{"id":13836,"date":"2018-08-24T21:59:44","date_gmt":"2018-08-24T21:59:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/precalcone\/?post_type=chapter&#038;p=13836"},"modified":"2020-05-21T04:52:46","modified_gmt":"2020-05-21T04:52:46","slug":"quadratic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/chapter\/quadratic-functions\/","title":{"raw":"Section 2.3: Quadratic Functions","rendered":"Section 2.3: Quadratic Functions"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify key characteristics of parabolas from the graph.<\/li>\r\n \t<li>Understand how the graph of a parabola is related to its quadratic function.<\/li>\r\n \t<li>Draw the graph of a quadratic function.<\/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<figure id=\"Figure_03_02_001\" class=\"medium\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010711\/CNX_Precalc_Figure_03_02_0012.jpg\" alt=\"Satellite dishes.\" width=\"731\" height=\"368\" \/> <b>Figure 1.<\/b> An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr)[\/caption]<\/figure>\r\n<p id=\"fs-id1165134339909\">Curved antennas, such as the ones shown in the photo, 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<h2>Recognize characteristics of parabolas<\/h2>\r\n<p id=\"fs-id1165137727999\">The graph of a quadratic function is a U-shaped curve called a <strong>parabola<\/strong>. One important feature of the graph is that it has an extreme point, called the <strong>vertex<\/strong>. If the parabola opens up, the vertex represents the lowest point on the graph, or the <strong>minimum value<\/strong> of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the <strong>maximum value<\/strong>. In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the <strong>axis of symmetry<\/strong>. These features are illustrated in Figure 2.<span id=\"fs-id1165134118332\">\r\n<\/span><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010711\/CNX_Precalc_Figure_03_02_0022.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are.\" width=\"487\" height=\"480\" \/> <b>Figure 2<\/b>[\/caption]\r\n<p id=\"fs-id1165137549127\">The <em>y<\/em>-intercept is the point at which the parabola crosses the <em>y<\/em>-axis. The <em>x<\/em>-intercepts are the points at which the parabola crosses the <em>x<\/em>-axis. If they exist, the <em>x<\/em>-intercepts represent the <strong>zeros<\/strong>, or <strong>roots<\/strong>, of the quadratic function, the values of <em>x<\/em>\u00a0at which <em>y\u00a0<\/em>= 0.<\/p>\r\n\r\n<div id=\"Example_03_02_01\" class=\"example\">\r\n<div id=\"fs-id1165131959514\" class=\"exercise\">\r\n<div id=\"fs-id1165135541748\" class=\"problem textbox shaded\">\r\n<h3>Example 1: Identifying the Characteristics of a Parabola<\/h3>\r\nDetermine the vertex, axis of symmetry, zeros, and <em>y<\/em>-intercept of the parabola shown in Figure 3.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010711\/CNX_Precalc_Figure_03_02_0032.jpg\" alt=\"Graph of a parabola with a vertex at (3, 1) and a y-intercept at (0, 7).\" width=\"487\" height=\"517\" \/> <b>Figure 3<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"300488\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"300488\"]\r\n\r\nThe vertex is the turning point of the graph. We can see that the vertex is at (3, 1). 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 <em>x\u00a0<\/em>= 3. This parabola does not cross the <em>x<\/em>-axis, so it has no real zeros. It crosses the <em>y<\/em>-axis at (0, 7) so this is the <em>y<\/em>-intercept.\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h2>\u00a0Understand how the graph of a parabola is related to its quadratic function<\/h2>\r\nThe <strong>general form of a quadratic function<\/strong> presents the function in the form\r\n<p style=\"text-align: center\">[latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex]<\/p>\r\n<p id=\"fs-id1165137544673\">where <em>a<\/em>,\u00a0<em>b<\/em>, and <em>c<\/em>\u00a0are real numbers and [latex]a\\ne 0[\/latex]. If [latex]a&gt;0[\/latex], the parabola opens upward. If [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 [latex]x=-\\frac{b}{2a}[\/latex]. If we use the quadratic formula, [latex]x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex], to solve [latex]a{x}^{2}+bx+c=0[\/latex] for the <em>x<\/em>-intercepts, or zeros, we find the value of\u00a0<em>x<\/em>\u00a0halfway between them is always [latex]x=-\\frac{b}{2a}[\/latex], the equation for the axis of symmetry.<\/p>\r\nFigure 4 shows\u00a0the graph of the quadratic function written in general form as [latex]y={x}^{2}+4x+3[\/latex]. In this form, [latex]a=1,\\text{ }b=4[\/latex], and [latex]c=3[\/latex]. Because [latex]a&gt;0[\/latex], the parabola opens upward. The axis of symmetry is [latex]x=-\\frac{4}{2\\left(1\\right)}=-2[\/latex]. This also makes sense because we can see from the graph that the vertical line [latex]x=-2[\/latex] divides the graph in half. The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph, in this instance, [latex]\\left(-2,-1\\right)[\/latex]. The <i>x<\/i>-intercepts, those points where the parabola crosses the <i>x<\/i>-axis, occur at [latex]\\left(-3,0\\right)[\/latex] and [latex]\\left(-1,0\\right)[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010711\/CNX_Precalc_Figure_03_02_0042.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are for the function y=x^2+4x+3.\" width=\"487\" height=\"555\" \/> <b>Figure 4<\/b>[\/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\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/p>\r\n<p id=\"fs-id1303104\">where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the <strong>vertex form of a quadratic function<\/strong>. The function above has the standard form:\u00a0 [latex]y=(x+2)^2-1[\/latex]<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010711\/CNX_Precalc_Figure_03_02_0052.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are for the function y=-3(x+2)^2+4.\" width=\"487\" height=\"630\" \/> <b>Figure 5<\/b>[\/caption]\r\n<p id=\"fs-id1165137894543\">As with the general form, if [latex]a&gt;0[\/latex], the parabola opens upward and the vertex is a minimum. If [latex]a&lt;0[\/latex], the parabola opens downward, and the vertex is a maximum. Figure 5\u00a0is the\u00a0graph of the quadratic function written in standard form as [latex]y=-3{\\left(x+2\\right)}^{2}+4[\/latex]. Since [latex]x-h=x+2[\/latex] in this example, [latex]h=-2[\/latex]. In this form, [latex]a=-3,\\text{ }h=-2[\/latex], and [latex]k=4[\/latex]. Because [latex]a&lt;0[\/latex], the parabola opens downward. The vertex is at [latex]\\left(-2,\\text{ 4}\\right)[\/latex].<span id=\"fs-id1165134252223\">\r\n<\/span><\/p>\r\nThe standard form is useful for determining how the graph is transformed from the graph of [latex]y={x}^{2}[\/latex]. Figure 6\u00a0is the graph of this basic function.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010712\/CNX_Precalc_Figure_03_02_0062.jpg\" alt=\"Graph of y=x^2.\" width=\"487\" height=\"480\" \/> <b>Figure 6<\/b>[\/caption]\r\n<p id=\"fs-id1165137770279\">If [latex]k&gt;0[\/latex], the graph shifts upward, whereas if [latex]k&lt;0[\/latex], the graph shifts downward. In Figure 5, [latex]k&gt;0[\/latex], so the graph is shifted 4 units upward. If [latex]h&gt;0[\/latex], the graph shifts toward the right and if [latex]h&lt;0[\/latex], the graph shifts to the left. In Figure 5, [latex]h&lt;0[\/latex], so the graph is shifted 2 units to the left. The magnitude of <em>a<\/em>\u00a0indicates the stretch of the graph. If [latex]|a|&gt;1[\/latex], the point associated with a particular <em>x<\/em>-value shifts farther from the <em>x-<\/em>axis, so the graph appears to become narrower, and there is a vertical stretch. But if [latex]|a|&lt;1[\/latex], the point associated with a particular <em>x<\/em>-value shifts closer to the <em>x-<\/em>axis, so the graph appears to become wider, but in fact there is a vertical compression. In Figure 5, [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\r\n<div id=\"eip-173\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{align}a{\\left(x-h\\right)}^{2}+k&amp;=a(x^2-2xh+h^2)+k \\\\ &amp;=a{x}^{2}-2ahx+a{h}^{2}+k=a{x}^{2}+bx+c \\end{align}[\/latex]<\/div>\r\n<p id=\"fs-id1165137409211\">For the quadratic expressions to be equal, the corresponding coefficients must be equal.<\/p>\r\n\r\n<div id=\"eip-144\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]-2ah=b,\\text{ so }h=-\\frac{b}{2a}[\/latex].<\/div>\r\n<p id=\"fs-id1165134118295\">This gives us the <strong>axis of symmetry<\/strong> we defined earlier. Setting the constant terms equal:<\/p>\r\n\r\n<div id=\"eip-313\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{align}a{h}^{2}+k&amp;=c \\\\ k&amp;=c-a{h}^{2} \\\\ &amp;=c-a{\\left(-\\frac{b}{2a}\\right)}^{2} \\\\ &amp;=c-\\frac{{b}^{2}}{4a} \\end{align}[\/latex]<\/div>\r\n<p id=\"fs-id1165137476446\">In practice, though, it is usually easier to remember that <em>k<\/em> is the output value of the function when the input is <em>h<\/em>, so [latex]f\\left(h\\right)=f\\left(-\\frac{b}{2a}\\right)=k[\/latex].<\/p>\r\n\r\n<div id=\"fs-id1165137749882\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Forms of Quadratic Functions<\/h3>\r\n<p id=\"fs-id1165135333154\">A quadratic function is a function of degree two. The graph of a <strong>quadratic function<\/strong> is a parabola. The <strong>general form of a quadratic function<\/strong> is [latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex] where <em>a,\u00a0b<\/em>, and <em>c<\/em>\u00a0are real numbers and [latex]a\\ne 0[\/latex].<\/p>\r\n<p id=\"fs-id1165137666538\">The <strong>standard form of a quadratic function<\/strong> is [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex].<\/p>\r\n<p id=\"fs-id1165137762385\">The vertex [latex]\\left(h,k\\right)[\/latex] is located at<\/p>\r\n<p style=\"text-align: center\">[latex]h=-\\frac{b}{2a},\\text{ }k=f\\left(h\\right)=f\\left(\\frac{-b}{2a}\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165131886746\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137650986\">How To: Given a graph of a quadratic function, write the equation of the function in general form.<\/h3>\r\n<ol id=\"fs-id1165134223276\">\r\n \t<li>Identify the horizontal shift of the parabola; this value is <em>h<\/em>. Identify the vertical shift of the parabola; this value is <em>k<\/em>.<\/li>\r\n \t<li>Substitute the values of the horizontal and vertical shift for <em>h<\/em>\u00a0and <em>k<\/em>. in the function [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{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 <em>x<\/em>\u00a0and [latex]f\\left(x\\right)[\/latex].<\/li>\r\n \t<li>Solve for the stretch factor, |<em>a<\/em>|.<\/li>\r\n \t<li>If the parabola opens up, [latex]a&gt;0[\/latex]. If the parabola opens down, [latex]a&lt;0[\/latex] since this means the graph was reflected about the <em>x<\/em>-axis.<\/li>\r\n \t<li>Expand and simplify to write in general form.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_02_02\" class=\"example\">\r\n<div id=\"fs-id1165135460939\" class=\"exercise\">\r\n<div id=\"fs-id1165135460941\" class=\"problem textbox shaded\">\r\n<h3>Example 2: Writing the Equation of a Quadratic Function from the Graph<\/h3>\r\n<p id=\"fs-id1165135532321\">Write an equation for the quadratic function <em>g<\/em>\u00a0in the graph below\u00a0as a transformation of [latex]f\\left(x\\right)={x}^{2}[\/latex], and then expand the formula, and simplify terms to write the equation in general form.<span id=\"fs-id1165137725791\">\r\n<\/span><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010712\/CNX_Precalc_Figure_03_02_0072.jpg\" alt=\"Graph of a parabola with its vertex at (-2, -3).\" width=\"487\" height=\"443\" \/> <b>Figure 7<\/b>[\/caption]\r\n\r\n[reveal-answer q=\"616320\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"616320\"]\r\n<p id=\"fs-id1165137742565\">We can see the graph of <em>g <\/em>is the graph of [latex]f\\left(x\\right)={x}^{2}[\/latex] shifted to the left 2 and down 3, giving a formula in the form [latex]g\\left(x\\right)=a{\\left(x+2\\right)}^{2}-3[\/latex].<\/p>\r\n<p id=\"fs-id1165134064001\">Substituting the coordinates of a point on the curve, such as [latex]\\left(0,-1\\right)[\/latex], we can solve for the stretch factor.<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{align}-1&amp;=a{\\left(0+2\\right)}^{2}-3 \\\\ 2&amp;=4a \\\\ a&amp;=\\frac{1}{2} \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137895371\">In standard form, the algebraic model for this graph is [latex]\\left(g\\right)x=\\frac{1}{2}{\\left(x+2\\right)}^{2}-3[\/latex].<\/p>\r\n<p id=\"fs-id1165137844164\">To write this in general polynomial form, we can expand the formula and simplify terms.<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{align}g\\left(x\\right)&amp;=\\frac{1}{2}{\\left(x+2\\right)}^{2}-3 \\\\ &amp;=\\frac{1}{2}\\left(x+2\\right)\\left(x+2\\right)-3 \\\\ &amp;=\\frac{1}{2}\\left({x}^{2}+4x+4\\right)-3 \\\\ &amp;=\\frac{1}{2}{x}^{2}+2x+2 - 3 \\\\ &amp;=\\frac{1}{2}{x}^{2}+2x - 1 \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137914060\">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>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165137803212\">We can check our work using the table feature on a graphing utility. First enter [latex]\\text{Y1}=\\frac{1}{2}{\\left(x+2\\right)}^{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}\\text{.}[\/latex]<\/p>\r\n\r\n<table id=\"Table_03_02_01\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>\u20136<\/td>\r\n<td>\u20134<\/td>\r\n<td>\u20132<\/td>\r\n<td>0<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em><strong>y<\/strong><\/em><\/td>\r\n<td>5<\/td>\r\n<td>\u20131<\/td>\r\n<td>\u20133<\/td>\r\n<td>\u20131<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id1165135570238\">The ordered pairs in the table correspond to points on the graph.<\/p>\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nA coordinate grid has been superimposed over the quadratic path of a basketball in the picture below. Find an equation for the path of the ball. Does the shooter make the basket?\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010712\/CNX_Precalc_Figure_03_02_0082.jpg\" alt=\"Stop motioned picture of a boy throwing a basketball into a hoop to show the parabolic curve it makes.\" width=\"487\" height=\"424\" \/> <b>Figure 8.<\/b> (credit: modification of work by Dan Meyer)[\/caption]\r\n\r\n[reveal-answer q=\"903323\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"903323\"]\r\n\r\nThe path passes through the origin and has vertex at [latex]\\left(-4,\\text{ }7\\right)[\/latex], so [latex]\\left(h\\right)x=-\\frac{7}{16}{\\left(x+4\\right)}^{2}+7[\/latex]. To make the shot, [latex]h\\left(-7.5\\right)[\/latex] would need to be about 4 but [latex]h\\left(-7.5\\right)\\approx 1.64[\/latex]; he doesn\u2019t make it.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135168275\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135574310\">How To: Given a quadratic function in general form, find the vertex of the parabola.<\/h3>\r\n<ol id=\"fs-id1165134108459\">\r\n \t<li>Identify <em>a<\/em>,\u00a0<em>b<\/em>, and\u00a0<em>c<\/em>.<\/li>\r\n \t<li>Find <em>h<\/em>, the <em>x<\/em>-coordinate of the vertex, by substituting <em>a<\/em> and <em>b<\/em>\u00a0into [latex]h=-\\frac{b}{2a}[\/latex].<\/li>\r\n \t<li>Find <em>k<\/em>, the <em>y<\/em>-coordinate of the vertex, by evaluating [latex]k=f\\left(h\\right)=f\\left(-\\frac{b}{2a}\\right)[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_02_03\" class=\"example\">\r\n<div id=\"fs-id1165137658566\" class=\"exercise\">\r\n<div id=\"fs-id1165137771901\" class=\"problem textbox shaded\">\r\n<h3>Example 3: Finding the Vertex of a Quadratic Function<\/h3>\r\n<p id=\"fs-id1165135173258\">Find the vertex of the quadratic function [latex]f\\left(x\\right)=2{x}^{2}-6x+7[\/latex]. Rewrite the quadratic in standard form (vertex form).<\/p>\r\n[reveal-answer q=\"648542\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"648542\"]\r\n<p id=\"fs-id1165137596323\">The horizontal coordinate of the vertex will be at<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{align}h&amp;=-\\frac{b}{2a} \\\\ &amp;=-\\frac{-6}{2\\left(2\\right)} \\\\ &amp;=\\frac{6}{4} \\\\ &amp;=\\frac{3}{2}\\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137465689\">The vertical coordinate of the vertex will be at<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{align}k&amp;=f\\left(h\\right) \\\\ &amp;=f\\left(\\frac{3}{2}\\right) \\\\ &amp;=2{\\left(\\frac{3}{2}\\right)}^{2}-6\\left(\\frac{3}{2}\\right)+7 \\\\ &amp;=\\frac{5}{2} \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165135177784\">Rewriting into standard form, the stretch factor will be the same as the [latex]a[\/latex] in the original quadratic.<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{gathered}f\\left(x\\right)=a{x}^{2}+bx+c \\\\ f\\left(x\\right)=2{x}^{2}-6x+7 \\end{gathered}[\/latex]<\/p>\r\n<p id=\"fs-id1165137653186\">Using the vertex to determine the shifts,<\/p>\r\n<p style=\"text-align: center\">[latex]f\\left(x\\right)=2{\\left(x-\\frac{3}{2}\\right)}^{2}+\\frac{5}{2}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n<p id=\"fs-id1165137638124\">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, (<em>k<\/em>), and where it occurs, (<em>x<\/em>).<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135193262\">Given the equation [latex]g\\left(x\\right)=13+{x}^{2}-6x[\/latex], write the equation in general form and then in standard form.<\/p>\r\n[reveal-answer q=\"312729\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"312729\"]\r\n\r\n[latex]g\\left(x\\right)={x}^{2}-6x+13[\/latex] in general form; [latex]g\\left(x\\right)={\\left(x - 3\\right)}^{2}+4[\/latex] in standard form\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>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>y<\/em>-values greater than or equal to the <em>y<\/em>-coordinate at the turning point or less than or equal to the <em>y<\/em>-coordinate at the turning point, depending on whether the parabola opens up or down.<\/p>\r\n\r\n<div id=\"fs-id1165135161405\" class=\"note textbox\">\r\n<h3 class=\"title\">A General Note: Domain and Range of a Quadratic Function<\/h3>\r\n<p id=\"fs-id1165135502927\">The domain of any <strong>quadratic function<\/strong> is all real numbers.<\/p>\r\n<p id=\"fs-id1165135502930\">The range of a quadratic function written in general form [latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex] with a positive <em>a<\/em>\u00a0value is [latex]f\\left(x\\right)\\ge f\\left(-\\frac{b}{2a}\\right)[\/latex], or [latex]\\left[f\\left(-\\frac{b}{2a}\\right),\\infty \\right)[\/latex]; the range of a quadratic function written in general form with a negative <em>a<\/em>\u00a0value is [latex]f\\left(x\\right)\\le f\\left(-\\frac{b}{2a}\\right)[\/latex], or [latex]\\left(-\\infty ,f\\left(-\\frac{b}{2a}\\right)\\right][\/latex].<\/p>\r\n<p id=\"fs-id1165137723229\">The range of a quadratic function written in standard form [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex] with a positive <em>a<\/em>\u00a0value is [latex]f\\left(x\\right)\\ge k[\/latex]; the range of a quadratic function written in standard form with a negative <em>a<\/em>\u00a0value is [latex]f\\left(x\\right)\\le k[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135205144\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137409165\">How To: Given a quadratic function, find the domain and range.<\/h3>\r\n<ol id=\"fs-id1165137843779\">\r\n \t<li>The domain of any quadratic function as all real numbers.<\/li>\r\n \t<li>Determine whether <em>a<\/em>\u00a0is positive or negative. If <em>a<\/em>\u00a0is positive, the parabola has a minimum. If <em>a<\/em>\u00a0is negative, the parabola has a maximum.<\/li>\r\n \t<li>Determine the maximum or minimum value of the parabola, <em>k<\/em>.<\/li>\r\n \t<li>If the parabola has a minimum, the range is given by [latex]f\\left(x\\right)\\ge k[\/latex], or [latex]\\left[k,\\infty \\right)[\/latex]. If the parabola has a maximum, the range is given by [latex]f\\left(x\\right)\\le k[\/latex], or [latex]\\left(-\\infty ,k\\right][\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_02_04\" class=\"example\">\r\n<div id=\"fs-id1165134257627\" class=\"exercise\">\r\n<div id=\"fs-id1165134257629\" class=\"problem textbox shaded\">\r\n<h3>Example 4: Finding the Domain and Range of a Quadratic Function<\/h3>\r\n<p id=\"fs-id1165137696393\">Find the domain and range of [latex]f\\left(x\\right)=-5{x}^{2}+9x - 1[\/latex].<\/p>\r\n[reveal-answer q=\"539374\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"539374\"]\r\n<p id=\"fs-id1165137837924\">As with any quadratic function, the domain is all real numbers.<\/p>\r\n<p id=\"fs-id1165137823619\">Because <em>a<\/em>\u00a0is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding the <em>x<\/em>-value of the vertex.<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{align}h&amp;=-\\frac{b}{2a} \\\\ &amp;=-\\frac{9}{2\\left(-5\\right)} \\\\ &amp;=\\frac{9}{10}&amp; \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137736576\">The maximum value is given by [latex]f\\left(h\\right)[\/latex].<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{align}f\\left(\\frac{9}{10}\\right)&amp;=5{\\left(\\frac{9}{10}\\right)}^{2}+9\\left(\\frac{9}{10}\\right)-1 \\\\ &amp;=\\frac{61}{20} \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137460169\">The range is [latex]f\\left(x\\right)\\le \\frac{61}{20}[\/latex], or [latex]\\left(-\\infty ,\\frac{61}{20}\\right][\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165135424650\">Find the domain and range of [latex]f\\left(x\\right)=2{\\left(x-\\frac{4}{7}\\right)}^{2}+\\frac{8}{11}[\/latex].<\/p>\r\n[reveal-answer q=\"345090\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"345090\"]\r\n\r\nThe domain is all real numbers. The range is [latex]f\\left(x\\right)\\ge \\frac{8}{11}[\/latex], or [latex]\\left[\\frac{8}{11},\\infty \\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>\u00a0Determine a quadratic function\u2019s minimum or maximum value<\/h2>\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\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010712\/CNX_Precalc_Figure_03_02_0092.jpg\" alt=\"Two graphs where the first graph shows the maximum value for f(x)=(x-2)^2+1 which occurs at (2, 1) and the second graph shows the minimum value for g(x)=-(x+3)^2+4 which occurs at (-3, 4).\" width=\"975\" height=\"558\" \/> <b>Figure 9<\/b>[\/caption]\r\n\r\n<div id=\"Example_03_02_05\" class=\"example\">\r\n<div id=\"fs-id1165134378616\" class=\"exercise\">\r\n<div id=\"fs-id1165134378618\" class=\"problem textbox shaded\">\r\n<h3>Example 5: Finding the Maximum Value of a Quadratic Function<\/h3>\r\n<p id=\"fs-id1165137653457\">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>\r\n\r\n<ol id=\"fs-id1165135640934\">\r\n \t<li>Find a formula for the area enclosed by the fence if the sides of fencing perpendicular to the existing fence have length <em>L<\/em>.<\/li>\r\n \t<li>What dimensions should she make her garden to maximize the enclosed area?<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"478551\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"478551\"]\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010713\/CNX_Precalc_Figure_03_02_0102.jpg\" alt=\"Diagram of the garden and the backyard.\" width=\"487\" height=\"310\" \/> <b>Figure 10<\/b>[\/caption]\r\n<p id=\"fs-id1165137836808\">Let\u2019s use a diagram such as the one in Figure 10\u00a0to record the given information. It is also helpful to introduce a temporary variable, <em>W<\/em>, to represent the width of the garden and the length of the fence section parallel to the backyard fence.<span id=\"fs-id1165135208803\">\r\n<\/span><\/p>\r\n\r\n<ol id=\"fs-id1165134363440\">\r\n \t<li>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, <em>W<\/em>, in terms of <em>L<\/em>.\r\n<div id=\"eip-id1165135697866\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]W=80 - 2L[\/latex]<\/div>\r\n<p id=\"fs-id1165135435476\">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>\r\n<p id=\"fs-id1165135435476\" style=\"text-align: center\">[latex]\\begin{align}A&amp;=LW \\\\ &amp;=L\\left(80 - 2L\\right) \\\\ &amp;=80L - 2{L}^{2} \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165135258914\">This formula represents the area of the fence in terms of the variable length <em>L<\/em>. The function, written in general form, is<\/p>\r\n\r\n<div id=\"eip-382\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]A\\left(L\\right)=-2{L}^{2}+80L[\/latex].<\/div><\/li>\r\n \t<li>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 <em>a<\/em>\u00a0is the coefficient of the squared term, [latex]a=-2,b=80[\/latex], and [latex]c=0[\/latex].<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1165137772015\">To find the vertex:<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{align}h&amp;=-\\dfrac{80}{2\\left(-2\\right)} &amp;&amp;&amp; k&amp;=A\\left(20\\right) \\\\ &amp;=20 &amp;&amp; \\text{and} &amp; &amp;=80\\left(20\\right)-2{\\left(20\\right)}^{2}\\\\ &amp;&amp;&amp;&amp;&amp;=800 \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165135174964\">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>\r\n\r\n<h4>Analysis of the Solution<\/h4>\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[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010713\/CNX_Precalc_Figure_03_02_0112.jpg\" alt=\"Graph of the parabolic function A(L)=-2L^2+80L, which the x-axis is labeled Length (L) and the y-axis is labeled Area (A). The vertex is at (20, 800).\" width=\"487\" height=\"476\" \/> <b>Figure 11<\/b>[\/caption]\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133340409\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137803708\">How To: Given an application involving revenue, use a quadratic equation to find the maximum.<\/h3>\r\n<ol id=\"fs-id1165135436584\">\r\n \t<li>Write a quadratic equation for revenue.<\/li>\r\n \t<li>Find the vertex of the quadratic equation.<\/li>\r\n \t<li>Determine the <em>y<\/em>-value of the vertex.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_02_06\" class=\"example\">\r\n<div id=\"fs-id1165134278696\" class=\"exercise\">\r\n<div id=\"fs-id1165137473136\" class=\"problem textbox shaded\">\r\n<h3>Example 6: Finding Maximum Revenue<\/h3>\r\n<p id=\"fs-id1165137473142\">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 local newspaper 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>\r\n[reveal-answer q=\"354854\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"354854\"]\r\n<p id=\"fs-id1165135389888\">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, <em>p<\/em>\u00a0for price per subscription and <em>Q<\/em>\u00a0for quantity, giving us the equation [latex]\\text{Revenue}=pQ[\/latex].<\/p>\r\n<p id=\"fs-id1165134232972\">Because the number of subscribers changes with the price, we 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 newspaper 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>\r\n<p style=\"text-align: center\">[latex]\\begin{align}m&amp;=\\frac{79,000 - 84,000}{32 - 30} \\\\ &amp;=\\frac{-5,000}{2} \\\\ &amp;=-2,500 \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165135559520\">This tells us the paper will lose 2,500 subscribers for each dollar they raise the price. We can then solve for the <em>y<\/em>-intercept.<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{align}Q&amp;=-2500p+b &amp;&amp; \\text{Substitute in the point }Q=84,000\\text{ and }p=30 \\\\ 84,000&amp;=-2500\\left(30\\right)+b &amp;&amp; \\text{Solve for }b \\\\ b&amp;=159,000 \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137933138\">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>\r\n<p style=\"text-align: center\">[latex]\\begin{align}\\text{Revenue}&amp;=pQ \\\\ &amp;=p\\left(-2,500p+159,000\\right) \\\\ &amp;=-2,500{p}^{2}+159,000p \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165135502033\">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 newspaper, we can find the vertex.<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{align}h&amp;=-\\frac{159,000}{2\\left(-2,500\\right)} \\\\ &amp;=31.8 \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137647087\">The model tells us that the maximum revenue will occur if the newspaper charges $31.80 for a subscription. To find what the maximum revenue is, we evaluate the revenue function.<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{align}\\text{maximum revenue}&amp;=-2,500{\\left(31.8\\right)}^{2}+159,000\\left(31.8\\right) \\\\ &amp;=2,528,100 \\end{align}[\/latex]<\/p>\r\nThe maximum revenue is $2,528,100.\r\n<h4>Analysis of the Solution<\/h4>\r\nThis could also be solved by graphing the quadratic. We can see the maximum revenue on a graph of the quadratic function.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010713\/CNX_Precalc_Figure_03_02_0122.jpg\" alt=\"Graph of the parabolic function which the x-axis is labeled Price (p) and the y-axis is labeled Revenue ($). The vertex is at (31.80, 258100).\" width=\"487\" height=\"327\" \/> <b>Figure 12<\/b>[\/caption]\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165135693703\">\r\n<h2 style=\"text-align: center\">Finding the <em>x<\/em>- and <em>y<\/em>-Intercepts of a Quadratic Function<\/h2>\r\nMuch 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\u00a0that the number of <em>x<\/em>-intercepts can vary depending upon the location of the graph.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010713\/CNX_Precalc_Figure_03_02_0132.jpg\" alt=\"Three graphs where the first graph shows a parabola with no x-intercept, the second is a parabola with one \u2013intercept, and the third parabola is of two x-intercepts.\" width=\"975\" height=\"317\" \/> <b>Figure 13.<\/b> Number of x-intercepts of a parabola[\/caption]\r\n\r\n<div id=\"fs-id1165137464602\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135638554\">How To: Given a quadratic function [latex]f\\left(x\\right)[\/latex], find the <em>y<\/em>-\u00a0and <em>x<\/em>-intercepts.<\/h3>\r\n<ol id=\"fs-id1165135378765\">\r\n \t<li>Evaluate [latex]f\\left(0\\right)[\/latex] to find the <em>y<\/em>-intercept.<\/li>\r\n \t<li>Solve the quadratic equation [latex]f\\left(x\\right)=0[\/latex] to find the <em>x<\/em>-intercepts.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_02_07\" class=\"example\">\r\n<div id=\"fs-id1165134129944\" class=\"exercise\">\r\n<div id=\"fs-id1165134129946\" class=\"problem textbox shaded\">\r\n<h3>Example 7: Finding the <em>y<\/em>- and <em>x<\/em>-Intercepts of a Parabola<\/h3>\r\n<p id=\"fs-id1165134138677\">Find the <em>y<\/em>- and <em>x<\/em>-intercepts of the quadratic [latex]f\\left(x\\right)=3{x}^{2}+5x - 2[\/latex].<\/p>\r\n[reveal-answer q=\"201680\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"201680\"]\r\n<p id=\"fs-id1165137901096\">We find the <em>y<\/em>-intercept by evaluating [latex]f\\left(0\\right)[\/latex].<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{align}f\\left(0\\right)&amp;=3{\\left(0\\right)}^{2}+5\\left(0\\right)-2 \\\\ &amp;=-2 \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165134232203\">So the <em>y<\/em>-intercept is at [latex]\\left(0,-2\\right)[\/latex].<\/p>\r\n<p id=\"fs-id1165135434816\">For the <em>x<\/em>-intercepts, we find all solutions of [latex]f\\left(x\\right)=0[\/latex].<\/p>\r\n<p style=\"text-align: center\">[latex]0=3{x}^{2}+5x - 2[\/latex]<\/p>\r\n<p id=\"fs-id1165135690677\">In this case, the quadratic can be factored easily, providing the simplest method for solution.<\/p>\r\n<p style=\"text-align: center\">[latex]0=\\left(3x - 1\\right)\\left(x+2\\right)[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{align}3x - 1&amp;=0\\\\x&amp;=\\frac{1}{3}&amp;&amp; \\end{align}[\/latex] or [latex]\\begin{align}&amp;&amp;x+2&amp;=0 \\\\ &amp;&amp;x&amp;=-2 \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165137644422\">So the <em>x<\/em>-intercepts are at [latex]\\left(\\frac{1}{3},0\\right)[\/latex] and [latex]\\left(-2,0\\right)[\/latex].<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nBy graphing the function, we can confirm that the graph crosses the <em>y<\/em>-axis at [latex]\\left(0,-2\\right)[\/latex]. We can also confirm that the graph crosses the <em>x<\/em>-axis at [latex]\\left(\\frac{1}{3},0\\right)[\/latex] and [latex]\\left(-2,0\\right)[\/latex].\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010713\/CNX_Precalc_Figure_03_02_0142.jpg\" alt=\"Graph of a parabola which has the following intercepts (-2, 0), (1\/3, 0), and (0, -2).\" width=\"487\" height=\"480\" \/> <b>Figure 14<\/b>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137911614\" class=\"commentary\">\r\n<div class=\"mceTemp\"><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<h2>\u00a0Solve problems involving a quadratic function\u2019s minimum or maximum value<\/h2>\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 id=\"fs-id1165133085664\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165135694488\">How To: Given a quadratic function, find the <em>x<\/em>-intercepts by rewriting in standard form.<\/h3>\r\n<ol id=\"fs-id1165134113976\">\r\n \t<li>Substitute <em>a<\/em>\u00a0and <em>b<\/em>\u00a0into [latex]h=-\\frac{b}{2a}[\/latex].<\/li>\r\n \t<li>Substitute <em>x<\/em> =\u00a0<em>h<\/em>\u00a0into the general form of the quadratic function to find <em>k<\/em>.<\/li>\r\n \t<li>Rewrite the quadratic in standard form using <em>h<\/em>\u00a0and <em>k<\/em>.<\/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 id=\"Example_03_02_08\" class=\"example\">\r\n<div id=\"fs-id1165134060458\" class=\"exercise\">\r\n<div id=\"fs-id1165134224010\" class=\"problem textbox shaded\">\r\n<h3>Example 8: Finding the <em>x<\/em>-Intercepts of a Parabola<\/h3>\r\n<p id=\"fs-id1165134224020\">Find the <em>x<\/em>-intercepts of the quadratic function [latex]f\\left(x\\right)=2{x}^{2}+4x - 4[\/latex].<\/p>\r\n[reveal-answer q=\"41935\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"41935\"]\r\n<p id=\"fs-id1165135524478\">We begin by solving for when the output will be zero.<\/p>\r\n<p style=\"text-align: center\">[latex]0=2{x}^{2}+4x - 4[\/latex]<\/p>\r\n<p id=\"fs-id1165135252139\">Because the quadratic is not easily factorable in this case, we solve for the intercepts by first rewriting the quadratic in standard form.<\/p>\r\n<p style=\"text-align: center\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/p>\r\n<p id=\"fs-id1165137925165\">We know that <em>a\u00a0<\/em>= 2. Then we solve for <em>h<\/em>\u00a0and <em>k<\/em>.<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{align}h&amp;=-\\frac{b}{2a} &amp;&amp;&amp; k&amp;=f\\left(-1\\right) \\\\ &amp;=-\\frac{4}{2\\left(2\\right)} &amp;&amp;&amp; &amp;=2{\\left(-1\\right)}^{2}+4\\left(-1\\right)-4 \\\\ &amp;=-1 &amp;&amp;&amp; &amp;=-6 \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165134031332\">So now we can rewrite in standard form.<\/p>\r\n<p style=\"text-align: center\">[latex]f\\left(x\\right)=2{\\left(x+1\\right)}^{2}-6[\/latex]<\/p>\r\n<p id=\"fs-id1165135381286\">We can now solve for when the output will be zero.<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{gathered}2{\\left(x+1\\right)}^{2}-6=0 \\\\ 2{\\left(x+1\\right)}^{2}=6 \\\\ {\\left(x+1\\right)}^{2}=3 \\\\ x+1=\\pm \\sqrt{3} \\\\ x=-1\\pm \\sqrt{3} \\end{gathered}[\/latex]<\/p>\r\n<p id=\"fs-id1165131959622\">The graph has <em>x-<\/em>intercepts at [latex]\\left(-1-\\sqrt{3},0\\right)[\/latex] and [latex]\\left(-1+\\sqrt{3},0\\right)[\/latex].<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010714\/CNX_Precalc_Figure_03_02_0152.jpg\" alt=\"Graph of a parabola which has the following x-intercepts (-2.732, 0) and (0.732, 0).\" width=\"487\" height=\"517\" \/> <b>Figure 15<\/b>[\/caption]\r\n<p id=\"fs-id1165137843092\">We can check our work by graphing the given function on a graphing utility and observing the <em>x-<\/em>intercepts.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137843086\" class=\"commentary\">\r\n<p id=\"fs-id1165137843092\"><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165134534213\">In Try It\u00a02, we found the standard and general form for the function [latex]g\\left(x\\right)=13+{x}^{2}-6x[\/latex]. Now find the <em>y<\/em>- and <em>x<\/em>-intercepts (if any).<\/p>\r\n[reveal-answer q=\"700527\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"700527\"]\r\n\r\n<em>y<\/em>-intercept at (0, 13), No <em>x-<\/em>intercepts\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"Example_03_02_09\" class=\"example\">\r\n<div id=\"fs-id1165135606916\" class=\"exercise\">\r\n<div id=\"fs-id1165135606918\" class=\"problem textbox shaded\">\r\n<h3>Example 9: Solving a Quadratic Equation with the Quadratic Formula<\/h3>\r\n<p id=\"fs-id1165135606924\">Solve [latex]{x}^{2}+x+2=0[\/latex].<\/p>\r\n[reveal-answer q=\"41618\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"41618\"]\r\n<p id=\"fs-id1165133036005\">Let\u2019s begin by writing the quadratic formula: [latex]x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex].<\/p>\r\n<p id=\"fs-id1165134102089\">When applying the <span class=\"no-emphasis\">quadratic formula<\/span>, we identify the coefficients <em>a<\/em>,\u00a0<em>b<\/em>, and\u00a0<em>c<\/em>. For the equation [latex]{x}^{2}+x+2=0[\/latex], we have <em>a<\/em>\u00a0=\u00a01,\u00a0<em>b<\/em>\u00a0=\u00a01, and <em>c<\/em>\u00a0=\u00a02.\u00a0Substituting these values into the formula we have:<\/p>\r\n<p style=\"text-align: center\">[latex]\\begin{align}x&amp;=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a} \\\\ &amp;=\\frac{-1\\pm \\sqrt{{1}^{2}-4\\cdot 1\\cdot \\left(2\\right)}}{2\\cdot 1} \\\\ &amp;=\\frac{-1\\pm \\sqrt{1 - 8}}{2} \\\\ &amp;=\\frac{-1\\pm \\sqrt{-7}}{2} \\\\ &amp;=\\frac{-1\\pm i\\sqrt{7}}{2} \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165134149125\">The solutions to the equation are [latex]x=\\frac{-1+i\\sqrt{7}}{2}[\/latex] and [latex]x=\\frac{-1-i\\sqrt{7}}{2}[\/latex] or [latex]x=\\frac{-1}{2}+\\frac{i\\sqrt{7}}{2}[\/latex] and [latex]x=\\frac{-1}{2}-\\frac{i\\sqrt{7}}{2}[\/latex]. Note that because of the\u00a0<em>i<\/em>, these are non-real zeros.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_03_02_10\" class=\"example\">\r\n<div id=\"fs-id1165134085786\" class=\"exercise\">\r\n<div id=\"fs-id1165134085788\" class=\"problem textbox shaded\">\r\n<h3>Example 10: Applying the Vertex and <em>x<\/em>-Intercepts of a Parabola<\/h3>\r\n<p id=\"fs-id1165134085798\">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\\left(t\\right)=-16{t}^{2}+80t+40[\/latex].<\/p>\r\n<p style=\"padding-left: 60px\">a. When does the ball reach the maximum height?<\/p>\r\n<p style=\"padding-left: 60px\">b. What is the maximum height of the ball?<\/p>\r\n<p style=\"padding-left: 60px\">c. When does the ball hit the ground?<\/p>\r\n[reveal-answer q=\"340065\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"340065\"]\r\n\r\na. The ball reaches the maximum height at the vertex of the parabola.\r\n<p style=\"text-align: center\">[latex]\\begin{align} h&amp;=-\\frac{80}{2\\left(-16\\right)} =\\frac{80}{32} \\\\ &amp;=\\frac{5}{2} \\\\ &amp;=2.5 \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165135528870\">The ball reaches a maximum height after 2.5 seconds.<\/p>\r\nb. To find the maximum height, find the <em>y\u00a0<\/em>coordinate of the vertex of the parabola.\r\n<p style=\"text-align: center\">[latex]\\begin{align}k&amp;=H\\left(-\\frac{b}{2a}\\right) \\\\ &amp;=H\\left(2.5\\right) \\\\ &amp;=-16{\\left(2.5\\right)}^{2}+80\\left(2.5\\right)+40 \\\\ &amp;=140 \\end{align}[\/latex]<\/p>\r\n<p id=\"fs-id1165135409750\">The ball reaches a maximum height of 140 feet.<\/p>\r\nc. To find when the ball hits the ground, we need to determine when the height is zero, [latex]H\\left(t\\right)=0[\/latex].\r\n\r\nWe use the quadratic formula.\r\n<p style=\"text-align: center\">[latex]\\begin{align} t&amp;=\\frac{-80\\pm \\sqrt{{80}^{2}-4\\left(-16\\right)\\left(40\\right)}}{2\\left(-16\\right)} \\\\ &amp;=\\frac{-80\\pm \\sqrt{8960}}{-32} \\end{align}[\/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]\\begin{align}t=\\frac{-80-\\sqrt{8960}}{-32}\\approx 5.458 &amp;&amp; \\text{or} &amp;&amp; t=\\frac{-80+\\sqrt{8960}}{-32}\\approx -0.458 \\end{align}[\/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.<span id=\"fs-id1165135580349\">\r\n<\/span>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010714\/CNX_Precalc_Figure_03_02_0162.jpg\" alt=\"Graph of a negative parabola where x goes from -1 to 6.\" width=\"487\" height=\"254\" \/> <b>Figure 16<\/b>[\/caption]\r\n\r\n[\/hidden-answer]<b><\/b>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165134081301\">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\\left(t\\right)=-16{t}^{2}+96t+112[\/latex].<\/p>\r\n<p style=\"padding-left: 60px\">a. When does the rock reach the maximum height?<\/p>\r\n<p style=\"padding-left: 60px\">b. What is the maximum height of the rock?<\/p>\r\n<p style=\"padding-left: 60px\">c. When does the rock hit the ocean?<\/p>\r\n[reveal-answer q=\"4901\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"4901\"]\r\n\r\na.\u00a03 seconds \u00a0b.\u00a0256 feet \u00a0c.\u00a07 seconds\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question hide_question_numbers=1]17065[\/ohm_question]\r\n\r\n<\/div>\r\n<span style=\"color: #077fab;font-size: 1.15em;font-weight: 600\">Key Equations<\/span>\r\n\r\n<section id=\"fs-id1165134205927\" class=\"key-equations\">\r\n<table id=\"eip-id1165137539373\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td>general form of a quadratic function<\/td>\r\n<td>[latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>the quadratic formula<\/td>\r\n<td>[latex]x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>standard form of a quadratic function<\/td>\r\n<td>[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><section id=\"fs-id1165135426424\" class=\"key-concepts\">\r\n<h2>Key Concepts<\/h2>\r\n<ul id=\"fs-id1165134570662\">\r\n \t<li>A polynomial function of degree two is called a quadratic function.<\/li>\r\n \t<li>The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.<\/li>\r\n \t<li>The axis of symmetry is the vertical line passing through the vertex. The zeros, or <em>x<\/em>-intercepts, are the points at which the parabola crosses the <em>x<\/em>-axis. The <em>y<\/em>-intercept is the point at which the parabola crosses the <em>y-<\/em>axis.<\/li>\r\n \t<li>Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph.<\/li>\r\n \t<li>The vertex can be found from an equation representing a quadratic function.<\/li>\r\n \t<li>The domain of a quadratic function is all real numbers. The range varies with the function.<\/li>\r\n \t<li>A quadratic function\u2019s minimum or maximum value is given by the <em>y<\/em>-value of the vertex.<\/li>\r\n \t<li>The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue.<\/li>\r\n \t<li>Some quadratic equations must be solved by using the quadratic formula.<\/li>\r\n \t<li>The vertex and the intercepts can be identified and interpreted to solve real-world problems.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165135449657\" class=\"definition\">\r\n \t<dt><strong>axis of symmetry<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135449662\">a vertical line drawn through the vertex of a parabola around which the parabola is symmetric; it is defined by [latex]x=-\\frac{b}{2a}[\/latex].<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135502777\" class=\"definition\">\r\n \t<dt><strong>general form of a quadratic function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135502783\">the function that describes a parabola, written in the form [latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex], where <em>a<\/em>,\u00a0<em>b<\/em>, and\u00a0<em>c<\/em>\u00a0are real numbers and [latex]a\\ne 0[\/latex].<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137931314\" class=\"definition\">\r\n \t<dt><strong>standard form of a quadratic function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137931319\">the function that describes a parabola, written in the form [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex], where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135623614\" class=\"definition\">\r\n \t<dt><strong>vertex<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135623619\">the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135623624\" class=\"definition\">\r\n \t<dt><strong>vertex form of a quadratic function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135623630\">another name for the standard form of a quadratic function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135623634\" class=\"definition\">\r\n \t<dt><strong>zeros<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135623639\">in a given function, the values of <em>x<\/em>\u00a0at which <em>y<\/em> = 0, also called roots<\/dd>\r\n<\/dl>\r\n<\/section>&nbsp;\r\n<h2 style=\"text-align: center\">Section 2.3 Homework Exercises<\/h2>\r\n1. Explain the advantage of writing a quadratic function in standard form.\r\n\r\n2. How can the vertex of a parabola be used in solving real world problems?\r\n\r\n3. Explain why the condition of [latex]a\\ne 0[\/latex] is imposed in the definition of the quadratic function.\r\n\r\n4.\u00a0What is another name for the standard form of a quadratic function?\r\n\r\n5. What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?\r\n\r\nFor the following exercises, rewrite the quadratic functions in standard form and give the vertex.\r\n\r\n6. [latex]f\\left(x\\right)={x}^{2}-12x+32[\/latex]\r\n\r\n7. [latex]g\\left(x\\right)={x}^{2}+2x - 3[\/latex]\r\n\r\n8.\u00a0[latex]f\\left(x\\right)={x}^{2}-x[\/latex]\r\n\r\n9. [latex]f\\left(x\\right)={x}^{2}+5x - 2[\/latex]\r\n\r\n10.\u00a0[latex]h\\left(x\\right)=2{x}^{2}+8x - 10[\/latex]\r\n\r\n11. [latex]k\\left(x\\right)=3{x}^{2}-6x - 9[\/latex]\r\n\r\n12.\u00a0[latex]f\\left(x\\right)=2{x}^{2}-6x[\/latex]\r\n\r\n13. [latex]f\\left(x\\right)=3{x}^{2}-5x - 1[\/latex]\r\n\r\nFor the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.\r\n\r\n14. [latex]y\\left(x\\right)=2{x}^{2}+10x+12[\/latex]\r\n\r\n15. [latex]f\\left(x\\right)=2{x}^{2}-10x+4[\/latex]\r\n\r\n16.\u00a0[latex]f\\left(x\\right)=-{x}^{2}+4x+3[\/latex]\r\n\r\n17. [latex]f\\left(x\\right)=4{x}^{2}+x - 1[\/latex]\r\n\r\n18. [latex]h\\left(t\\right)=-4{t}^{2}+6t - 1[\/latex]\r\n\r\n19. [latex]f\\left(x\\right)=\\frac{1}{2}{x}^{2}+3x+1[\/latex]\r\n\r\n20. [latex]f\\left(x\\right)=-\\frac{1}{3}{x}^{2}-2x+3[\/latex]\r\n\r\nFor the following exercises, determine the domain and range of the quadratic function.\r\n\r\n21. [latex]f\\left(x\\right)={\\left(x - 3\\right)}^{2}+2[\/latex]\r\n\r\n22.\u00a0[latex]f\\left(x\\right)=-2{\\left(x+3\\right)}^{2}-6[\/latex]\r\n\r\n23. [latex]f\\left(x\\right)={x}^{2}+6x+4[\/latex]\r\n\r\n24.\u00a0[latex]f\\left(x\\right)=2{x}^{2}-4x+2[\/latex]\r\n\r\n25. [latex]k\\left(x\\right)=3{x}^{2}-6x - 9[\/latex]\r\n\r\nFor the following exercises, solve the equations over the complex numbers.\r\n\r\n26. [latex]{x}^{2}=-25[\/latex]\r\n\r\n27. [latex]{x}^{2}=-8[\/latex]\r\n\r\n28.\u00a0[latex]{x}^{2}+36=0[\/latex]\r\n\r\n29. [latex]{x}^{2}+27=0[\/latex]\r\n\r\n30.\u00a0[latex]{x}^{2}+2x+5=0[\/latex]\r\n\r\n31. [latex]{x}^{2}-4x+5=0[\/latex]\r\n\r\n32.\u00a0[latex]{x}^{2}+8x+25=0[\/latex]\r\n\r\n33. [latex]{x}^{2}-4x+13=0[\/latex]\r\n\r\n34.\u00a0[latex]{x}^{2}+6x+25=0[\/latex]\r\n\r\n35. [latex]{x}^{2}-10x+26=0[\/latex]\r\n\r\n36.\u00a0[latex]{x}^{2}-6x+10=0[\/latex]\r\n\r\n37. [latex]x\\left(x - 4\\right)=20[\/latex]\r\n\r\n38.\u00a0[latex]x\\left(x - 2\\right)=10[\/latex]\r\n\r\n39. [latex]2{x}^{2}+2x+5=0[\/latex]\r\n\r\n40.\u00a0[latex]5{x}^{2}-8x+5=0[\/latex]\r\n\r\n41. [latex]5{x}^{2}+6x+2=0[\/latex]\r\n\r\n42. [latex]2{x}^{2}-6x+5=0[\/latex]\r\n\r\n43. [latex]{x}^{2}+x+2=0[\/latex]\r\n\r\n44.\u00a0[latex]{x}^{2}-2x+4=0[\/latex]\r\n\r\nFor the following exercises, use the vertex (<em>h<\/em>, <em>k<\/em>) and a point on the graph (<em>x<\/em>, <em>y<\/em>) to find the general form of the equation of the quadratic function.\r\n\r\n45. [latex]\\left(h,k\\right)=\\left(2,0\\right),\\left(x,y\\right)=\\left(4,4\\right)[\/latex]\r\n\r\n46.\u00a0[latex]\\left(h,k\\right)=\\left(-2,-1\\right),\\left(x,y\\right)=\\left(-4,3\\right)[\/latex]\r\n\r\n47. [latex]\\left(h,k\\right)=\\left(0,1\\right),\\left(x,y\\right)=\\left(2,5\\right)[\/latex]\r\n\r\n48.\u00a0[latex]\\left(h,k\\right)=\\left(2,3\\right),\\left(x,y\\right)=\\left(5,12\\right)[\/latex]\r\n\r\n49. [latex]\\left(h,k\\right)=\\left(-5,3\\right),\\left(x,y\\right)=\\left(2,9\\right)[\/latex]\r\n\r\n50.\u00a0[latex]\\left(h,k\\right)=\\left(3,2\\right),\\left(x,y\\right)=\\left(10,1\\right)[\/latex]\r\n\r\n51. [latex]\\left(h,k\\right)=\\left(0,1\\right),\\left(x,y\\right)=\\left(1,0\\right)[\/latex]\r\n\r\n52.\u00a0[latex]\\left(h,k\\right)=\\left(1,0\\right),\\left(x,y\\right)=\\left(0,1\\right)[\/latex]\r\n\r\nFor the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.\r\n\r\n53. [latex]f\\left(x\\right)={x}^{2}-2x[\/latex]\r\n\r\n54.\u00a0[latex]f\\left(x\\right)={x}^{2}-6x - 1[\/latex]\r\n\r\n55. [latex]f\\left(x\\right)={x}^{2}-5x - 6[\/latex]\r\n\r\n56.\u00a0[latex]f\\left(x\\right)={x}^{2}-7x+3[\/latex]\r\n\r\n57. [latex]f\\left(x\\right)=-2{x}^{2}+5x - 8[\/latex]\r\n\r\n58. [latex]f\\left(x\\right)=4{x}^{2}-12x - 3[\/latex]\r\n\r\nFor the following exercises, write the equation for the graphed function.\r\n\r\n59.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005210\/CNX_Precalc_Figure_03_02_207.jpg\" alt=\"Graph of a positive parabola with a vertex at (2, -3) and y-intercept at (0, 1).\" \/>\r\n\r\n60.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005210\/CNX_Precalc_Figure_03_02_208.jpg\" alt=\"Graph of a positive parabola with a vertex at (-1, 2) and y-intercept at (0, 3)\" \/>\r\n\r\n61.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005210\/CNX_Precalc_Figure_03_02_209.jpg\" alt=\"Graph of a negative parabola with a vertex at (2, 7).\" \/>\r\n\r\n62.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005211\/CNX_Precalc_Figure_03_02_210.jpg\" alt=\"Graph of a negative parabola with a vertex at (-1, 2).\" \/>\r\n\r\n63.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005211\/CNX_Precalc_Figure_03_02_211n.jpg\" alt=\"Graph of a positive parabola with a vertex at (3, -1) and y-intercept at (0, 3.5).\" \/>\r\n\r\n64.\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005211\/CNX_Precalc_Figure_03_02_212.jpg\" alt=\"Graph of a negative parabola with a vertex at (-2, 3).\" \/>\r\n\r\nFor 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.\r\n\r\n65.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>\u20132<\/td>\r\n<td>\u20131<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em><strong>y<\/strong><\/em><\/td>\r\n<td>5<\/td>\r\n<td>2<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n66.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>\u20132<\/td>\r\n<td>\u20131<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em><strong>y<\/strong><\/em><\/td>\r\n<td>1<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>4<\/td>\r\n<td>9<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n67.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>\u20132<\/td>\r\n<td>\u20131<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em><strong>y<\/strong><\/em><\/td>\r\n<td>\u20132<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>1<\/td>\r\n<td>\u20132<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n68.\r\n<table id=\"fs-id1165134138604\" class=\"unnumbered\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>\u20132<\/td>\r\n<td>\u20131<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em><strong>y<\/strong><\/em><\/td>\r\n<td>\u20138<\/td>\r\n<td>\u20133<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n69.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>\u20132<\/td>\r\n<td>\u20131<\/td>\r\n<td>0<\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em><strong>y<\/strong><\/em><\/td>\r\n<td>8<\/td>\r\n<td>2<\/td>\r\n<td>0<\/td>\r\n<td>2<\/td>\r\n<td>8<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFor the following exercises, use a calculator to find the answer.\r\n\r\n70. Graph on the same set of axes the functions [latex]f\\left(x\\right)={x}^{2},f\\left(x\\right)=2{x}^{2},\\text{ and }f\\left(x\\right)=\\frac{1}{3}{x}^{2}[\/latex].\u00a0What appears to be the effect of changing the coefficient?\r\n\r\n71. Graph on the same set of axes [latex]f\\left(x\\right)={x}^{2},f\\left(x\\right)={x}^{2}+2[\/latex] and [latex]f\\left(x\\right)={x}^{2},f\\left(x\\right)={x}^{2}+5[\/latex] and [latex]f\\left(x\\right)={x}^{2}-3[\/latex]. What appears to be the effect of adding a constant?\r\n\r\n72.\u00a0Graph on the same set of axes [latex]f\\left(x\\right)={x}^{2},f\\left(x\\right)={\\left(x - 2\\right)}^{2},f{\\left(x - 3\\right)}^{2},\\text{ and }f\\left(x\\right)={\\left(x+4\\right)}^{2}[\/latex].\u00a0What appears to be the effect of adding or subtracting those numbers?\r\n\r\n73. 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\\left(x\\right)=\\frac{-32}{{\\left(80\\right)}^{2}}{x}^{2}+x[\/latex] where <em>x<\/em>\u00a0is the horizontal distance traveled and [latex]h\\left(x\\right)[\/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\n74.\u00a0A suspension bridge can be modeled by the quadratic function [latex]h\\left(x\\right)=.0001{x}^{2}[\/latex] with [latex]-2000\\le x\\le 2000[\/latex] where |<em>x<\/em>| is the number of feet from the center and [latex]h\\left(x\\right)[\/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\nFor 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.\r\n\r\n75. Vertex [latex]\\left(1,-2\\right)[\/latex], opens up.\r\n\r\n76.\u00a0Vertex [latex]\\left(-1,2\\right)[\/latex] opens down.\r\n\r\n77. Vertex [latex]\\left(-5,11\\right)[\/latex], opens down.\r\n\r\n78.\u00a0Vertex [latex]\\left(-100,100\\right)[\/latex], opens up.\r\n\r\nFor the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function.\r\n\r\n79. Contains (1, 1) and has shape of [latex]f\\left(x\\right)=2{x}^{2}[\/latex]. Vertex is on the <em>y<\/em>-axis.\r\n\r\n80.\u00a0Contains (-1, 4) and has the shape of [latex]f\\left(x\\right)=2{x}^{2}[\/latex]. Vertex is on the <em>y<\/em>-axis.\r\n\r\n81. Contains (2, 3) and has the shape of [latex]f\\left(x\\right)=3{x}^{2}[\/latex]. Vertex is on the\u00a0<em>y<\/em>-axis.\r\n\r\n82.\u00a0Contains (1, \u20133) and has the shape of [latex]f\\left(x\\right)=-{x}^{2}[\/latex]. Vertex is on the\u00a0<em>y<\/em>-axis.\r\n\r\n83. Contains (4, 3) and has the shape of [latex]f\\left(x\\right)=5{x}^{2}[\/latex]. Vertex is on the\u00a0<em>y<\/em>-axis.\r\n\r\n84.\u00a0Contains (1, \u20136) has the shape of [latex]f\\left(x\\right)=3{x}^{2}[\/latex]. Vertex has <em>x<\/em>-coordinate of \u20131.\r\n\r\n85.\u00a0Find the dimensions of the rectangular corral producing the greatest enclosed area given 200 feet of fencing.\r\n\r\n86.\u00a0Find the dimensions of the rectangular corral split into 2 pens of the same size producing the greatest possible enclosed area given 300 feet of fencing.\r\n\r\n87. Find the dimensions of the rectangular corral producing the greatest enclosed area split into 3 pens of the same size given 500 feet of fencing.\r\n\r\n88.\u00a0Among all of the pairs of numbers whose sum is 6, find the pair with the largest product. What is the product?\r\n\r\n89. 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\n90.\u00a0Suppose that the price per unit in dollars of a cell phone production is modeled by [latex]p=$45 - 0.0125x[\/latex], where <em>x<\/em>\u00a0is in thousands of phones produced, and the revenue represented by thousands of dollars is [latex]R=x\\cdot p[\/latex]. Find the production level that will maximize revenue.\r\n\r\n91. 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\\left(t\\right)=-4.9{t}^{2}+229t+234[\/latex]. Find the maximum height the rocket attains.\r\n\r\n92.\u00a0A 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\\left(t\\right)=-4.9{t}^{2}+24t+8[\/latex]. How long does it take to reach maximum height?\r\n\r\n93. A soccer stadium 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\n94. 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?","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify key characteristics of parabolas from the graph.<\/li>\n<li>Understand how the graph of a parabola is related to its quadratic function.<\/li>\n<li>Draw the graph of a quadratic function.<\/li>\n<li>Solve problems involving a quadratic function\u2019s minimum or maximum value.<\/li>\n<\/ul>\n<\/div>\n<figure id=\"Figure_03_02_001\" class=\"medium\">\n<div style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010711\/CNX_Precalc_Figure_03_02_0012.jpg\" alt=\"Satellite dishes.\" width=\"731\" height=\"368\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 1.<\/b> An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr)<\/p>\n<\/div>\n<\/figure>\n<p id=\"fs-id1165134339909\">Curved antennas, such as the ones shown in the photo, 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<h2>Recognize characteristics of parabolas<\/h2>\n<p id=\"fs-id1165137727999\">The graph of a quadratic function is a U-shaped curve called a <strong>parabola<\/strong>. One important feature of the graph is that it has an extreme point, called the <strong>vertex<\/strong>. If the parabola opens up, the vertex represents the lowest point on the graph, or the <strong>minimum value<\/strong> of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the <strong>maximum value<\/strong>. In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the <strong>axis of symmetry<\/strong>. These features are illustrated in Figure 2.<span id=\"fs-id1165134118332\"><br \/>\n<\/span><\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010711\/CNX_Precalc_Figure_03_02_0022.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are.\" width=\"487\" height=\"480\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 2<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137549127\">The <em>y<\/em>-intercept is the point at which the parabola crosses the <em>y<\/em>-axis. The <em>x<\/em>-intercepts are the points at which the parabola crosses the <em>x<\/em>-axis. If they exist, the <em>x<\/em>-intercepts represent the <strong>zeros<\/strong>, or <strong>roots<\/strong>, of the quadratic function, the values of <em>x<\/em>\u00a0at which <em>y\u00a0<\/em>= 0.<\/p>\n<div id=\"Example_03_02_01\" class=\"example\">\n<div id=\"fs-id1165131959514\" class=\"exercise\">\n<div id=\"fs-id1165135541748\" class=\"problem textbox shaded\">\n<h3>Example 1: Identifying the Characteristics of a Parabola<\/h3>\n<p>Determine the vertex, axis of symmetry, zeros, and <em>y<\/em>-intercept of the parabola shown in Figure 3.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010711\/CNX_Precalc_Figure_03_02_0032.jpg\" alt=\"Graph of a parabola with a vertex at (3, 1) and a y-intercept at (0, 7).\" width=\"487\" height=\"517\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 3<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q300488\">Show Solution<\/span><\/p>\n<div id=\"q300488\" class=\"hidden-answer\" style=\"display: none\">\n<p>The vertex is the turning point of the graph. We can see that the vertex is at (3, 1). 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 <em>x\u00a0<\/em>= 3. This parabola does not cross the <em>x<\/em>-axis, so it has no real zeros. It crosses the <em>y<\/em>-axis at (0, 7) so this is the <em>y<\/em>-intercept.<\/p>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>\u00a0Understand how the graph of a parabola is related to its quadratic function<\/h2>\n<p>The <strong>general form of a quadratic function<\/strong> presents the function in the form<\/p>\n<p style=\"text-align: center\">[latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex]<\/p>\n<p id=\"fs-id1165137544673\">where <em>a<\/em>,\u00a0<em>b<\/em>, and <em>c<\/em>\u00a0are real numbers and [latex]a\\ne 0[\/latex]. If [latex]a>0[\/latex], the parabola opens upward. If [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 [latex]x=-\\frac{b}{2a}[\/latex]. If we use the quadratic formula, [latex]x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex], to solve [latex]a{x}^{2}+bx+c=0[\/latex] for the <em>x<\/em>-intercepts, or zeros, we find the value of\u00a0<em>x<\/em>\u00a0halfway between them is always [latex]x=-\\frac{b}{2a}[\/latex], the equation for the axis of symmetry.<\/p>\n<p>Figure 4 shows\u00a0the graph of the quadratic function written in general form as [latex]y={x}^{2}+4x+3[\/latex]. In this form, [latex]a=1,\\text{ }b=4[\/latex], and [latex]c=3[\/latex]. Because [latex]a>0[\/latex], the parabola opens upward. The axis of symmetry is [latex]x=-\\frac{4}{2\\left(1\\right)}=-2[\/latex]. This also makes sense because we can see from the graph that the vertical line [latex]x=-2[\/latex] divides the graph in half. The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph, in this instance, [latex]\\left(-2,-1\\right)[\/latex]. The <i>x<\/i>-intercepts, those points where the parabola crosses the <i>x<\/i>-axis, occur at [latex]\\left(-3,0\\right)[\/latex] and [latex]\\left(-1,0\\right)[\/latex].<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010711\/CNX_Precalc_Figure_03_02_0042.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are for the function y=x^2+4x+3.\" width=\"487\" height=\"555\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 4<\/b><\/p>\n<\/div>\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\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/p>\n<p id=\"fs-id1303104\">where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the <strong>vertex form of a quadratic function<\/strong>. The function above has the standard form:\u00a0 [latex]y=(x+2)^2-1[\/latex]<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010711\/CNX_Precalc_Figure_03_02_0052.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are for the function y=-3(x+2)^2+4.\" width=\"487\" height=\"630\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 5<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137894543\">As with the general form, if [latex]a>0[\/latex], the parabola opens upward and the vertex is a minimum. If [latex]a<0[\/latex], the parabola opens downward, and the vertex is a maximum. Figure 5\u00a0is the\u00a0graph of the quadratic function written in standard form as [latex]y=-3{\\left(x+2\\right)}^{2}+4[\/latex]. Since [latex]x-h=x+2[\/latex] in this example, [latex]h=-2[\/latex]. In this form, [latex]a=-3,\\text{ }h=-2[\/latex], and [latex]k=4[\/latex]. Because [latex]a<0[\/latex], the parabola opens downward. The vertex is at [latex]\\left(-2,\\text{ 4}\\right)[\/latex].<span id=\"fs-id1165134252223\"><br \/>\n<\/span><\/p>\n<p>The standard form is useful for determining how the graph is transformed from the graph of [latex]y={x}^{2}[\/latex]. Figure 6\u00a0is the graph of this basic function.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010712\/CNX_Precalc_Figure_03_02_0062.jpg\" alt=\"Graph of y=x^2.\" width=\"487\" height=\"480\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 6<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137770279\">If [latex]k>0[\/latex], the graph shifts upward, whereas if [latex]k<0[\/latex], the graph shifts downward. In Figure 5, [latex]k>0[\/latex], so the graph is shifted 4 units upward. If [latex]h>0[\/latex], the graph shifts toward the right and if [latex]h<0[\/latex], the graph shifts to the left. In Figure 5, [latex]h<0[\/latex], so the graph is shifted 2 units to the left. The magnitude of <em>a<\/em>\u00a0indicates the stretch of the graph. If [latex]|a|>1[\/latex], the point associated with a particular <em>x<\/em>-value shifts farther from the <em>x-<\/em>axis, so the graph appears to become narrower, and there is a vertical stretch. But if [latex]|a|<1[\/latex], the point associated with a particular <em>x<\/em>-value shifts closer to the <em>x-<\/em>axis, so the graph appears to become wider, but in fact there is a vertical compression. In Figure 5, [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<div id=\"eip-173\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{align}a{\\left(x-h\\right)}^{2}+k&=a(x^2-2xh+h^2)+k \\\\ &=a{x}^{2}-2ahx+a{h}^{2}+k=a{x}^{2}+bx+c \\end{align}[\/latex]<\/div>\n<p id=\"fs-id1165137409211\">For the quadratic expressions to be equal, the corresponding coefficients must be equal.<\/p>\n<div id=\"eip-144\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]-2ah=b,\\text{ so }h=-\\frac{b}{2a}[\/latex].<\/div>\n<p id=\"fs-id1165134118295\">This gives us the <strong>axis of symmetry<\/strong> we defined earlier. Setting the constant terms equal:<\/p>\n<div id=\"eip-313\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]\\begin{align}a{h}^{2}+k&=c \\\\ k&=c-a{h}^{2} \\\\ &=c-a{\\left(-\\frac{b}{2a}\\right)}^{2} \\\\ &=c-\\frac{{b}^{2}}{4a} \\end{align}[\/latex]<\/div>\n<p id=\"fs-id1165137476446\">In practice, though, it is usually easier to remember that <em>k<\/em> is the output value of the function when the input is <em>h<\/em>, so [latex]f\\left(h\\right)=f\\left(-\\frac{b}{2a}\\right)=k[\/latex].<\/p>\n<div id=\"fs-id1165137749882\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Forms of Quadratic Functions<\/h3>\n<p id=\"fs-id1165135333154\">A quadratic function is a function of degree two. The graph of a <strong>quadratic function<\/strong> is a parabola. The <strong>general form of a quadratic function<\/strong> is [latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex] where <em>a,\u00a0b<\/em>, and <em>c<\/em>\u00a0are real numbers and [latex]a\\ne 0[\/latex].<\/p>\n<p id=\"fs-id1165137666538\">The <strong>standard form of a quadratic function<\/strong> is [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex].<\/p>\n<p id=\"fs-id1165137762385\">The vertex [latex]\\left(h,k\\right)[\/latex] is located at<\/p>\n<p style=\"text-align: center\">[latex]h=-\\frac{b}{2a},\\text{ }k=f\\left(h\\right)=f\\left(\\frac{-b}{2a}\\right)[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165131886746\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137650986\">How To: Given a graph of a quadratic function, write the equation of the function in general form.<\/h3>\n<ol id=\"fs-id1165134223276\">\n<li>Identify the horizontal shift of the parabola; this value is <em>h<\/em>. Identify the vertical shift of the parabola; this value is <em>k<\/em>.<\/li>\n<li>Substitute the values of the horizontal and vertical shift for <em>h<\/em>\u00a0and <em>k<\/em>. in the function [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex].<\/li>\n<li>Substitute the values of any point, other than the vertex, on the graph of the parabola for <em>x<\/em>\u00a0and [latex]f\\left(x\\right)[\/latex].<\/li>\n<li>Solve for the stretch factor, |<em>a<\/em>|.<\/li>\n<li>If the parabola opens up, [latex]a>0[\/latex]. If the parabola opens down, [latex]a<0[\/latex] since this means the graph was reflected about the <em>x<\/em>-axis.<\/li>\n<li>Expand and simplify to write in general form.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_02_02\" class=\"example\">\n<div id=\"fs-id1165135460939\" class=\"exercise\">\n<div id=\"fs-id1165135460941\" class=\"problem textbox shaded\">\n<h3>Example 2: Writing the Equation of a Quadratic Function from the Graph<\/h3>\n<p id=\"fs-id1165135532321\">Write an equation for the quadratic function <em>g<\/em>\u00a0in the graph below\u00a0as a transformation of [latex]f\\left(x\\right)={x}^{2}[\/latex], and then expand the formula, and simplify terms to write the equation in general form.<span id=\"fs-id1165137725791\"><br \/>\n<\/span><\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010712\/CNX_Precalc_Figure_03_02_0072.jpg\" alt=\"Graph of a parabola with its vertex at (-2, -3).\" width=\"487\" height=\"443\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 7<\/b><\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q616320\">Show Solution<\/span><\/p>\n<div id=\"q616320\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137742565\">We can see the graph of <em>g <\/em>is the graph of [latex]f\\left(x\\right)={x}^{2}[\/latex] shifted to the left 2 and down 3, giving a formula in the form [latex]g\\left(x\\right)=a{\\left(x+2\\right)}^{2}-3[\/latex].<\/p>\n<p id=\"fs-id1165134064001\">Substituting the coordinates of a point on the curve, such as [latex]\\left(0,-1\\right)[\/latex], we can solve for the stretch factor.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}-1&=a{\\left(0+2\\right)}^{2}-3 \\\\ 2&=4a \\\\ a&=\\frac{1}{2} \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137895371\">In standard form, the algebraic model for this graph is [latex]\\left(g\\right)x=\\frac{1}{2}{\\left(x+2\\right)}^{2}-3[\/latex].<\/p>\n<p id=\"fs-id1165137844164\">To write this in general polynomial form, we can expand the formula and simplify terms.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}g\\left(x\\right)&=\\frac{1}{2}{\\left(x+2\\right)}^{2}-3 \\\\ &=\\frac{1}{2}\\left(x+2\\right)\\left(x+2\\right)-3 \\\\ &=\\frac{1}{2}\\left({x}^{2}+4x+4\\right)-3 \\\\ &=\\frac{1}{2}{x}^{2}+2x+2 - 3 \\\\ &=\\frac{1}{2}{x}^{2}+2x - 1 \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137914060\">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<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165137803212\">We can check our work using the table feature on a graphing utility. First enter [latex]\\text{Y1}=\\frac{1}{2}{\\left(x+2\\right)}^{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}\\text{.}[\/latex]<\/p>\n<table id=\"Table_03_02_01\" summary=\"..\">\n<tbody>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>\u20136<\/td>\n<td>\u20134<\/td>\n<td>\u20132<\/td>\n<td>0<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td><em><strong>y<\/strong><\/em><\/td>\n<td>5<\/td>\n<td>\u20131<\/td>\n<td>\u20133<\/td>\n<td>\u20131<\/td>\n<td>5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165135570238\">The ordered pairs in the table correspond to points on the graph.<\/p>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>A coordinate grid has been superimposed over the quadratic path of a basketball in the picture below. Find an equation for the path of the ball. Does the shooter make the basket?<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010712\/CNX_Precalc_Figure_03_02_0082.jpg\" alt=\"Stop motioned picture of a boy throwing a basketball into a hoop to show the parabolic curve it makes.\" width=\"487\" height=\"424\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 8.<\/b> (credit: modification of work by Dan Meyer)<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q903323\">Show Solution<\/span><\/p>\n<div id=\"q903323\" class=\"hidden-answer\" style=\"display: none\">\n<p>The path passes through the origin and has vertex at [latex]\\left(-4,\\text{ }7\\right)[\/latex], so [latex]\\left(h\\right)x=-\\frac{7}{16}{\\left(x+4\\right)}^{2}+7[\/latex]. To make the shot, [latex]h\\left(-7.5\\right)[\/latex] would need to be about 4 but [latex]h\\left(-7.5\\right)\\approx 1.64[\/latex]; he doesn\u2019t make it.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135168275\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135574310\">How To: Given a quadratic function in general form, find the vertex of the parabola.<\/h3>\n<ol id=\"fs-id1165134108459\">\n<li>Identify <em>a<\/em>,\u00a0<em>b<\/em>, and\u00a0<em>c<\/em>.<\/li>\n<li>Find <em>h<\/em>, the <em>x<\/em>-coordinate of the vertex, by substituting <em>a<\/em> and <em>b<\/em>\u00a0into [latex]h=-\\frac{b}{2a}[\/latex].<\/li>\n<li>Find <em>k<\/em>, the <em>y<\/em>-coordinate of the vertex, by evaluating [latex]k=f\\left(h\\right)=f\\left(-\\frac{b}{2a}\\right)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_02_03\" class=\"example\">\n<div id=\"fs-id1165137658566\" class=\"exercise\">\n<div id=\"fs-id1165137771901\" class=\"problem textbox shaded\">\n<h3>Example 3: Finding the Vertex of a Quadratic Function<\/h3>\n<p id=\"fs-id1165135173258\">Find the vertex of the quadratic function [latex]f\\left(x\\right)=2{x}^{2}-6x+7[\/latex]. Rewrite the quadratic in standard form (vertex form).<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q648542\">Show Solution<\/span><\/p>\n<div id=\"q648542\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137596323\">The horizontal coordinate of the vertex will be at<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}h&=-\\frac{b}{2a} \\\\ &=-\\frac{-6}{2\\left(2\\right)} \\\\ &=\\frac{6}{4} \\\\ &=\\frac{3}{2}\\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137465689\">The vertical coordinate of the vertex will be at<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}k&=f\\left(h\\right) \\\\ &=f\\left(\\frac{3}{2}\\right) \\\\ &=2{\\left(\\frac{3}{2}\\right)}^{2}-6\\left(\\frac{3}{2}\\right)+7 \\\\ &=\\frac{5}{2} \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165135177784\">Rewriting into standard form, the stretch factor will be the same as the [latex]a[\/latex] in the original quadratic.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{gathered}f\\left(x\\right)=a{x}^{2}+bx+c \\\\ f\\left(x\\right)=2{x}^{2}-6x+7 \\end{gathered}[\/latex]<\/p>\n<p id=\"fs-id1165137653186\">Using the vertex to determine the shifts,<\/p>\n<p style=\"text-align: center\">[latex]f\\left(x\\right)=2{\\left(x-\\frac{3}{2}\\right)}^{2}+\\frac{5}{2}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p id=\"fs-id1165137638124\">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, (<em>k<\/em>), and where it occurs, (<em>x<\/em>).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135193262\">Given the equation [latex]g\\left(x\\right)=13+{x}^{2}-6x[\/latex], write the equation in general form and then in standard form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q312729\">Show Solution<\/span><\/p>\n<div id=\"q312729\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]g\\left(x\\right)={x}^{2}-6x+13[\/latex] in general form; [latex]g\\left(x\\right)={\\left(x - 3\\right)}^{2}+4[\/latex] in standard form<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>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>y<\/em>-values greater than or equal to the <em>y<\/em>-coordinate at the turning point or less than or equal to the <em>y<\/em>-coordinate at the turning point, depending on whether the parabola opens up or down.<\/p>\n<div id=\"fs-id1165135161405\" class=\"note textbox\">\n<h3 class=\"title\">A General Note: Domain and Range of a Quadratic Function<\/h3>\n<p id=\"fs-id1165135502927\">The domain of any <strong>quadratic function<\/strong> is all real numbers.<\/p>\n<p id=\"fs-id1165135502930\">The range of a quadratic function written in general form [latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex] with a positive <em>a<\/em>\u00a0value is [latex]f\\left(x\\right)\\ge f\\left(-\\frac{b}{2a}\\right)[\/latex], or [latex]\\left[f\\left(-\\frac{b}{2a}\\right),\\infty \\right)[\/latex]; the range of a quadratic function written in general form with a negative <em>a<\/em>\u00a0value is [latex]f\\left(x\\right)\\le f\\left(-\\frac{b}{2a}\\right)[\/latex], or [latex]\\left(-\\infty ,f\\left(-\\frac{b}{2a}\\right)\\right][\/latex].<\/p>\n<p id=\"fs-id1165137723229\">The range of a quadratic function written in standard form [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex] with a positive <em>a<\/em>\u00a0value is [latex]f\\left(x\\right)\\ge k[\/latex]; the range of a quadratic function written in standard form with a negative <em>a<\/em>\u00a0value is [latex]f\\left(x\\right)\\le k[\/latex].<\/p>\n<\/div>\n<div id=\"fs-id1165135205144\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137409165\">How To: Given a quadratic function, find the domain and range.<\/h3>\n<ol id=\"fs-id1165137843779\">\n<li>The domain of any quadratic function as all real numbers.<\/li>\n<li>Determine whether <em>a<\/em>\u00a0is positive or negative. If <em>a<\/em>\u00a0is positive, the parabola has a minimum. If <em>a<\/em>\u00a0is negative, the parabola has a maximum.<\/li>\n<li>Determine the maximum or minimum value of the parabola, <em>k<\/em>.<\/li>\n<li>If the parabola has a minimum, the range is given by [latex]f\\left(x\\right)\\ge k[\/latex], or [latex]\\left[k,\\infty \\right)[\/latex]. If the parabola has a maximum, the range is given by [latex]f\\left(x\\right)\\le k[\/latex], or [latex]\\left(-\\infty ,k\\right][\/latex].<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_02_04\" class=\"example\">\n<div id=\"fs-id1165134257627\" class=\"exercise\">\n<div id=\"fs-id1165134257629\" class=\"problem textbox shaded\">\n<h3>Example 4: Finding the Domain and Range of a Quadratic Function<\/h3>\n<p id=\"fs-id1165137696393\">Find the domain and range of [latex]f\\left(x\\right)=-5{x}^{2}+9x - 1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q539374\">Show Solution<\/span><\/p>\n<div id=\"q539374\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137837924\">As with any quadratic function, the domain is all real numbers.<\/p>\n<p id=\"fs-id1165137823619\">Because <em>a<\/em>\u00a0is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding the <em>x<\/em>-value of the vertex.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}h&=-\\frac{b}{2a} \\\\ &=-\\frac{9}{2\\left(-5\\right)} \\\\ &=\\frac{9}{10}& \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137736576\">The maximum value is given by [latex]f\\left(h\\right)[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}f\\left(\\frac{9}{10}\\right)&=5{\\left(\\frac{9}{10}\\right)}^{2}+9\\left(\\frac{9}{10}\\right)-1 \\\\ &=\\frac{61}{20} \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137460169\">The range is [latex]f\\left(x\\right)\\le \\frac{61}{20}[\/latex], or [latex]\\left(-\\infty ,\\frac{61}{20}\\right][\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165135424650\">Find the domain and range of [latex]f\\left(x\\right)=2{\\left(x-\\frac{4}{7}\\right)}^{2}+\\frac{8}{11}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q345090\">Show Solution<\/span><\/p>\n<div id=\"q345090\" class=\"hidden-answer\" style=\"display: none\">\n<p>The domain is all real numbers. The range is [latex]f\\left(x\\right)\\ge \\frac{8}{11}[\/latex], or [latex]\\left[\\frac{8}{11},\\infty \\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>\u00a0Determine a quadratic function\u2019s minimum or maximum value<\/h2>\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 style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010712\/CNX_Precalc_Figure_03_02_0092.jpg\" alt=\"Two graphs where the first graph shows the maximum value for f(x)=(x-2)^2+1 which occurs at (2, 1) and the second graph shows the minimum value for g(x)=-(x+3)^2+4 which occurs at (-3, 4).\" width=\"975\" height=\"558\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 9<\/b><\/p>\n<\/div>\n<div id=\"Example_03_02_05\" class=\"example\">\n<div id=\"fs-id1165134378616\" class=\"exercise\">\n<div id=\"fs-id1165134378618\" class=\"problem textbox shaded\">\n<h3>Example 5: Finding the Maximum Value of a Quadratic Function<\/h3>\n<p id=\"fs-id1165137653457\">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<ol id=\"fs-id1165135640934\">\n<li>Find a formula for the area enclosed by the fence if the sides of fencing perpendicular to the existing fence have length <em>L<\/em>.<\/li>\n<li>What dimensions should she make her garden to maximize the enclosed area?<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q478551\">Show Solution<\/span><\/p>\n<div id=\"q478551\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010713\/CNX_Precalc_Figure_03_02_0102.jpg\" alt=\"Diagram of the garden and the backyard.\" width=\"487\" height=\"310\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 10<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137836808\">Let\u2019s use a diagram such as the one in Figure 10\u00a0to record the given information. It is also helpful to introduce a temporary variable, <em>W<\/em>, to represent the width of the garden and the length of the fence section parallel to the backyard fence.<span id=\"fs-id1165135208803\"><br \/>\n<\/span><\/p>\n<ol id=\"fs-id1165134363440\">\n<li>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, <em>W<\/em>, in terms of <em>L<\/em>.\n<div id=\"eip-id1165135697866\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]W=80 - 2L[\/latex]<\/div>\n<p id=\"fs-id1165135435476\">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 id=\"fs-id1165135435476\" style=\"text-align: center\">[latex]\\begin{align}A&=LW \\\\ &=L\\left(80 - 2L\\right) \\\\ &=80L - 2{L}^{2} \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165135258914\">This formula represents the area of the fence in terms of the variable length <em>L<\/em>. The function, written in general form, is<\/p>\n<div id=\"eip-382\" class=\"equation unnumbered\" style=\"text-align: center\">[latex]A\\left(L\\right)=-2{L}^{2}+80L[\/latex].<\/div>\n<\/li>\n<li>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 <em>a<\/em>\u00a0is the coefficient of the squared term, [latex]a=-2,b=80[\/latex], and [latex]c=0[\/latex].<\/li>\n<\/ol>\n<p id=\"fs-id1165137772015\">To find the vertex:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}h&=-\\dfrac{80}{2\\left(-2\\right)} &&& k&=A\\left(20\\right) \\\\ &=20 && \\text{and} & &=80\\left(20\\right)-2{\\left(20\\right)}^{2}\\\\ &&&&&=800 \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165135174964\">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<h4>Analysis of the Solution<\/h4>\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<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010713\/CNX_Precalc_Figure_03_02_0112.jpg\" alt=\"Graph of the parabolic function A(L)=-2L^2+80L, which the x-axis is labeled Length (L) and the y-axis is labeled Area (A). The vertex is at (20, 800).\" width=\"487\" height=\"476\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 11<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133340409\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137803708\">How To: Given an application involving revenue, use a quadratic equation to find the maximum.<\/h3>\n<ol id=\"fs-id1165135436584\">\n<li>Write a quadratic equation for revenue.<\/li>\n<li>Find the vertex of the quadratic equation.<\/li>\n<li>Determine the <em>y<\/em>-value of the vertex.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_02_06\" class=\"example\">\n<div id=\"fs-id1165134278696\" class=\"exercise\">\n<div id=\"fs-id1165137473136\" class=\"problem textbox shaded\">\n<h3>Example 6: Finding Maximum Revenue<\/h3>\n<p id=\"fs-id1165137473142\">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 local newspaper 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<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q354854\">Show Solution<\/span><\/p>\n<div id=\"q354854\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135389888\">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, <em>p<\/em>\u00a0for price per subscription and <em>Q<\/em>\u00a0for quantity, giving us the equation [latex]\\text{Revenue}=pQ[\/latex].<\/p>\n<p id=\"fs-id1165134232972\">Because the number of subscribers changes with the price, we 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 newspaper 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{align}m&=\\frac{79,000 - 84,000}{32 - 30} \\\\ &=\\frac{-5,000}{2} \\\\ &=-2,500 \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165135559520\">This tells us the paper 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{align}Q&=-2500p+b && \\text{Substitute in the point }Q=84,000\\text{ and }p=30 \\\\ 84,000&=-2500\\left(30\\right)+b && \\text{Solve for }b \\\\ b&=159,000 \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137933138\">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{align}\\text{Revenue}&=pQ \\\\ &=p\\left(-2,500p+159,000\\right) \\\\ &=-2,500{p}^{2}+159,000p \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165135502033\">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 newspaper, we can find the vertex.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}h&=-\\frac{159,000}{2\\left(-2,500\\right)} \\\\ &=31.8 \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137647087\">The model tells us that the maximum revenue will occur if the newspaper 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{align}\\text{maximum revenue}&=-2,500{\\left(31.8\\right)}^{2}+159,000\\left(31.8\\right) \\\\ &=2,528,100 \\end{align}[\/latex]<\/p>\n<p>The maximum revenue is $2,528,100.<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>This could also be solved by graphing the quadratic. We can see the maximum revenue on a graph of the quadratic function.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010713\/CNX_Precalc_Figure_03_02_0122.jpg\" alt=\"Graph of the parabolic function which the x-axis is labeled Price (p) and the y-axis is labeled Revenue ($). The vertex is at (31.80, 258100).\" width=\"487\" height=\"327\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 12<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<section id=\"fs-id1165135693703\">\n<h2 style=\"text-align: center\">Finding the <em>x<\/em>&#8211; and <em>y<\/em>-Intercepts of a Quadratic Function<\/h2>\n<p>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\u00a0that the number of <em>x<\/em>-intercepts can vary depending upon the location of the graph.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010713\/CNX_Precalc_Figure_03_02_0132.jpg\" alt=\"Three graphs where the first graph shows a parabola with no x-intercept, the second is a parabola with one \u2013intercept, and the third parabola is of two x-intercepts.\" width=\"975\" height=\"317\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 13.<\/b> Number of x-intercepts of a parabola<\/p>\n<\/div>\n<div id=\"fs-id1165137464602\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135638554\">How To: Given a quadratic function [latex]f\\left(x\\right)[\/latex], find the <em>y<\/em>&#8211;\u00a0and <em>x<\/em>-intercepts.<\/h3>\n<ol id=\"fs-id1165135378765\">\n<li>Evaluate [latex]f\\left(0\\right)[\/latex] to find the <em>y<\/em>-intercept.<\/li>\n<li>Solve the quadratic equation [latex]f\\left(x\\right)=0[\/latex] to find the <em>x<\/em>-intercepts.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_02_07\" class=\"example\">\n<div id=\"fs-id1165134129944\" class=\"exercise\">\n<div id=\"fs-id1165134129946\" class=\"problem textbox shaded\">\n<h3>Example 7: Finding the <em>y<\/em>&#8211; and <em>x<\/em>-Intercepts of a Parabola<\/h3>\n<p id=\"fs-id1165134138677\">Find the <em>y<\/em>&#8211; and <em>x<\/em>-intercepts of the quadratic [latex]f\\left(x\\right)=3{x}^{2}+5x - 2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q201680\">Show Solution<\/span><\/p>\n<div id=\"q201680\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137901096\">We find the <em>y<\/em>-intercept by evaluating [latex]f\\left(0\\right)[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}f\\left(0\\right)&=3{\\left(0\\right)}^{2}+5\\left(0\\right)-2 \\\\ &=-2 \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165134232203\">So the <em>y<\/em>-intercept is at [latex]\\left(0,-2\\right)[\/latex].<\/p>\n<p id=\"fs-id1165135434816\">For the <em>x<\/em>-intercepts, we find all solutions of [latex]f\\left(x\\right)=0[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]0=3{x}^{2}+5x - 2[\/latex]<\/p>\n<p id=\"fs-id1165135690677\">In this case, the quadratic can be factored easily, providing the simplest method for solution.<\/p>\n<p style=\"text-align: center\">[latex]0=\\left(3x - 1\\right)\\left(x+2\\right)[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}3x - 1&=0\\\\x&=\\frac{1}{3}&& \\end{align}[\/latex] or [latex]\\begin{align}&&x+2&=0 \\\\ &&x&=-2 \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165137644422\">So the <em>x<\/em>-intercepts are at [latex]\\left(\\frac{1}{3},0\\right)[\/latex] and [latex]\\left(-2,0\\right)[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>By graphing the function, we can confirm that the graph crosses the <em>y<\/em>-axis at [latex]\\left(0,-2\\right)[\/latex]. We can also confirm that the graph crosses the <em>x<\/em>-axis at [latex]\\left(\\frac{1}{3},0\\right)[\/latex] and [latex]\\left(-2,0\\right)[\/latex].<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010713\/CNX_Precalc_Figure_03_02_0142.jpg\" alt=\"Graph of a parabola which has the following intercepts (-2, 0), (1\/3, 0), and (0, -2).\" width=\"487\" height=\"480\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 14<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137911614\" class=\"commentary\">\n<div class=\"mceTemp\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2>\u00a0Solve problems involving a quadratic function\u2019s minimum or maximum value<\/h2>\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 id=\"fs-id1165133085664\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165135694488\">How To: Given a quadratic function, find the <em>x<\/em>-intercepts by rewriting in standard form.<\/h3>\n<ol id=\"fs-id1165134113976\">\n<li>Substitute <em>a<\/em>\u00a0and <em>b<\/em>\u00a0into [latex]h=-\\frac{b}{2a}[\/latex].<\/li>\n<li>Substitute <em>x<\/em> =\u00a0<em>h<\/em>\u00a0into the general form of the quadratic function to find <em>k<\/em>.<\/li>\n<li>Rewrite the quadratic in standard form using <em>h<\/em>\u00a0and <em>k<\/em>.<\/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 id=\"Example_03_02_08\" class=\"example\">\n<div id=\"fs-id1165134060458\" class=\"exercise\">\n<div id=\"fs-id1165134224010\" class=\"problem textbox shaded\">\n<h3>Example 8: Finding the <em>x<\/em>-Intercepts of a Parabola<\/h3>\n<p id=\"fs-id1165134224020\">Find the <em>x<\/em>-intercepts of the quadratic function [latex]f\\left(x\\right)=2{x}^{2}+4x - 4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q41935\">Show Solution<\/span><\/p>\n<div id=\"q41935\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135524478\">We begin by solving for when the output will be zero.<\/p>\n<p style=\"text-align: center\">[latex]0=2{x}^{2}+4x - 4[\/latex]<\/p>\n<p id=\"fs-id1165135252139\">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\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/p>\n<p id=\"fs-id1165137925165\">We know that <em>a\u00a0<\/em>= 2. Then we solve for <em>h<\/em>\u00a0and <em>k<\/em>.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}h&=-\\frac{b}{2a} &&& k&=f\\left(-1\\right) \\\\ &=-\\frac{4}{2\\left(2\\right)} &&& &=2{\\left(-1\\right)}^{2}+4\\left(-1\\right)-4 \\\\ &=-1 &&& &=-6 \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165134031332\">So now we can rewrite in standard form.<\/p>\n<p style=\"text-align: center\">[latex]f\\left(x\\right)=2{\\left(x+1\\right)}^{2}-6[\/latex]<\/p>\n<p id=\"fs-id1165135381286\">We can now solve for when the output will be zero.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{gathered}2{\\left(x+1\\right)}^{2}-6=0 \\\\ 2{\\left(x+1\\right)}^{2}=6 \\\\ {\\left(x+1\\right)}^{2}=3 \\\\ x+1=\\pm \\sqrt{3} \\\\ x=-1\\pm \\sqrt{3} \\end{gathered}[\/latex]<\/p>\n<p id=\"fs-id1165131959622\">The graph has <em>x-<\/em>intercepts at [latex]\\left(-1-\\sqrt{3},0\\right)[\/latex] and [latex]\\left(-1+\\sqrt{3},0\\right)[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010714\/CNX_Precalc_Figure_03_02_0152.jpg\" alt=\"Graph of a parabola which has the following x-intercepts (-2.732, 0) and (0.732, 0).\" width=\"487\" height=\"517\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 15<\/b><\/p>\n<\/div>\n<p id=\"fs-id1165137843092\">We can check our work by graphing the given function on a graphing utility and observing the <em>x-<\/em>intercepts.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137843086\" class=\"commentary\">\n<p id=\"fs-id1165137843092\">\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165134534213\">In Try It\u00a02, we found the standard and general form for the function [latex]g\\left(x\\right)=13+{x}^{2}-6x[\/latex]. Now find the <em>y<\/em>&#8211; and <em>x<\/em>-intercepts (if any).<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q700527\">Show Solution<\/span><\/p>\n<div id=\"q700527\" class=\"hidden-answer\" style=\"display: none\">\n<p><em>y<\/em>-intercept at (0, 13), No <em>x-<\/em>intercepts<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_02_09\" class=\"example\">\n<div id=\"fs-id1165135606916\" class=\"exercise\">\n<div id=\"fs-id1165135606918\" class=\"problem textbox shaded\">\n<h3>Example 9: Solving a Quadratic Equation with the Quadratic Formula<\/h3>\n<p id=\"fs-id1165135606924\">Solve [latex]{x}^{2}+x+2=0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q41618\">Show Solution<\/span><\/p>\n<div id=\"q41618\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165133036005\">Let\u2019s begin by writing the quadratic formula: [latex]x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex].<\/p>\n<p id=\"fs-id1165134102089\">When applying the <span class=\"no-emphasis\">quadratic formula<\/span>, we identify the coefficients <em>a<\/em>,\u00a0<em>b<\/em>, and\u00a0<em>c<\/em>. For the equation [latex]{x}^{2}+x+2=0[\/latex], we have <em>a<\/em>\u00a0=\u00a01,\u00a0<em>b<\/em>\u00a0=\u00a01, and <em>c<\/em>\u00a0=\u00a02.\u00a0Substituting these values into the formula we have:<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}x&=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a} \\\\ &=\\frac{-1\\pm \\sqrt{{1}^{2}-4\\cdot 1\\cdot \\left(2\\right)}}{2\\cdot 1} \\\\ &=\\frac{-1\\pm \\sqrt{1 - 8}}{2} \\\\ &=\\frac{-1\\pm \\sqrt{-7}}{2} \\\\ &=\\frac{-1\\pm i\\sqrt{7}}{2} \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165134149125\">The solutions to the equation are [latex]x=\\frac{-1+i\\sqrt{7}}{2}[\/latex] and [latex]x=\\frac{-1-i\\sqrt{7}}{2}[\/latex] or [latex]x=\\frac{-1}{2}+\\frac{i\\sqrt{7}}{2}[\/latex] and [latex]x=\\frac{-1}{2}-\\frac{i\\sqrt{7}}{2}[\/latex]. Note that because of the\u00a0<em>i<\/em>, these are non-real zeros.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_02_10\" class=\"example\">\n<div id=\"fs-id1165134085786\" class=\"exercise\">\n<div id=\"fs-id1165134085788\" class=\"problem textbox shaded\">\n<h3>Example 10: Applying the Vertex and <em>x<\/em>-Intercepts of a Parabola<\/h3>\n<p id=\"fs-id1165134085798\">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\\left(t\\right)=-16{t}^{2}+80t+40[\/latex].<\/p>\n<p style=\"padding-left: 60px\">a. When does the ball reach the maximum height?<\/p>\n<p style=\"padding-left: 60px\">b. What is the maximum height of the ball?<\/p>\n<p style=\"padding-left: 60px\">c. When does the ball hit the ground?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q340065\">Show Solution<\/span><\/p>\n<div id=\"q340065\" class=\"hidden-answer\" style=\"display: none\">\n<p>a. The ball reaches the maximum height at the vertex of the parabola.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align} h&=-\\frac{80}{2\\left(-16\\right)} =\\frac{80}{32} \\\\ &=\\frac{5}{2} \\\\ &=2.5 \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165135528870\">The ball reaches a maximum height after 2.5 seconds.<\/p>\n<p>b. To find the maximum height, find the <em>y\u00a0<\/em>coordinate of the vertex of the parabola.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}k&=H\\left(-\\frac{b}{2a}\\right) \\\\ &=H\\left(2.5\\right) \\\\ &=-16{\\left(2.5\\right)}^{2}+80\\left(2.5\\right)+40 \\\\ &=140 \\end{align}[\/latex]<\/p>\n<p id=\"fs-id1165135409750\">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\\left(t\\right)=0[\/latex].<\/p>\n<p>We use the quadratic formula.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align} t&=\\frac{-80\\pm \\sqrt{{80}^{2}-4\\left(-16\\right)\\left(40\\right)}}{2\\left(-16\\right)} \\\\ &=\\frac{-80\\pm \\sqrt{8960}}{-32} \\end{align}[\/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]\\begin{align}t=\\frac{-80-\\sqrt{8960}}{-32}\\approx 5.458 && \\text{or} && t=\\frac{-80+\\sqrt{8960}}{-32}\\approx -0.458 \\end{align}[\/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.<span id=\"fs-id1165135580349\"><br \/>\n<\/span><\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03010714\/CNX_Precalc_Figure_03_02_0162.jpg\" alt=\"Graph of a negative parabola where x goes from -1 to 6.\" width=\"487\" height=\"254\" \/><\/p>\n<p class=\"wp-caption-text\"><b>Figure 16<\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><b><\/b><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165134081301\">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\\left(t\\right)=-16{t}^{2}+96t+112[\/latex].<\/p>\n<p style=\"padding-left: 60px\">a. When does the rock reach the maximum height?<\/p>\n<p style=\"padding-left: 60px\">b. What is the maximum height of the rock?<\/p>\n<p style=\"padding-left: 60px\">c. When does the rock hit the ocean?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q4901\">Show Solution<\/span><\/p>\n<div id=\"q4901\" class=\"hidden-answer\" style=\"display: none\">\n<p>a.\u00a03 seconds \u00a0b.\u00a0256 feet \u00a0c.\u00a07 seconds<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm17065\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=17065&theme=oea&iframe_resize_id=ohm17065\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p><span style=\"color: #077fab;font-size: 1.15em;font-weight: 600\">Key Equations<\/span><\/p>\n<section id=\"fs-id1165134205927\" class=\"key-equations\">\n<table id=\"eip-id1165137539373\" summary=\"..\">\n<tbody>\n<tr>\n<td>general form of a quadratic function<\/td>\n<td>[latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>the quadratic formula<\/td>\n<td>[latex]x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>standard form of a quadratic function<\/td>\n<td>[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section id=\"fs-id1165135426424\" class=\"key-concepts\">\n<h2>Key Concepts<\/h2>\n<ul id=\"fs-id1165134570662\">\n<li>A polynomial function of degree two is called a quadratic function.<\/li>\n<li>The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.<\/li>\n<li>The axis of symmetry is the vertical line passing through the vertex. The zeros, or <em>x<\/em>-intercepts, are the points at which the parabola crosses the <em>x<\/em>-axis. The <em>y<\/em>-intercept is the point at which the parabola crosses the <em>y-<\/em>axis.<\/li>\n<li>Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph.<\/li>\n<li>The vertex can be found from an equation representing a quadratic function.<\/li>\n<li>The domain of a quadratic function is all real numbers. The range varies with the function.<\/li>\n<li>A quadratic function\u2019s minimum or maximum value is given by the <em>y<\/em>-value of the vertex.<\/li>\n<li>The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue.<\/li>\n<li>Some quadratic equations must be solved by using the quadratic formula.<\/li>\n<li>The vertex and the intercepts can be identified and interpreted to solve real-world problems.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165135449657\" class=\"definition\">\n<dt><strong>axis of symmetry<\/strong><\/dt>\n<dd id=\"fs-id1165135449662\">a vertical line drawn through the vertex of a parabola around which the parabola is symmetric; it is defined by [latex]x=-\\frac{b}{2a}[\/latex].<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135502777\" class=\"definition\">\n<dt><strong>general form of a quadratic function<\/strong><\/dt>\n<dd id=\"fs-id1165135502783\">the function that describes a parabola, written in the form [latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex], where <em>a<\/em>,\u00a0<em>b<\/em>, and\u00a0<em>c<\/em>\u00a0are real numbers and [latex]a\\ne 0[\/latex].<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137931314\" class=\"definition\">\n<dt><strong>standard form of a quadratic function<\/strong><\/dt>\n<dd id=\"fs-id1165137931319\">the function that describes a parabola, written in the form [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex], where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135623614\" class=\"definition\">\n<dt><strong>vertex<\/strong><\/dt>\n<dd id=\"fs-id1165135623619\">the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135623624\" class=\"definition\">\n<dt><strong>vertex form of a quadratic function<\/strong><\/dt>\n<dd id=\"fs-id1165135623630\">another name for the standard form of a quadratic function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135623634\" class=\"definition\">\n<dt><strong>zeros<\/strong><\/dt>\n<dd id=\"fs-id1165135623639\">in a given function, the values of <em>x<\/em>\u00a0at which <em>y<\/em> = 0, also called roots<\/dd>\n<\/dl>\n<\/section>\n<p>&nbsp;<\/p>\n<h2 style=\"text-align: center\">Section 2.3 Homework Exercises<\/h2>\n<p>1. Explain the advantage of writing a quadratic function in standard form.<\/p>\n<p>2. How can the vertex of a parabola be used in solving real world problems?<\/p>\n<p>3. Explain why the condition of [latex]a\\ne 0[\/latex] is imposed in the definition of the quadratic function.<\/p>\n<p>4.\u00a0What is another name for the standard form of a quadratic function?<\/p>\n<p>5. What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?<\/p>\n<p>For the following exercises, rewrite the quadratic functions in standard form and give the vertex.<\/p>\n<p>6. [latex]f\\left(x\\right)={x}^{2}-12x+32[\/latex]<\/p>\n<p>7. [latex]g\\left(x\\right)={x}^{2}+2x - 3[\/latex]<\/p>\n<p>8.\u00a0[latex]f\\left(x\\right)={x}^{2}-x[\/latex]<\/p>\n<p>9. [latex]f\\left(x\\right)={x}^{2}+5x - 2[\/latex]<\/p>\n<p>10.\u00a0[latex]h\\left(x\\right)=2{x}^{2}+8x - 10[\/latex]<\/p>\n<p>11. [latex]k\\left(x\\right)=3{x}^{2}-6x - 9[\/latex]<\/p>\n<p>12.\u00a0[latex]f\\left(x\\right)=2{x}^{2}-6x[\/latex]<\/p>\n<p>13. [latex]f\\left(x\\right)=3{x}^{2}-5x - 1[\/latex]<\/p>\n<p>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<p>14. [latex]y\\left(x\\right)=2{x}^{2}+10x+12[\/latex]<\/p>\n<p>15. [latex]f\\left(x\\right)=2{x}^{2}-10x+4[\/latex]<\/p>\n<p>16.\u00a0[latex]f\\left(x\\right)=-{x}^{2}+4x+3[\/latex]<\/p>\n<p>17. [latex]f\\left(x\\right)=4{x}^{2}+x - 1[\/latex]<\/p>\n<p>18. [latex]h\\left(t\\right)=-4{t}^{2}+6t - 1[\/latex]<\/p>\n<p>19. [latex]f\\left(x\\right)=\\frac{1}{2}{x}^{2}+3x+1[\/latex]<\/p>\n<p>20. [latex]f\\left(x\\right)=-\\frac{1}{3}{x}^{2}-2x+3[\/latex]<\/p>\n<p>For the following exercises, determine the domain and range of the quadratic function.<\/p>\n<p>21. [latex]f\\left(x\\right)={\\left(x - 3\\right)}^{2}+2[\/latex]<\/p>\n<p>22.\u00a0[latex]f\\left(x\\right)=-2{\\left(x+3\\right)}^{2}-6[\/latex]<\/p>\n<p>23. [latex]f\\left(x\\right)={x}^{2}+6x+4[\/latex]<\/p>\n<p>24.\u00a0[latex]f\\left(x\\right)=2{x}^{2}-4x+2[\/latex]<\/p>\n<p>25. [latex]k\\left(x\\right)=3{x}^{2}-6x - 9[\/latex]<\/p>\n<p>For the following exercises, solve the equations over the complex numbers.<\/p>\n<p>26. [latex]{x}^{2}=-25[\/latex]<\/p>\n<p>27. [latex]{x}^{2}=-8[\/latex]<\/p>\n<p>28.\u00a0[latex]{x}^{2}+36=0[\/latex]<\/p>\n<p>29. [latex]{x}^{2}+27=0[\/latex]<\/p>\n<p>30.\u00a0[latex]{x}^{2}+2x+5=0[\/latex]<\/p>\n<p>31. [latex]{x}^{2}-4x+5=0[\/latex]<\/p>\n<p>32.\u00a0[latex]{x}^{2}+8x+25=0[\/latex]<\/p>\n<p>33. [latex]{x}^{2}-4x+13=0[\/latex]<\/p>\n<p>34.\u00a0[latex]{x}^{2}+6x+25=0[\/latex]<\/p>\n<p>35. [latex]{x}^{2}-10x+26=0[\/latex]<\/p>\n<p>36.\u00a0[latex]{x}^{2}-6x+10=0[\/latex]<\/p>\n<p>37. [latex]x\\left(x - 4\\right)=20[\/latex]<\/p>\n<p>38.\u00a0[latex]x\\left(x - 2\\right)=10[\/latex]<\/p>\n<p>39. [latex]2{x}^{2}+2x+5=0[\/latex]<\/p>\n<p>40.\u00a0[latex]5{x}^{2}-8x+5=0[\/latex]<\/p>\n<p>41. [latex]5{x}^{2}+6x+2=0[\/latex]<\/p>\n<p>42. [latex]2{x}^{2}-6x+5=0[\/latex]<\/p>\n<p>43. [latex]{x}^{2}+x+2=0[\/latex]<\/p>\n<p>44.\u00a0[latex]{x}^{2}-2x+4=0[\/latex]<\/p>\n<p>For the following exercises, use the vertex (<em>h<\/em>, <em>k<\/em>) and a point on the graph (<em>x<\/em>, <em>y<\/em>) to find the general form of the equation of the quadratic function.<\/p>\n<p>45. [latex]\\left(h,k\\right)=\\left(2,0\\right),\\left(x,y\\right)=\\left(4,4\\right)[\/latex]<\/p>\n<p>46.\u00a0[latex]\\left(h,k\\right)=\\left(-2,-1\\right),\\left(x,y\\right)=\\left(-4,3\\right)[\/latex]<\/p>\n<p>47. [latex]\\left(h,k\\right)=\\left(0,1\\right),\\left(x,y\\right)=\\left(2,5\\right)[\/latex]<\/p>\n<p>48.\u00a0[latex]\\left(h,k\\right)=\\left(2,3\\right),\\left(x,y\\right)=\\left(5,12\\right)[\/latex]<\/p>\n<p>49. [latex]\\left(h,k\\right)=\\left(-5,3\\right),\\left(x,y\\right)=\\left(2,9\\right)[\/latex]<\/p>\n<p>50.\u00a0[latex]\\left(h,k\\right)=\\left(3,2\\right),\\left(x,y\\right)=\\left(10,1\\right)[\/latex]<\/p>\n<p>51. [latex]\\left(h,k\\right)=\\left(0,1\\right),\\left(x,y\\right)=\\left(1,0\\right)[\/latex]<\/p>\n<p>52.\u00a0[latex]\\left(h,k\\right)=\\left(1,0\\right),\\left(x,y\\right)=\\left(0,1\\right)[\/latex]<\/p>\n<p>For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.<\/p>\n<p>53. [latex]f\\left(x\\right)={x}^{2}-2x[\/latex]<\/p>\n<p>54.\u00a0[latex]f\\left(x\\right)={x}^{2}-6x - 1[\/latex]<\/p>\n<p>55. [latex]f\\left(x\\right)={x}^{2}-5x - 6[\/latex]<\/p>\n<p>56.\u00a0[latex]f\\left(x\\right)={x}^{2}-7x+3[\/latex]<\/p>\n<p>57. [latex]f\\left(x\\right)=-2{x}^{2}+5x - 8[\/latex]<\/p>\n<p>58. [latex]f\\left(x\\right)=4{x}^{2}-12x - 3[\/latex]<\/p>\n<p>For the following exercises, write the equation for the graphed function.<\/p>\n<p>59.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005210\/CNX_Precalc_Figure_03_02_207.jpg\" alt=\"Graph of a positive parabola with a vertex at (2, -3) and y-intercept at (0, 1).\" \/><\/p>\n<p>60.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005210\/CNX_Precalc_Figure_03_02_208.jpg\" alt=\"Graph of a positive parabola with a vertex at (-1, 2) and y-intercept at (0, 3)\" \/><\/p>\n<p>61.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005210\/CNX_Precalc_Figure_03_02_209.jpg\" alt=\"Graph of a negative parabola with a vertex at (2, 7).\" \/><\/p>\n<p>62.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005211\/CNX_Precalc_Figure_03_02_210.jpg\" alt=\"Graph of a negative parabola with a vertex at (-1, 2).\" \/><\/p>\n<p>63.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005211\/CNX_Precalc_Figure_03_02_211n.jpg\" alt=\"Graph of a positive parabola with a vertex at (3, -1) and y-intercept at (0, 3.5).\" \/><\/p>\n<p>64.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1227\/2015\/04\/03005211\/CNX_Precalc_Figure_03_02_212.jpg\" alt=\"Graph of a negative parabola with a vertex at (-2, 3).\" \/><\/p>\n<p>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<p>65.<\/p>\n<table>\n<tbody>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>\u20132<\/td>\n<td>\u20131<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td><em><strong>y<\/strong><\/em><\/td>\n<td>5<\/td>\n<td>2<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>66.<\/p>\n<table>\n<tbody>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>\u20132<\/td>\n<td>\u20131<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td><em><strong>y<\/strong><\/em><\/td>\n<td>1<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>4<\/td>\n<td>9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>67.<\/p>\n<table>\n<tbody>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>\u20132<\/td>\n<td>\u20131<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td><em><strong>y<\/strong><\/em><\/td>\n<td>\u20132<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>1<\/td>\n<td>\u20132<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>68.<\/p>\n<table id=\"fs-id1165134138604\" class=\"unnumbered\" summary=\"..\">\n<tbody>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>\u20132<\/td>\n<td>\u20131<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td><em><strong>y<\/strong><\/em><\/td>\n<td>\u20138<\/td>\n<td>\u20133<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>69.<\/p>\n<table>\n<tbody>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>\u20132<\/td>\n<td>\u20131<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<td>2<\/td>\n<\/tr>\n<tr>\n<td><em><strong>y<\/strong><\/em><\/td>\n<td>8<\/td>\n<td>2<\/td>\n<td>0<\/td>\n<td>2<\/td>\n<td>8<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>For the following exercises, use a calculator to find the answer.<\/p>\n<p>70. Graph on the same set of axes the functions [latex]f\\left(x\\right)={x}^{2},f\\left(x\\right)=2{x}^{2},\\text{ and }f\\left(x\\right)=\\frac{1}{3}{x}^{2}[\/latex].\u00a0What appears to be the effect of changing the coefficient?<\/p>\n<p>71. Graph on the same set of axes [latex]f\\left(x\\right)={x}^{2},f\\left(x\\right)={x}^{2}+2[\/latex] and [latex]f\\left(x\\right)={x}^{2},f\\left(x\\right)={x}^{2}+5[\/latex] and [latex]f\\left(x\\right)={x}^{2}-3[\/latex]. What appears to be the effect of adding a constant?<\/p>\n<p>72.\u00a0Graph on the same set of axes [latex]f\\left(x\\right)={x}^{2},f\\left(x\\right)={\\left(x - 2\\right)}^{2},f{\\left(x - 3\\right)}^{2},\\text{ and }f\\left(x\\right)={\\left(x+4\\right)}^{2}[\/latex].\u00a0What appears to be the effect of adding or subtracting those numbers?<\/p>\n<p>73. 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\\left(x\\right)=\\frac{-32}{{\\left(80\\right)}^{2}}{x}^{2}+x[\/latex] where <em>x<\/em>\u00a0is the horizontal distance traveled and [latex]h\\left(x\\right)[\/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<p>74.\u00a0A suspension bridge can be modeled by the quadratic function [latex]h\\left(x\\right)=.0001{x}^{2}[\/latex] with [latex]-2000\\le x\\le 2000[\/latex] where |<em>x<\/em>| is the number of feet from the center and [latex]h\\left(x\\right)[\/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<p>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<p>75. Vertex [latex]\\left(1,-2\\right)[\/latex], opens up.<\/p>\n<p>76.\u00a0Vertex [latex]\\left(-1,2\\right)[\/latex] opens down.<\/p>\n<p>77. Vertex [latex]\\left(-5,11\\right)[\/latex], opens down.<\/p>\n<p>78.\u00a0Vertex [latex]\\left(-100,100\\right)[\/latex], opens up.<\/p>\n<p>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<p>79. Contains (1, 1) and has shape of [latex]f\\left(x\\right)=2{x}^{2}[\/latex]. Vertex is on the <em>y<\/em>-axis.<\/p>\n<p>80.\u00a0Contains (-1, 4) and has the shape of [latex]f\\left(x\\right)=2{x}^{2}[\/latex]. Vertex is on the <em>y<\/em>-axis.<\/p>\n<p>81. Contains (2, 3) and has the shape of [latex]f\\left(x\\right)=3{x}^{2}[\/latex]. Vertex is on the\u00a0<em>y<\/em>-axis.<\/p>\n<p>82.\u00a0Contains (1, \u20133) and has the shape of [latex]f\\left(x\\right)=-{x}^{2}[\/latex]. Vertex is on the\u00a0<em>y<\/em>-axis.<\/p>\n<p>83. Contains (4, 3) and has the shape of [latex]f\\left(x\\right)=5{x}^{2}[\/latex]. Vertex is on the\u00a0<em>y<\/em>-axis.<\/p>\n<p>84.\u00a0Contains (1, \u20136) has the shape of [latex]f\\left(x\\right)=3{x}^{2}[\/latex]. Vertex has <em>x<\/em>-coordinate of \u20131.<\/p>\n<p>85.\u00a0Find the dimensions of the rectangular corral producing the greatest enclosed area given 200 feet of fencing.<\/p>\n<p>86.\u00a0Find the dimensions of the rectangular corral split into 2 pens of the same size producing the greatest possible enclosed area given 300 feet of fencing.<\/p>\n<p>87. Find the dimensions of the rectangular corral producing the greatest enclosed area split into 3 pens of the same size given 500 feet of fencing.<\/p>\n<p>88.\u00a0Among all of the pairs of numbers whose sum is 6, find the pair with the largest product. What is the product?<\/p>\n<p>89. Among all of the pairs of numbers whose difference is 12, find the pair with the smallest product. What is the product?<\/p>\n<p>90.\u00a0Suppose that the price per unit in dollars of a cell phone production is modeled by [latex]p=$45 - 0.0125x[\/latex], where <em>x<\/em>\u00a0is in thousands of phones produced, and the revenue represented by thousands of dollars is [latex]R=x\\cdot p[\/latex]. Find the production level that will maximize revenue.<\/p>\n<p>91. 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\\left(t\\right)=-4.9{t}^{2}+229t+234[\/latex]. Find the maximum height the rocket attains.<\/p>\n<p>92.\u00a0A 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\\left(t\\right)=-4.9{t}^{2}+24t+8[\/latex]. How long does it take to reach maximum height?<\/p>\n<p>93. A soccer stadium 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<p>94. 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\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-13836\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":97803,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-13836","chapter","type-chapter","status-publish","hentry"],"part":10733,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters\/13836","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/users\/97803"}],"version-history":[{"count":12,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters\/13836\/revisions"}],"predecessor-version":[{"id":17571,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters\/13836\/revisions\/17571"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/parts\/10733"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapters\/13836\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/media?parent=13836"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=13836"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/contributor?post=13836"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/wp-json\/wp\/v2\/license?post=13836"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}