{"id":264,"date":"2025-04-09T17:37:04","date_gmt":"2025-04-09T17:37:04","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/5-1-quadratic-functions-college-algebra-2e-openstax\/"},"modified":"2025-08-20T14:59:10","modified_gmt":"2025-08-20T14:59:10","slug":"5-1-quadratic-functions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/5-1-quadratic-functions\/","title":{"raw":"5.1 Quadratic Functions","rendered":"5.1 Quadratic Functions"},"content":{"raw":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\r\n<div id=\"page_55f2e8ec-a982-4586-9d48-a2f43d7b4107\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Recognize characteristics of parabolas.<\/li>\r\n \t<li>Understand how the graph of a parabola is related to its quadratic function.<\/li>\r\n \t<li>Determine a quadratic function\u2019s minimum or maximum value.<\/li>\r\n \t<li>Solve problems involving a quadratic function\u2019s minimum or maximum value.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<section>\r\n<ul id=\"list-00001\"><\/ul>\r\n<\/section><\/div>\r\n<div id=\"Figure_03_02_001\" class=\"os-figure\">\r\n<div class=\"os-caption-container\">\r\n\r\n[caption id=\"attachment_713\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-713 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-1-300x151.jpeg\" alt=\"\" width=\"300\" height=\"151\" \/> Figure 1. An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr)[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165134339909\">Curved antennas, such as the ones shown in Figure 1, are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function.<\/p>\r\n<p id=\"fs-id1165134081264\">In this section, we will investigate quadratic functions, which frequently model problems involving area and projectile motion. Working with quadratic functions can be less complex than working with higher degree functions, so they provide a good opportunity for a detailed study of function behavior.<\/p>\r\n\r\n<section id=\"fs-id1165137762207\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Recognizing Characteristics of Parabolas<\/h2>\r\n<p id=\"fs-id1165137727999\">The graph of a quadratic function is a U-shaped curve called a <span id=\"term-00007\" class=\"no-emphasis\" data-type=\"term\">parabola<\/span>. One important feature of the graph is that it has an extreme point, called the <strong><span id=\"term-00008\" data-type=\"term\">vertex<\/span><\/strong>. If the parabola opens up, the vertex represents the lowest point on the graph, or the <span id=\"term-00009\" class=\"no-emphasis\" data-type=\"term\">minimum value<\/span> of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the <span id=\"term-00010\" class=\"no-emphasis\" data-type=\"term\">maximum value<\/span>. 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 <strong><span id=\"term-00011\" data-type=\"term\">axis of symmetry<\/span><\/strong>. These features are illustrated in Figure 2.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_715\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-715\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-2-300x283.jpeg\" alt=\"\" width=\"300\" height=\"283\" \/> Figure 2[\/caption]\r\n<p id=\"fs-id1165137549127\">The <em data-effect=\"italics\">y<\/em>-intercept is the point at which the parabola crosses the <em data-effect=\"italics\">y<\/em>-axis. The <em data-effect=\"italics\">x<\/em>-intercepts are the points at which the parabola crosses the <em data-effect=\"italics\">x<\/em>-axis. If they exist, the <em data-effect=\"italics\">x<\/em>-intercepts represent the <strong><span id=\"term-00012\" data-type=\"term\">zeros<\/span>, <\/strong>or <strong><span id=\"term-00013\" data-type=\"term\">roots<\/span><\/strong>, of the quadratic function, the values of\u00a0[latex] x [\/latex] at which [latex] y=0. [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 1: Identifying the Characteristics of a Parabola<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nDetermine the vertex, axis of symmetry, zeros, and y-intercept of the parabola shown in Figure 3.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_716\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-716\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-3-300x292.jpeg\" alt=\"\" width=\"300\" height=\"292\" \/> Figure 3[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The vertex is the turning point of the graph. We can see that the vertex is at\u00a0[latex] (3, 1). [\/latex] Because this parabola opens upward, the axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is\u00a0[latex] x=3. [\/latex] This parabola does not cross the <em>x-<\/em>axis, so it has no zeros. It crosses the y-axis at\u00a0[latex] (0, 7) [\/latex] so this is the<em> y-<\/em>intercept.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165137641326\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Understanding How the Graphs of Parabolas are Related to Their Quadratic Functions<\/h2>\r\n<p id=\"fs-id1165137652877\">The <span id=\"term-00014\" data-type=\"term\">general form<\/span> <strong>of a quadratic function <\/strong>presents the function in the form<\/p>\r\n<p style=\"text-align: center;\">[latex] f(x)=ax^2+bx+c [\/latex]<\/p>\r\n<p id=\"fs-id1165137544673\">where\u00a0[latex] a, b, [\/latex] and\u00a0[latex] c [\/latex] are real numbers and\u00a0[latex] a\\not=0. [\/latex] If\u00a0[latex] a&gt; 0, [\/latex] the parabola opens upward. If\u00a0[latex] a&lt; 0, [\/latex] the parabola opens downward. We can use the general form of a parabola to find the equation for the axis of symmetry.<\/p>\r\n<p id=\"fs-id1165133234001\">The axis of symmetry is defined by\u00a0[latex] x=-\\frac{b}{2a}. [\/latex] If we use the quadratic formula,\u00a0[latex] x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}, [\/latex] to solve\u00a0[latex] ax^2+bx+c=0 [\/latex] for the\u00a0 <em>x-<\/em>intercepts, or zeros, we find the value of\u00a0[latex] x [\/latex] halfway between them is always\u00a0[latex] x==\\frac{b}{2a}, [\/latex] the equation for the axis of symmetry.<\/p>\r\n<p id=\"fs-id1165135190920\">Figure 4 represents the graph of the quadratic function written in general form as\u00a0[latex]- y=x^2+4x+3. [\/latex] In this form,\u00a0[latex] a=1, b=4, [\/latex] and\u00a0[latex] c=3. [\/latex] Because\u00a0[latex] a&gt; 0, [\/latex] the parabola opens upward. The axis of symmetry is\u00a0[latex] x=-\\frac{4}{2(1)}=-2. [\/latex] This also makes sense because we can see from the graph that the vertical line\u00a0[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,\u00a0[latex] (-2, -1). [\/latex] The <em>x-<\/em>intercepts, those points where the parabola crosses the <em>x-<\/em>axis, occur at\u00a0[latex]- (-3, 0) [\/latex] and [latex] (-1, 0). [\/latex]<\/p>\r\n\r\n\r\n[caption id=\"attachment_717\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-717\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-4-300x279.jpeg\" alt=\"\" width=\"300\" height=\"279\" \/> Figure 4[\/caption]\r\n<p id=\"fs-id1165137676320\">The <strong>standard form of a quadratic function<\/strong> presents the function in the form<\/p>\r\n<p style=\"text-align: center;\">[latex] f(x)=a(x-h)^2+k [\/latex]<\/p>\r\n<p id=\"fs-id1303104\">where\u00a0[latex] (h, k) [\/latex] is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the <strong><span id=\"term-00015\" data-type=\"term\">vertex form of a quadratic function<\/span><\/strong>.<\/p>\r\n<p id=\"fs-id1165137894543\">As with the general form, if\u00a0[latex] a&gt; 0, [\/latex] the parabola opens upward and the vertex is a minimum. If\u00a0[latex] a&lt; 0, [\/latex] the parabola opens downward, and the vertex is a maximum. Figure 5 represents the graph of the quadratic function written in standard form as\u00a0[latex] y=-3(x+2)^2+4. [\/latex] Since\u00a0[ x-h=x+2latex] [\/latex] in this example,\u00a0[latex] h=-2. [\/latex] In this form,\u00a0[latex] a=-3, h=-2, [\/latex] and\u00a0[latex] k=4. [\/latex] Because\u00a0[latex] a&lt; 0, [\/latex] the parabola opens downward. The vertex is at [latex] (-2, 4). [\/latex]<\/p>\r\n\r\n<\/section><section data-depth=\"1\">\r\n\r\n[caption id=\"attachment_718\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-718\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-5-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" \/> Figure 5[\/caption]\r\n<p id=\"fs-id1165137453489\">The standard form is useful for determining how the graph is transformed from the graph of\u00a0[latex] y=x^2. [\/latex] Figure 6 is the graph of this basic function.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_719\" align=\"aligncenter\" width=\"280\"]<img class=\"size-medium wp-image-719\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-6-280x300.jpeg\" alt=\"\" width=\"280\" height=\"300\" \/> Figure 6[\/caption]\r\n<p id=\"fs-id1165137770279\">If\u00a0[latex] k&gt; 0, [\/latex] the graph shifts upward, whereas if\u00a0[latex] k&lt; 0, [\/latex] the graph shifts downward. In Figure 5,\u00a0[latex] k&gt; 0m [\/latex] so the graph is shifted 4 units upward. If\u00a0[latex] h&gt; 0, [\/latex] the graph shifts toward the right and if\u00a0[latex] h&lt; 0, [\/latex] the graph shifts to the left. In Figure 5,\u00a0[latex] h&lt; 0, [\/latex] so the graph is shifted 2 units to the left. The magnitude of\u00a0[latex] a [\/latex] indicates the stretch of the graph. If\u00a0[latex] |a|&gt; 1, [\/latex] the point associated with a particular<em> x-<\/em>value shifts farther from the <em data-effect=\"italics\">x-<\/em>axis, so the graph appears to become narrower, and there is a vertical stretch. But if\u00a0[latex] |a|&lt; 1, [\/latex] the point associated with a particular <em>x-<\/em>value shifts closer to the <em data-effect=\"italics\">x-<\/em>axis, so the graph appears to become wider, but in fact there is a vertical compression. In Figure 5,\u00a0[latex] |a|&gt; 1, [\/latex] so the graph becomes narrower.<\/p>\r\n<p id=\"fs-id1165135353112\">The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} a(x-h)^2+k &amp;=&amp; ax^2+bx+c \\\\ ax^2-2ahx+(ah^2+k) &amp;=&amp; ax^2+bx+c \\end{array} [\/latex]<\/p>\r\nFor the linear terms to be equal, the coefficients must be equal.\r\n<p style=\"text-align: center;\">[latex] -2ah=b, \\ \\ \\text{so} \\ \\\u00a0 h=-\\frac{b}{2a} [\/latex]<\/p>\r\n<p id=\"fs-id1165134118295\">This is the axis of symmetry we defined earlier. Setting the constant terms equal:<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} ah^2+k &amp;=&amp; c \\\\ \\quad \\quad \\ \\ \\ k &amp;=&amp; c-ah^2 \\\\ &amp;=&amp; c-a-\\left(\\frac{b}{2a}\\right)^2 \\\\ &amp;=&amp; c-\\frac{b^2}{4a} \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1165137476446\">In practice, though, it is usually easier to remember that <em data-effect=\"italics\">k<\/em> is the output value of the function when the input is\u00a0[latex] h, [\/latex] so [latex] f(h)=k. [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Forms of Quadratic Functions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA quadratic function is a polynomial function of degree two. The graph of a <span id=\"term-00017\" class=\"no-emphasis\" data-type=\"term\">quadratic function<\/span> is a parabola.\r\n\r\nThe\u00a0<strong>general form of a quadratic function<\/strong> is\u00a0[latex] f(x)=ax^2+bx+c [\/latex] where\u00a0[latex] a, b, [\/latex] and\u00a0[latex] c [\/latex] are real numbers and [latex] a\\not=0. [\/latex]\r\n\r\nThe\u00a0<strong>standard form of a quadratic function<\/strong> is\u00a0[latex] f(x)=a(x-h)^2+k [\/latex] where [latex] a\\not=0. [\/latex]\r\n\r\nThe vertex\u00a0[latex] (h, k) [\/latex] is located at\r\n<p style=\"text-align: center;\">[latex] h=-\\frac{b}{2a}, k=f(h)=f(\\frac{-b}{2a}) [\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a graph of a quadratic function, write the equation of the function in general form.<\/strong>\r\n<ol>\r\n \t<li>Identify the horizontal shift of the parabola; this value is\u00a0[latex] h. [\/latex] Identify the vertical shift of the parabola; this value is [latex] k. [\/latex]<\/li>\r\n \t<li>Substitute the values of horizontal and vertical shift for\u00a0[latex] h [\/latex] and\u00a0[latex] k. [\/latex] in the function [latex] f(x)=a(x-h)^2+k. [\/latex]<\/li>\r\n \t<li>Substitute the values of any point, other than the vertex, on the graph of the parabola for\u00a0[latex] x [\/latex] and [latex] f(x). [\/latex]<\/li>\r\n \t<li>Solve for the stretch factor, [latex] |a|. [\/latex]<\/li>\r\n \t<li>Expand and simplify to write in general form.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 2: Writing the Equation of a Quadratic Function from the Graph<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite an equation for the quadratic function [latex] g [\/latex] in Figure 7 as a transformation of [latex] f(x)=x^2, [\/latex] and then expand the formula, and simplify terms to write the equation in general form.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_720\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-720\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-7-300x273.jpeg\" alt=\"\" width=\"300\" height=\"273\" \/> Figure 7[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>We can see the graph of [latex] g [\/latex] is the graph of [latex] f(x)=x^2 [\/latex] shifted to the left 2 and down 3, giving a formula in the form\r\n<p style=\"text-align: center;\">[latex] g(x)=a(x-(-2))^2-3=a(x+2)^2-3. [\/latex]<\/p>\r\nSubstituting the coordinates of a point on the curve, such as [latex] (0, -1), [\/latex] we can solve for the stretch factor.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} -1 &amp;=&amp; a(0+2)^2-3 \\\\ \\ \\ \\ 2 &amp;=&amp; 4a \\\\ \\ \\ \\ a &amp;=&amp; \\frac{1}{2} \\end{array} [\/latex]<\/p>\r\nIn standard form, the algebraic model for this graph is\r\n<p style=\"text-align: center;\">[latex] g(x)=\\frac{1}{2}(x+2)^2-3. [\/latex]<\/p>\r\nTo write this in general polynomial form, we can expand the formula and simplify terms.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} g(x) &amp;=&amp; \\frac{1}{2}(x+2)^2-3 \\\\ &amp;=&amp; \\frac{1}{2}(x+2)(x+2)-3 \\\\ &amp;=&amp; \\frac{1}{2}(x^2+4x+4)-3 \\\\ &amp;=&amp; \\frac{1}{2}x^2+2x+2-3 \\\\ &amp;=&amp; \\frac{1}{2}x^2+2x-1 \\end{array}\u00a0[\/latex]<\/p>\r\nNotice that the horizontal and vertical shifts of the basic graph of the quadratic function determine the location of the vertex of the parabola; the vertex is unaffected by stretches and compressions.\r\n<h3>Analysis<\/h3>\r\nWe can check our work using the table feature on a graphing utility. First enter [latex] Y1-\\frac{1}{2}(x+2)^2-3. [\/latex] Next, select [latex] \\text{TBLSET}, [\/latex] then use [latex] \\text{TblStart}=-6 [\/latex] and [latex] \\Delta \\text{Tbl}=2, [\/latex] and select [latex] \\text{TABLE}. [\/latex] See Table 1.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 1<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 16.6667%;\">[latex] x [\/latex]<\/td>\r\n<td style=\"width: 16.6667%;\">-6<\/td>\r\n<td style=\"width: 16.6667%;\">-4<\/td>\r\n<td style=\"width: 16.6667%;\">-2<\/td>\r\n<td style=\"width: 16.6667%;\">0<\/td>\r\n<td style=\"width: 16.6667%;\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.6667%;\">[latex] y [\/latex]<\/td>\r\n<td style=\"width: 16.6667%;\">5<\/td>\r\n<td style=\"width: 16.6667%;\">-1<\/td>\r\n<td style=\"width: 16.6667%;\">-3<\/td>\r\n<td style=\"width: 16.6667%;\">-1<\/td>\r\n<td style=\"width: 16.6667%;\">5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe ordered pairs in the table correspond to points on the graph.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1165135460939\" class=\"unnumbered\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137838619\" data-type=\"commentary\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA coordinate grid has been superimposed over the quadratic path of a basketball in Figure 8. Assume that the point\u00a0[latex] (-4, 7) [\/latex] is the highest point of the basketball\u2019s trajectory. Find an equation for the path of the ball. Does the shooter make the basket?\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_721\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-721\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-8-300x261.jpeg\" alt=\"\" width=\"300\" height=\"261\" \/> Figure 8. (credit: modification of work by Dan Meyer)[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a quadratic function in general form, find the vertex of the parabola.<\/strong>\r\n<ol>\r\n \t<li>Identify\u00a0[latex] a, b, [\/latex] and [latex] c. [\/latex]<\/li>\r\n \t<li>Find\u00a0[latex] h, [\/latex] the <em>x-<\/em>coordinate of the vertex, by substituting\u00a0[latex] a [\/latex] and\u00a0[latex] b [\/latex] into [latex] h=-\\frac{b}{2a}. [\/latex]<\/li>\r\n \t<li>Find\u00a0[latex] k, [\/latex] the <em>y-<\/em>coordinate of the vertex, by evaluating [latex] k=f(h)=f(-\\frac{b}{2a}). [\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3: Finding the Vertex of a Quadratic Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the vertex of the quadratic function\u00a0[latex] f(x)=2x^2-6x+7. [\/latex] Rewrite the quadratic in standard form (vertex form).\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The horizontal coordinate of the vertex will be at\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} h &amp;=&amp; -\\frac{b}{2a} \\\\ &amp;=&amp; -\\frac{-6}{2(2)} \\\\ &amp;=&amp; \\frac{6}{4} \\\\ &amp;=&amp; \\frac{3}{2} \\end{array} [\/latex]<\/p>\r\nThe vertical coordinate of the vertex will be at\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} k &amp;=&amp; f(h) \\\\ &amp;=&amp; f\\left(\\frac{3}{2}\\right) \\\\ &amp;=&amp; 2\\left(\\frac{3}{2}\\right)^2-6\\left(\\frac{3}{2}\\right)+7 \\\\ &amp;=&amp; \\frac{5}{2} \\end{array}\u00a0[\/latex]<\/p>\r\nRewriting into standard form, the stretch factor will be same as the\u00a0[latex] a [\/latex] in the original quadratic. First, find the horizontal coordinate of the vertex. Then find the vertical coordinate of the vertex. Substitute the values into standard form, using the \"[latex] a [\/latex]\" from the general form.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} f(x) &amp;=&amp; ax^2+bx+c \\\\ f(x) &amp;=&amp; 2x^2-6x+7 \\end{array} [\/latex]<\/p>\r\nThe standard form of a quadratic function prior to writing the function then becomes the following:\r\n<p style=\"text-align: center;\">[latex] f(x)=2(x-\\frac{3}{2})^2+\\frac{5}{2} [\/latex]<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\nOne reason we may want to identify the vertex 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] x. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1165137658566\" class=\"unnumbered\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137591920\" data-type=\"commentary\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGiven the equation\u00a0[latex] g(x)=13+x^2-6x, [\/latex] write the equation in general form and then in standard form.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165133436210\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Finding the Domain and Range of a Quadratic Function<\/h2>\r\n<p id=\"fs-id1165135596509\">Any number can be the input value of a quadratic function. Therefore, the domain of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum point, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the range will consist of all <em data-effect=\"italics\">y<\/em>-values greater than or equal to the <em data-effect=\"italics\">y<\/em>-coordinate at the turning point or less than or equal to the <em data-effect=\"italics\">y<\/em>-coordinate at the turning point, depending on whether the parabola opens up or down.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Domain and Range of a Quadratic Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe domain of any quadratic function is all real numbers unless the context of the function presents some restrictions.\r\n\r\nThe range of a quadratic function written in general form [latex] f(x)=ax^2+bx+c [\/latex] with a positive [latex] a [\/latex] value is [latex] f(x)\\ge f\\left(-\\frac{b}{2a}\\right), [\/latex] or [latex] [f\\left(-\\frac{b}{2a}), \\infty\\right); [\/latex] the range of a quadratic function written in general form with a negative [latex] a [\/latex] value is [latex] f(x)\\le f\\left(-\\frac{b}{2a}\\right), [\/latex] or [latex] (-\\infty, f\\left(-\\frac{b}{2a}\\right)]. [\/latex]\r\n\r\nThe range of a quadratic function written in standard form\u00a0[latex] f(x)=a(x-h)^2+k [\/latex] with a positive\u00a0[latex] a [\/latex] value is\u00a0[latex] f(x)\\ge k; [\/latex] the range of a quadratic function written in standard form with a negative\u00a0[latex] a [\/latex] value is [latex] f(x)\\le k. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a quadratic function, find the domain and range.<\/strong>\r\n<ol>\r\n \t<li>Identify the domain of any quadratic function as all real numbers.<\/li>\r\n \t<li>Determine whether\u00a0[latex] a [\/latex] is positive or negative. If [latex] a [\/latex] is positive, the parabola has a minimum. If\u00a0[latex] a [\/latex] is negative, the parabola has a maximum.<\/li>\r\n \t<li>Determine the maximum or minimum value of the parabola, [latex] k. [\/latex]<\/li>\r\n \t<li>If the parabola has a minimum, the range is given by\u00a0[latex] f(x)\\ge k, [\/latex] or\u00a0[latex] [k, \\infty). [\/latex] If the parabola has a maximum, the range is given by\u00a0[latex] f(x)\\le k, [\/latex] or [latex] (-\\infty, k]. [\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 4: Finding the Domain and Range of a Quadratic Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the domain and range of [latex] f(x)=-5x^2+9x-1. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>As with any quadratic function, the domain is all real numbers.\r\n\r\nBecause [latex] a [\/latex] is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding the x-value of the vertex.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} h &amp;=&amp; -\\frac{b}{2a} \\\\ &amp;=&amp; -\\frac{9}{2(-5)} \\\\ &amp;=&amp; \\frac{9}{10} \\end{array} [\/latex]<\/p>\r\nThe maximum value is given by [latex] f(h). [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} f\\left(\\frac{9}{10}\\right) &amp;=&amp; -5\\left(\\frac{9}{10}\\right)+9\\left(\\frac{9}{10}\\right)-1 \\\\ &amp;=&amp; \\frac{61}{20}\\end{array} [\/latex]<\/p>\r\nThe range is [latex] f(x)\\le \\frac{61}{20}, [\/latex] or [latex] (-\\infty, \\frac{61}{20}). [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the domain and range of [latex] f(x)=2(x-\\frac{4}{7})^2+\\frac{8}{11}. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165137736541\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Determining the Maximum and Minimum Values of Quadratic Functions<\/h2>\r\n<p id=\"fs-id1165137442167\">The output of the quadratic function at the vertex is the maximum or minimum value of the function, depending on the orientation of the <span id=\"term-00021\" class=\"no-emphasis\" data-type=\"term\">parabola<\/span>. We can see the maximum and minimum values in Figure 9.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_724\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-724\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-9-300x172.jpeg\" alt=\"\" width=\"300\" height=\"172\" \/> Figure 9[\/caption]\r\n<p id=\"fs-id1165137431411\">There are many real-world scenarios that involve finding the maximum or minimum value of a quadratic function, such as applications involving area and revenue.<\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 5: Finding the Maximum Value of a Quadratic Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA backyard farmer wants to enclose a rectangular space for a new garden within her fenced backyard. She has purchased 80 feet of wire fencing to enclose three sides, and she will use a section of the backyard fence as the fourth side.\r\n\r\n(a) Find a formula for the area enclosed by the fence if the sides of fencing perpendicular to the existing fence have length [latex] L. [\/latex]\r\n\r\n(b) What dimensions should she make her garden to maximize the enclosed area?\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Let's use a diagram such as Figure 10 to record the given information. It is also helpful to introduce a temporary variable, [latex] W, [\/latex] to represent the width of the garden and the length of the fence section parallel to the backyard fence.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_725\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-725\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-10-300x191.jpeg\" alt=\"\" width=\"300\" height=\"191\" \/> Figure 10[\/caption]\r\n\r\n(a) We know we have only 80 feet of fence available, and [latex] L+W+L=80, [\/latex] or more simply [latex] 2L+W=80. [\/latex] This allows us to represent the width, [latex] W, [\/latex] in terms of [latex] L. [\/latex]\r\n<p style=\"text-align: center;\">[latex] W=80-2L [\/latex]<\/p>\r\nNow we are ready to write an equation for the area the fence encloses. We know the area of a rectangle is length multiplied by width, so\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{lll} A &amp;=&amp; LW &amp;=&amp; L(80-2L) \\\\ A(L) &amp;=&amp; 80L-2L^2 \\end{array} [\/latex]<\/p>\r\nThis formula represents the area of the fence in terms of the variable length [latex] L. [\/latex] The function, written in general form, is\r\n<p style=\"text-align: center;\">[latex] A(L)=-2L^2+80L [\/latex]<\/p>\r\n(b) The quadratic has a negative leading coefficient, so the graph will open downward, and the vertex will be the maximum value for the area. In finding the vertex, we must be careful because the equation is not written in standard polynomial form with decreasing powers. This is why we rewrote the function in general form above. Since [latex] a [\/latex] is the coefficient of the squared term, [latex] a=-2, b=80, [\/latex] and [latex] c-0. [\/latex]\r\n\r\nTo find the vertex:\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rclcrcl}h &amp;=&amp; -\\frac{b}{2a} &amp;&amp; \\quad k &amp;=&amp; A(20) \\\\&amp;=&amp; -\\frac{80}{2(-2)} &amp; \\quad \\text{and} &amp;&amp; =&amp; 80(20) - 2(20)^2 \\\\&amp;=&amp; 20 &amp;&amp;&amp;=&amp; 800\\end{array} [\/latex]<\/p>\r\nThe maximum value of the function is an area of 800 square feet, which occurs when [latex] L=20 [\/latex] feet. When the shorter sides are 20 feet, there is 40 feet of fencing left for the longer side. To maximize the area, she should enclose the garden so the two shorter sides have length 20 feet and the longer side parallel to the existing fence has length 40 feet.\r\n<h3>Analysis<\/h3>\r\nThis problem also could be solved by graphing the quadratic function. We can see where the maximum area occurs on a graph of the quadratic function in Figure 11.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_726\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-726\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-11-300x247.jpeg\" alt=\"\" width=\"300\" height=\"247\" \/> Figure 11[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given an application involving revenue, use a quadratic equation to find the maximum.<\/strong>\r\n<ol>\r\n \t<li>Write a quadratic equation for a revenue function.<\/li>\r\n \t<li>Find the vertex of the quadratic equation.<\/li>\r\n \t<li>Determine the <em data-effect=\"italics\">y<\/em>-value of the vertex.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 6: Finding Maximum Revenue<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe unit price of an item affects its supply and demand. That is, if the unit price goes up, the demand for the item will usually decrease. For example, a streaming service currently has 84,000 subscribers at a quarterly charge of $30. Market research has suggested that if the owners raise the price to $32, they would lose 5,000 subscribers. Assuming that subscriptions are linearly related to the price, what price should the newspaper charge for a quarterly subscription to maximize their revenue?\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Revenue is the amount of money a company brings in. In this case, the revenue can be found by multiplying the price per subscription times the number of subscribers, or quantity. We can introduce variables, [latex] p [\/latex] for price per subscription and [latex] Q [\/latex] for quantity, giving us the equation [latex] Revenue=pQ. [\/latex]\r\n\r\nBecause the number of subscribers changes with the price, as need to find a relationship between the variables. We know that currently [latex] p=30 [\/latex] and [latex] Q=84,000. [\/latex] We also know that if the price rises to $32, the streaming service would lose 5,000 subscribers, giving a second pair of values, [latex] p=32 [\/latex] and [latex] Q=79,000. [\/latex] From this we can find a linear equation relating the two quantities. The slope will be\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} m &amp;=&amp; \\frac{79,000-84,000}{32-30} \\\\ &amp;=&amp; \\frac{-5,000}{2} \\\\ &amp;=&amp; -2,500\\end{array} [\/latex]<\/p>\r\nThis tells us the service will lose 2,500 subscribers for each dollar they raise the price. We can then solve for the <em>y-<\/em>intercept.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{lll} Q &amp;=&amp; -2,500p+b &amp; \\text{Substitute in the point} \\ Q=84,000 \\ \\text{and} \\ p=30 \\\\ 84,000 &amp;=&amp; -2,500(30)=b &amp; \\text{Solve for} \\ b \\\\ b &amp;=&amp; 159,000 \\end{array} [\/latex]<\/p>\r\nThis gives us the linear equation [latex] Q=-2,500p+159,000 [\/latex] relating cost and subscribers. We now return to our revenue equation\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} \\text{Revenue} &amp;=&amp; pQ \\\\ \\text{Revenue} &amp;=&amp; p(-2,500p+159,000) \\\\ \\text{Revenue} &amp;=&amp; -2,500p^2+159,000p \\end{array} [\/latex]<\/p>\r\nWe now have a quadratic function for revenue as a function of the subscription charge. To find the price that will maximize revenue for the service, we can find the vertex.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} h &amp;=&amp; \\frac{159,000}{2(-2,500)} \\\\ &amp;=&amp; 31.8 \\end{array} [\/latex]<\/p>\r\nThe model tells us that the maximum revenue will occur if the streaming service charges $31.80 for a subscription. To find what the maximum revenue is, we evaluate the revenue function.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} \\text{maximum revenue} &amp;=&amp; -2,500(31.8)^2+159,000(31.8) \\\\ &amp;=&amp; 2,528,100 \\end{array} [\/latex]<\/p>\r\n\r\n<h3>Analysis<\/h3>\r\nThis could also be solved by graphing the quadratic as in Figure 12. We can see the maximum revenue on a graph of the quadratic function.\r\n\r\n[caption id=\"attachment_728\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-728\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-12-300x231.jpeg\" alt=\"\" width=\"300\" height=\"231\" \/> Figure 12[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<section id=\"fs-id1165135693703\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Finding the <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-Intercepts of a Quadratic Function<\/h3>\r\n<p id=\"fs-id1165134569121\">Much as we did in the application problems above, we also need to find intercepts of quadratic equations for graphing parabolas. Recall that we find the <em>y-<\/em>intercept of a quadratic by evaluating the function at an input of zero, and we find the <em>x-<\/em>intercepts at locations where the output is zero. Notice in Figure 13 that the number of <em>x-<\/em>intercepts can vary depending upon the location of the graph.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_729\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-729\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-13-300x114.jpeg\" alt=\"\" width=\"300\" height=\"114\" \/> Figure 13. Number of x-intercepts of a parabola[\/caption]\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Examples<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a quadratic function\u00a0[latex] f(x) [\/latex] find the y- and x-intercepts.<\/strong>\r\n<ol>\r\n \t<li>Evaluate [latex] f(0) [\/latex] to find the <em>y-<\/em>intercept.<\/li>\r\n \t<li>Solve the quadratic equation [latex] f(x)=0 [\/latex]to find the <em>x-<\/em>intercept.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 7: Finding the <em>y-<\/em> and <em>x-<\/em>intercepts of a Parabola<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the y- and x-intercepts of the quadratic [latex] f(x)=3x^2+5x-2. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>We find the y-intercept by evaluating [latex] f(0). [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} f(0) &amp;=&amp; 3(0)^2+5(0)-2 \\\\ &amp;=&amp; -2 \\end{array} [\/latex]<\/p>\r\nSo the <em>y-<\/em>intercept is at [latex] (0, -2). [\/latex]\r\n\r\nFor the <em>x-<\/em>intercepts, we find all solutions of [latex] f(x)=0. [\/latex]\r\n<p style=\"text-align: center;\">[latex] 0=3x^2+5x-2 [\/latex]<\/p>\r\nIn this case, the quadratic can be factored easily, providing the simplest method for solution.\r\n<p style=\"text-align: center;\">[latex] 0=(3x-1)(x+2) [\/latex]<\/p>\r\nSo the<em> x-<\/em>intercepts are at [latex] (\\frac{1}{3}, 0) [\/latex] and [latex] (-2, 0). [\/latex]\r\n<h3>Analysis<\/h3>\r\nBy graphing the function, we can confirm that the graph crosses the <em data-effect=\"italics\">y<\/em>-axis at [latex] (0, -2). [\/latex] We can also confirm that the graph crosses the <em>x-<\/em>axis at [latex] (\\frac{1}{3}, 0) [\/latex] and [latex] (-2, 0). [\/latex] See Figure 14.\r\n\r\n[caption id=\"attachment_730\" align=\"aligncenter\" width=\"293\"]<img class=\"size-medium wp-image-730\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-14-293x300.jpeg\" alt=\"\" width=\"293\" height=\"300\" \/> Figure 14[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135381309\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Rewriting Quadratics in Standard Form<\/h3>\r\n<p id=\"fs-id1165135381314\">In Example 7, the quadratic was easily solved by factoring. However, there are many quadratics that cannot be factored. We can solve these quadratics by first rewriting them in standard form.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a quadratic function, find the x-intercepts by rewriting in standard form.<\/strong>\r\n<ol>\r\n \t<li>Substitute [latex] a [\/latex] and [latex] b [\/latex] into [latex] h=-\\frac{b}{2a}. [\/latex]<\/li>\r\n \t<li>Substitute [latex] x=h [\/latex] into the general form of the quadratic function to find [latex] k. [\/latex]<\/li>\r\n \t<li>Rewrite the quadratic in standard form using [latex] h [\/latex]and [latex] k. [\/latex]<\/li>\r\n \t<li>Solve for when the output of the function will be zero to find the <em>x-<\/em>intercepts.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"os-note-body\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 8: Finding the <em>x-<\/em>intercepts of a Parabola<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind the x-intercepts of the quadratic function [latex] f(x)=2x^2+4x-4. [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>We begin by solving for when the output will be zero.\r\n<p style=\"text-align: center;\">[latex] 0=2x^2+4x-4 [\/latex]<\/p>\r\nBecause the quadratic is not easily factorable in this case, we solve for the intercepts by first rewriting the quadratic in standard form.\r\n<p style=\"text-align: center;\">[latex] f(x)=a(x-h)^2+k [\/latex]<\/p>\r\nWe know that [latex] a=2. [\/latex] Then we solve for [latex] h [\/latex] and [latex] k. [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rclcrcl}h &amp;=&amp; -\\frac{b}{2a} &amp; \\quad k &amp;=&amp; f(-1) \\\\&amp;=&amp; -\\frac{4}{2(2)} &amp;&amp;=&amp; 2(-1)^2 + 4(-1) - 4 \\\\&amp;=&amp; -1 &amp;&amp;=&amp; -6\\end{array} [\/latex]<\/p>\r\nSo now we can rewrite in standard form.\r\n<p style=\"text-align: center;\">[latex] f(x)=2(x+1)^2-6 [\/latex]<\/p>\r\nWe can now solve for when the output will be zero.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} 0 &amp;=&amp; 2(x+1)^2-6 \\\\ 6 &amp;=&amp; 2(x+1)^2 \\\\ 3 &amp;=&amp; (x+1)^2 \\\\ x+1 &amp;=&amp; \\pm\\sqrt{3} \\\\ x &amp;=&amp; -1\\pm\\sqrt{3} \\end{array} [\/latex]<\/p>\r\nThe graph has<em> x-<\/em>intercepts at [latex] (-1-\\sqrt{3}, 0) [\/latex] and [latex] (-1+\\sqrt{3}, 0). [\/latex]\r\n\r\nWe can check out work by graphing the given function on a graphing utility and observing the <em>x-<\/em>intercepts. See Figure 15.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_731\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-731\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-15-300x293.jpeg\" alt=\"\" width=\"300\" height=\"293\" \/> Figure 15[\/caption]\r\n<h3>Analysis<\/h3>\r\nWe could have achieved the same results using the quadratic formula. Identify [latex] a=2, b=4 [\/latex] and [latex] c=-4. [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} x&amp;=&amp; \\frac{-b\\pm\\sqrt{b^2-4ac}}{2a} \\\\ &amp;=&amp; \\frac{-4\\pm\\sqrt{4^2-4(2)(-4)}}{2(2)} \\\\ &amp;=&amp; \\frac{-4\\pm\\sqrt{48}}{4} \\\\ &amp;=&amp; \\frac{-4\\pm\\sqrt{3(16)}}{4} \\\\ &amp;=&amp; -1\\pm\\sqrt{3} \\end{array} [\/latex]<\/p>\r\nSo the <em>x-<\/em>intercepts occur at [latex] (-1-\\sqrt{3}, 0) [\/latex] and [latex] (-1+\\sqrt{3}, 0). [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIn Try It #2, we found the standard and general form for the function\u00a0[latex] g(x)=13+x^2-6x. [\/latex] Now find the <em>y-<\/em> and <em>x-<\/em>intercepts (if any).\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 9: Applying the Vertex and x-intercepts of a Parabola<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA ball is thrown upward from the top of a 40 foot high building at a speed of 80 feet per second. The ball\u2019s height above ground can be modeled by the equation [latex] H(t)=-16t^2+80t+40. [\/latex]\r\n\r\n(a) When does the ball reach the maximum height?\r\n\r\n(b) What is the maximum height of the ball?\r\n\r\n(c) When does the ball hit the ground?\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>(a) The ball reached the maximum height at the vertex of the parabola.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} h &amp;=&amp; -\\frac{8}{2(-16)} \\\\ &amp;=&amp; \\frac{80}{32} \\\\ &amp;=&amp; \\frac{5}{2} \\\\ &amp;=&amp; 2.5 \\end{array} [\/latex]<\/p>\r\nThe ball reached a maximum height after 2.5 seconds.\r\n\r\n(b) To find the maximum height, find the <em>y-<\/em>coordinate of the vertex of the parabola.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} k &amp;=&amp; H\\left(-\\frac{b}{2a}\\right) \\\\ &amp;=&amp; H(2.5) \\\\ &amp;=&amp; -16(2.5)^2+80(2.5)+40 \\\\ &amp;=&amp; 140 \\end{array} [\/latex]<\/p>\r\nThe ball reaches a maximum height of 140 feet.\r\n\r\n(c) To find when the ball hits the ground, we need to determine when the height is zero, [latex] H(t)=0. [\/latex]\r\n\r\nWe use the quadratic formula.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ll} t &amp;=&amp; \\frac{-80\\pm\\sqrt{80^2-4(-16)(40)}}{2(-16)} \\\\ &amp;=&amp; \\frac{-80\\pm\\sqrt{8960}}{-32} \\end{array} [\/latex]<\/p>\r\nBecause the square root does not simplify nicely, we can use a calculator to approximate the values of the solutions.\r\n<p style=\"text-align: center;\">[latex] t=\\frac{-80-\\sqrt{8960}}{-32}\\approx 5.458 \\ \\ \\text{or} \\ \\ t=\\frac{-80+\\sqrt{8960}}{-32}\\approx -0.458 [\/latex]<\/p>\r\nThe second answer is outside the reasonable domain of our model, so we conclude the ball will hit the ground after about 5.458 seconds. See Figure 16.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_732\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-732\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-16-300x252.jpeg\" alt=\"\" width=\"300\" height=\"252\" \/> Figure 16[\/caption]\r\n\r\nNote that the graph does not represent the physical path of the ball upward and downward. Keep the quantities on each axis in mind while interpreting the graph.\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA rock is thrown upward from the top of a 112-foot high cliff overlooking the ocean at a speed of 96 feet per second. The rock\u2019s height above ocean can be modeled by the equation [latex] H(t)=-16t^2+96t+112. [\/latex]\r\n\r\n(a) When does the rock reach the maximum height?\r\n\r\n(b) What is the maximum height of the rock?\r\n\r\n(c) When does the rock hit the ocean?\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Media<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAccess these online resources for additional instruction and practice with quadratic equations.\r\n<ul>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=oH6eZ6oDRBI\">Graphing Quadratic Functions in General Form<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=L8-7QepRSi8\">Graphing Quadratic Functions in Standard Form<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=kFw3XU0wisU\">Quadratic Function Review<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=H46QVUdbBn0\">Characteristics of a Quadratic Function<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\r\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">5.1 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id1165135361324\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1165135361327\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1165135361332\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135361334\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135361332-solution\">1<\/a><span class=\"os-divider\">. <\/span>Explain the advantage of writing a quadratic function in standard form.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134367971\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134367972\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>How can the vertex of a parabola be used in solving real-world problems?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134367976\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134367977\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134367976-solution\">3<\/a><span class=\"os-divider\">. <\/span>Explain why the condition of [latex] a\\not=0 [\/latex] is imposed in the definition of the quadratic function.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>\r\n<div id=\"fs-id1165134094470\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134094472\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>What is another name for the standard form of a quadratic function?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134094476\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134094477\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134094476-solution\">5<\/a><span class=\"os-divider\">. <\/span>What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165133276257\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id1165133276262\">For the following exercises, rewrite the quadratic functions in standard form and give the vertex.<\/p>\r\n\r\n<div id=\"fs-id1165133276266\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133276268\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=x^2-12x+32 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135336013\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133065125\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135336013-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex] g(x)=x^2+2x-3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134199469\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134199470\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=x^2-x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135536371\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135536373\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135536371-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=x^2+5x-2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134476611\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134476612\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex] h(x)=2x^2+8x-10 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133328085\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133328087\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165133328085-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex] k(x)=3x^2-6x-9 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134422869\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134422870\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=2x^2-6x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135409784\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135409786\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135409784-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=3x^2-5x-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135258891\">For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.<\/p>\r\n\r\n<div id=\"fs-id1165135258896\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135258897\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex] y(x)=2x^2+10x+12 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133344089\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133344091\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165133344089-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=2x^2-10x+4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134091244\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134091247\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=-x^2+4x+3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134278554\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134278556\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134278554-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=4x^2+x-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135193269\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135193271\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex] h(t)=-4t^2+6t-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135380708\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135380710\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135380708-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\frac{1}{2}x^2+3x+1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134362815\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134362817\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=-\\frac{1}{3}x^2-2x+3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135419821\">For the following exercises, determine the domain and range of the quadratic function.<\/p>\r\n\r\n<div id=\"fs-id1165135419824\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135419827\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135419824-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=(x-3)^2+2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137681943\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137681945\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=-2(x+3)^2-6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135528310\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135528312\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135528310-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=x^2+6x+4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134199532\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134199534\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=2x^2-4x+2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134138647\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134138649\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134138647-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex] k(x)=3x^2-6x-9 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165133252496\">For the following exercises, use the vertex [latex] (h, k) [\/latex] and a point on the graph [latex] (x, y) [\/latex] to find the general form of the equation of the quadratic function.<\/p>\r\n\r\n<div id=\"fs-id1165137898126\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137898128\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex] (h, k)=(2, 0), (x, y)=(4, 4) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134079556\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134079558\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134079556-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex] (h, k)=(-2, -1), (x, y)=(-4, 3) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134485654\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134485656\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex] (h, k)=(0, 1), (x, y)=(2, 5) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133077523\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133077525\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165133077523-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex] (h, k)=(2, 3), (x, y)=(5, 12) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132912701\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132912704\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex] (h, k)=(-5, 3), (x, y)=(2, 9) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133252567\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135536384\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165133252567-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex] (h, k)=(3, 2), (x, y)=(10, 1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135412141\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135412143\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex] (h, k)=(0, 1), (x, y)=(1, 0) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134479000\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134479002\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134479000-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex] (h, k)=(1, 0), (x, y)=(0, 1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165133095093\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphical<\/h3>\r\n<p id=\"fs-id1165133095098\">For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.<\/p>\r\n\r\n<div id=\"fs-id1165133095103\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133095105\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=x^2-2x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135646115\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135646117\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135646115-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=x^2-6x-1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135692195\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135692197\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=^2-5x-6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135205053\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135205055\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135205053-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=x^2-7x+3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135500665\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135500667\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=-2x^2+5x-8 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135639341\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135639343\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135639341-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=4x^2-12x-3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165134391625\">For the following exercises, write the equation for the graphed quadratic function.<\/p>\r\n\r\n<div id=\"fs-id1165134391628\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134391630\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165134391635\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a positive parabola with a vertex at (2, -3) and y-intercept at (0, 1).\">\r\n<img class=\"alignnone size-medium wp-image-733\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.40-298x300.jpeg\" alt=\"\" width=\"298\" height=\"300\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135262595\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135262597\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135262595-solution\">41<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165135262603\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a positive parabola with a vertex at (-1, 2) and y-intercept at (0, 3)\">\r\n<img class=\"alignnone size-medium wp-image-734\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.41-296x300.jpeg\" alt=\"\" width=\"296\" height=\"300\" \/>\r\n<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135262616\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135262618\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165135262624\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a negative parabola with a vertex at (2, 7).\">\r\n<img class=\"alignnone size-medium wp-image-735\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.42-276x300.jpeg\" alt=\"\" width=\"276\" height=\"300\" \/>\r\n<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134162152\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134162154\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134162152-solution\">43<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n\r\n<span id=\"fs-id1165134162160\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a negative parabola with a vertex at (-1, 2).\">\r\n<img class=\"alignnone size-medium wp-image-736\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.43-298x300.jpeg\" alt=\"\" width=\"298\" height=\"300\" \/>\r\n<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134162173\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134162175\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165134196136\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a positive parabola with a vertex at (3, -1) and y-intercept at (0, 3.5).\">\r\n<img class=\"alignnone size-medium wp-image-737\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.44-300x194.jpeg\" alt=\"\" width=\"300\" height=\"194\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137940515\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137940518\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137940515-solution\">45<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165131959568\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a negative parabola with a vertex at (-2, 3).\">\r\n<img class=\"alignnone size-medium wp-image-738\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.45-277x300.jpeg\" alt=\"\" width=\"277\" height=\"300\" \/>\r\n<\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165131959583\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Numeric<\/h3>\r\n<p id=\"fs-id1165131959588\">For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.<\/p>\r\n\r\n<div id=\"fs-id1165131959593\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165131959596\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n<div id=\"fs-id1165131959598\" class=\"os-table first-element\">\r\n<table class=\"grid\" style=\"height: 154px; width: 307px;\" data-id=\"fs-id1165131959598\" data-label=\"\">\r\n<tbody>\r\n<tr style=\"height: 66px;\">\r\n<td style=\"width: 53.5px; height: 66px; text-align: center;\" data-align=\"center\"><em>x<\/em><\/td>\r\n<td style=\"width: 45.3167px; height: 66px;\" data-align=\"center\">\u20132<\/td>\r\n<td style=\"width: 51.2px; height: 66px;\" data-align=\"center\">\u20131<\/td>\r\n<td style=\"width: 43.5667px; height: 66px;\" data-align=\"center\">0<\/td>\r\n<td style=\"width: 35.8667px; height: 66px;\" data-align=\"center\">1<\/td>\r\n<td style=\"width: 37.55px; height: 66px;\" data-align=\"center\">2<\/td>\r\n<\/tr>\r\n<tr style=\"height: 88px;\">\r\n<td style=\"width: 53.5px; height: 88px;\" data-align=\"center\">\r\n<p style=\"text-align: center;\"><em>y<\/em><\/p>\r\n<\/td>\r\n<td style=\"width: 45.3167px; height: 88px;\" data-align=\"center\">5<\/td>\r\n<td style=\"width: 51.2px; height: 88px;\" data-align=\"center\">2<\/td>\r\n<td style=\"width: 43.5667px; height: 88px;\" data-align=\"center\">1<\/td>\r\n<td style=\"width: 35.8667px; height: 88px;\" data-align=\"center\">2<\/td>\r\n<td style=\"width: 37.55px; height: 88px;\" data-align=\"center\">5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\">\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135485986\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135485987\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135485986-solution\">47<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n<div id=\"fs-id1165135485989\" class=\"os-table first-element\">\r\n<table class=\"grid\" style=\"width: 259px;\" data-id=\"fs-id1165135485989\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 43.45px;\" data-align=\"center\">\r\n<p style=\"text-align: center;\"><em>x<\/em><\/p>\r\n<\/td>\r\n<td style=\"width: 32.5px;\" data-align=\"center\">\u20132<\/td>\r\n<td style=\"width: 37.3167px;\" data-align=\"center\">\u20131<\/td>\r\n<td style=\"width: 39.6167px;\" data-align=\"center\">0<\/td>\r\n<td style=\"width: 30.65px;\" data-align=\"center\">1<\/td>\r\n<td style=\"width: 35.4667px;\" data-align=\"center\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 43.45px;\" data-align=\"center\">\r\n<p style=\"text-align: center;\"><em>y<\/em><\/p>\r\n<\/td>\r\n<td style=\"width: 32.5px;\" data-align=\"center\">1<\/td>\r\n<td style=\"width: 37.3167px;\" data-align=\"center\">0<\/td>\r\n<td style=\"width: 39.6167px;\" data-align=\"center\">1<\/td>\r\n<td style=\"width: 30.65px;\" data-align=\"center\">4<\/td>\r\n<td style=\"width: 35.4667px;\" data-align=\"center\">9<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165133248546\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133248548\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n<div id=\"fs-id1165133248550\" class=\"os-table first-element\">\r\n<table class=\"grid\" style=\"width: 217px;\" data-id=\"fs-id1165133248550\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 33.7667px;\" data-align=\"center\">\r\n<p style=\"text-align: center;\"><em>x<\/em><\/p>\r\n<\/td>\r\n<td style=\"width: 29.5833px;\" data-align=\"center\">\u20132<\/td>\r\n<td style=\"width: 26.5833px;\" data-align=\"center\">\u20131<\/td>\r\n<td style=\"width: 29.4833px;\" data-align=\"center\">0<\/td>\r\n<td style=\"width: 28px;\" data-align=\"center\">1<\/td>\r\n<td style=\"width: 29.5833px;\" data-align=\"center\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.7667px;\" data-align=\"center\">\r\n<p style=\"text-align: center;\"><em>y<\/em><\/p>\r\n<\/td>\r\n<td style=\"width: 29.5833px;\" data-align=\"center\">\u20132<\/td>\r\n<td style=\"width: 26.5833px;\" data-align=\"center\">1<\/td>\r\n<td style=\"width: 29.4833px;\" data-align=\"center\">2<\/td>\r\n<td style=\"width: 28px;\" data-align=\"center\">1<\/td>\r\n<td style=\"width: 29.5833px;\" data-align=\"center\">\u20132<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\">\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134138600\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134138602\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134138600-solution\">49<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n<div id=\"fs-id1165134138604\" class=\"os-table first-element\">\r\n<table class=\"grid\" data-id=\"fs-id1165134138604\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 104px;\" data-align=\"center\">\r\n<p style=\"text-align: center;\"><em>x<\/em><\/p>\r\n<\/td>\r\n<td style=\"width: 143px;\" data-align=\"center\">\u20132<\/td>\r\n<td style=\"width: 141px;\" data-align=\"center\">\u20131<\/td>\r\n<td style=\"width: 106px;\" data-align=\"center\">0<\/td>\r\n<td style=\"width: 88px;\" data-align=\"center\">1<\/td>\r\n<td style=\"width: 108px;\" data-align=\"center\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 104px;\" data-align=\"center\">\r\n<p style=\"text-align: center;\"><em>y<\/em><\/p>\r\n<\/td>\r\n<td style=\"width: 143px;\" data-align=\"center\">\u20138<\/td>\r\n<td style=\"width: 141px;\" data-align=\"center\">\u20133<\/td>\r\n<td style=\"width: 106px;\" data-align=\"center\">0<\/td>\r\n<td style=\"width: 88px;\" data-align=\"center\">1<\/td>\r\n<td style=\"width: 108px;\" data-align=\"center\">0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134328259\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134328261\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134328259-solution\">50<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container has-first-element\">\r\n<div id=\"fs-id1165134328263\" class=\"os-table first-element\">\r\n<table class=\"grid\" style=\"width: 688px;\" data-id=\"fs-id1165134328263\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 89.5333px;\" data-align=\"center\">\r\n<p style=\"text-align: center;\"><em>x<\/em><\/p>\r\n<\/td>\r\n<td style=\"width: 125.217px;\" data-align=\"center\">\u20132<\/td>\r\n<td style=\"width: 138.317px;\" data-align=\"center\">\u20131<\/td>\r\n<td style=\"width: 102.7px;\" data-align=\"center\">0<\/td>\r\n<td style=\"width: 96.1px;\" data-align=\"center\">1<\/td>\r\n<td style=\"width: 96.1333px;\" data-align=\"center\">2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 89.5333px;\" data-align=\"center\">\r\n<p style=\"text-align: center;\"><em>y<\/em><\/p>\r\n<\/td>\r\n<td style=\"width: 125.217px;\" data-align=\"center\">8<\/td>\r\n<td style=\"width: 138.317px;\" data-align=\"center\">2<\/td>\r\n<td style=\"width: 102.7px;\" data-align=\"center\">0<\/td>\r\n<td style=\"width: 96.1px;\" data-align=\"center\">2<\/td>\r\n<td style=\"width: 96.1333px;\" data-align=\"center\">8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165135513626\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Technology<\/h3>\r\n<p id=\"fs-id1165135513632\">For the following exercises, use a calculator to find the answer.<\/p>\r\n\r\n<div id=\"fs-id1165135513635\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135513637\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">51<\/span><span class=\"os-divider\">. <\/span>Graph on the same set of axes the functions [latex] f(x)=x^2, f(x)=2x^2 [\/latex] and [latex] f(x)=\\frac{1}{3}x^2. [\/latex] What appears to be the effect of changing the coefficient?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165133402109\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133402111\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span>Graph on the same set of axes [latex] f(x)=x^2, f(x)=x^2+2 [\/latex] and [latex] f(x)=x^2, f(x)=x^2+5 [\/latex] and [latex] f(x)=^2-3. [\/latex] What appears to be the effect of adding a constant?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137843146\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137843148\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137843146-solution\">53<\/a><span class=\"os-divider\">. <\/span>Graph on the same set of axes [latex] f(x)=x^2, f(x)=(x-2)^2, f(x-3)^2, [\/latex] and [latex] f(x)=(x+4)^2. [\/latex] What appears to be the effect of adding or subtracting those numbers?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137697912\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137697915\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span>The path of an object projected at a 45 degree angle with initial velocity of 80 feet per second is given by the function [latex] h(x)=\\frac{-32}{(80)^2}x^2+x [\/latex] where [latex] x [\/latex] is the horizontal distance traveled and [latex] h(x) [\/latex] is the height in feet. Use the TRACE feature of your calculator to determine the height of the object when it has traveled 100 feet away horizontally.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135533793\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135533795\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135533793-solution\">55<\/a><span class=\"os-divider\">. <\/span>A suspension bridge can be modeled by the quadratic function [latex] h(x)=.0001x^2 [\/latex] with [latex] -2000\\le x\\le 2000 [\/latex] where [latex] |x| [\/latex] is the number of feet from the center and [latex] h(x) [\/latex] is height in feet. Use the TRACE feature of your calculator to estimate how far from the center does the bridge have a height of 100 feet.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165135629622\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extensions<\/h3>\r\n<p id=\"fs-id1165135523290\">For the following exercises, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function.<\/p>\r\n\r\n<div id=\"fs-id1165135523294\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135523296\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>Vertex [latex] (1, -2), [\/latex] opens up.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165137897840\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137897842\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137897840-solution\">57<\/a><span class=\"os-divider\">. <\/span>Vertex [latex] (-1, 2), [\/latex] opens down.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135609192\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135609194\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>Vertex [latex] (-5, 11), [\/latex] opens down.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135501919\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135501921\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135501919-solution\">59<\/a><span class=\"os-divider\">. <\/span>Vertex [latex] (-100, 100), [\/latex] opens up.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135501959\">For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function.<\/p>\r\n\r\n<div id=\"fs-id1165135501964\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135501966\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span>Contains [latex] (1, 1) [\/latex] and has shape of [latex] f(x)=2x^2. [\/latex] Vertex is on the <em>y-<\/em>axis.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137642716\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165133289615\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137642716-solution\">61<\/a><span class=\"os-divider\">. <\/span>Contains [latex] (-1, 4) [\/latex] and has the shape of [latex] f(x)=2x^2. [\/latex] Vertex is on the <em>y-<\/em>axis.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135551155\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135551157\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span>Contains [latex] (2, 3) [\/latex] and has the shape of [latex] f(x)=3x^2. [\/latex] Vertex is on the <em>y-<\/em>axis.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135525847\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135525850\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135525847-solution\">63<\/a><span class=\"os-divider\">. <\/span>Contains [latex] (1, -3) [\/latex] and has the shape of [latex] f(x)=-x^2. [\/latex] Vertex is on the <em>y-<\/em>axis.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135442559\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135442561\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span>Contains [latex] (4, 3) [\/latex] and has the shape of [latex] f(x)=5x^2. [\/latex] Vertex is on the <em>y-<\/em>axis.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137901057\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137901059\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137901057-solution\">65<\/a><span class=\"os-divider\">. <\/span>Contains [latex] (1, -6) [\/latex] has the shape of [latex] f(x)=3x^2. [\/latex] Vertex has<em> x-<\/em>coordinate of [latex] -1. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165131857387\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Real-World Applications<\/h3>\r\n<div id=\"fs-id1165131857392\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165131857394\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span>Find the dimensions of the rectangular dog park, Bicentennial Park, producing the greatest enclosed area given 200 feet of fencing.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165131857448\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137940524\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165131857448-solution\">67<\/a><span class=\"os-divider\">. <\/span>Find the dimensions of the rectangular dog park, Bicentennial Park, split into 2 pens of the same size producing the greatest possible enclosed area given 300 feet of fencing.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137940531\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137940533\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">68<\/span><span class=\"os-divider\">. <\/span>Find the dimensions of the rectangular dog park, Bicentennial Park, producing the greatest enclosed area split into 3 sections of the same size given 500 feet of fencing.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137940588\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134089391\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137940588-solution\">69<\/a><span class=\"os-divider\">. <\/span>Among all of the pairs of numbers whose sum is 6, find the pair with the largest product. What is the product?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134089397\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134089399\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">70<\/span><span class=\"os-divider\">. <\/span>Among all of the pairs of numbers whose difference is 12, find the pair with the smallest product. What is the product?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135364089\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135364091\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135364089-solution\">71<\/a><span class=\"os-divider\">. <\/span>Suppose that the price per unit in dollars of a cell phone production is modeled by [latex] p=\\$45-0.0125x, [\/latex] where [latex] x [\/latex] is in thousands of phones produced, and the revenue represented by thousands of dollars is [latex] R=x*p. [\/latex] Find the production level that will maximize revenue.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165135571732\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135571734\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">72<\/span><span class=\"os-divider\">. <\/span>A rocket is launched in the air. Its height, in meters above sea level, as a function of time, in seconds, is given by [latex] h(t)=-4.9t^2+229t+234. [\/latex] Find the maximum height the rocket attains.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135394035\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135394038\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135394035-solution\">73<\/a><span class=\"os-divider\">. <\/span>A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time, in seconds, is given by [latex] h(t)=-4.9t^2+24t+8. [\/latex] How long does it take to reach maximum height?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135449627\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135449629\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">74<\/span><span class=\"os-divider\">. <\/span>Dick's Sporting Goods Park holds 62,000 spectators. With a ticket price of $11, the average attendance has been 26,000. When the price dropped to $9, the average attendance rose to 31,000. Assuming that attendance is linearly related to ticket price, what ticket price would maximize revenue?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135449642\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135449644\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135449642-solution\">75<\/a><span class=\"os-divider\">. <\/span>A farmer finds that if she plants 75 trees per acre, each tree will yield 20 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest?\r\n\r\n<\/div>\r\n<\/section><\/div>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_55f2e8ec-a982-4586-9d48-a2f43d7b4107\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>In this section, you will:<\/p>\n<ul>\n<li>Recognize characteristics of parabolas.<\/li>\n<li>Understand how the graph of a parabola is related to its quadratic function.<\/li>\n<li>Determine a quadratic function\u2019s minimum or maximum value.<\/li>\n<li>Solve problems involving a quadratic function\u2019s minimum or maximum value.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<section>\n<ul id=\"list-00001\"><\/ul>\n<\/section>\n<\/div>\n<div id=\"Figure_03_02_001\" class=\"os-figure\">\n<div class=\"os-caption-container\">\n<figure id=\"attachment_713\" aria-describedby=\"caption-attachment-713\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-713 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-1-300x151.jpeg\" alt=\"\" width=\"300\" height=\"151\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-1-300x151.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-1-65x33.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-1-225x113.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-1-350x176.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-1.jpeg 731w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-713\" class=\"wp-caption-text\">Figure 1. An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr)<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165134339909\">Curved antennas, such as the ones shown in Figure 1, are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function.<\/p>\n<p id=\"fs-id1165134081264\">In this section, we will investigate quadratic functions, which frequently model problems involving area and projectile motion. Working with quadratic functions can be less complex than working with higher degree functions, so they provide a good opportunity for a detailed study of function behavior.<\/p>\n<section id=\"fs-id1165137762207\" data-depth=\"1\">\n<h2 data-type=\"title\">Recognizing Characteristics of Parabolas<\/h2>\n<p id=\"fs-id1165137727999\">The graph of a quadratic function is a U-shaped curve called a <span id=\"term-00007\" class=\"no-emphasis\" data-type=\"term\">parabola<\/span>. One important feature of the graph is that it has an extreme point, called the <strong><span id=\"term-00008\" data-type=\"term\">vertex<\/span><\/strong>. If the parabola opens up, the vertex represents the lowest point on the graph, or the <span id=\"term-00009\" class=\"no-emphasis\" data-type=\"term\">minimum value<\/span> of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the <span id=\"term-00010\" class=\"no-emphasis\" data-type=\"term\">maximum value<\/span>. 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 <strong><span id=\"term-00011\" data-type=\"term\">axis of symmetry<\/span><\/strong>. These features are illustrated in Figure 2.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_715\" aria-describedby=\"caption-attachment-715\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-715\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-2-300x283.jpeg\" alt=\"\" width=\"300\" height=\"283\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-2-300x283.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-2-65x61.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-2-225x212.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-2-350x330.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-2.jpeg 397w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-715\" class=\"wp-caption-text\">Figure 2<\/figcaption><\/figure>\n<p id=\"fs-id1165137549127\">The <em data-effect=\"italics\">y<\/em>-intercept is the point at which the parabola crosses the <em data-effect=\"italics\">y<\/em>-axis. The <em data-effect=\"italics\">x<\/em>-intercepts are the points at which the parabola crosses the <em data-effect=\"italics\">x<\/em>-axis. If they exist, the <em data-effect=\"italics\">x<\/em>-intercepts represent the <strong><span id=\"term-00012\" data-type=\"term\">zeros<\/span>, <\/strong>or <strong><span id=\"term-00013\" data-type=\"term\">roots<\/span><\/strong>, of the quadratic function, the values of\u00a0[latex]x[\/latex] at which [latex]y=0.[\/latex]<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 1: Identifying the Characteristics of a Parabola<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Determine the vertex, axis of symmetry, zeros, and y-intercept of the parabola shown in Figure 3.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_716\" aria-describedby=\"caption-attachment-716\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-716\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-3-300x292.jpeg\" alt=\"\" width=\"300\" height=\"292\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-3-300x292.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-3-65x63.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-3-225x219.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-3-350x341.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-3.jpeg 357w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-716\" class=\"wp-caption-text\">Figure 3<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The vertex is the turning point of the graph. We can see that the vertex is at\u00a0[latex](3, 1).[\/latex] Because this parabola opens upward, the axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is\u00a0[latex]x=3.[\/latex] This parabola does not cross the <em>x-<\/em>axis, so it has no zeros. It crosses the y-axis at\u00a0[latex](0, 7)[\/latex] so this is the<em> y-<\/em>intercept.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137641326\" data-depth=\"1\">\n<h2 data-type=\"title\">Understanding How the Graphs of Parabolas are Related to Their Quadratic Functions<\/h2>\n<p id=\"fs-id1165137652877\">The <span id=\"term-00014\" data-type=\"term\">general form<\/span> <strong>of a quadratic function <\/strong>presents the function in the form<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=ax^2+bx+c[\/latex]<\/p>\n<p id=\"fs-id1165137544673\">where\u00a0[latex]a, b,[\/latex] and\u00a0[latex]c[\/latex] are real numbers and\u00a0[latex]a\\not=0.[\/latex] If\u00a0[latex]a> 0,[\/latex] the parabola opens upward. If\u00a0[latex]a< 0,[\/latex] the parabola opens downward. We can use the general form of a parabola to find the equation for the axis of symmetry.<\/p>\n<p id=\"fs-id1165133234001\">The axis of symmetry is defined by\u00a0[latex]x=-\\frac{b}{2a}.[\/latex] If we use the quadratic formula,\u00a0[latex]x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a},[\/latex] to solve\u00a0[latex]ax^2+bx+c=0[\/latex] for the\u00a0 <em>x-<\/em>intercepts, or zeros, we find the value of\u00a0[latex]x[\/latex] halfway between them is always\u00a0[latex]x==\\frac{b}{2a},[\/latex] the equation for the axis of symmetry.<\/p>\n<p id=\"fs-id1165135190920\">Figure 4 represents the graph of the quadratic function written in general form as\u00a0[latex]- y=x^2+4x+3.[\/latex] In this form,\u00a0[latex]a=1, b=4,[\/latex] and\u00a0[latex]c=3.[\/latex] Because\u00a0[latex]a> 0,[\/latex] the parabola opens upward. The axis of symmetry is\u00a0[latex]x=-\\frac{4}{2(1)}=-2.[\/latex] This also makes sense because we can see from the graph that the vertical line\u00a0[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,\u00a0[latex](-2, -1).[\/latex] The <em>x-<\/em>intercepts, those points where the parabola crosses the <em>x-<\/em>axis, occur at\u00a0[latex]- (-3, 0)[\/latex] and [latex](-1, 0).[\/latex]<\/p>\n<figure id=\"attachment_717\" aria-describedby=\"caption-attachment-717\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-717\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-4-300x279.jpeg\" alt=\"\" width=\"300\" height=\"279\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-4-300x279.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-4-65x60.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-4-225x209.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-4-350x325.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-4.jpeg 380w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-717\" class=\"wp-caption-text\">Figure 4<\/figcaption><\/figure>\n<p id=\"fs-id1165137676320\">The <strong>standard form of a quadratic function<\/strong> presents the function in the form<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=a(x-h)^2+k[\/latex]<\/p>\n<p id=\"fs-id1303104\">where\u00a0[latex](h, k)[\/latex] is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the <strong><span id=\"term-00015\" data-type=\"term\">vertex form of a quadratic function<\/span><\/strong>.<\/p>\n<p id=\"fs-id1165137894543\">As with the general form, if\u00a0[latex]a> 0,[\/latex] the parabola opens upward and the vertex is a minimum. If\u00a0[latex]a< 0,[\/latex] the parabola opens downward, and the vertex is a maximum. Figure 5 represents the graph of the quadratic function written in standard form as\u00a0[latex]y=-3(x+2)^2+4.[\/latex] Since\u00a0[ x-h=x+2latex] [\/latex] in this example,\u00a0[latex]h=-2.[\/latex] In this form,\u00a0[latex]a=-3, h=-2,[\/latex] and\u00a0[latex]k=4.[\/latex] Because\u00a0[latex]a< 0,[\/latex] the parabola opens downward. The vertex is at [latex](-2, 4).[\/latex]<\/p>\n<\/section>\n<section data-depth=\"1\">\n<figure id=\"attachment_718\" aria-describedby=\"caption-attachment-718\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-718\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-5-300x300.jpeg\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-5-300x300.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-5-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-5-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-5-225x224.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-5-350x349.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-5.jpeg 386w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-718\" class=\"wp-caption-text\">Figure 5<\/figcaption><\/figure>\n<p id=\"fs-id1165137453489\">The standard form is useful for determining how the graph is transformed from the graph of\u00a0[latex]y=x^2.[\/latex] Figure 6 is the graph of this basic function.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_719\" aria-describedby=\"caption-attachment-719\" style=\"width: 280px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-719\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-6-280x300.jpeg\" alt=\"\" width=\"280\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-6-280x300.jpeg 280w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-6-65x70.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-6-225x241.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-6-350x375.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-6.jpeg 373w\" sizes=\"auto, (max-width: 280px) 100vw, 280px\" \/><figcaption id=\"caption-attachment-719\" class=\"wp-caption-text\">Figure 6<\/figcaption><\/figure>\n<p id=\"fs-id1165137770279\">If\u00a0[latex]k> 0,[\/latex] the graph shifts upward, whereas if\u00a0[latex]k< 0,[\/latex] the graph shifts downward. In Figure 5,\u00a0[latex]k> 0m[\/latex] so the graph is shifted 4 units upward. If\u00a0[latex]h> 0,[\/latex] the graph shifts toward the right and if\u00a0[latex]h< 0,[\/latex] the graph shifts to the left. In Figure 5,\u00a0[latex]h< 0,[\/latex] so the graph is shifted 2 units to the left. The magnitude of\u00a0[latex]a[\/latex] indicates the stretch of the graph. If\u00a0[latex]|a|> 1,[\/latex] the point associated with a particular<em> x-<\/em>value shifts farther from the <em data-effect=\"italics\">x-<\/em>axis, so the graph appears to become narrower, and there is a vertical stretch. But if\u00a0[latex]|a|< 1,[\/latex] the point associated with a particular <em>x-<\/em>value shifts closer to the <em data-effect=\"italics\">x-<\/em>axis, so the graph appears to become wider, but in fact there is a vertical compression. In Figure 5,\u00a0[latex]|a|> 1,[\/latex] so the graph becomes narrower.<\/p>\n<p id=\"fs-id1165135353112\">The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} a(x-h)^2+k &=& ax^2+bx+c \\\\ ax^2-2ahx+(ah^2+k) &=& ax^2+bx+c \\end{array}[\/latex]<\/p>\n<p>For the linear terms to be equal, the coefficients must be equal.<\/p>\n<p style=\"text-align: center;\">[latex]-2ah=b, \\ \\ \\text{so} \\ \\\u00a0 h=-\\frac{b}{2a}[\/latex]<\/p>\n<p id=\"fs-id1165134118295\">This is the axis of symmetry we defined earlier. Setting the constant terms equal:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} ah^2+k &=& c \\\\ \\quad \\quad \\ \\ \\ k &=& c-ah^2 \\\\ &=& c-a-\\left(\\frac{b}{2a}\\right)^2 \\\\ &=& c-\\frac{b^2}{4a} \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165137476446\">In practice, though, it is usually easier to remember that <em data-effect=\"italics\">k<\/em> is the output value of the function when the input is\u00a0[latex]h,[\/latex] so [latex]f(h)=k.[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Forms of Quadratic Functions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A quadratic function is a polynomial function of degree two. The graph of a <span id=\"term-00017\" class=\"no-emphasis\" data-type=\"term\">quadratic function<\/span> is a parabola.<\/p>\n<p>The\u00a0<strong>general form of a quadratic function<\/strong> is\u00a0[latex]f(x)=ax^2+bx+c[\/latex] where\u00a0[latex]a, b,[\/latex] and\u00a0[latex]c[\/latex] are real numbers and [latex]a\\not=0.[\/latex]<\/p>\n<p>The\u00a0<strong>standard form of a quadratic function<\/strong> is\u00a0[latex]f(x)=a(x-h)^2+k[\/latex] where [latex]a\\not=0.[\/latex]<\/p>\n<p>The vertex\u00a0[latex](h, k)[\/latex] is located at<\/p>\n<p style=\"text-align: center;\">[latex]h=-\\frac{b}{2a}, k=f(h)=f(\\frac{-b}{2a})[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a graph of a quadratic function, write the equation of the function in general form.<\/strong><\/p>\n<ol>\n<li>Identify the horizontal shift of the parabola; this value is\u00a0[latex]h.[\/latex] Identify the vertical shift of the parabola; this value is [latex]k.[\/latex]<\/li>\n<li>Substitute the values of horizontal and vertical shift for\u00a0[latex]h[\/latex] and\u00a0[latex]k.[\/latex] in the function [latex]f(x)=a(x-h)^2+k.[\/latex]<\/li>\n<li>Substitute the values of any point, other than the vertex, on the graph of the parabola for\u00a0[latex]x[\/latex] and [latex]f(x).[\/latex]<\/li>\n<li>Solve for the stretch factor, [latex]|a|.[\/latex]<\/li>\n<li>Expand and simplify to write in general form.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 2: Writing the Equation of a Quadratic Function from the Graph<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write an equation for the quadratic function [latex]g[\/latex] in Figure 7 as a transformation of [latex]f(x)=x^2,[\/latex] and then expand the formula, and simplify terms to write the equation in general form.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_720\" aria-describedby=\"caption-attachment-720\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-720\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-7-300x273.jpeg\" alt=\"\" width=\"300\" height=\"273\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-7-300x273.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-7-65x59.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-7-225x205.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-7-350x318.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-7.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-720\" class=\"wp-caption-text\">Figure 7<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>We can see the graph of [latex]g[\/latex] is the graph of [latex]f(x)=x^2[\/latex] shifted to the left 2 and down 3, giving a formula in the form<\/p>\n<p style=\"text-align: center;\">[latex]g(x)=a(x-(-2))^2-3=a(x+2)^2-3.[\/latex]<\/p>\n<p>Substituting the coordinates of a point on the curve, such as [latex](0, -1),[\/latex] we can solve for the stretch factor.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} -1 &=& a(0+2)^2-3 \\\\ \\ \\ \\ 2 &=& 4a \\\\ \\ \\ \\ a &=& \\frac{1}{2} \\end{array}[\/latex]<\/p>\n<p>In standard form, the algebraic model for this graph is<\/p>\n<p style=\"text-align: center;\">[latex]g(x)=\\frac{1}{2}(x+2)^2-3.[\/latex]<\/p>\n<p>To write this in general polynomial form, we can expand the formula and simplify terms.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} g(x) &=& \\frac{1}{2}(x+2)^2-3 \\\\ &=& \\frac{1}{2}(x+2)(x+2)-3 \\\\ &=& \\frac{1}{2}(x^2+4x+4)-3 \\\\ &=& \\frac{1}{2}x^2+2x+2-3 \\\\ &=& \\frac{1}{2}x^2+2x-1 \\end{array}\u00a0[\/latex]<\/p>\n<p>Notice that the horizontal and vertical shifts of the basic graph of the quadratic function determine the location of the vertex of the parabola; the vertex is unaffected by stretches and compressions.<\/p>\n<h3>Analysis<\/h3>\n<p>We can check our work using the table feature on a graphing utility. First enter [latex]Y1-\\frac{1}{2}(x+2)^2-3.[\/latex] Next, select [latex]\\text{TBLSET},[\/latex] then use [latex]\\text{TblStart}=-6[\/latex] and [latex]\\Delta \\text{Tbl}=2,[\/latex] and select [latex]\\text{TABLE}.[\/latex] See Table 1.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 1<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 16.6667%;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 16.6667%;\">-6<\/td>\n<td style=\"width: 16.6667%;\">-4<\/td>\n<td style=\"width: 16.6667%;\">-2<\/td>\n<td style=\"width: 16.6667%;\">0<\/td>\n<td style=\"width: 16.6667%;\">2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.6667%;\">[latex]y[\/latex]<\/td>\n<td style=\"width: 16.6667%;\">5<\/td>\n<td style=\"width: 16.6667%;\">-1<\/td>\n<td style=\"width: 16.6667%;\">-3<\/td>\n<td style=\"width: 16.6667%;\">-1<\/td>\n<td style=\"width: 16.6667%;\">5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The ordered pairs in the table correspond to points on the graph.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1165135460939\" class=\"unnumbered\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137838619\" data-type=\"commentary\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A coordinate grid has been superimposed over the quadratic path of a basketball in Figure 8. Assume that the point\u00a0[latex](-4, 7)[\/latex] is the highest point of the basketball\u2019s trajectory. Find an equation for the path of the ball. Does the shooter make the basket?<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_721\" aria-describedby=\"caption-attachment-721\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-721\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-8-300x261.jpeg\" alt=\"\" width=\"300\" height=\"261\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-8-300x261.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-8-65x56.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-8-225x195.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-8-350x304.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-8.jpeg 488w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-721\" class=\"wp-caption-text\">Figure 8. (credit: modification of work by Dan Meyer)<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a quadratic function in general form, find the vertex of the parabola.<\/strong><\/p>\n<ol>\n<li>Identify\u00a0[latex]a, b,[\/latex] and [latex]c.[\/latex]<\/li>\n<li>Find\u00a0[latex]h,[\/latex] the <em>x-<\/em>coordinate of the vertex, by substituting\u00a0[latex]a[\/latex] and\u00a0[latex]b[\/latex] into [latex]h=-\\frac{b}{2a}.[\/latex]<\/li>\n<li>Find\u00a0[latex]k,[\/latex] the <em>y-<\/em>coordinate of the vertex, by evaluating [latex]k=f(h)=f(-\\frac{b}{2a}).[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3: Finding the Vertex of a Quadratic Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the vertex of the quadratic function\u00a0[latex]f(x)=2x^2-6x+7.[\/latex] Rewrite the quadratic in standard form (vertex form).<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The horizontal coordinate of the vertex will be at<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} h &=& -\\frac{b}{2a} \\\\ &=& -\\frac{-6}{2(2)} \\\\ &=& \\frac{6}{4} \\\\ &=& \\frac{3}{2} \\end{array}[\/latex]<\/p>\n<p>The vertical coordinate of the vertex will be at<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} k &=& f(h) \\\\ &=& f\\left(\\frac{3}{2}\\right) \\\\ &=& 2\\left(\\frac{3}{2}\\right)^2-6\\left(\\frac{3}{2}\\right)+7 \\\\ &=& \\frac{5}{2} \\end{array}\u00a0[\/latex]<\/p>\n<p>Rewriting into standard form, the stretch factor will be same as the\u00a0[latex]a[\/latex] in the original quadratic. First, find the horizontal coordinate of the vertex. Then find the vertical coordinate of the vertex. Substitute the values into standard form, using the &#8220;[latex]a[\/latex]&#8221; from the general form.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} f(x) &=& ax^2+bx+c \\\\ f(x) &=& 2x^2-6x+7 \\end{array}[\/latex]<\/p>\n<p>The standard form of a quadratic function prior to writing the function then becomes the following:<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=2(x-\\frac{3}{2})^2+\\frac{5}{2}[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>One reason we may want to identify the vertex 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]x.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1165137658566\" class=\"unnumbered\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137591920\" data-type=\"commentary\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Given the equation\u00a0[latex]g(x)=13+x^2-6x,[\/latex] write the equation in general form and then in standard form.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165133436210\" data-depth=\"1\">\n<h2 data-type=\"title\">Finding the Domain and Range of a Quadratic Function<\/h2>\n<p id=\"fs-id1165135596509\">Any number can be the input value of a quadratic function. Therefore, the domain of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum point, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the range will consist of all <em data-effect=\"italics\">y<\/em>-values greater than or equal to the <em data-effect=\"italics\">y<\/em>-coordinate at the turning point or less than or equal to the <em data-effect=\"italics\">y<\/em>-coordinate at the turning point, depending on whether the parabola opens up or down.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Domain and Range of a Quadratic Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The domain of any quadratic function is all real numbers unless the context of the function presents some restrictions.<\/p>\n<p>The range of a quadratic function written in general form [latex]f(x)=ax^2+bx+c[\/latex] with a positive [latex]a[\/latex] value is [latex]f(x)\\ge f\\left(-\\frac{b}{2a}\\right),[\/latex] or [latex][f\\left(-\\frac{b}{2a}), \\infty\\right);[\/latex] the range of a quadratic function written in general form with a negative [latex]a[\/latex] value is [latex]f(x)\\le f\\left(-\\frac{b}{2a}\\right),[\/latex] or [latex](-\\infty, f\\left(-\\frac{b}{2a}\\right)].[\/latex]<\/p>\n<p>The range of a quadratic function written in standard form\u00a0[latex]f(x)=a(x-h)^2+k[\/latex] with a positive\u00a0[latex]a[\/latex] value is\u00a0[latex]f(x)\\ge k;[\/latex] the range of a quadratic function written in standard form with a negative\u00a0[latex]a[\/latex] value is [latex]f(x)\\le k.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a quadratic function, find the domain and range.<\/strong><\/p>\n<ol>\n<li>Identify the domain of any quadratic function as all real numbers.<\/li>\n<li>Determine whether\u00a0[latex]a[\/latex] is positive or negative. If [latex]a[\/latex] is positive, the parabola has a minimum. If\u00a0[latex]a[\/latex] is negative, the parabola has a maximum.<\/li>\n<li>Determine the maximum or minimum value of the parabola, [latex]k.[\/latex]<\/li>\n<li>If the parabola has a minimum, the range is given by\u00a0[latex]f(x)\\ge k,[\/latex] or\u00a0[latex][k, \\infty).[\/latex] If the parabola has a maximum, the range is given by\u00a0[latex]f(x)\\le k,[\/latex] or [latex](-\\infty, k].[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 4: Finding the Domain and Range of a Quadratic Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the domain and range of [latex]f(x)=-5x^2+9x-1.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>As with any quadratic function, the domain is all real numbers.<\/p>\n<p>Because [latex]a[\/latex] is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding the x-value of the vertex.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} h &=& -\\frac{b}{2a} \\\\ &=& -\\frac{9}{2(-5)} \\\\ &=& \\frac{9}{10} \\end{array}[\/latex]<\/p>\n<p>The maximum value is given by [latex]f(h).[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} f\\left(\\frac{9}{10}\\right) &=& -5\\left(\\frac{9}{10}\\right)+9\\left(\\frac{9}{10}\\right)-1 \\\\ &=& \\frac{61}{20}\\end{array}[\/latex]<\/p>\n<p>The range is [latex]f(x)\\le \\frac{61}{20},[\/latex] or [latex](-\\infty, \\frac{61}{20}).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the domain and range of [latex]f(x)=2(x-\\frac{4}{7})^2+\\frac{8}{11}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137736541\" data-depth=\"1\">\n<h2 data-type=\"title\">Determining the Maximum and Minimum Values of Quadratic Functions<\/h2>\n<p id=\"fs-id1165137442167\">The output of the quadratic function at the vertex is the maximum or minimum value of the function, depending on the orientation of the <span id=\"term-00021\" class=\"no-emphasis\" data-type=\"term\">parabola<\/span>. We can see the maximum and minimum values in Figure 9.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_724\" aria-describedby=\"caption-attachment-724\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-724\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-9-300x172.jpeg\" alt=\"\" width=\"300\" height=\"172\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-9-300x172.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-9-768x440.jpeg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-9-65x37.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-9-225x129.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-9-350x200.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-9.jpeg 975w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-724\" class=\"wp-caption-text\">Figure 9<\/figcaption><\/figure>\n<p id=\"fs-id1165137431411\">There are many real-world scenarios that involve finding the maximum or minimum value of a quadratic function, such as applications involving area and revenue.<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 5: Finding the Maximum Value of a Quadratic Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A backyard farmer wants to enclose a rectangular space for a new garden within her fenced backyard. She has purchased 80 feet of wire fencing to enclose three sides, and she will use a section of the backyard fence as the fourth side.<\/p>\n<p>(a) Find a formula for the area enclosed by the fence if the sides of fencing perpendicular to the existing fence have length [latex]L.[\/latex]<\/p>\n<p>(b) What dimensions should she make her garden to maximize the enclosed area?<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Let&#8217;s use a diagram such as Figure 10 to record the given information. It is also helpful to introduce a temporary variable, [latex]W,[\/latex] to represent the width of the garden and the length of the fence section parallel to the backyard fence.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_725\" aria-describedby=\"caption-attachment-725\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-725\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-10-300x191.jpeg\" alt=\"\" width=\"300\" height=\"191\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-10-300x191.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-10-65x41.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-10-225x143.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-10-350x223.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-10.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-725\" class=\"wp-caption-text\">Figure 10<\/figcaption><\/figure>\n<p>(a) We know we have only 80 feet of fence available, and [latex]L+W+L=80,[\/latex] or more simply [latex]2L+W=80.[\/latex] This allows us to represent the width, [latex]W,[\/latex] in terms of [latex]L.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]W=80-2L[\/latex]<\/p>\n<p>Now we are ready to write an equation for the area the fence encloses. We know the area of a rectangle is length multiplied by width, so<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll} A &=& LW &=& L(80-2L) \\\\ A(L) &=& 80L-2L^2 \\end{array}[\/latex]<\/p>\n<p>This formula represents the area of the fence in terms of the variable length [latex]L.[\/latex] The function, written in general form, is<\/p>\n<p style=\"text-align: center;\">[latex]A(L)=-2L^2+80L[\/latex]<\/p>\n<p>(b) The quadratic has a negative leading coefficient, so the graph will open downward, and the vertex will be the maximum value for the area. In finding the vertex, we must be careful because the equation is not written in standard polynomial form with decreasing powers. This is why we rewrote the function in general form above. Since [latex]a[\/latex] is the coefficient of the squared term, [latex]a=-2, b=80,[\/latex] and [latex]c-0.[\/latex]<\/p>\n<p>To find the vertex:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rclcrcl}h &=& -\\frac{b}{2a} && \\quad k &=& A(20) \\\\&=& -\\frac{80}{2(-2)} & \\quad \\text{and} && =& 80(20) - 2(20)^2 \\\\&=& 20 &&&=& 800\\end{array}[\/latex]<\/p>\n<p>The maximum value of the function is an area of 800 square feet, which occurs when [latex]L=20[\/latex] feet. When the shorter sides are 20 feet, there is 40 feet of fencing left for the longer side. To maximize the area, she should enclose the garden so the two shorter sides have length 20 feet and the longer side parallel to the existing fence has length 40 feet.<\/p>\n<h3>Analysis<\/h3>\n<p>This problem also could be solved by graphing the quadratic function. We can see where the maximum area occurs on a graph of the quadratic function in Figure 11.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_726\" aria-describedby=\"caption-attachment-726\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-726\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-11-300x247.jpeg\" alt=\"\" width=\"300\" height=\"247\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-11-300x247.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-11-65x54.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-11-225x186.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-11-350x289.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-11.jpeg 497w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-726\" class=\"wp-caption-text\">Figure 11<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given an application involving revenue, use a quadratic equation to find the maximum.<\/strong><\/p>\n<ol>\n<li>Write a quadratic equation for a revenue function.<\/li>\n<li>Find the vertex of the quadratic equation.<\/li>\n<li>Determine the <em data-effect=\"italics\">y<\/em>-value of the vertex.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 6: Finding Maximum Revenue<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The unit price of an item affects its supply and demand. That is, if the unit price goes up, the demand for the item will usually decrease. For example, a streaming service currently has 84,000 subscribers at a quarterly charge of $30. Market research has suggested that if the owners raise the price to $32, they would lose 5,000 subscribers. Assuming that subscriptions are linearly related to the price, what price should the newspaper charge for a quarterly subscription to maximize their revenue?<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Revenue is the amount of money a company brings in. In this case, the revenue can be found by multiplying the price per subscription times the number of subscribers, or quantity. We can introduce variables, [latex]p[\/latex] for price per subscription and [latex]Q[\/latex] for quantity, giving us the equation [latex]Revenue=pQ.[\/latex]<\/p>\n<p>Because the number of subscribers changes with the price, as need to find a relationship between the variables. We know that currently [latex]p=30[\/latex] and [latex]Q=84,000.[\/latex] We also know that if the price rises to $32, the streaming service would lose 5,000 subscribers, giving a second pair of values, [latex]p=32[\/latex] and [latex]Q=79,000.[\/latex] From this we can find a linear equation relating the two quantities. The slope will be<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} m &=& \\frac{79,000-84,000}{32-30} \\\\ &=& \\frac{-5,000}{2} \\\\ &=& -2,500\\end{array}[\/latex]<\/p>\n<p>This tells us the service will lose 2,500 subscribers for each dollar they raise the price. We can then solve for the <em>y-<\/em>intercept.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll} Q &=& -2,500p+b & \\text{Substitute in the point} \\ Q=84,000 \\ \\text{and} \\ p=30 \\\\ 84,000 &=& -2,500(30)=b & \\text{Solve for} \\ b \\\\ b &=& 159,000 \\end{array}[\/latex]<\/p>\n<p>This gives us the linear equation [latex]Q=-2,500p+159,000[\/latex] relating cost and subscribers. We now return to our revenue equation<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} \\text{Revenue} &=& pQ \\\\ \\text{Revenue} &=& p(-2,500p+159,000) \\\\ \\text{Revenue} &=& -2,500p^2+159,000p \\end{array}[\/latex]<\/p>\n<p>We now have a quadratic function for revenue as a function of the subscription charge. To find the price that will maximize revenue for the service, we can find the vertex.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} h &=& \\frac{159,000}{2(-2,500)} \\\\ &=& 31.8 \\end{array}[\/latex]<\/p>\n<p>The model tells us that the maximum revenue will occur if the streaming service charges $31.80 for a subscription. To find what the maximum revenue is, we evaluate the revenue function.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} \\text{maximum revenue} &=& -2,500(31.8)^2+159,000(31.8) \\\\ &=& 2,528,100 \\end{array}[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>This could also be solved by graphing the quadratic as in Figure 12. We can see the maximum revenue on a graph of the quadratic function.<\/p>\n<figure id=\"attachment_728\" aria-describedby=\"caption-attachment-728\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-728\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-12-300x231.jpeg\" alt=\"\" width=\"300\" height=\"231\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-12-300x231.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-12-65x50.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-12-225x173.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-12-350x269.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-12.jpeg 486w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-728\" class=\"wp-caption-text\">Figure 12<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<section id=\"fs-id1165135693703\" data-depth=\"2\">\n<h3 data-type=\"title\">Finding the <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-Intercepts of a Quadratic Function<\/h3>\n<p id=\"fs-id1165134569121\">Much as we did in the application problems above, we also need to find intercepts of quadratic equations for graphing parabolas. Recall that we find the <em>y-<\/em>intercept of a quadratic by evaluating the function at an input of zero, and we find the <em>x-<\/em>intercepts at locations where the output is zero. Notice in Figure 13 that the number of <em>x-<\/em>intercepts can vary depending upon the location of the graph.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_729\" aria-describedby=\"caption-attachment-729\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-729\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-13-300x114.jpeg\" alt=\"\" width=\"300\" height=\"114\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-13-300x114.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-13-768x292.jpeg 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-13-65x25.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-13-225x86.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-13-350x133.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-13.jpeg 891w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-729\" class=\"wp-caption-text\">Figure 13. Number of x-intercepts of a parabola<\/figcaption><\/figure>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Examples<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a quadratic function\u00a0[latex]f(x)[\/latex] find the y- and x-intercepts.<\/strong><\/p>\n<ol>\n<li>Evaluate [latex]f(0)[\/latex] to find the <em>y-<\/em>intercept.<\/li>\n<li>Solve the quadratic equation [latex]f(x)=0[\/latex]to find the <em>x-<\/em>intercept.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 7: Finding the <em>y-<\/em> and <em>x-<\/em>intercepts of a Parabola<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the y- and x-intercepts of the quadratic [latex]f(x)=3x^2+5x-2.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>We find the y-intercept by evaluating [latex]f(0).[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} f(0) &=& 3(0)^2+5(0)-2 \\\\ &=& -2 \\end{array}[\/latex]<\/p>\n<p>So the <em>y-<\/em>intercept is at [latex](0, -2).[\/latex]<\/p>\n<p>For the <em>x-<\/em>intercepts, we find all solutions of [latex]f(x)=0.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]0=3x^2+5x-2[\/latex]<\/p>\n<p>In this case, the quadratic can be factored easily, providing the simplest method for solution.<\/p>\n<p style=\"text-align: center;\">[latex]0=(3x-1)(x+2)[\/latex]<\/p>\n<p>So the<em> x-<\/em>intercepts are at [latex](\\frac{1}{3}, 0)[\/latex] and [latex](-2, 0).[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>By graphing the function, we can confirm that the graph crosses the <em data-effect=\"italics\">y<\/em>-axis at [latex](0, -2).[\/latex] We can also confirm that the graph crosses the <em>x-<\/em>axis at [latex](\\frac{1}{3}, 0)[\/latex] and [latex](-2, 0).[\/latex] See Figure 14.<\/p>\n<figure id=\"attachment_730\" aria-describedby=\"caption-attachment-730\" style=\"width: 293px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-730\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-14-293x300.jpeg\" alt=\"\" width=\"293\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-14-293x300.jpeg 293w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-14-65x67.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-14-225x231.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-14-350x359.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-14.jpeg 398w\" sizes=\"auto, (max-width: 293px) 100vw, 293px\" \/><figcaption id=\"caption-attachment-730\" class=\"wp-caption-text\">Figure 14<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135381309\" data-depth=\"2\">\n<h3 data-type=\"title\">Rewriting Quadratics in Standard Form<\/h3>\n<p id=\"fs-id1165135381314\">In Example 7, the quadratic was easily solved by factoring. However, there are many quadratics that cannot be factored. We can solve these quadratics by first rewriting them in standard form.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a quadratic function, find the x-intercepts by rewriting in standard form.<\/strong><\/p>\n<ol>\n<li>Substitute [latex]a[\/latex] and [latex]b[\/latex] into [latex]h=-\\frac{b}{2a}.[\/latex]<\/li>\n<li>Substitute [latex]x=h[\/latex] into the general form of the quadratic function to find [latex]k.[\/latex]<\/li>\n<li>Rewrite the quadratic in standard form using [latex]h[\/latex]and [latex]k.[\/latex]<\/li>\n<li>Solve for when the output of the function will be zero to find the <em>x-<\/em>intercepts.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"os-note-body\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 8: Finding the <em>x-<\/em>intercepts of a Parabola<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find the x-intercepts of the quadratic function [latex]f(x)=2x^2+4x-4.[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>We begin by solving for when the output will be zero.<\/p>\n<p style=\"text-align: center;\">[latex]0=2x^2+4x-4[\/latex]<\/p>\n<p>Because the quadratic is not easily factorable in this case, we solve for the intercepts by first rewriting the quadratic in standard form.<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=a(x-h)^2+k[\/latex]<\/p>\n<p>We know that [latex]a=2.[\/latex] Then we solve for [latex]h[\/latex] and [latex]k.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rclcrcl}h &=& -\\frac{b}{2a} & \\quad k &=& f(-1) \\\\&=& -\\frac{4}{2(2)} &&=& 2(-1)^2 + 4(-1) - 4 \\\\&=& -1 &&=& -6\\end{array}[\/latex]<\/p>\n<p>So now we can rewrite in standard form.<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=2(x+1)^2-6[\/latex]<\/p>\n<p>We can now solve for when the output will be zero.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} 0 &=& 2(x+1)^2-6 \\\\ 6 &=& 2(x+1)^2 \\\\ 3 &=& (x+1)^2 \\\\ x+1 &=& \\pm\\sqrt{3} \\\\ x &=& -1\\pm\\sqrt{3} \\end{array}[\/latex]<\/p>\n<p>The graph has<em> x-<\/em>intercepts at [latex](-1-\\sqrt{3}, 0)[\/latex] and [latex](-1+\\sqrt{3}, 0).[\/latex]<\/p>\n<p>We can check out work by graphing the given function on a graphing utility and observing the <em>x-<\/em>intercepts. See Figure 15.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_731\" aria-describedby=\"caption-attachment-731\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-731\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-15-300x293.jpeg\" alt=\"\" width=\"300\" height=\"293\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-15-300x293.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-15-65x63.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-15-225x220.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-15-350x342.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-15.jpeg 381w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-731\" class=\"wp-caption-text\">Figure 15<\/figcaption><\/figure>\n<h3>Analysis<\/h3>\n<p>We could have achieved the same results using the quadratic formula. Identify [latex]a=2, b=4[\/latex] and [latex]c=-4.[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} x&=& \\frac{-b\\pm\\sqrt{b^2-4ac}}{2a} \\\\ &=& \\frac{-4\\pm\\sqrt{4^2-4(2)(-4)}}{2(2)} \\\\ &=& \\frac{-4\\pm\\sqrt{48}}{4} \\\\ &=& \\frac{-4\\pm\\sqrt{3(16)}}{4} \\\\ &=& -1\\pm\\sqrt{3} \\end{array}[\/latex]<\/p>\n<p>So the <em>x-<\/em>intercepts occur at [latex](-1-\\sqrt{3}, 0)[\/latex] and [latex](-1+\\sqrt{3}, 0).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>In Try It #2, we found the standard and general form for the function\u00a0[latex]g(x)=13+x^2-6x.[\/latex] Now find the <em>y-<\/em> and <em>x-<\/em>intercepts (if any).<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 9: Applying the Vertex and x-intercepts of a Parabola<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A ball is thrown upward from the top of a 40 foot high building at a speed of 80 feet per second. The ball\u2019s height above ground can be modeled by the equation [latex]H(t)=-16t^2+80t+40.[\/latex]<\/p>\n<p>(a) When does the ball reach the maximum height?<\/p>\n<p>(b) What is the maximum height of the ball?<\/p>\n<p>(c) When does the ball hit the ground?<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>(a) The ball reached the maximum height at the vertex of the parabola.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} h &=& -\\frac{8}{2(-16)} \\\\ &=& \\frac{80}{32} \\\\ &=& \\frac{5}{2} \\\\ &=& 2.5 \\end{array}[\/latex]<\/p>\n<p>The ball reached a maximum height after 2.5 seconds.<\/p>\n<p>(b) To find the maximum height, find the <em>y-<\/em>coordinate of the vertex of the parabola.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} k &=& H\\left(-\\frac{b}{2a}\\right) \\\\ &=& H(2.5) \\\\ &=& -16(2.5)^2+80(2.5)+40 \\\\ &=& 140 \\end{array}[\/latex]<\/p>\n<p>The ball reaches a maximum height of 140 feet.<\/p>\n<p>(c) To find when the ball hits the ground, we need to determine when the height is zero, [latex]H(t)=0.[\/latex]<\/p>\n<p>We use the quadratic formula.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll} t &=& \\frac{-80\\pm\\sqrt{80^2-4(-16)(40)}}{2(-16)} \\\\ &=& \\frac{-80\\pm\\sqrt{8960}}{-32} \\end{array}[\/latex]<\/p>\n<p>Because the square root does not simplify nicely, we can use a calculator to approximate the values of the solutions.<\/p>\n<p style=\"text-align: center;\">[latex]t=\\frac{-80-\\sqrt{8960}}{-32}\\approx 5.458 \\ \\ \\text{or} \\ \\ t=\\frac{-80+\\sqrt{8960}}{-32}\\approx -0.458[\/latex]<\/p>\n<p>The second answer is outside the reasonable domain of our model, so we conclude the ball will hit the ground after about 5.458 seconds. See Figure 16.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_732\" aria-describedby=\"caption-attachment-732\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-732\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-16-300x252.jpeg\" alt=\"\" width=\"300\" height=\"252\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-16-300x252.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-16-65x55.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-16-225x189.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-16-350x295.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1-fig-16.jpeg 442w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-732\" class=\"wp-caption-text\">Figure 16<\/figcaption><\/figure>\n<p>Note that the graph does not represent the physical path of the ball upward and downward. Keep the quantities on each axis in mind while interpreting the graph.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A rock is thrown upward from the top of a 112-foot high cliff overlooking the ocean at a speed of 96 feet per second. The rock\u2019s height above ocean can be modeled by the equation [latex]H(t)=-16t^2+96t+112.[\/latex]<\/p>\n<p>(a) When does the rock reach the maximum height?<\/p>\n<p>(b) What is the maximum height of the rock?<\/p>\n<p>(c) When does the rock hit the ocean?<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Media<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Access these online resources for additional instruction and practice with quadratic equations.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=oH6eZ6oDRBI\">Graphing Quadratic Functions in General Form<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=L8-7QepRSi8\">Graphing Quadratic Functions in Standard Form<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=kFw3XU0wisU\">Quadratic Function Review<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=H46QVUdbBn0\">Characteristics of a Quadratic Function<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">5.1 Section Exercises<\/span><\/h2>\n<section id=\"fs-id1165135361324\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1165135361327\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1165135361332\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135361334\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135361332-solution\">1<\/a><span class=\"os-divider\">. <\/span>Explain the advantage of writing a quadratic function in standard form.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134367971\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134367972\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>How can the vertex of a parabola be used in solving real-world problems?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134367976\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134367977\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134367976-solution\">3<\/a><span class=\"os-divider\">. <\/span>Explain why the condition of [latex]a\\not=0[\/latex] is imposed in the definition of the quadratic function.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134094470\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134094472\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>What is another name for the standard form of a quadratic function?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134094476\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134094477\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134094476-solution\">5<\/a><span class=\"os-divider\">. <\/span>What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165133276257\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id1165133276262\">For the following exercises, rewrite the quadratic functions in standard form and give the vertex.<\/p>\n<div id=\"fs-id1165133276266\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133276268\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=x^2-12x+32[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135336013\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133065125\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135336013-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex]g(x)=x^2+2x-3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134199469\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134199470\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=x^2-x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135536371\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135536373\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135536371-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=x^2+5x-2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134476611\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134476612\" data-type=\"problem\">\n<p><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex]h(x)=2x^2+8x-10[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133328085\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133328087\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165133328085-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex]k(x)=3x^2-6x-9[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134422869\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134422870\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=2x^2-6x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135409784\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135409786\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135409784-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=3x^2-5x-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135258891\">For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.<\/p>\n<div id=\"fs-id1165135258896\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135258897\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex]y(x)=2x^2+10x+12[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133344089\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133344091\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165133344089-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=2x^2-10x+4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134091244\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134091247\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=-x^2+4x+3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134278554\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134278556\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134278554-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=4x^2+x-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135193269\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135193271\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex]h(t)=-4t^2+6t-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135380708\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135380710\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135380708-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\frac{1}{2}x^2+3x+1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134362815\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134362817\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=-\\frac{1}{3}x^2-2x+3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135419821\">For the following exercises, determine the domain and range of the quadratic function.<\/p>\n<div id=\"fs-id1165135419824\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135419827\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135419824-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=(x-3)^2+2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137681943\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137681945\" data-type=\"problem\">\n<p><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=-2(x+3)^2-6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135528310\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135528312\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135528310-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=x^2+6x+4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134199532\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134199534\" data-type=\"problem\">\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=2x^2-4x+2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134138647\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134138649\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134138647-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex]k(x)=3x^2-6x-9[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165133252496\">For the following exercises, use the vertex [latex](h, k)[\/latex] and a point on the graph [latex](x, y)[\/latex] to find the general form of the equation of the quadratic function.<\/p>\n<div id=\"fs-id1165137898126\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137898128\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex](h, k)=(2, 0), (x, y)=(4, 4)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134079556\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134079558\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134079556-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex](h, k)=(-2, -1), (x, y)=(-4, 3)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134485654\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134485656\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex](h, k)=(0, 1), (x, y)=(2, 5)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133077523\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133077525\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165133077523-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex](h, k)=(2, 3), (x, y)=(5, 12)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132912701\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132912704\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex](h, k)=(-5, 3), (x, y)=(2, 9)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133252567\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135536384\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165133252567-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex](h, k)=(3, 2), (x, y)=(10, 1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135412141\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135412143\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex](h, k)=(0, 1), (x, y)=(1, 0)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134479000\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134479002\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134479000-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex](h, k)=(1, 0), (x, y)=(0, 1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165133095093\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphical<\/h3>\n<p id=\"fs-id1165133095098\">For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.<\/p>\n<div id=\"fs-id1165133095103\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133095105\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=x^2-2x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135646115\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135646117\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135646115-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=x^2-6x-1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135692195\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135692197\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=^2-5x-6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135205053\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135205055\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135205053-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=x^2-7x+3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135500665\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135500667\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=-2x^2+5x-8[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135639341\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135639343\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135639341-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=4x^2-12x-3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165134391625\">For the following exercises, write the equation for the graphed quadratic function.<\/p>\n<div id=\"fs-id1165134391628\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134391630\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165134391635\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a positive parabola with a vertex at (2, -3) and y-intercept at (0, 1).\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-733\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.40-298x300.jpeg\" alt=\"\" width=\"298\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.40-298x300.jpeg 298w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.40-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.40-65x65.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.40-225x226.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.40-350x352.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.40.jpeg 377w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135262595\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135262597\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135262595-solution\">41<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165135262603\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a positive parabola with a vertex at (-1, 2) and y-intercept at (0, 3)\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-734\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.41-296x300.jpeg\" alt=\"\" width=\"296\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.41-296x300.jpeg 296w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.41-65x66.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.41-225x228.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.41-350x355.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.41.jpeg 374w\" sizes=\"auto, (max-width: 296px) 100vw, 296px\" \/><br \/>\n<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135262616\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135262618\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165135262624\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a negative parabola with a vertex at (2, 7).\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-735\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.42-276x300.jpeg\" alt=\"\" width=\"276\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.42-276x300.jpeg 276w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.42-65x71.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.42-225x245.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.42-350x381.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.42.jpeg 376w\" sizes=\"auto, (max-width: 276px) 100vw, 276px\" \/><br \/>\n<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134162152\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134162154\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134162152-solution\">43<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<p><span id=\"fs-id1165134162160\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a negative parabola with a vertex at (-1, 2).\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-736\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.43-298x300.jpeg\" alt=\"\" width=\"298\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.43-298x300.jpeg 298w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.43-150x150.jpeg 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.43-65x66.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.43-225x227.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.43-350x353.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.43.jpeg 376w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><br \/>\n<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134162173\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134162175\" data-type=\"problem\">\n<p><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165134196136\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a positive parabola with a vertex at (3, -1) and y-intercept at (0, 3.5).\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-737\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.44-300x194.jpeg\" alt=\"\" width=\"300\" height=\"194\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.44-300x194.jpeg 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.44-65x42.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.44-225x146.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.44-350x226.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.44.jpeg 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137940515\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137940518\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137940515-solution\">45<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\"><span id=\"fs-id1165131959568\" class=\"first-element\" data-type=\"media\" data-alt=\"Graph of a negative parabola with a vertex at (-2, 3).\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-738\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.45-277x300.jpeg\" alt=\"\" width=\"277\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.45-277x300.jpeg 277w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.45-65x70.jpeg 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.45-225x244.jpeg 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.45-350x379.jpeg 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/5.1.45.jpeg 407w\" sizes=\"auto, (max-width: 277px) 100vw, 277px\" \/><br \/>\n<\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165131959583\" data-depth=\"2\">\n<h3 data-type=\"title\">Numeric<\/h3>\n<p id=\"fs-id1165131959588\">For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.<\/p>\n<div id=\"fs-id1165131959593\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165131959596\" data-type=\"problem\">\n<p><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<div id=\"fs-id1165131959598\" class=\"os-table first-element\">\n<table class=\"grid\" style=\"height: 154px; width: 307px;\" data-id=\"fs-id1165131959598\" data-label=\"\">\n<tbody>\n<tr style=\"height: 66px;\">\n<td style=\"width: 53.5px; height: 66px; text-align: center;\" data-align=\"center\"><em>x<\/em><\/td>\n<td style=\"width: 45.3167px; height: 66px;\" data-align=\"center\">\u20132<\/td>\n<td style=\"width: 51.2px; height: 66px;\" data-align=\"center\">\u20131<\/td>\n<td style=\"width: 43.5667px; height: 66px;\" data-align=\"center\">0<\/td>\n<td style=\"width: 35.8667px; height: 66px;\" data-align=\"center\">1<\/td>\n<td style=\"width: 37.55px; height: 66px;\" data-align=\"center\">2<\/td>\n<\/tr>\n<tr style=\"height: 88px;\">\n<td style=\"width: 53.5px; height: 88px;\" data-align=\"center\">\n<p style=\"text-align: center;\"><em>y<\/em><\/p>\n<\/td>\n<td style=\"width: 45.3167px; height: 88px;\" data-align=\"center\">5<\/td>\n<td style=\"width: 51.2px; height: 88px;\" data-align=\"center\">2<\/td>\n<td style=\"width: 43.5667px; height: 88px;\" data-align=\"center\">1<\/td>\n<td style=\"width: 35.8667px; height: 88px;\" data-align=\"center\">2<\/td>\n<td style=\"width: 37.55px; height: 88px;\" data-align=\"center\">5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\">\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135485986\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135485987\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135485986-solution\">47<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<div id=\"fs-id1165135485989\" class=\"os-table first-element\">\n<table class=\"grid\" style=\"width: 259px;\" data-id=\"fs-id1165135485989\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 43.45px;\" data-align=\"center\">\n<p style=\"text-align: center;\"><em>x<\/em><\/p>\n<\/td>\n<td style=\"width: 32.5px;\" data-align=\"center\">\u20132<\/td>\n<td style=\"width: 37.3167px;\" data-align=\"center\">\u20131<\/td>\n<td style=\"width: 39.6167px;\" data-align=\"center\">0<\/td>\n<td style=\"width: 30.65px;\" data-align=\"center\">1<\/td>\n<td style=\"width: 35.4667px;\" data-align=\"center\">2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 43.45px;\" data-align=\"center\">\n<p style=\"text-align: center;\"><em>y<\/em><\/p>\n<\/td>\n<td style=\"width: 32.5px;\" data-align=\"center\">1<\/td>\n<td style=\"width: 37.3167px;\" data-align=\"center\">0<\/td>\n<td style=\"width: 39.6167px;\" data-align=\"center\">1<\/td>\n<td style=\"width: 30.65px;\" data-align=\"center\">4<\/td>\n<td style=\"width: 35.4667px;\" data-align=\"center\">9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165133248546\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133248548\" data-type=\"problem\">\n<p><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<div id=\"fs-id1165133248550\" class=\"os-table first-element\">\n<table class=\"grid\" style=\"width: 217px;\" data-id=\"fs-id1165133248550\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 33.7667px;\" data-align=\"center\">\n<p style=\"text-align: center;\"><em>x<\/em><\/p>\n<\/td>\n<td style=\"width: 29.5833px;\" data-align=\"center\">\u20132<\/td>\n<td style=\"width: 26.5833px;\" data-align=\"center\">\u20131<\/td>\n<td style=\"width: 29.4833px;\" data-align=\"center\">0<\/td>\n<td style=\"width: 28px;\" data-align=\"center\">1<\/td>\n<td style=\"width: 29.5833px;\" data-align=\"center\">2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.7667px;\" data-align=\"center\">\n<p style=\"text-align: center;\"><em>y<\/em><\/p>\n<\/td>\n<td style=\"width: 29.5833px;\" data-align=\"center\">\u20132<\/td>\n<td style=\"width: 26.5833px;\" data-align=\"center\">1<\/td>\n<td style=\"width: 29.4833px;\" data-align=\"center\">2<\/td>\n<td style=\"width: 28px;\" data-align=\"center\">1<\/td>\n<td style=\"width: 29.5833px;\" data-align=\"center\">\u20132<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\">\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134138600\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134138602\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134138600-solution\">49<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<div id=\"fs-id1165134138604\" class=\"os-table first-element\">\n<table class=\"grid\" data-id=\"fs-id1165134138604\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 104px;\" data-align=\"center\">\n<p style=\"text-align: center;\"><em>x<\/em><\/p>\n<\/td>\n<td style=\"width: 143px;\" data-align=\"center\">\u20132<\/td>\n<td style=\"width: 141px;\" data-align=\"center\">\u20131<\/td>\n<td style=\"width: 106px;\" data-align=\"center\">0<\/td>\n<td style=\"width: 88px;\" data-align=\"center\">1<\/td>\n<td style=\"width: 108px;\" data-align=\"center\">2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 104px;\" data-align=\"center\">\n<p style=\"text-align: center;\"><em>y<\/em><\/p>\n<\/td>\n<td style=\"width: 143px;\" data-align=\"center\">\u20138<\/td>\n<td style=\"width: 141px;\" data-align=\"center\">\u20133<\/td>\n<td style=\"width: 106px;\" data-align=\"center\">0<\/td>\n<td style=\"width: 88px;\" data-align=\"center\">1<\/td>\n<td style=\"width: 108px;\" data-align=\"center\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134328259\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134328261\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165134328259-solution\">50<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container has-first-element\">\n<div id=\"fs-id1165134328263\" class=\"os-table first-element\">\n<table class=\"grid\" style=\"width: 688px;\" data-id=\"fs-id1165134328263\" data-label=\"\">\n<tbody>\n<tr>\n<td style=\"width: 89.5333px;\" data-align=\"center\">\n<p style=\"text-align: center;\"><em>x<\/em><\/p>\n<\/td>\n<td style=\"width: 125.217px;\" data-align=\"center\">\u20132<\/td>\n<td style=\"width: 138.317px;\" data-align=\"center\">\u20131<\/td>\n<td style=\"width: 102.7px;\" data-align=\"center\">0<\/td>\n<td style=\"width: 96.1px;\" data-align=\"center\">1<\/td>\n<td style=\"width: 96.1333px;\" data-align=\"center\">2<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 89.5333px;\" data-align=\"center\">\n<p style=\"text-align: center;\"><em>y<\/em><\/p>\n<\/td>\n<td style=\"width: 125.217px;\" data-align=\"center\">8<\/td>\n<td style=\"width: 138.317px;\" data-align=\"center\">2<\/td>\n<td style=\"width: 102.7px;\" data-align=\"center\">0<\/td>\n<td style=\"width: 96.1px;\" data-align=\"center\">2<\/td>\n<td style=\"width: 96.1333px;\" data-align=\"center\">8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135513626\" data-depth=\"2\">\n<h3 data-type=\"title\">Technology<\/h3>\n<p id=\"fs-id1165135513632\">For the following exercises, use a calculator to find the answer.<\/p>\n<div id=\"fs-id1165135513635\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135513637\" data-type=\"problem\">\n<p><span class=\"os-number\">51<\/span><span class=\"os-divider\">. <\/span>Graph on the same set of axes the functions [latex]f(x)=x^2, f(x)=2x^2[\/latex] and [latex]f(x)=\\frac{1}{3}x^2.[\/latex] What appears to be the effect of changing the coefficient?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165133402109\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133402111\" data-type=\"problem\">\n<p><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span>Graph on the same set of axes [latex]f(x)=x^2, f(x)=x^2+2[\/latex] and [latex]f(x)=x^2, f(x)=x^2+5[\/latex] and [latex]f(x)=^2-3.[\/latex] What appears to be the effect of adding a constant?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137843146\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137843148\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137843146-solution\">53<\/a><span class=\"os-divider\">. <\/span>Graph on the same set of axes [latex]f(x)=x^2, f(x)=(x-2)^2, f(x-3)^2,[\/latex] and [latex]f(x)=(x+4)^2.[\/latex] What appears to be the effect of adding or subtracting those numbers?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137697912\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137697915\" data-type=\"problem\">\n<p><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span>The path of an object projected at a 45 degree angle with initial velocity of 80 feet per second is given by the function [latex]h(x)=\\frac{-32}{(80)^2}x^2+x[\/latex] where [latex]x[\/latex] is the horizontal distance traveled and [latex]h(x)[\/latex] is the height in feet. Use the TRACE feature of your calculator to determine the height of the object when it has traveled 100 feet away horizontally.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135533793\" class=\"material-set-2 os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135533795\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135533793-solution\">55<\/a><span class=\"os-divider\">. <\/span>A suspension bridge can be modeled by the quadratic function [latex]h(x)=.0001x^2[\/latex] with [latex]-2000\\le x\\le 2000[\/latex] where [latex]|x|[\/latex] is the number of feet from the center and [latex]h(x)[\/latex] is height in feet. Use the TRACE feature of your calculator to estimate how far from the center does the bridge have a height of 100 feet.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165135629622\" data-depth=\"2\">\n<h3 data-type=\"title\">Extensions<\/h3>\n<p id=\"fs-id1165135523290\">For the following exercises, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function.<\/p>\n<div id=\"fs-id1165135523294\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135523296\" data-type=\"problem\">\n<p><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>Vertex [latex](1, -2),[\/latex] opens up.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165137897840\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137897842\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137897840-solution\">57<\/a><span class=\"os-divider\">. <\/span>Vertex [latex](-1, 2),[\/latex] opens down.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135609192\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135609194\" data-type=\"problem\">\n<p><span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>Vertex [latex](-5, 11),[\/latex] opens down.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135501919\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135501921\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135501919-solution\">59<\/a><span class=\"os-divider\">. <\/span>Vertex [latex](-100, 100),[\/latex] opens up.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135501959\">For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function.<\/p>\n<div id=\"fs-id1165135501964\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135501966\" data-type=\"problem\">\n<p><span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span>Contains [latex](1, 1)[\/latex] and has shape of [latex]f(x)=2x^2.[\/latex] Vertex is on the <em>y-<\/em>axis.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137642716\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165133289615\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137642716-solution\">61<\/a><span class=\"os-divider\">. <\/span>Contains [latex](-1, 4)[\/latex] and has the shape of [latex]f(x)=2x^2.[\/latex] Vertex is on the <em>y-<\/em>axis.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135551155\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135551157\" data-type=\"problem\">\n<p><span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span>Contains [latex](2, 3)[\/latex] and has the shape of [latex]f(x)=3x^2.[\/latex] Vertex is on the <em>y-<\/em>axis.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135525847\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135525850\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135525847-solution\">63<\/a><span class=\"os-divider\">. <\/span>Contains [latex](1, -3)[\/latex] and has the shape of [latex]f(x)=-x^2.[\/latex] Vertex is on the <em>y-<\/em>axis.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135442559\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135442561\" data-type=\"problem\">\n<p><span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span>Contains [latex](4, 3)[\/latex] and has the shape of [latex]f(x)=5x^2.[\/latex] Vertex is on the <em>y-<\/em>axis.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137901057\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137901059\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137901057-solution\">65<\/a><span class=\"os-divider\">. <\/span>Contains [latex](1, -6)[\/latex] has the shape of [latex]f(x)=3x^2.[\/latex] Vertex has<em> x-<\/em>coordinate of [latex]-1.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165131857387\" data-depth=\"2\">\n<h3 data-type=\"title\">Real-World Applications<\/h3>\n<div id=\"fs-id1165131857392\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165131857394\" data-type=\"problem\">\n<p><span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span>Find the dimensions of the rectangular dog park, Bicentennial Park, producing the greatest enclosed area given 200 feet of fencing.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165131857448\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137940524\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165131857448-solution\">67<\/a><span class=\"os-divider\">. <\/span>Find the dimensions of the rectangular dog park, Bicentennial Park, split into 2 pens of the same size producing the greatest possible enclosed area given 300 feet of fencing.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137940531\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137940533\" data-type=\"problem\">\n<p><span class=\"os-number\">68<\/span><span class=\"os-divider\">. <\/span>Find the dimensions of the rectangular dog park, Bicentennial Park, producing the greatest enclosed area split into 3 sections of the same size given 500 feet of fencing.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137940588\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134089391\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165137940588-solution\">69<\/a><span class=\"os-divider\">. <\/span>Among all of the pairs of numbers whose sum is 6, find the pair with the largest product. What is the product?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134089397\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134089399\" data-type=\"problem\">\n<p><span class=\"os-number\">70<\/span><span class=\"os-divider\">. <\/span>Among all of the pairs of numbers whose difference is 12, find the pair with the smallest product. What is the product?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135364089\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135364091\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135364089-solution\">71<\/a><span class=\"os-divider\">. <\/span>Suppose that the price per unit in dollars of a cell phone production is modeled by [latex]p=\\$45-0.0125x,[\/latex] where [latex]x[\/latex] is in thousands of phones produced, and the revenue represented by thousands of dollars is [latex]R=x*p.[\/latex] Find the production level that will maximize revenue.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165135571732\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135571734\" data-type=\"problem\">\n<p><span class=\"os-number\">72<\/span><span class=\"os-divider\">. <\/span>A rocket is launched in the air. Its height, in meters above sea level, as a function of time, in seconds, is given by [latex]h(t)=-4.9t^2+229t+234.[\/latex] Find the maximum height the rocket attains.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135394035\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135394038\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135394035-solution\">73<\/a><span class=\"os-divider\">. <\/span>A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time, in seconds, is given by [latex]h(t)=-4.9t^2+24t+8.[\/latex] How long does it take to reach maximum height?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135449627\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135449629\" data-type=\"problem\">\n<p><span class=\"os-number\">74<\/span><span class=\"os-divider\">. <\/span>Dick&#8217;s Sporting Goods Park holds 62,000 spectators. With a ticket price of $11, the average attendance has been 26,000. When the price dropped to $9, the average attendance rose to 31,000. Assuming that attendance is linearly related to ticket price, what ticket price would maximize revenue?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135449642\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135449644\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-5\" data-page-slug=\"chapter-5\" data-page-uuid=\"a3e49083-ba8c-53a6-bba4-458d10c86585\" data-page-fragment=\"fs-id1165135449642-solution\">75<\/a><span class=\"os-divider\">. <\/span>A farmer finds that if she plants 75 trees per acre, each tree will yield 20 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest?<\/p>\n<\/div>\n<\/section>\n<\/div>\n","protected":false},"author":158,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-264","chapter","type-chapter","status-publish","hentry"],"part":158,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/264","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/users\/158"}],"version-history":[{"count":29,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/264\/revisions"}],"predecessor-version":[{"id":1513,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/264\/revisions\/1513"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/parts\/158"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/264\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=264"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=264"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=264"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=264"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}