{"id":122974,"date":"2022-05-30T09:16:51","date_gmt":"2022-05-30T14:16:51","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=122974"},"modified":"2026-03-28T11:53:19","modified_gmt":"2026-03-28T16:53:19","slug":"converting-between-standard-form-and-vertex-form","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/converting-between-standard-form-and-vertex-form\/","title":{"rendered":"Converting Between Standard Form and Vertex Form"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_PQY7eTWBp9w\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_PQY7eTWBp9w\" data-source-videoID=\"PQY7eTWBp9w\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Converting Between Standard Form and Vertex Form Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Converting Between Standard Form and Vertex Form\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_PQY7eTWBp9w:hover {cursor:pointer;} img#videoThumbnailImage_PQY7eTWBp9w {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/2471-thumbnail-1-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_PQY7eTWBp9w\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_PQY7eTWBp9w\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Converting Between Standard Form and Vertex Form\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_PQY7eTWBp9w\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_PQY7eTWBp9w\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_PQY7eTWBp9w\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction JuC_Function() {\n  var x = document.getElementById(\"JuC\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"JuC_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"JuC\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Standard_Form\" class=\"smooth-scroll\">Standard Form<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Vertex_Form\" class=\"smooth-scroll\">Vertex Form<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Converting_Between_Forms\" class=\"smooth-scroll\">Converting Between Forms<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hello! Welcome to this video on converting between <strong>standard form and vertex form of a quadratic equation<\/strong>. Before we start converting, let\u2019s review what these forms look like.<\/p>\n<h2><span id=\"Standard_Form\" class=\"m-toc-anchor\"><\/span>Standard Form<\/h2>\n<p>\nStandard form of a quadratic equation is:<\/p>\n<div class=\"examplesentence\">\\(y=ax^{2}+bx+c\\)<\/div>\n<p>\n&nbsp;<br \/>\nWhere \\(a\\), \\(b\\), and \\(c\\) are real numbers, and \\(a\\neq  0\\).<\/p>\n<h2><span id=\"Vertex_Form\" class=\"m-toc-anchor\"><\/span>Vertex Form<\/h2>\n<p>\nVertex form of a quadratic equation is:<\/p>\n<div class=\"examplesentence\">\\(y=a(x-h)^{2}+k\\)<\/div>\n<p>\n&nbsp;<br \/>\nWhere \\(a\\), \\(h\\), and \\(k\\) are real numbers, \\(a\\neq 0\\), and \\((h,k)\\) is the <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/vertex-of-a-parabola\/\">vertex of the parabola<\/a>.<\/p>\n<h2><span id=\"Converting_Between_Forms\" class=\"m-toc-anchor\"><\/span>Converting Between Forms<\/h2>\n<p>\nNow that we\u2019ve reviewed these forms, let\u2019s start by converting a standard form equation to vertex form. <\/p>\n<h3><span id=\"Example_1\" class=\"m-toc-anchor\"><\/span>Example #1<\/h3>\n<p>\nConvert the standard form equation \\(y=x^{2}+14x-9\\) to vertex form.<\/p>\n<p>In order to do this, we need to complete the square. To complete the square, \\(a\\) must equal 1. In this equation, \\(a\\) is already 1, so we can move on to the next step.<\/p>\n<p>Find the \\(b\\)-value, divide it by 2, and square it.<\/p>\n<div class=\"examplesentence\">\n\\((\\frac{b}{2})^{2}=(\\frac{14}{2})^{2}=(7)^{2}=49\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow, add and subtract this value after the \\(x\\)-term.<\/p>\n<div class=\"examplesentence\">\\(y=x^{2}+14x+49-49-9\\)<\/div>\n<p>\n&nbsp;<br \/>\nI\u2019m going to add parentheses around these first three terms because this will be our perfect square trinomial. <\/p>\n<div class=\"examplesentence\">\\(y=(x^{2}+14x+49)-49-9\\)<\/div>\n<p>\n&nbsp;<br \/>\nThis trinomial factors to the perfect square \\((x+7)^{2}\\).<\/p>\n<div class=\"examplesentence\">\\(y=(x+7)^{2}-49-9\\)<\/div>\n<p>\n&nbsp;<br \/>\nRemember, the point of completing the square is to get it to factor into something that looks like this. So now, all we have to do is combine our like terms right here.<\/p>\n<div class=\"examplesentence\">\\(y=(x+7)^{2}-58\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow, notice that we went through this <a class=\"ylist\" class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/completing-the-square\/\">completing the square<\/a> process fairly quickly. If you want some more help on this, see one of our other videos where we go more in depth over how to do this process.<\/p>\n<h3><span id=\"Example_2\" class=\"m-toc-anchor\"><\/span>Example #2<\/h3>\n<p>\nLet&#8217;s try another problem where we convert a standard form equation to vertex form.<\/p>\n<div class=\"examplesentence\">\\(y=-2x^{2}-4x+8\\)<\/div>\n<p>\n&nbsp;<br \/>\nRemember, in order to complete the square, the value of a must be 1. Factor a \u20132 out of the right side of the equation.<\/p>\n<div class=\"examplesentence\">\\(y=-2(x^{2}+2x-4)\\)<\/div>\n<p>\n&nbsp;<br \/>\nRemember, we factor \u20132 out of each of the terms. Now, identify the \\(b\\)-value, halve it, and square it. So, we want the \\(b\\)-value of this trinomial where a is 1 \\((x^{2}+2x-4)\\), so our \\(b\\)-value is 2.<\/p>\n<div class=\"examplesentence\">\\((\\frac{b}{2})^{2}=(\\frac{2}{2})^{2}=(1)^{2}=1\\)<\/div>\n<p>\n&nbsp;<br \/>\nAdd and subtract this value after the \\(x\\)-term.<\/p>\n<div class=\"examplesentence\">\\(y=-2(x^{2}+2x+1-1-4)\\)<\/div>\n<p>\n&nbsp;<br \/>\nPut parentheses around the first three terms to identify the perfect square trinomial.<\/p>\n<div class=\"examplesentence\">\\(y=\\)\\(-2((x^{2}+2x+1)-1-4)\\)<\/div>\n<p>\n&nbsp;<br \/>\nFactor the perfect square trinomial.<\/p>\n<div class=\"examplesentence\">\\(y=-2((x+1)^{2}-1-4)\\)<\/div>\n<p>\n&nbsp;<br \/>\nCombine like terms inside the parentheses.<\/p>\n<div class=\"examplesentence\">\\(y=-2((x+1)^{2}-5)\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow, the final step in simplifying this just a little bit more, is distributing the \u20132 inside this larger set of parentheses, so to the \\((x+1)^{2}\\) and to the \u20135.<\/p>\n<div class=\"examplesentence\">\\(y=-2(x+1)^{2}+10\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow let\u2019s move on to converting a vertex form equation to standard form. This conversion process is much simpler.<\/p>\n<p>Convert the equation \\(y=3(x-4)^{2}+7\\) to standard form.<\/p>\n<p>The first thing to do is expand the squared term.<\/p>\n<div class=\"examplesentence\">\\(y=3(x-4)(x-4)+7\\)<\/div>\n<p>\n&nbsp;<br \/>\nAnd then we can <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/multiplying-terms-using-the-foil-method\/\">FOIL<\/a> to simplify this part.<\/p>\n<div class=\"examplesentence\">\\(y=3(x^{2}-4x-4x+16)+7\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow, we can combine like terms.<\/p>\n<div class=\"examplesentence\">\\(y=3(x^{2}-8x+16)+7\\)<\/div>\n<p>\n&nbsp;<br \/>\nNow, distribute 3 to the trinomial.<\/p>\n<div class=\"examplesentence\">\\(y=3x^{2}-24x+48+7\\)<\/div>\n<p>\n&nbsp;<br \/>\nFinally, combine like terms.<\/p>\n<div class=\"examplesentence\">\\(y=3x^{2}-24x+55\\)<\/div>\n<p>\n&nbsp;<br \/>\nAnd there\u2019s our equation in standard form.<\/p>\n<p>Let\u2019s try one more example before we go.<\/p>\n<p>Convert the equation \\(y=-11(x+8)2-9\\) to standard form.<\/p>\n<p>Start by expanding the squared binomial.<\/p>\n<div class=\"examplesentence\">\\(y=-11(x+8)(x+8)-9\\)<\/div>\n<p>\n&nbsp;<br \/>\nAnd again, we can FOIL these two terms.<\/p>\n<div class=\"examplesentence\">\\(y=\\)\\(-11(x^{2}+8x+8x+64)-9\\)<\/div>\n<p>\n&nbsp;<br \/>\nWe can simplify by combining like terms.<\/p>\n<div class=\"examplesentence\">\\(y=-11(x^{2}+16x+64)-9\\)<\/div>\n<p>\n&nbsp;<br \/>\nThen, distribute \u201311 to the trinomial.<\/p>\n<div class=\"examplesentence\">\\(y=-11x^{2}-176x-704-9\\)<\/div>\n<p>\n&nbsp;<br \/>\nFinally, combine like terms.<\/p>\n<div class=\"examplesentence\">\\(y=-11x^{2}-176x-713\\)<\/div>\n<p>\n&nbsp;<br \/>\nAnd there you have it!<\/p>\n<p>I hope this video on converting between standard form and vertex form of a quadratic equation was helpful. 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