameterize a curve is because the parametric equations yield more information: specifically, the direction of the object\u2019s motion over time.\r\n\r\nWhen we graph parametric equations, we can observe the individual behaviors of [latex]x[\/latex] and of [latex]y[\/latex]. There are a number of shapes that cannot be represented in the form [latex]y=f\\left(x\\right)[\/latex], meaning that they are not functions. For example, consider the graph of a circle, given as [latex]{r}^{2}={x}^{2}+{y}^{2}[\/latex]. Solving for [latex]y[\/latex] gives [latex]y=\\pm \\sqrt{{r}^{2}-{x}^{2}}[\/latex], or two equations: [latex]{y}_{1}=\\sqrt{{r}^{2}-{x}^{2}}[\/latex] and [latex]{y}_{2}=-\\sqrt{{r}^{2}-{x}^{2}}[\/latex]. If we graph [latex]{y}_{1}[\/latex] and [latex]{y}_{2}[\/latex] together, the graph will not pass the vertical line test. Thus, the equation for the graph of a circle is not a function.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180917\/CNX_Precalc_Figure_08_06_0022.jpg\" alt=\"Graph of a circle in the rectangular coordinate system - the vertical line test shows that the circle r^2 = x^2 + y^2 is not a function. The dotted red vertical line intersects the function in two places - it should only intersect in one place to be a function.\" width=\"487\" height=\"291\" \/>\r\n\r\nHowever, if we were to graph each equation on its own, each one would pass the vertical line test and therefore would represent a function. In some instances, the concept of breaking up the equation for a circle into two functions is similar to the concept of creating parametric equations, as we use two functions to produce a non-function. This will become clearer as we move forward.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>parametric equations<\/h3>\r\nSuppose [latex]t[\/latex] is a number on an interval, [latex]I[\/latex]. The set of ordered pairs, [latex]\\left(x\\left(t\\right),y\\left(t\\right)\\right)[\/latex], where [latex]x=f\\left(t\\right)[\/latex] and [latex]y=g\\left(t\\right)[\/latex], forms a plane curve based on the parameter [latex]t[\/latex]. The equations [latex]x=f\\left(t\\right)[\/latex] and [latex]y=g\\left(t\\right)[\/latex] are the parametric equations.\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<h3><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; font-size: 16px; font-weight: 400;\">Parameterize the curve [latex]y={x}^{2}-1[\/latex] letting [latex]x\\left(t\\right)=t[\/latex]. Graph both equations.<\/span><\/h3>\r\n[reveal-answer q=\"910055\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"910055\"]\r\n\r\nIf [latex]x\\left(t\\right)=t[\/latex], then to find [latex]y\\left(t\\right)[\/latex] we replace the variable [latex]x[\/latex] with the expression given in [latex]x\\left(t\\right)[\/latex]. In other words, [latex]y\\left(t\\right)={t}^{2}-1[\/latex]. Make a table of values similar to the table below, and sketch the graph.\r\n<table id=\"Table_08_06_001\" summary=\"Ten rows and three columns. First column is labeled t, second column is labeled x(t), third column is labeled y(t). The table has ordered triples of each of these row values: (-4,-4, y(-4)=(-4)^2 - 1 = 15), (-3,-3, y(-3)= (-3)^2 -1 = 8), (-2,-2, y(-2) = (-2)^2 -1 = 3), (-1,-1, y(-1)= (-1)^2 -1 = 0), (0,0, y(0) = (0)^2 -1 = -1), (1,1, y(1) = (1)^2 -1 = 0), (2,2, y(2) = (2)^2 -1 =3), (3,3, y(3) = (3)^2 - 1 = 8), (4,4, y(4) = (4)^2 - 1 = 15).\">\r\n<thead>\r\n<tr>\r\n<th>[latex]t[\/latex]<\/th>\r\n<th>[latex]x\\left(t\\right)[\/latex]<\/th>\r\n<th>[latex]y\\left(t\\right)[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]-4[\/latex]<\/td>\r\n<td>[latex]-4[\/latex]<\/td>\r\n<td>[latex]y\\left(-4\\right)={\\left(-4\\right)}^{2}-1=15[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]y\\left(-3\\right)={\\left(-3\\right)}^{2}-1=8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]y\\left(-2\\right)={\\left(-2\\right)}^{2}-1=3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]y\\left(-1\\right)={\\left(-1\\right)}^{2}-1=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]y\\left(0\\right)={\\left(0\\right)}^{2}-1=-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]y\\left(1\\right)={\\left(1\\right)}^{2}-1=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]y\\left(2\\right)={\\left(2\\right)}^{2}-1=3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]y\\left(3\\right)={\\left(3\\right)}^{2}-1=8[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]y\\left(4\\right)={\\left(4\\right)}^{2}-1=15[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIt may be helpful to use the TRACE feature of a graphing calculator to see how the points are generated as [latex]t[\/latex] increases.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180919\/CNX_Precalc_Figure_08_06_0152.jpg\" alt=\"Graph of a parabola in two forms: a parametric equation and rectangular coordinates. It is the same function, just different ways of writing it.\" width=\"731\" height=\"291\" \/> (a) Parametric [latex]y\\left(t\\right)={t}^{2}-1[\/latex] (b) Rectangular [latex]y={x}^{2}-1[\/latex][\/caption]<strong>Analysis of the Solution<\/strong>The arrows indicate the direction in which the curve is generated. Notice the curve is identical to the curve of [latex]y={x}^{2}-1[\/latex].[\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">Construct a table of values and plot the parametric equations: [latex]x\\left(t\\right)=t - 3,y\\left(t\\right)=2t+4;-1\\le t\\le 2[\/latex].[reveal-answer q=\"217668\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"217668\"]\r\n<table id=\"fs-id1165137810323\" class=\"unnumbered\" summary=\"Five rows and three columns. First column is labeled t, second column is labeled x(t), third column is labeled y(t). The table has ordered triples of each of these row values: (-1, -4, 2), (0,-3,4), (1,-2,6), (2,-1,8).\">\r\n<tbody>\r\n<tr>\r\n<td>[latex]t[\/latex]<\/td>\r\n<td>[latex]x\\left(t\\right)[\/latex]<\/td>\r\n<td>[latex]y\\left(t\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]-4[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180940\/CNX_Precalc_Figure_08_06_0062.jpg\" alt=\"\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<dl id=\"fs-id1165135186874\" class=\"definition\">\r\n \t<dd id=\"fs-id1165134357588\"><\/dd>\r\n<\/dl>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find a rectangular equation for a curve defined parametrically.<\/li>\n<li>Find parametric equations for curves defined by rectangular equations.<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<p>Consider sets of equations given by [latex]x\\left(t\\right)[\/latex] and [latex]y\\left(t\\right)[\/latex] where [latex]t[\/latex] is the independent variable of time. We can use these parametric equations in a number of applications when we are looking for not only a particular position but also the direction of the movement. As we trace out successive values of [latex]t[\/latex], the orientation of the curve becomes clear. This is one of the primary advantages of using parametric equations: we are able to trace the movement of an object along a path according to time.<\/p>\n<h2>Parameterizing a Curve<\/h2>\n<p>When an object moves along a curve\u2014or <strong>curvilinear path<\/strong>\u2014in a given direction and in a given amount of time, the position of the object in the plane is given by the <em>x-<\/em>coordinate and the <em>y-<\/em>coordinate. However, both [latex]x[\/latex] and [latex]y[\/latex] vary over time and so are functions of time. For this reason, we add another variable, the <strong>parameter<\/strong>, upon which both [latex]x[\/latex] and [latex]y[\/latex] are dependent functions. Parametric equations primarily describe motion and direction.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<div>\n<h3>parameter<\/h3>\n<p>The variable that [latex]x[\/latex] and [latex]y[\/latex] are both dependent on.<\/p>\n<\/div>\n<\/section>\n<p>When we parameterize a curve, we are translating a single equation in two variables, such as [latex]x[\/latex] and [latex]y[\/latex], into an equivalent pair of equations in three variables, [latex]x,y[\/latex], and [latex]t[\/latex]. One of the reasons we parameterize a curve is because the parametric equations yield more information: specifically, the direction of the object\u2019s motion over time.<\/p>\n<p>When we graph parametric equations, we can observe the individual behaviors of [latex]x[\/latex] and of [latex]y[\/latex]. There are a number of shapes that cannot be represented in the form [latex]y=f\\left(x\\right)[\/latex], meaning that they are not functions. For example, consider the graph of a circle, given as [latex]{r}^{2}={x}^{2}+{y}^{2}[\/latex]. Solving for [latex]y[\/latex] gives [latex]y=\\pm \\sqrt{{r}^{2}-{x}^{2}}[\/latex], or two equations: [latex]{y}_{1}=\\sqrt{{r}^{2}-{x}^{2}}[\/latex] and [latex]{y}_{2}=-\\sqrt{{r}^{2}-{x}^{2}}[\/latex]. If we graph [latex]{y}_{1}[\/latex] and [latex]{y}_{2}[\/latex] together, the graph will not pass the vertical line test. Thus, the equation for the graph of a circle is not a function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180917\/CNX_Precalc_Figure_08_06_0022.jpg\" alt=\"Graph of a circle in the rectangular coordinate system - the vertical line test shows that the circle r^2 = x^2 + y^2 is not a function. The dotted red vertical line intersects the function in two places - it should only intersect in one place to be a function.\" width=\"487\" height=\"291\" \/><\/p>\n<p>However, if we were to graph each equation on its own, each one would pass the vertical line test and therefore would represent a function. In some instances, the concept of breaking up the equation for a circle into two functions is similar to the concept of creating parametric equations, as we use two functions to produce a non-function. This will become clearer as we move forward.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>parametric equations<\/h3>\n<p>Suppose [latex]t[\/latex] is a number on an interval, [latex]I[\/latex]. The set of ordered pairs, [latex]\\left(x\\left(t\\right),y\\left(t\\right)\\right)[\/latex], where [latex]x=f\\left(t\\right)[\/latex] and [latex]y=g\\left(t\\right)[\/latex], forms a plane curve based on the parameter [latex]t[\/latex]. The equations [latex]x=f\\left(t\\right)[\/latex] and [latex]y=g\\left(t\\right)[\/latex] are the parametric equations.<\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<h3><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; font-size: 16px; font-weight: 400;\">Parameterize the curve [latex]y={x}^{2}-1[\/latex] letting [latex]x\\left(t\\right)=t[\/latex]. Graph both equations.<\/span><\/h3>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q910055\">Show Solution<\/button><\/p>\n<div id=\"q910055\" class=\"hidden-answer\" style=\"display: none\">\n<p>If [latex]x\\left(t\\right)=t[\/latex], then to find [latex]y\\left(t\\right)[\/latex] we replace the variable [latex]x[\/latex] with the expression given in [latex]x\\left(t\\right)[\/latex]. In other words, [latex]y\\left(t\\right)={t}^{2}-1[\/latex]. Make a table of values similar to the table below, and sketch the graph.<\/p>\n<table id=\"Table_08_06_001\" summary=\"Ten rows and three columns. First column is labeled t, second column is labeled x(t), third column is labeled y(t). The table has ordered triples of each of these row values: (-4,-4, y(-4)=(-4)^2 - 1 = 15), (-3,-3, y(-3)= (-3)^2 -1 = 8), (-2,-2, y(-2) = (-2)^2 -1 = 3), (-1,-1, y(-1)= (-1)^2 -1 = 0), (0,0, y(0) = (0)^2 -1 = -1), (1,1, y(1) = (1)^2 -1 = 0), (2,2, y(2) = (2)^2 -1 =3), (3,3, y(3) = (3)^2 - 1 = 8), (4,4, y(4) = (4)^2 - 1 = 15).\">\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x\\left(t\\right)[\/latex]<\/th>\n<th>[latex]y\\left(t\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]-4[\/latex]<\/td>\n<td>[latex]-4[\/latex]<\/td>\n<td>[latex]y\\left(-4\\right)={\\left(-4\\right)}^{2}-1=15[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]y\\left(-3\\right)={\\left(-3\\right)}^{2}-1=8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]y\\left(-2\\right)={\\left(-2\\right)}^{2}-1=3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]y\\left(-1\\right)={\\left(-1\\right)}^{2}-1=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]y\\left(0\\right)={\\left(0\\right)}^{2}-1=-1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]y\\left(1\\right)={\\left(1\\right)}^{2}-1=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]y\\left(2\\right)={\\left(2\\right)}^{2}-1=3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]y\\left(3\\right)={\\left(3\\right)}^{2}-1=8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]y\\left(4\\right)={\\left(4\\right)}^{2}-1=15[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>It may be helpful to use the TRACE feature of a graphing calculator to see how the points are generated as [latex]t[\/latex] increases.<\/p>\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180919\/CNX_Precalc_Figure_08_06_0152.jpg\" alt=\"Graph of a parabola in two forms: a parametric equation and rectangular coordinates. It is the same function, just different ways of writing it.\" width=\"731\" height=\"291\" \/><figcaption class=\"wp-caption-text\">(a) Parametric [latex]y\\left(t\\right)={t}^{2}-1[\/latex] (b) Rectangular [latex]y={x}^{2}-1[\/latex]<\/figcaption><\/figure>\n<p><strong>Analysis of the Solution<\/strong>The arrows indicate the direction in which the curve is generated. Notice the curve is identical to the curve of [latex]y={x}^{2}-1[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">Construct a table of values and plot the parametric equations: [latex]x\\left(t\\right)=t - 3,y\\left(t\\right)=2t+4;-1\\le t\\le 2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q217668\">Show Solution<\/button><\/p>\n<div id=\"q217668\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"fs-id1165137810323\" class=\"unnumbered\" summary=\"Five rows and three columns. First column is labeled t, second column is labeled x(t), third column is labeled y(t). The table has ordered triples of each of these row values: (-1, -4, 2), (0,-3,4), (1,-2,6), (2,-1,8).\">\n<tbody>\n<tr>\n<td>[latex]t[\/latex]<\/td>\n<td>[latex]x\\left(t\\right)[\/latex]<\/td>\n<td>[latex]y\\left(t\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]-4[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180940\/CNX_Precalc_Figure_08_06_0062.jpg\" alt=\"\" \/><\/p>\n<\/div>\n<\/div>\n<\/section>\n<dl id=\"fs-id1165135186874\" class=\"definition\">\n<dd id=\"fs-id1165134357588\"><\/dd>\n<\/dl>\n","protected":false},"author":6,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":520,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/227"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/227\/revisions"}],"predecessor-version":[{"id":4784,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/227\/revisions\/4784"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/520"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/227\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=227"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=227"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=227"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=227"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}