{"id":33906,"date":"2022-04-27T22:12:42","date_gmt":"2022-04-28T02:12:42","guid":{"rendered":"https:\/\/nstem.org\/staging\/?p=33906"},"modified":"2022-04-27T22:14:15","modified_gmt":"2022-04-28T02:14:15","slug":"a-comprehensive-guide-to-parametric-equations","status":"publish","type":"post","link":"https:\/\/nstem.org\/staging\/2022\/04\/a-comprehensive-guide-to-parametric-equations\/","title":{"rendered":"A Comprehensive Guide to Parametric Equations"},"content":{"rendered":"

In mathematics, parametric equations <\/span>describe <\/span>a pair of quantities as explicit functions of independent variables regarded as \u201cparameters.\u201d <\/span>They are most commonly used to define the coordinates of points that comprise a geometric artifact such as a <\/span>curve<\/span><\/a> or surface, in which case the equations are referred to as a \u201cparametric representation\u201d or \u201cparameterization of an object.\u201d In this article, we will learn to graph plane curves and eliminate the parameter.<\/span><\/p>\n

Although, before all else, we will consider the empirical use of parametric equations in order to gain a conceptual understanding of it outside of purely abstract notions.<\/span><\/p>\n

Imagine a particle moving along a curve <\/span>C<\/span><\/i>, shown below.<\/span><\/p>\n

\"\"
Image Credit-Estefania Olaiz<\/figcaption><\/figure>\n

It is virtually impossible to describe <\/span>C<\/span><\/i> with an equation in the form of <\/span>y = f(x)<\/span><\/i>. This is because it fails the Vertical Line Test, which allows us to determine whether a curve in a graph represents a function. However, the <\/span>x<\/span><\/i> and <\/span>y<\/span><\/i> coordinates of the particle are considered functions of time. Therefore, we can consider it as <\/span>x = f(t) and y = g(t)<\/span><\/i>. This should give you a practical idea of our topic of discussion.<\/span><\/p>\n

Graphing plane curves<\/b><\/p>\n

Graphing a plane curve involves plotting points in the rectangular coordinate system and connecting them with a smooth curve. To do this, we need to select values for<\/span> t<\/span><\/i> on a given interval, use the given parametric equations to calculate <\/span>x<\/span><\/i> and <\/span>y<\/span><\/i> for each value of <\/span>t<\/span><\/i>, and plot <\/span>(x, y)<\/span><\/i> in increasing order, as well as connect them with the smooth curve.<\/span><\/p>\n

Example:<\/b> Graph the plane curve defined by the parametric equations <\/span>x = t\u00b2 – 1, y = 2t, <\/span><\/i>and<\/span> -2 \u2264 t \u2264 2<\/span><\/i>.\u00a0<\/span><\/p>\n

Step 1.\u00a0 Select values from the given interval.<\/b> We will now choose integral values of <\/span>t<\/span><\/i> on the interval <\/span>-2 \u2264 t \u2264 2<\/span><\/i>. Let <\/span>t = -2, -1, 0, 1,<\/span><\/i> and <\/span>2<\/span><\/i>.<\/span><\/p>\n

Step 2.\u00a0 For each value of <\/b>t<\/i><\/b>, use the listed parametric equations to calculate <\/b>x<\/i><\/b> and <\/b>y<\/i><\/b>.<\/b> This is demonstrated by the table below, where the first column details the choices for parameter <\/span>t<\/span><\/i>, the following two columns depict the values for <\/span>x<\/span><\/i> and <\/span>y<\/span><\/i>, and the last column delineates the ordered pair <\/span>(x, y)<\/span><\/i>.<\/span><\/p>\n

\"\"
Image Credit-Estefania Olaiz<\/figcaption><\/figure>\n

Step 3. Plot the points <\/b>(x, y)<\/i><\/b> in order of increasing <\/b>t<\/i><\/b> and adjoin them with a smooth curve. <\/b>The plane curve defined by the parametric equations on the interval is loosely represented below.<\/span><\/p>\n

\"\"
Image Credit-Estefania Olaiz<\/figcaption><\/figure>\n

Note: The line is meant to be smoothed.\u00a0<\/span><\/i><\/p>\n

Eliminating the parameter<\/b><\/p>\n

The aforementioned graph shows <\/span>x = t\u00b2 – 1, y = 2t, -2 \u2264 t \u2264 2<\/span><\/i>. Even if we carefully scrutinize the equations, we may not be able to identify that a corresponding plane curve is a section of the parabola. By removing the parameter, we can create an equation that is equivalent to the parametric equations. To do this, we need to solve one equation for parameter <\/span>t<\/span><\/i> and substitute the value in the second equation.<\/span><\/p>\n

Example:<\/b> x = t\u00b2 – 1 <\/span><\/i>and<\/span> y = 2t.<\/span><\/i><\/p>\n

Step 1. Solve for parameter <\/b>t<\/i><\/b>.<\/b><\/p>\n

Solving for<\/span> y = 2t<\/span><\/i>, we get a result of <\/span>t = y\/2<\/span><\/i>.<\/span><\/p>\n

Step 2. Substitute for <\/b>t<\/i><\/b>.<\/b><\/p>\n

Using the value <\/span>t = y\/2<\/span><\/i> and the equation<\/span> x = t\u00b2 – 1<\/span><\/i>, we substitute. Therefore, we end up with <\/span>x = (y\/2)\u00b2 – 1<\/span><\/i>.<\/span><\/p>\n

In conclusion, we have just learned how to graph and eliminate parametric equations. A comprehensive approach to the subject will prove itself particularly relevant in a calculus class, more specifically, in an AP Calculus BC course.<\/span><\/p>\n

 <\/p>\n

Written By: Estefania Olaiz<\/span><\/a><\/p>\n

Date: February 28, 2022<\/span><\/p>\n

Sources:<\/b><\/p>\n

1. https:\/\/www.britannica.com\/science\/parametric-equation<\/span><\/a><\/p>\n

2. https:\/\/en.wikipedia.org\/wiki\/Parametric_equation<\/span><\/a><\/p>\n

3. https:\/\/tutorial.math.lamar.edu\/classes\/calcii\/parametriceqn.aspx<\/span><\/a><\/p>\n

4. http:\/\/rfrith.uaa.alaska.edu\/M201\/Chapter10\/Chap10_Sec1.pdf<\/span>\u00a0<\/span><\/a><\/p>\n

5. https:\/\/fl01000126.schoolwires.net\/cms\/lib\/FL01000126\/Centricity\/Domain\/261\/Ch9_Section5.pdf<\/span> <\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

In mathematics, parametric equations describe a pair of quantities as explicit functions of independent variables regarded as \u201cparameters.\u201d They are most commonly used to define the coordinates of points that comprise a geometric artifact such as a curve or surface, in which case the equations are referred to as a \u201cparametric representation\u201d or \u201cparameterization of […]<\/p>\n","protected":false},"author":9,"featured_media":33907,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"pmpro_default_level":"","_kad_blocks_custom_css":"","_kad_blocks_head_custom_js":"","_kad_blocks_body_custom_js":"","_kad_blocks_footer_custom_js":"","_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[198],"tags":[492,491,211,199],"persona":[],"class_list":{"0":"post-33906","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-blog","8":"tag-college-and-university","9":"tag-high-school-9th-through-12th-grade","10":"tag-stem","11":"tag-stem-education","12":"pmpro-has-access","13":"entry"},"yoast_head":"\nA Comprehensive Guide to Parametric Equations National STEM\u2122 Honor Society<\/title>\n<meta name=\"description\" content=\"In mathematics, parametric equations describe a pair of quantities as explicit functions of independent variables regarded as \u201cparameters.\u201d They are most\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/nstem.org\/staging\/2022\/04\/a-comprehensive-guide-to-parametric-equations\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"A Comprehensive Guide to Parametric Equations\" \/>\n<meta property=\"og:description\" content=\"In mathematics, parametric equations describe a pair of quantities as explicit functions of independent variables regarded as \u201cparameters.\u201d They are most\" \/>\n<meta property=\"og:url\" content=\"https:\/\/nstem.org\/staging\/2022\/04\/a-comprehensive-guide-to-parametric-equations\/\" \/>\n<meta property=\"og:site_name\" content=\"Stage Site\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/NSTEMHS\/\" \/>\n<meta property=\"article:author\" content=\"https:\/\/www.facebook.com\/GloucesterHighNSW\/\" \/>\n<meta property=\"article:published_time\" content=\"2022-04-28T02:12:42+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2022-04-28T02:14:15+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/nstem.org\/staging\/wp-content\/uploads\/2022\/04\/1.png\" \/>\n\t<meta property=\"og:image:width\" content=\"512\" \/>\n\t<meta property=\"og:image:height\" content=\"334\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/png\" \/>\n<meta name=\"author\" content=\"Eric Magers\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@https:\/\/twitter.com\/ghs_info?lang=en\" \/>\n<meta name=\"twitter:site\" content=\"@NSTEMHS\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Eric Magers\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/nstem.org\\\/staging\\\/2022\\\/04\\\/a-comprehensive-guide-to-parametric-equations\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/nstem.org\\\/staging\\\/2022\\\/04\\\/a-comprehensive-guide-to-parametric-equations\\\/\"},\"author\":{\"name\":\"Eric Magers\",\"@id\":\"https:\\\/\\\/nstem.org\\\/staging\\\/#\\\/schema\\\/person\\\/7215c76e7aff48d0e1e8488b15d70adf\"},\"headline\":\"A Comprehensive Guide to Parametric Equations\",\"datePublished\":\"2022-04-28T02:12:42+00:00\",\"dateModified\":\"2022-04-28T02:14:15+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/nstem.org\\\/staging\\\/2022\\\/04\\\/a-comprehensive-guide-to-parametric-equations\\\/\"},\"wordCount\":639,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/nstem.org\\\/staging\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/nstem.org\\\/staging\\\/2022\\\/04\\\/a-comprehensive-guide-to-parametric-equations\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/nstem.org\\\/staging\\\/wp-content\\\/uploads\\\/2022\\\/04\\\/1.png\",\"keywords\":[\"College and University\",\"High School - 9th through 12th Grade\",\"STEM\",\"STEM Education\"],\"articleSection\":[\"Blog\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/nstem.org\\\/staging\\\/2022\\\/04\\\/a-comprehensive-guide-to-parametric-equations\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/nstem.org\\\/staging\\\/2022\\\/04\\\/a-comprehensive-guide-to-parametric-equations\\\/\",\"url\":\"https:\\\/\\\/nstem.org\\\/staging\\\/2022\\\/04\\\/a-comprehensive-guide-to-parametric-equations\\\/\",\"name\":\"A Comprehensive Guide to Parametric Equations National STEM\u2122 Honor Society\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/nstem.org\\\/staging\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/nstem.org\\\/staging\\\/2022\\\/04\\\/a-comprehensive-guide-to-parametric-equations\\\/#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/nstem.org\\\/staging\\\/2022\\\/04\\\/a-comprehensive-guide-to-parametric-equations\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/nstem.org\\\/staging\\\/wp-content\\\/uploads\\\/2022\\\/04\\\/1.png\",\"datePublished\":\"2022-04-28T02:12:42+00:00\",\"dateModified\":\"2022-04-28T02:14:15+00:00\",\"description\":\"In mathematics, parametric equations describe a pair of quantities as explicit functions of independent variables regarded as \u201cparameters.\u201d They are most\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/nstem.org\\\/staging\\\/2022\\\/04\\\/a-comprehensive-guide-to-parametric-equations\\\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/nstem.org\\\/staging\\\/2022\\\/04\\\/a-comprehensive-guide-to-parametric-equations\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/nstem.org\\\/staging\\\/2022\\\/04\\\/a-comprehensive-guide-to-parametric-equations\\\/#primaryimage\",\"url\":\"https:\\\/\\\/nstem.org\\\/staging\\\/wp-content\\\/uploads\\\/2022\\\/04\\\/1.png\",\"contentUrl\":\"https:\\\/\\\/nstem.org\\\/staging\\\/wp-content\\\/uploads\\\/2022\\\/04\\\/1.png\",\"width\":512,\"height\":334},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/nstem.org\\\/staging\\\/2022\\\/04\\\/a-comprehensive-guide-to-parametric-equations\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/nstem.org\\\/staging\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Blog\",\"item\":\"https:\\\/\\\/nstem.org\\\/staging\\\/category\\\/blog\\\/\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"A Comprehensive Guide to Parametric Equations\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/nstem.org\\\/staging\\\/#website\",\"url\":\"https:\\\/\\\/nstem.org\\\/staging\\\/\",\"name\":\"Stage Site\",\"description\":\"From Pre-K to Career\u2122\",\"publisher\":{\"@id\":\"https:\\\/\\\/nstem.org\\\/staging\\\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/nstem.org\\\/staging\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\\\/\\\/nstem.org\\\/staging\\\/#organization\",\"name\":\"National STEM Honor Society\u00ae\",\"url\":\"https:\\\/\\\/nstem.org\\\/staging\\\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/nstem.org\\\/staging\\\/#\\\/schema\\\/logo\\\/image\\\/\",\"url\":\"https:\\\/\\\/nstem.org\\\/staging\\\/wp-content\\\/uploads\\\/2018\\\/08\\\/cropped-logo-512.png\",\"contentUrl\":\"https:\\\/\\\/nstem.org\\\/staging\\\/wp-content\\\/uploads\\\/2018\\\/08\\\/cropped-logo-512.png\",\"width\":512,\"height\":512,\"caption\":\"National STEM Honor Society\u00ae\"},\"image\":{\"@id\":\"https:\\\/\\\/nstem.org\\\/staging\\\/#\\\/schema\\\/logo\\\/image\\\/\"},\"sameAs\":[\"https:\\\/\\\/www.facebook.com\\\/NSTEMHS\\\/\",\"https:\\\/\\\/x.com\\\/NSTEMHS\",\"https:\\\/\\\/www.instagram.com\\\/nstemhs\\\/\",\"https:\\\/\\\/www.linkedin.com\\\/company\\\/national-stem-honor-society\\\/\",\"https:\\\/\\\/www.youtube.com\\\/channel\\\/UCxLcsf1yw90kOZfbP8byp_w\\\/featured\"]},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/nstem.org\\\/staging\\\/#\\\/schema\\\/person\\\/7215c76e7aff48d0e1e8488b15d70adf\",\"name\":\"Eric Magers\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/a8c81c21c7bf0764319fee9105deac7f01749db5200ca37030bd70457351556d?s=96&d=mm&r=g\",\"url\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/a8c81c21c7bf0764319fee9105deac7f01749db5200ca37030bd70457351556d?s=96&d=mm&r=g\",\"contentUrl\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/a8c81c21c7bf0764319fee9105deac7f01749db5200ca37030bd70457351556d?s=96&d=mm&r=g\",\"caption\":\"Eric Magers\"},\"sameAs\":[\"https:\\\/\\\/ghs.gc.k12.va.us\\\/\",\"https:\\\/\\\/www.facebook.com\\\/GloucesterHighNSW\\\/\",\"https:\\\/\\\/www.instagram.com\\\/sphs6thform\\\/\",\"https:\\\/\\\/www.pinterest.com\\\/pin\\\/346214290110403010\\\/\",\"https:\\\/\\\/x.com\\\/https:\\\/\\\/twitter.com\\\/ghs_info?lang=en\",\"https:\\\/\\\/www.snapchat.com\\\/add\\\/basketballrecap?web_client_id=\",\"https:\\\/\\\/www.youtube.com\\\/@gloucesterhs4504\",\"https:\\\/\\\/www.tiktok.com\\\/discover\\\/gloucester-high-school-ottawa?lang=en\",\"https:\\\/\\\/www.youtube.com\\\/channel\\\/UCxLcsf1yw90kOZfbP8byp_w\"]}]}<\/script>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"A Comprehensive Guide to Parametric Equations National STEM\u2122 Honor Society","description":"In mathematics, parametric equations describe a pair of quantities as explicit functions of independent variables regarded as \u201cparameters.\u201d They are most","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/nstem.org\/staging\/2022\/04\/a-comprehensive-guide-to-parametric-equations\/","og_locale":"en_US","og_type":"article","og_title":"A Comprehensive Guide to Parametric Equations","og_description":"In mathematics, parametric equations describe a pair of quantities as explicit functions of independent variables regarded as \u201cparameters.\u201d They are most","og_url":"https:\/\/nstem.org\/staging\/2022\/04\/a-comprehensive-guide-to-parametric-equations\/","og_site_name":"Stage Site","article_publisher":"https:\/\/www.facebook.com\/NSTEMHS\/","article_author":"https:\/\/www.facebook.com\/GloucesterHighNSW\/","article_published_time":"2022-04-28T02:12:42+00:00","article_modified_time":"2022-04-28T02:14:15+00:00","og_image":[{"width":512,"height":334,"url":"https:\/\/nstem.org\/staging\/wp-content\/uploads\/2022\/04\/1.png","type":"image\/png"}],"author":"Eric Magers","twitter_card":"summary_large_image","twitter_creator":"@https:\/\/twitter.com\/ghs_info?lang=en","twitter_site":"@NSTEMHS","twitter_misc":{"Written by":"Eric Magers","Est. reading time":"4 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/nstem.org\/staging\/2022\/04\/a-comprehensive-guide-to-parametric-equations\/#article","isPartOf":{"@id":"https:\/\/nstem.org\/staging\/2022\/04\/a-comprehensive-guide-to-parametric-equations\/"},"author":{"name":"Eric Magers","@id":"https:\/\/nstem.org\/staging\/#\/schema\/person\/7215c76e7aff48d0e1e8488b15d70adf"},"headline":"A Comprehensive Guide to Parametric Equations","datePublished":"2022-04-28T02:12:42+00:00","dateModified":"2022-04-28T02:14:15+00:00","mainEntityOfPage":{"@id":"https:\/\/nstem.org\/staging\/2022\/04\/a-comprehensive-guide-to-parametric-equations\/"},"wordCount":639,"commentCount":0,"publisher":{"@id":"https:\/\/nstem.org\/staging\/#organization"},"image":{"@id":"https:\/\/nstem.org\/staging\/2022\/04\/a-comprehensive-guide-to-parametric-equations\/#primaryimage"},"thumbnailUrl":"https:\/\/nstem.org\/staging\/wp-content\/uploads\/2022\/04\/1.png","keywords":["College and University","High School - 9th through 12th Grade","STEM","STEM Education"],"articleSection":["Blog"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/nstem.org\/staging\/2022\/04\/a-comprehensive-guide-to-parametric-equations\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/nstem.org\/staging\/2022\/04\/a-comprehensive-guide-to-parametric-equations\/","url":"https:\/\/nstem.org\/staging\/2022\/04\/a-comprehensive-guide-to-parametric-equations\/","name":"A Comprehensive Guide to Parametric Equations National STEM\u2122 Honor Society","isPartOf":{"@id":"https:\/\/nstem.org\/staging\/#website"},"primaryImageOfPage":{"@id":"https:\/\/nstem.org\/staging\/2022\/04\/a-comprehensive-guide-to-parametric-equations\/#primaryimage"},"image":{"@id":"https:\/\/nstem.org\/staging\/2022\/04\/a-comprehensive-guide-to-parametric-equations\/#primaryimage"},"thumbnailUrl":"https:\/\/nstem.org\/staging\/wp-content\/uploads\/2022\/04\/1.png","datePublished":"2022-04-28T02:12:42+00:00","dateModified":"2022-04-28T02:14:15+00:00","description":"In mathematics, parametric equations describe a pair of quantities as explicit functions of independent variables regarded as \u201cparameters.\u201d They are most","breadcrumb":{"@id":"https:\/\/nstem.org\/staging\/2022\/04\/a-comprehensive-guide-to-parametric-equations\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/nstem.org\/staging\/2022\/04\/a-comprehensive-guide-to-parametric-equations\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/nstem.org\/staging\/2022\/04\/a-comprehensive-guide-to-parametric-equations\/#primaryimage","url":"https:\/\/nstem.org\/staging\/wp-content\/uploads\/2022\/04\/1.png","contentUrl":"https:\/\/nstem.org\/staging\/wp-content\/uploads\/2022\/04\/1.png","width":512,"height":334},{"@type":"BreadcrumbList","@id":"https:\/\/nstem.org\/staging\/2022\/04\/a-comprehensive-guide-to-parametric-equations\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/nstem.org\/staging\/"},{"@type":"ListItem","position":2,"name":"Blog","item":"https:\/\/nstem.org\/staging\/category\/blog\/"},{"@type":"ListItem","position":3,"name":"A Comprehensive Guide to Parametric Equations"}]},{"@type":"WebSite","@id":"https:\/\/nstem.org\/staging\/#website","url":"https:\/\/nstem.org\/staging\/","name":"Stage Site","description":"From Pre-K to Career\u2122","publisher":{"@id":"https:\/\/nstem.org\/staging\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/nstem.org\/staging\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/nstem.org\/staging\/#organization","name":"National STEM Honor Society\u00ae","url":"https:\/\/nstem.org\/staging\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/nstem.org\/staging\/#\/schema\/logo\/image\/","url":"https:\/\/nstem.org\/staging\/wp-content\/uploads\/2018\/08\/cropped-logo-512.png","contentUrl":"https:\/\/nstem.org\/staging\/wp-content\/uploads\/2018\/08\/cropped-logo-512.png","width":512,"height":512,"caption":"National STEM Honor Society\u00ae"},"image":{"@id":"https:\/\/nstem.org\/staging\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/NSTEMHS\/","https:\/\/x.com\/NSTEMHS","https:\/\/www.instagram.com\/nstemhs\/","https:\/\/www.linkedin.com\/company\/national-stem-honor-society\/","https:\/\/www.youtube.com\/channel\/UCxLcsf1yw90kOZfbP8byp_w\/featured"]},{"@type":"Person","@id":"https:\/\/nstem.org\/staging\/#\/schema\/person\/7215c76e7aff48d0e1e8488b15d70adf","name":"Eric Magers","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/secure.gravatar.com\/avatar\/a8c81c21c7bf0764319fee9105deac7f01749db5200ca37030bd70457351556d?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/a8c81c21c7bf0764319fee9105deac7f01749db5200ca37030bd70457351556d?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/a8c81c21c7bf0764319fee9105deac7f01749db5200ca37030bd70457351556d?s=96&d=mm&r=g","caption":"Eric Magers"},"sameAs":["https:\/\/ghs.gc.k12.va.us\/","https:\/\/www.facebook.com\/GloucesterHighNSW\/","https:\/\/www.instagram.com\/sphs6thform\/","https:\/\/www.pinterest.com\/pin\/346214290110403010\/","https:\/\/x.com\/https:\/\/twitter.com\/ghs_info?lang=en","https:\/\/www.snapchat.com\/add\/basketballrecap?web_client_id=","https:\/\/www.youtube.com\/@gloucesterhs4504","https:\/\/www.tiktok.com\/discover\/gloucester-high-school-ottawa?lang=en","https:\/\/www.youtube.com\/channel\/UCxLcsf1yw90kOZfbP8byp_w"]}]}},"taxonomy_info":{"category":[{"value":198,"label":"Blog"}],"post_tag":[{"value":492,"label":"College and University"},{"value":491,"label":"High School - 9th through 12th Grade"},{"value":211,"label":"STEM"},{"value":199,"label":"STEM Education"}]},"featured_image_src_large":["https:\/\/nstem.org\/staging\/wp-content\/uploads\/2022\/04\/1.png",512,334,false],"author_info":{"display_name":"Eric Magers","author_link":"https:\/\/nstem.org\/staging\/author\/magerse\/"},"comment_info":0,"category_info":[{"term_id":198,"name":"Blog","slug":"blog","term_group":0,"term_taxonomy_id":198,"taxonomy":"category","description":"","parent":0,"count":274,"filter":"raw","meta":[],"cat_ID":198,"category_count":274,"category_description":"","cat_name":"Blog","category_nicename":"blog","category_parent":0}],"tag_info":[{"term_id":492,"name":"College and University","slug":"college-and-university","term_group":0,"term_taxonomy_id":492,"taxonomy":"post_tag","description":"","parent":0,"count":51,"filter":"raw","meta":[]},{"term_id":491,"name":"High School - 9th through 12th Grade","slug":"high-school-9th-through-12th-grade","term_group":0,"term_taxonomy_id":491,"taxonomy":"post_tag","description":"","parent":0,"count":87,"filter":"raw","meta":[]},{"term_id":211,"name":"STEM","slug":"stem","term_group":0,"term_taxonomy_id":211,"taxonomy":"post_tag","description":"","parent":0,"count":57,"filter":"raw","meta":[]},{"term_id":199,"name":"STEM Education","slug":"stem-education","term_group":0,"term_taxonomy_id":199,"taxonomy":"post_tag","description":"","parent":0,"count":80,"filter":"raw","meta":[]}],"featured_image_src":"https:\/\/nstem.org\/staging\/wp-content\/uploads\/2022\/04\/1.png","featured_image_src_square":"https:\/\/nstem.org\/staging\/wp-content\/uploads\/2022\/04\/1.png","_links":{"self":[{"href":"https:\/\/nstem.org\/staging\/wp-json\/wp\/v2\/posts\/33906","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/nstem.org\/staging\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/nstem.org\/staging\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/nstem.org\/staging\/wp-json\/wp\/v2\/users\/9"}],"replies":[{"embeddable":true,"href":"https:\/\/nstem.org\/staging\/wp-json\/wp\/v2\/comments?post=33906"}],"version-history":[{"count":0,"href":"https:\/\/nstem.org\/staging\/wp-json\/wp\/v2\/posts\/33906\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/nstem.org\/staging\/wp-json\/wp\/v2\/media\/33907"}],"wp:attachment":[{"href":"https:\/\/nstem.org\/staging\/wp-json\/wp\/v2\/media?parent=33906"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/nstem.org\/staging\/wp-json\/wp\/v2\/categories?post=33906"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/nstem.org\/staging\/wp-json\/wp\/v2\/tags?post=33906"},{"taxonomy":"persona","embeddable":true,"href":"https:\/\/nstem.org\/staging\/wp-json\/wp\/v2\/persona?post=33906"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}