{"id":8152,"date":"2023-09-20T17:51:07","date_gmt":"2023-09-20T17:51:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=8152"},"modified":"2025-08-28T04:39:21","modified_gmt":"2025-08-28T04:39:21","slug":"quadratic-functions-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/quadratic-functions-learn-it-1\/","title":{"raw":"Quadratic Functions: Learn It 1","rendered":"Quadratic Functions: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Identify quadratic functions in both general and standard form<\/li>\r\n\t<li>Determine the domain and range of a quadratic function by recognizing whether the vertex represents a maximum or minimum point<\/li>\r\n\t<li>Recognize key features of a parabola's graph: vertex, axis of symmetry, y-intercept, and minimum or maximum value<\/li>\r\n\t<li>Create graphs of quadratic functions using tables and transformations<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Equations of Quadratic Functions<\/h2>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>general form of a quadratic function<\/h3>\r\n<p>The <strong>general form of a quadratic function<\/strong> presents the function in the form<\/p>\r\n<p>&nbsp;<\/p>\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex]<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>where [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex]\u00a0are real numbers and [latex]a\\ne 0[\/latex].<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>If [latex]a&gt;0[\/latex], the parabola opens upward. If [latex]a&lt;0[\/latex], the parabola opens downward.<\/p>\r\n<\/section>\r\n<p>We can use the general form of a parabola to find the equation for the <strong>axis of symmetry<\/strong>.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>axis of symmetry<\/h3>\r\n<p>The axis of symmetry is defined by [latex]x=-\\dfrac{b}{2a}[\/latex]. If we use the quadratic formula, [latex]x=\\dfrac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex], to solve [latex]a{x}^{2}+bx+c=0[\/latex] for the [latex]x[\/latex]-intercepts, or zeros, we find the value of [latex]x[\/latex]\u00a0halfway between them is always [latex]x=-\\dfrac{b}{2a}[\/latex], the equation for the axis of symmetry.<\/p>\r\n<\/section>\r\n<p>The figure below shows\u00a0the graph of the quadratic function written in general form as [latex]y={x}^{2}+4x+3[\/latex]. In this form, [latex]a=1,\\text{ }b=4[\/latex], and [latex]c=3[\/latex]. Because [latex]a&gt;0[\/latex], the parabola opens upward. The axis of symmetry is [latex]x=-\\dfrac{4}{2\\left(1\\right)}=-2[\/latex]. This also makes sense because we can see from the graph that the vertical line [latex]x=-2[\/latex] divides the graph in half. The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph, in this instance, [latex]\\left(-2,-1\\right)[\/latex]. The [latex]x[\/latex]-intercepts, those points where the parabola crosses the [latex]x[\/latex]-axis, occur at [latex]\\left(-3,0\\right)[\/latex] and [latex]\\left(-1,0\\right)[\/latex].<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170335\/CNX_Precalc_Figure_03_02_0042.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are for the function y=x^2+4x+3.\" width=\"487\" height=\"555\" \/> Figure 1. A graph of the function [latex]y={x}^{2}+4x+3[\/latex][\/caption]\r\n<\/center>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>standard form of a quadratic function<\/h3>\r\n<p>The <strong>standard form of a quadratic function<\/strong> presents the function in the form<\/p>\r\n<p>&nbsp;<\/p>\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the <strong>vertex<\/strong>.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>Because the vertex appears in the standard form of the quadratic function, this form is also known as the <strong>vertex form of a quadratic function<\/strong>.<\/p>\r\n<\/section>\r\n<h3>Given a quadratic function in general form, find the vertex of the parabola.<\/h3>\r\n<p>One reason we may want to identify the <strong>vertex <\/strong>of the parabola is that this point will inform us where the maximum or minimum value of the output occurs, [latex]k[\/latex], and where it occurs, [latex]h[\/latex].<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>vertex of the parabola<\/h3>\r\n<p>If we are given the general form of a quadratic function:<\/p>\r\n<p>&nbsp;<\/p>\r\n<p style=\"text-align: center;\">[latex]f(x)=ax^2+bx+c[\/latex]<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>We can define the vertex, [latex](h,k)[\/latex], by doing the following:<\/p>\r\n<p>&nbsp;<\/p>\r\n<ul>\r\n\t<li>Identify [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex].<\/li>\r\n\t<li>Find [latex]h[\/latex], the [latex]x[\/latex]-coordinate of the vertex, by substituting [latex]a[\/latex] and [latex]b[\/latex]\u00a0into [latex]h=-\\dfrac{b}{2a}[\/latex].<\/li>\r\n\t<li>Find [latex]k[\/latex], the [latex]y[\/latex]-coordinate of the vertex, by evaluating [latex]k=f\\left(h\\right)=f\\left(-\\dfrac{b}{2a}\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Find the vertex of the quadratic function [latex]f\\left(x\\right)=2{x}^{2}-6x+7[\/latex]. Rewrite the quadratic in standard form (vertex form).<\/p>\r\n<p>[reveal-answer q=\"466886\"]Show Solution[\/reveal-answer]<\/p>\r\n<p>[hidden-answer a=\"466886\"]<\/p>\r\n<p>The horizontal coordinate of the vertex will be at<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}h&amp;=-\\dfrac{b}{2a}\\ \\\\[2mm] &amp;=-\\dfrac{-6}{2\\left(2\\right)} \\\\[2mm]&amp;=\\dfrac{6}{4} \\\\[2mm]&amp;=\\dfrac{3}{2} \\end{align}[\/latex]<\/p>\r\n<p>The vertical coordinate of the vertex will be at<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}k&amp;=f\\left(h\\right) \\\\[2mm]&amp;=f\\left(\\dfrac{3}{2}\\right) \\\\[2mm]&amp;=2{\\left(\\dfrac{3}{2}\\right)}^{2}-6\\left(\\dfrac{3}{2}\\right)+7 \\\\[2mm]&amp;=\\dfrac{5}{2}\\end{align}[\/latex]<\/p>\r\n<p>So the vertex is [latex]\\left(\\dfrac{3}{2},\\dfrac{5}{2}\\right)[\/latex]<\/p>\r\n<p>Rewriting into standard form, the stretch factor will be the same as the [latex]a[\/latex] in the original quadratic.<\/p>\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=2{\\left(x-\\frac{3}{2}\\right)}^{2}+\\frac{5}{2}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]13811[\/ohm2_question]<\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Identify quadratic functions in both general and standard form<\/li>\n<li>Determine the domain and range of a quadratic function by recognizing whether the vertex represents a maximum or minimum point<\/li>\n<li>Recognize key features of a parabola&#8217;s graph: vertex, axis of symmetry, y-intercept, and minimum or maximum value<\/li>\n<li>Create graphs of quadratic functions using tables and transformations<\/li>\n<\/ul>\n<\/section>\n<h2>Equations of Quadratic Functions<\/h2>\n<section class=\"textbox keyTakeaway\">\n<h3>general form of a quadratic function<\/h3>\n<p>The <strong>general form of a quadratic function<\/strong> presents the function in the form<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>where [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex]\u00a0are real numbers and [latex]a\\ne 0[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>If [latex]a>0[\/latex], the parabola opens upward. If [latex]a<0[\/latex], the parabola opens downward.<\/p>\n<\/section>\n<p>We can use the general form of a parabola to find the equation for the <strong>axis of symmetry<\/strong>.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>axis of symmetry<\/h3>\n<p>The axis of symmetry is defined by [latex]x=-\\dfrac{b}{2a}[\/latex]. If we use the quadratic formula, [latex]x=\\dfrac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex], to solve [latex]a{x}^{2}+bx+c=0[\/latex] for the [latex]x[\/latex]-intercepts, or zeros, we find the value of [latex]x[\/latex]\u00a0halfway between them is always [latex]x=-\\dfrac{b}{2a}[\/latex], the equation for the axis of symmetry.<\/p>\n<\/section>\n<p>The figure below shows\u00a0the graph of the quadratic function written in general form as [latex]y={x}^{2}+4x+3[\/latex]. In this form, [latex]a=1,\\text{ }b=4[\/latex], and [latex]c=3[\/latex]. Because [latex]a>0[\/latex], the parabola opens upward. The axis of symmetry is [latex]x=-\\dfrac{4}{2\\left(1\\right)}=-2[\/latex]. This also makes sense because we can see from the graph that the vertical line [latex]x=-2[\/latex] divides the graph in half. The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph, in this instance, [latex]\\left(-2,-1\\right)[\/latex]. The [latex]x[\/latex]-intercepts, those points where the parabola crosses the [latex]x[\/latex]-axis, occur at [latex]\\left(-3,0\\right)[\/latex] and [latex]\\left(-1,0\\right)[\/latex].<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170335\/CNX_Precalc_Figure_03_02_0042.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are for the function y=x^2+4x+3.\" width=\"487\" height=\"555\" \/><figcaption class=\"wp-caption-text\">Figure 1. A graph of the function [latex]y={x}^{2}+4x+3[\/latex]<\/figcaption><\/figure>\n<\/div>\n<section class=\"textbox keyTakeaway\">\n<h3>standard form of a quadratic function<\/h3>\n<p>The <strong>standard form of a quadratic function<\/strong> presents the function in the form<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the <strong>vertex<\/strong>.<\/p>\n<p>&nbsp;<\/p>\n<p>Because the vertex appears in the standard form of the quadratic function, this form is also known as the <strong>vertex form of a quadratic function<\/strong>.<\/p>\n<\/section>\n<h3>Given a quadratic function in general form, find the vertex of the parabola.<\/h3>\n<p>One reason we may want to identify the <strong>vertex <\/strong>of the parabola is that this point will inform us where the maximum or minimum value of the output occurs, [latex]k[\/latex], and where it occurs, [latex]h[\/latex].<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>vertex of the parabola<\/h3>\n<p>If we are given the general form of a quadratic function:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=ax^2+bx+c[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>We can define the vertex, [latex](h,k)[\/latex], by doing the following:<\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li>Identify [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex].<\/li>\n<li>Find [latex]h[\/latex], the [latex]x[\/latex]-coordinate of the vertex, by substituting [latex]a[\/latex] and [latex]b[\/latex]\u00a0into [latex]h=-\\dfrac{b}{2a}[\/latex].<\/li>\n<li>Find [latex]k[\/latex], the [latex]y[\/latex]-coordinate of the vertex, by evaluating [latex]k=f\\left(h\\right)=f\\left(-\\dfrac{b}{2a}\\right)[\/latex]<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">\n<p>Find the vertex of the quadratic function [latex]f\\left(x\\right)=2{x}^{2}-6x+7[\/latex]. Rewrite the quadratic in standard form (vertex form).<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q466886\">Show Solution<\/button><\/p>\n<div id=\"q466886\" class=\"hidden-answer\" style=\"display: none\">\n<p>The horizontal coordinate of the vertex will be at<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}h&=-\\dfrac{b}{2a}\\ \\\\[2mm] &=-\\dfrac{-6}{2\\left(2\\right)} \\\\[2mm]&=\\dfrac{6}{4} \\\\[2mm]&=\\dfrac{3}{2} \\end{align}[\/latex]<\/p>\n<p>The vertical coordinate of the vertex will be at<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}k&=f\\left(h\\right) \\\\[2mm]&=f\\left(\\dfrac{3}{2}\\right) \\\\[2mm]&=2{\\left(\\dfrac{3}{2}\\right)}^{2}-6\\left(\\dfrac{3}{2}\\right)+7 \\\\[2mm]&=\\dfrac{5}{2}\\end{align}[\/latex]<\/p>\n<p>So the vertex is [latex]\\left(\\dfrac{3}{2},\\dfrac{5}{2}\\right)[\/latex]<\/p>\n<p>Rewriting into standard form, the stretch factor will be the same as the [latex]a[\/latex] in the original quadratic.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=2{\\left(x-\\frac{3}{2}\\right)}^{2}+\\frac{5}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm13811\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13811&theme=lumen&iframe_resize_id=ohm13811&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":15,"menu_order":30,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Jay Abramson\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":72,"module-header":"learn_it","content_attributions":[{"type":"original","description":"","author":"","organization":"Lumen Learning","url":"","project":"","license":"cc-by","license_terms":""},{"type":"cc","description":"College Algebra","author":"Jay Abramson","organization":"OpenStax","url":"","project":"","license":"cc-by","license_terms":"Access for free at https:\/\/openstax.org\/books\/college-algebra\/pages\/1-introduction-to-prerequisites"}],"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\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8152"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":24,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8152\/revisions"}],"predecessor-version":[{"id":15850,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8152\/revisions\/15850"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/72"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/8152\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=8152"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=8152"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=8152"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=8152"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}