{"id":1581,"date":"2025-07-25T03:55:38","date_gmt":"2025-07-25T03:55:38","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1581"},"modified":"2026-03-12T07:15:53","modified_gmt":"2026-03-12T07:15:53","slug":"graphing-parametric-equations-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphing-parametric-equations-fresh-take\/","title":{"raw":"Graphing Parametric Equations: Fresh Take","rendered":"Graphing Parametric Equations: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Graph plane curves described by parametric equations by plotting points.<\/li>\r\n \t<li>Graph parametric equations.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Parametric Equations and Their Graphs<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n<p class=\"whitespace-normal break-words\">Traditional functions [latex]y = f(x)[\/latex] work great for many curves, but what about loops, vertical lines, or paths that double back on themselves? Parametric equations solve this by letting both [latex]x[\/latex] and [latex]y[\/latex] depend on a third variable called a parameter.<\/p>\r\n<p class=\"whitespace-normal break-words\">In parametric equations, we write [latex]x = x(t)[\/latex] and [latex]y = y(t)[\/latex], where [latex]t[\/latex] is the parameter. Think of [latex]t[\/latex] as time on a stopwatch\u2014as [latex]t[\/latex] changes, both coordinates change simultaneously, tracing out a path through the plane.<\/p>\r\n<p class=\"whitespace-normal break-words\">The parameter [latex]t[\/latex] gives curves an orientation\u2014a direction of travel. As [latex]t[\/latex] increases, you can follow the path from start to finish with arrows showing which way you're moving.<\/p>\r\n<strong>The Process:<\/strong> Make a table with [latex]t[\/latex] values, calculate corresponding [latex]x[\/latex] and [latex]y[\/latex] coordinates, then plot and connect the points. The arrows show the direction as [latex]t[\/latex] increases.\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\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 <em data-effect=\"italics\">t<\/em> values from \u22123 to 2.<\/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\">\r\n\r\n<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 t = \u22123, the point (2, \u22121) is marked t = 0, and the point (8, 3) is marked t = 2. On the graph there are also written three equations: x(t) = 3t + 2, y(t) = t2 \u2212 1, and \u22123 \u2264 t \u2264 2.\" width=\"642\" height=\"423\" data-media-type=\"image\/jpeg\" \/>\r\n\r\n<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ggagfdee-tsnHL1Lb5MU\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/tsnHL1Lb5MU?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ggagfdee-tsnHL1Lb5MU\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661453&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ggagfdee-tsnHL1Lb5MU&vembed=0&video_id=tsnHL1Lb5MU&video_target=tpm-plugin-ggagfdee-tsnHL1Lb5MU'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Parametric+Curves+-+Basic+Graphing_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cParametric Curves - Basic Graphing\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Graph plane curves described by parametric equations by plotting points.<\/li>\n<li>Graph parametric equations.<\/li>\n<\/ul>\n<\/section>\n<h2>Parametric Equations and Their Graphs<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Traditional functions [latex]y = f(x)[\/latex] work great for many curves, but what about loops, vertical lines, or paths that double back on themselves? Parametric equations solve this by letting both [latex]x[\/latex] and [latex]y[\/latex] depend on a third variable called a parameter.<\/p>\n<p class=\"whitespace-normal break-words\">In parametric equations, we write [latex]x = x(t)[\/latex] and [latex]y = y(t)[\/latex], where [latex]t[\/latex] is the parameter. Think of [latex]t[\/latex] as time on a stopwatch\u2014as [latex]t[\/latex] changes, both coordinates change simultaneously, tracing out a path through the plane.<\/p>\n<p class=\"whitespace-normal break-words\">The parameter [latex]t[\/latex] gives curves an orientation\u2014a direction of travel. As [latex]t[\/latex] increases, you can follow the path from start to finish with arrows showing which way you&#8217;re moving.<\/p>\n<p><strong>The Process:<\/strong> Make a table with [latex]t[\/latex] values, calculate corresponding [latex]x[\/latex] and [latex]y[\/latex] coordinates, then plot and connect the points. The arrows show the direction as [latex]t[\/latex] increases.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558898\">Hint<\/button><\/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 <em data-effect=\"italics\">t<\/em> values from \u22123 to 2.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558897\">Show Solution<\/button><\/p>\n<div id=\"q44558897\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1169295453647\" data-type=\"solution\">\n<p><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 t = \u22123, the point (2, \u22121) is marked t = 0, and the point (8, 3) is marked t = 2. On the graph there are also written three equations: x(t) = 3t + 2, y(t) = t2 \u2212 1, and \u22123 \u2264 t \u2264 2.\" width=\"642\" height=\"423\" data-media-type=\"image\/jpeg\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ggagfdee-tsnHL1Lb5MU\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/tsnHL1Lb5MU?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ggagfdee-tsnHL1Lb5MU\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14661453&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ggagfdee-tsnHL1Lb5MU&#38;vembed=0&#38;video_id=tsnHL1Lb5MU&#38;video_target=tpm-plugin-ggagfdee-tsnHL1Lb5MU\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Parametric+Curves+-+Basic+Graphing_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cParametric Curves &#8211; Basic Graphing\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":14,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Parametric Curves - Basic Graphing\",\"author\":\"Patrick J\",\"organization\":\"Patrick JMT\",\"url\":\"https:\/\/youtu.be\/tsnHL1Lb5MU\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":520,"module-header":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"Parametric Curves - Basic Graphing","author":"Patrick J","organization":"Patrick JMT","url":"https:\/\/youtu.be\/tsnHL1Lb5MU","project":"","license":"arr","license_terms":"Standard YouTube License"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661453&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ggagfdee-tsnHL1Lb5MU&vembed=0&video_id=tsnHL1Lb5MU&video_target=tpm-plugin-ggagfdee-tsnHL1Lb5MU'><\/script>\n","media_targets":["tpm-plugin-ggagfdee-tsnHL1Lb5MU"]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1581"}],"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":11,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1581\/revisions"}],"predecessor-version":[{"id":5826,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1581\/revisions\/5826"}],"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\/1581\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1581"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1581"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1581"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1581"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}