{"id":1578,"date":"2025-07-25T03:54:53","date_gmt":"2025-07-25T03:54:53","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1578"},"modified":"2026-03-12T07:08:09","modified_gmt":"2026-03-12T07:08:09","slug":"parametric-equations-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/parametric-equations-fresh-take\/","title":{"raw":"Parametric Equations: Fresh Take","rendered":"Parametric Equations: Fresh Take"},"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 data-type=\"title\">Eliminating the Parameter<\/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\">Sometimes you want to figure out what familiar curve your parametric equations represent. Eliminating the parameter means getting rid of [latex]t[\/latex] to find a direct relationship between [latex]x[\/latex] and [latex]y[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\"><strong>The Basic Strategy:<\/strong> Solve one parametric equation for [latex]t[\/latex], then substitute that expression into the other equation. Choose whichever equation makes the algebra easier.<\/p>\r\n\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\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 <em data-effect=\"italics\">t<\/em> 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<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 t = 2 and the point (2.5, 5) is marked t = 6. On the graph there are also written three equations: x(t) = 2 + 3\/t, y(t) = t \u2212 1, and 2 \u2264 t \u2264 6.\" width=\"330\" height=\"310\" 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-hbaagdhg-IZ0fwB22tbs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/IZ0fwB22tbs?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hbaagdhg-IZ0fwB22tbs\" 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=14661432&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hbaagdhg-IZ0fwB22tbs&vembed=0&video_id=IZ0fwB22tbs&video_target=tpm-plugin-hbaagdhg-IZ0fwB22tbs'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/BC+Calculus+-+Eliminating+the+Parameter+with+Parametric+Equations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cBC Calculus: Eliminating the Parameter with Parametric Equations\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\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<\/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-ebhbfddc-v5dPU__12XA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/v5dPU__12XA?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ebhbfddc-v5dPU__12XA\" 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=14661433&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ebhbfddc-v5dPU__12XA&vembed=0&video_id=v5dPU__12XA&video_target=tpm-plugin-ebhbfddc-v5dPU__12XA'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Find+Two+Parametric+Equations+for+the+Given+Rectangular+Equation_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFind Two Parametric Equations for the Given Rectangular Equation\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>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 data-type=\"title\">Eliminating the Parameter<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">Sometimes you want to figure out what familiar curve your parametric equations represent. Eliminating the parameter means getting rid of [latex]t[\/latex] to find a direct relationship between [latex]x[\/latex] and [latex]y[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\"><strong>The Basic Strategy:<\/strong> Solve one parametric equation for [latex]t[\/latex], then substitute that expression into the other equation. Choose whichever equation makes the algebra easier.<\/p>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558893\">Hint<\/button><\/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 <em data-effect=\"italics\">t<\/em> and substitute into the other equation.<\/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=\"q44558894\">Show Solution<\/button><\/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<p><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 t = 2 and the point (2.5, 5) is marked t = 6. On the graph there are also written three equations: x(t) = 2 + 3\/t, y(t) = t \u2212 1, and 2 \u2264 t \u2264 6.\" width=\"330\" height=\"310\" 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-hbaagdhg-IZ0fwB22tbs\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/IZ0fwB22tbs?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hbaagdhg-IZ0fwB22tbs\" 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=14661432&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-hbaagdhg-IZ0fwB22tbs&#38;vembed=0&#38;video_id=IZ0fwB22tbs&#38;video_target=tpm-plugin-hbaagdhg-IZ0fwB22tbs\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/BC+Calculus+-+Eliminating+the+Parameter+with+Parametric+Equations_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cBC Calculus: Eliminating the Parameter with Parametric Equations\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558890\">Hint<\/button><\/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\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q44558891\">Show Solution<\/button><\/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<\/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-ebhbfddc-v5dPU__12XA\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/v5dPU__12XA?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ebhbfddc-v5dPU__12XA\" 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=14661433&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ebhbfddc-v5dPU__12XA&#38;vembed=0&#38;video_id=v5dPU__12XA&#38;video_target=tpm-plugin-ebhbfddc-v5dPU__12XA\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Find+Two+Parametric+Equations+for+the+Given+Rectangular+Equation_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cFind Two Parametric Equations for the Given Rectangular Equation\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"BC Calculus: Eliminating the Parameter with Parametric 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License"},{"type":"copyrighted_video","description":"Find Two Parametric Equations for the Given Rectangular Equation","author":"","organization":"Math and Stats Help","url":"https:\/\/youtu.be\/v5dPU__12XA","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=14661432&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hbaagdhg-IZ0fwB22tbs&vembed=0&video_id=IZ0fwB22tbs&video_target=tpm-plugin-hbaagdhg-IZ0fwB22tbs'><\/script>\n<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=14661433&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ebhbfddc-v5dPU__12XA&vembed=0&video_id=v5dPU__12XA&video_target=tpm-plugin-ebhbfddc-v5dPU__12XA'><\/script>\n","media_targets":["tpm-plugin-hbaagdhg-IZ0fwB22tbs","tpm-plugin-ebhbfddc-v5dPU__12XA"]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1578"}],"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":10,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1578\/revisions"}],"predecessor-version":[{"id":5824,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1578\/revisions\/5824"}],"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\/1578\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1578"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1578"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1578"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1578"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}