{"id":3279,"date":"2025-08-16T01:46:51","date_gmt":"2025-08-16T01:46:51","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3279"},"modified":"2025-10-21T18:00:38","modified_gmt":"2025-10-21T18:00:38","slug":"parametric-equations-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/parametric-equations-apply-it\/","title":{"raw":"Parametric Equations: Apply It","rendered":"Parametric Equations: Apply It"},"content":{"raw":"<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>\r\n<h2>Understanding Motion with Parametric Functions<\/h2>\r\n<div class=\"bcc-box bcc-success\"><section class=\"textbox example\" aria-label=\"Example\">An object travels at a steady rate along a straight path [latex]\\left(-5,3\\right)[\/latex] to [latex]\\left(3,-1\\right)[\/latex] in the same plane in four seconds. The coordinates are measured in meters. Find parametric equations for the position of the object.[reveal-answer q=\"419034\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"419034\"]\r\n\r\nThe parametric equations are simple linear expressions, but we need to view this problem in a step-by-step fashion. The <em>x<\/em>-value of the object starts at [latex]-5[\/latex] meters and goes to 3 meters. This means the distance <em>x<\/em> has changed by 8 meters in 4 seconds, which is a rate of [latex]\\frac{\\text{8 m}}{4\\text{ s}}[\/latex], or [latex]2\\text{m}\/\\text{s}[\/latex]. We can write the <em>x<\/em>-coordinate as a linear function with respect to time as [latex]x\\left(t\\right)=2t - 5[\/latex]. In the linear function template [latex]y=mx+b,2t=mx[\/latex] and [latex]-5=b[\/latex].\r\n\r\nSimilarly, the <em>y<\/em>-value of the object starts at 3 and goes to [latex]-1[\/latex], which is a change in the distance <em>y<\/em> of \u22124 meters in 4 seconds, which is a rate of [latex]\\frac{-4\\text{ m}}{4\\text{ s}}[\/latex], or [latex]-1\\text{m}\/\\text{s}[\/latex]. We can also write the <em>y<\/em>-coordinate as the linear function [latex]y\\left(t\\right)=-t+3[\/latex]. Together, these are the parametric equations for the position of the object, where [latex]x[\/latex]\u00a0and [latex]y[\/latex]\u00a0are expressed in meters and [latex]t[\/latex]\u00a0represents time:\r\n<p style=\"text-align: center;\">[latex]\\begin{align}x\\left(t\\right)&amp;=2t - 5 \\\\ y\\left(t\\right)&amp;=-t+3 \\end{align}[\/latex]<\/p>\r\nUsing these equations, we can build a table of values for [latex]t,x[\/latex], and [latex]y[\/latex]. In this example, we limited values of [latex]t[\/latex] to non-negative numbers. In general, any value of [latex]t[\/latex] can be used.\r\n<table id=\"Table_08_06_03\" summary=\"Six rows and three columns. First column is labeled t, second column is labeled x(t)=2t-5, third column is labeled y(t)=-t+3. The table has ordered triples of each of these row values: (0, x=2(0)-5 = -5, y=-(0) +3 = 3), (1, x=2(1)-5 = -3, y=-(1) + 3 = 2), (2, x=2(2) - 5 = -1, y=-(2) + 3 = 1), (3, x=2(3) - 5 = 1, y = -(3) + 3 =0), (4, x=2(4) -5 = 3, y=-(4) + 3 = -1).\">\r\n<thead>\r\n<tr>\r\n<th>[latex]t[\/latex]<\/th>\r\n<th>[latex]x\\left(t\\right)=2t - 5[\/latex]<\/th>\r\n<th>[latex]y\\left(t\\right)=-t+3[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]x=2\\left(0\\right)-5=-5[\/latex]<\/td>\r\n<td>[latex]y=-\\left(0\\right)+3=3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]x=2\\left(1\\right)-5=-3[\/latex]<\/td>\r\n<td>[latex]y=-\\left(1\\right)+3=2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]x=2\\left(2\\right)-5=-1[\/latex]<\/td>\r\n<td>[latex]y=-\\left(2\\right)+3=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]x=2\\left(3\\right)-5=1[\/latex]<\/td>\r\n<td>[latex]y=-\\left(3\\right)+3=0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]4[\/latex]<\/td>\r\n<td>[latex]x=2\\left(4\\right)-5=3[\/latex]<\/td>\r\n<td>[latex]y=-\\left(4\\right)+3=-1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFrom this table, we can create three graphs.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27180925\/CNX_Precalc_Figure_08_06_0032.jpg\" alt=\"Three graphs side by side. (A) has the horizontal position over time, (B) has the vertical position over time, and (C) has the position of the object in the plane at time t. See caption for more information.\" width=\"975\" height=\"514\" \/> (a) A graph of [latex]x[\/latex] vs. [latex]t[\/latex], representing the horizontal position over time. (b) A graph of [latex]y[\/latex] vs. [latex]t[\/latex], representing the vertical position over time. (c) A graph of [latex]y[\/latex] vs. [latex]x[\/latex], representing the position of the object in the plane at time [latex]t[\/latex].[\/caption]<strong>Analysis of the Solution<\/strong>\r\n\r\nAgain, we see that when the parameter represents time, we can indicate the movement of the object along the path with arrows.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">A robot vacuum moves in a straight line from position [latex]\\left(2, 1\\right)[\/latex] to position [latex]\\left(8, 4\\right)[\/latex] in 3 seconds. Coordinates are measured in feet. Find parametric equations for the robot's position.<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]x(t) =[\/latex] [response area]<\/p>\r\n<p class=\"whitespace-normal break-words\">[latex]y(t) =[\/latex] [response area]<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>Correct answer:<\/strong> [latex]x(t) = 2t + 2[\/latex], [latex]y(t) = t + 1[\/latex]<\/p>\r\n\r\n<\/section><\/div>","rendered":"<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<h2>Understanding Motion with Parametric Functions<\/h2>\n<div class=\"bcc-box bcc-success\">\n<section class=\"textbox example\" aria-label=\"Example\">An object travels at a steady rate along a straight path [latex]\\left(-5,3\\right)[\/latex] to [latex]\\left(3,-1\\right)[\/latex] in the same plane in four seconds. The coordinates are measured in meters. Find parametric equations for the position of the object.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q419034\">Show Solution<\/button><\/p>\n<div id=\"q419034\" class=\"hidden-answer\" style=\"display: none\">\n<p>The parametric equations are simple linear expressions, but we need to view this problem in a step-by-step fashion. The <em>x<\/em>-value of the object starts at [latex]-5[\/latex] meters and goes to 3 meters. This means the distance <em>x<\/em> has changed by 8 meters in 4 seconds, which is a rate of [latex]\\frac{\\text{8 m}}{4\\text{ s}}[\/latex], or [latex]2\\text{m}\/\\text{s}[\/latex]. We can write the <em>x<\/em>-coordinate as a linear function with respect to time as [latex]x\\left(t\\right)=2t - 5[\/latex]. In the linear function template [latex]y=mx+b,2t=mx[\/latex] and [latex]-5=b[\/latex].<\/p>\n<p>Similarly, the <em>y<\/em>-value of the object starts at 3 and goes to [latex]-1[\/latex], which is a change in the distance <em>y<\/em> of \u22124 meters in 4 seconds, which is a rate of [latex]\\frac{-4\\text{ m}}{4\\text{ s}}[\/latex], or [latex]-1\\text{m}\/\\text{s}[\/latex]. We can also write the <em>y<\/em>-coordinate as the linear function [latex]y\\left(t\\right)=-t+3[\/latex]. Together, these are the parametric equations for the position of the object, where [latex]x[\/latex]\u00a0and [latex]y[\/latex]\u00a0are expressed in meters and [latex]t[\/latex]\u00a0represents time:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}x\\left(t\\right)&=2t - 5 \\\\ y\\left(t\\right)&=-t+3 \\end{align}[\/latex]<\/p>\n<p>Using these equations, we can build a table of values for [latex]t,x[\/latex], and [latex]y[\/latex]. In this example, we limited values of [latex]t[\/latex] to non-negative numbers. In general, any value of [latex]t[\/latex] can be used.<\/p>\n<table id=\"Table_08_06_03\" summary=\"Six rows and three columns. First column is labeled t, second column is labeled x(t)=2t-5, third column is labeled y(t)=-t+3. The table has ordered triples of each of these row values: (0, x=2(0)-5 = -5, y=-(0) +3 = 3), (1, x=2(1)-5 = -3, y=-(1) + 3 = 2), (2, x=2(2) - 5 = -1, y=-(2) + 3 = 1), (3, x=2(3) - 5 = 1, y = -(3) + 3 =0), (4, x=2(4) -5 = 3, y=-(4) + 3 = -1).\">\n<thead>\n<tr>\n<th>[latex]t[\/latex]<\/th>\n<th>[latex]x\\left(t\\right)=2t - 5[\/latex]<\/th>\n<th>[latex]y\\left(t\\right)=-t+3[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]x=2\\left(0\\right)-5=-5[\/latex]<\/td>\n<td>[latex]y=-\\left(0\\right)+3=3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]x=2\\left(1\\right)-5=-3[\/latex]<\/td>\n<td>[latex]y=-\\left(1\\right)+3=2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]x=2\\left(2\\right)-5=-1[\/latex]<\/td>\n<td>[latex]y=-\\left(2\\right)+3=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]x=2\\left(3\\right)-5=1[\/latex]<\/td>\n<td>[latex]y=-\\left(3\\right)+3=0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]x=2\\left(4\\right)-5=3[\/latex]<\/td>\n<td>[latex]y=-\\left(4\\right)+3=-1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>From this table, we can create three graphs.<\/p>\n<figure style=\"width: 975px\" 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\/27180925\/CNX_Precalc_Figure_08_06_0032.jpg\" alt=\"Three graphs side by side. (A) has the horizontal position over time, (B) has the vertical position over time, and (C) has the position of the object in the plane at time t. See caption for more information.\" width=\"975\" height=\"514\" \/><figcaption class=\"wp-caption-text\">(a) A graph of [latex]x[\/latex] vs. [latex]t[\/latex], representing the horizontal position over time. (b) A graph of [latex]y[\/latex] vs. [latex]t[\/latex], representing the vertical position over time. (c) A graph of [latex]y[\/latex] vs. [latex]x[\/latex], representing the position of the object in the plane at time [latex]t[\/latex].<\/figcaption><\/figure>\n<p><strong>Analysis of the Solution<\/strong><\/p>\n<p>Again, we see that when the parameter represents time, we can indicate the movement of the object along the path with arrows.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-normal break-words\">A robot vacuum moves in a straight line from position [latex]\\left(2, 1\\right)[\/latex] to position [latex]\\left(8, 4\\right)[\/latex] in 3 seconds. Coordinates are measured in feet. Find parametric equations for the robot&#8217;s position.<\/p>\n<p class=\"whitespace-normal break-words\">[latex]x(t) =[\/latex] [response area]<\/p>\n<p class=\"whitespace-normal break-words\">[latex]y(t) =[\/latex] [response area]<\/p>\n<p class=\"whitespace-normal break-words\"><strong>Correct answer:<\/strong> [latex]x(t) = 2t + 2[\/latex], [latex]y(t) = t + 1[\/latex]<\/p>\n<\/section>\n<\/div>\n","protected":false},"author":67,"menu_order":9,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":520,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3279"}],"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\/67"}],"version-history":[{"count":3,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3279\/revisions"}],"predecessor-version":[{"id":4794,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3279\/revisions\/4794"}],"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\/3279\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3279"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3279"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3279"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3279"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}