{"id":2310,"date":"2024-07-22T20:46:57","date_gmt":"2024-07-22T20:46:57","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2310"},"modified":"2025-08-15T15:31:34","modified_gmt":"2025-08-15T15:31:34","slug":"module-13-background-youll-need-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/module-13-background-youll-need-2\/","title":{"raw":"Systems of Equations and Inequalities: Background You'll Need 2","rendered":"Systems of Equations and Inequalities: Background You&#8217;ll Need 2"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li><span data-sheets-root=\"1\">Graph a quadratic function<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2><strong>Quadratic Functions<\/strong><\/h2>\r\nThe graph of a quadratic function is a U-shaped curve called a\u00a0<strong>parabola<\/strong>. One important feature of the graph is that it has an extreme point, called the\u00a0<strong>vertex<\/strong>. If the parabola opens up, the vertex represents the lowest point on the graph, or the\u00a0<strong>minimum value<\/strong>\u00a0of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the\u00a0<strong>maximum value<\/strong>. In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the\u00a0<strong>axis of symmetry<\/strong>.\r\n\r\n<center>\r\n\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\/02170330\/CNX_Precalc_Figure_03_02_0022.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are.\" width=\"487\" height=\"480\" \/> Graph of a parabola with key features labeled[\/caption]\r\n\r\n<\/center><center><\/center>&nbsp;\r\n\r\nThe [latex]y[\/latex]-intercept is the point at which the parabola crosses the [latex]y[\/latex]-axis. The [latex]x[\/latex]-intercepts are the points at which the parabola crosses the [latex]x[\/latex]-axis. If they exist, the [latex]x[\/latex]-intercepts represent the <strong>zeros<\/strong>, or <strong>roots<\/strong>, of the quadratic function, the values of [latex]x[\/latex]\u00a0at which [latex]y=0[\/latex].\r\n<table style=\"border-collapse: collapse; width: 100%;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 33.3333%;\"><\/td>\r\n<td style=\"width: 33.3333%;\">General Form: [latex]f(x) = ax^2 +bx +c[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">Standard Form:[latex]f(x) = a(x-h)^2+k[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">Orientation<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]a &gt; 0[\/latex] opens up;\r\n\r\n[latex]a &lt; 0[\/latex] opens down<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]a &gt; 0[\/latex] opens up;\r\n\r\n[latex]a &lt; 0[\/latex] opens down<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">Axis of symmetry<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]x = -\\frac{b}{2a}[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]x = h[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">Vertex<\/td>\r\n<td style=\"width: 33.3333%;\">[latex](-\\frac{b}{2a}, f(-\\frac{b}{2a}))[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex](h,k)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[caption id=\"attachment_2425\" align=\"aligncenter\" width=\"762\"]<img class=\"wp-image-2425 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/25214100\/Screenshot-2024-07-25-at-2.40.57%E2%80%AFPM.png\" alt=\"\" width=\"762\" height=\"332\" \/> Graph of two parabolas with axis of symmetry, orientation, and vertex labeled[\/caption]\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Graph the parabola [latex]f(x) = -x^2+4x-3[\/latex].[reveal-answer q=\"50705\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"50705\"]\r\n<ul>\r\n \t<li>Since [latex]a = -1[\/latex], the parabola opens downward.<\/li>\r\n \t<li>The axis of symmetry is [latex]x = -\\frac{b}{2a} = -\\frac{4}{2(-1)} = 2[\/latex].<\/li>\r\n \t<li>Vertex:\r\nLet's substitute [latex]x=2[\/latex] into the function: [latex]f(2)=-(2)^2+4(2)-3 = -4+8-3 = 1[\/latex].\r\nThus, the vertex is [latex](2, 1)[\/latex].<\/li>\r\n \t<li>We can also find additional points to help us graph. For example, the [latex]y[\/latex]-intercept is [latex]f(0) = -0^2+4(0)-3 = -3[\/latex]<\/li>\r\n<\/ul>\r\nGraph:\r\n\r\n[caption id=\"attachment_2426\" align=\"aligncenter\" width=\"399\"]<img class=\"wp-image-2426\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/25214550\/Screenshot-2024-07-25-at-2.45.46%E2%80%AFPM.png\" alt=\"\" width=\"399\" height=\"276\" \/> Graph of a parabola[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24834[\/ohm2_question]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm2_question hide_question_numbers=1]24835[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li><span data-sheets-root=\"1\">Graph a quadratic function<\/span><\/li>\n<\/ul>\n<\/section>\n<h2><strong>Quadratic Functions<\/strong><\/h2>\n<p>The graph of a quadratic function is a U-shaped curve called a\u00a0<strong>parabola<\/strong>. One important feature of the graph is that it has an extreme point, called the\u00a0<strong>vertex<\/strong>. If the parabola opens up, the vertex represents the lowest point on the graph, or the\u00a0<strong>minimum value<\/strong>\u00a0of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the\u00a0<strong>maximum value<\/strong>. In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the\u00a0<strong>axis of symmetry<\/strong>.<\/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\/02170330\/CNX_Precalc_Figure_03_02_0022.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are.\" width=\"487\" height=\"480\" \/><figcaption class=\"wp-caption-text\">Graph of a parabola with key features labeled<\/figcaption><\/figure>\n<\/div>\n<div style=\"text-align: center;\"><\/div>\n<p>&nbsp;<\/p>\n<p>The [latex]y[\/latex]-intercept is the point at which the parabola crosses the [latex]y[\/latex]-axis. The [latex]x[\/latex]-intercepts are the points at which the parabola crosses the [latex]x[\/latex]-axis. If they exist, the [latex]x[\/latex]-intercepts represent the <strong>zeros<\/strong>, or <strong>roots<\/strong>, of the quadratic function, the values of [latex]x[\/latex]\u00a0at which [latex]y=0[\/latex].<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<td style=\"width: 33.3333%;\"><\/td>\n<td style=\"width: 33.3333%;\">General Form: [latex]f(x) = ax^2 +bx +c[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">Standard Form:[latex]f(x) = a(x-h)^2+k[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">Orientation<\/td>\n<td style=\"width: 33.3333%;\">[latex]a > 0[\/latex] opens up;<\/p>\n<p>[latex]a < 0[\/latex] opens down<\/td>\n<td style=\"width: 33.3333%;\">[latex]a > 0[\/latex] opens up;<\/p>\n<p>[latex]a < 0[\/latex] opens down<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">Axis of symmetry<\/td>\n<td style=\"width: 33.3333%;\">[latex]x = -\\frac{b}{2a}[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]x = h[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">Vertex<\/td>\n<td style=\"width: 33.3333%;\">[latex](-\\frac{b}{2a}, f(-\\frac{b}{2a}))[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex](h,k)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<figure id=\"attachment_2425\" aria-describedby=\"caption-attachment-2425\" style=\"width: 762px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2425 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/25214100\/Screenshot-2024-07-25-at-2.40.57%E2%80%AFPM.png\" alt=\"\" width=\"762\" height=\"332\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/25214100\/Screenshot-2024-07-25-at-2.40.57%E2%80%AFPM.png 762w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/25214100\/Screenshot-2024-07-25-at-2.40.57%E2%80%AFPM-300x131.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/25214100\/Screenshot-2024-07-25-at-2.40.57%E2%80%AFPM-65x28.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/25214100\/Screenshot-2024-07-25-at-2.40.57%E2%80%AFPM-225x98.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/25214100\/Screenshot-2024-07-25-at-2.40.57%E2%80%AFPM-350x152.png 350w\" sizes=\"(max-width: 762px) 100vw, 762px\" \/><figcaption id=\"caption-attachment-2425\" class=\"wp-caption-text\">Graph of two parabolas with axis of symmetry, orientation, and vertex labeled<\/figcaption><\/figure>\n<section class=\"textbox example\" aria-label=\"Example\">Graph the parabola [latex]f(x) = -x^2+4x-3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q50705\">Show Answer<\/button><\/p>\n<div id=\"q50705\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>Since [latex]a = -1[\/latex], the parabola opens downward.<\/li>\n<li>The axis of symmetry is [latex]x = -\\frac{b}{2a} = -\\frac{4}{2(-1)} = 2[\/latex].<\/li>\n<li>Vertex:<br \/>\nLet&#8217;s substitute [latex]x=2[\/latex] into the function: [latex]f(2)=-(2)^2+4(2)-3 = -4+8-3 = 1[\/latex].<br \/>\nThus, the vertex is [latex](2, 1)[\/latex].<\/li>\n<li>We can also find additional points to help us graph. For example, the [latex]y[\/latex]-intercept is [latex]f(0) = -0^2+4(0)-3 = -3[\/latex]<\/li>\n<\/ul>\n<p>Graph:<\/p>\n<figure id=\"attachment_2426\" aria-describedby=\"caption-attachment-2426\" style=\"width: 399px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2426\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/25214550\/Screenshot-2024-07-25-at-2.45.46%E2%80%AFPM.png\" alt=\"\" width=\"399\" height=\"276\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/25214550\/Screenshot-2024-07-25-at-2.45.46%E2%80%AFPM.png 762w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/25214550\/Screenshot-2024-07-25-at-2.45.46%E2%80%AFPM-300x207.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/25214550\/Screenshot-2024-07-25-at-2.45.46%E2%80%AFPM-65x45.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/25214550\/Screenshot-2024-07-25-at-2.45.46%E2%80%AFPM-225x155.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/07\/25214550\/Screenshot-2024-07-25-at-2.45.46%E2%80%AFPM-350x242.png 350w\" sizes=\"(max-width: 399px) 100vw, 399px\" \/><figcaption id=\"caption-attachment-2426\" class=\"wp-caption-text\">Graph of a parabola<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24834\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24834&theme=lumen&iframe_resize_id=ohm24834&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm24835\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=24835&theme=lumen&iframe_resize_id=ohm24835&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":300,"module-header":"background_you_need","content_attributions":[],"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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2310"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":9,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2310\/revisions"}],"predecessor-version":[{"id":7843,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2310\/revisions\/7843"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/300"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2310\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=2310"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2310"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=2310"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=2310"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}