{"id":2319,"date":"2018-07-24T18:05:56","date_gmt":"2018-07-24T18:05:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/?post_type=chapter&#038;p=2319"},"modified":"2018-07-24T18:05:56","modified_gmt":"2018-07-24T18:05:56","slug":"quadratic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/chapter\/quadratic-functions\/","title":{"raw":"Quadratic Functions","rendered":"Quadratic Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Recognize characteristics of parabolas.<\/li>\r\n \t<li>Understand how the graph of a parabola is related to its quadratic function.<\/li>\r\n \t<li>Determine a quadratic function\u2019s minimum or maximum value.<\/li>\r\n \t<li>Solve problems involving a quadratic function\u2019s minimum or maximum value.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div 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\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135358\/CNX_Precalc_Figure_03_02_001.jpg\" alt=\"Satellite dishes.\" width=\"731\" height=\"368\" \/> <strong>Figure 1. <\/strong>An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr)[\/caption]\r\n\r\n<div class=\"wp-caption-text\"><\/div>\r\n<\/div>\r\n<p id=\"fs-id1165134339909\">Curved antennas, such as the ones shown in <a class=\"autogenerated-content\" href=\"#Figure_03_02_001\">(Figure)<\/a>, 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<div id=\"fs-id1165137762207\" class=\"bc-section section\">\r\n<h3>Recognizing Characteristics of Parabolas<\/h3>\r\n<p id=\"fs-id1165137727999\">The graph of a quadratic function is a U-shaped curve called a <span class=\"no-emphasis\">parabola<\/span>. One important feature of the graph is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph, or the <span class=\"no-emphasis\">minimum value<\/span> of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the <span class=\"no-emphasis\">maximum value<\/span>. In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the axis of symmetry. These features are illustrated in <a class=\"autogenerated-content\" href=\"#Figure_03_02_002\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_03_02_002\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135405\/CNX_Precalc_Figure_03_02_002.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are.\" width=\"487\" height=\"480\" \/> <strong>Figure 2.<\/strong>[\/caption]\r\n\r\n<\/div>\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 zeros<strong>, <\/strong>or roots, of the quadratic function, the values of[latex]\\,x\\,[\/latex]at which[latex]\\,y=0.[\/latex]<\/p>\r\n\r\n<div id=\"Example_03_02_01\" class=\"textbox examples\">\r\n<div id=\"fs-id1165131959514\">\r\n<div id=\"fs-id1165135541748\">\r\n<h3>Identifying the Characteristics of a Parabola<\/h3>\r\n<p id=\"fs-id1165135366534\">Determine the vertex, axis of symmetry, zeros, and[latex]\\,y\\text{-}[\/latex]intercept of the parabola shown in <a class=\"autogenerated-content\" href=\"#Figure_03_02_003\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_03_02_003\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135407\/CNX_Precalc_Figure_03_02_003.jpg\" alt=\"Graph of a parabola with a vertex at (3, 1) and a y-intercept at (0, 7).\" width=\"487\" height=\"517\" \/> <strong>Figure 3.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137892270\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137892270\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137892270\"]\r\n<p id=\"fs-id1165137695151\">The vertex is the turning point of the graph. We can see that the vertex is at[latex]\\,\\left(3,1\\right).\\,[\/latex]Because this parabola opens upward, the axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is[latex]\\,x=3.\\,[\/latex]This parabola does not cross the[latex]\\,x\\text{-}[\/latex]axis, so it has no zeros. It crosses the[latex]\\,y\\text{-}[\/latex]axis at[latex]\\,\\left(0,7\\right)\\,[\/latex]so this is the <em>y<\/em>-intercept.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137641326\" class=\"bc-section section\">\r\n<h3>Understanding How the Graphs of Parabolas are Related to Their Quadratic Functions<\/h3>\r\n<p id=\"fs-id1165137652877\">The general form <strong>of a quadratic function <\/strong>presents the function in the form<\/p>\r\n\r\n<div id=\"fs-id1165137422466\">[latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex]<\/div>\r\n<p id=\"fs-id1165137544673\">where[latex]\\,a,b,\\,[\/latex]and[latex]\\,c\\,[\/latex]are 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\u00b1\\sqrt{{b}^{2}-4ac}}{2a},\\,[\/latex]to solve[latex]\\,a{x}^{2}+bx+c=0\\,[\/latex]for the[latex]\\,x\\text{-}[\/latex]intercepts, or zeros, we find the value of[latex]\\,x\\,[\/latex]halfway between them is always[latex]\\,x=-\\frac{b}{2a},\\,[\/latex]the equation for the axis of symmetry.<\/p>\r\n<p id=\"fs-id1165135190920\"><a class=\"autogenerated-content\" href=\"#Figure_03_02_004\">(Figure)<\/a> represents the graph of the quadratic function written in general form as[latex]\\,y={x}^{2}+4x+3.\\,[\/latex]In this form,[latex]\\,a=1,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[latex]\\,x\\text{-}[\/latex]intercepts, those points where the parabola crosses the[latex]\\,x\\text{-}[\/latex]axis, occur at[latex]\\,\\left(-3,0\\right)\\,[\/latex]and[latex]\\,\\left(-1,0\\right).[\/latex]<\/p>\r\n\r\n<div id=\"Figure_03_02_004\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135417\/CNX_Precalc_Figure_03_02_004.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\" \/> <strong>Figure 4.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137676320\">The <strong>standard form of a quadratic function<\/strong> presents the function in the form<\/p>\r\n\r\n<div id=\"fs-id1165135320100\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/div>\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 vertex form of a quadratic function.<\/p>\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. <a class=\"autogenerated-content\" href=\"#Figure_03_02_005\">(Figure)<\/a> represents the graph of the quadratic function written in standard form as[latex]\\,y=-3{\\left(x+2\\right)}^{2}+4.\\,[\/latex]Since[latex]\\,x\u2013h=x+2\\,[\/latex]in this example,[latex]\\,h=\u20132.\\,[\/latex]In this form,[latex]\\,a=-3,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]<\/p>\r\n\r\n<div id=\"Figure_03_02_005\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135423\/CNX_Precalc_Figure_03_02_005.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\" \/> <strong>Figure 5.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137453489\">The standard form is useful for determining how the graph is transformed from the graph of[latex]\\,y={x}^{2}.\\,[\/latex]<a class=\"autogenerated-content\" href=\"#Figure_03_02_006\">(Figure)<\/a> is the graph of this basic function.<\/p>\r\n\r\n<div id=\"Figure_03_02_006\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135436\/CNX_Precalc_Figure_03_02_006.jpg\" alt=\"Graph of y=x^2.\" width=\"487\" height=\"480\" \/> <strong>Figure 6.<\/strong>[\/caption]\r\n\r\n<\/div>\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 <a class=\"autogenerated-content\" href=\"#Figure_03_02_005\">(Figure)<\/a>,[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 <a class=\"autogenerated-content\" href=\"#Figure_03_02_005\">(Figure)<\/a>,[latex]\\,h&lt;0,\\,[\/latex]so the graph is shifted 2 units to the left. The magnitude of[latex]\\,a\\,[\/latex]indicates the stretch of the graph. If[latex]|a|&gt;1,[\/latex] the point associated with a particular[latex]\\,x\\text{-}[\/latex]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[latex]\\,x\\text{-}[\/latex]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 <a class=\"autogenerated-content\" href=\"#Figure_03_02_005\">(Figure)<\/a>,[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=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill a{\\left(x-h\\right)}^{2}+k&amp; =&amp; a{x}^{2}+bx+c\\hfill \\\\ \\hfill a{x}^{2}-2ahx+\\left(a{h}^{2}+k\\right)&amp; =&amp; a{x}^{2}+bx+c\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137409211\">For the linear terms to be equal, the coefficients must be equal.<\/p>\r\n\r\n<div id=\"eip-144\" class=\"unnumbered aligncenter\">[latex]\u20132ah=b,\\text{ so }h=-\\frac{b}{2a}[\/latex]<\/div>\r\n<p id=\"fs-id1165134118295\">This is the <span class=\"no-emphasis\">axis of symmetry<\/span> we defined earlier. Setting the constant terms equal:<\/p>\r\n\r\n<div id=\"eip-313\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill a{h}^{2}+k&amp; =&amp; c\\hfill \\\\ \\hfill k&amp; =&amp; c-a{h}^{2}\\hfill \\\\ &amp; =&amp; c-a-{\\left(\\frac{b}{2a}\\right)}^{2}\\hfill \\\\ &amp; =&amp; c-\\frac{{b}^{2}}{4a}\\hfill \\end{array}[\/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[latex]\\,h,\\,[\/latex]so[latex]\\,f\\left(h\\right)=k.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1165137749882\" class=\"textbox key-takeaways\">\r\n<h3>Forms of Quadratic Functions<\/h3>\r\n<p id=\"fs-id1165135333154\">A quadratic function is a polynomial function of degree two. The graph of a <span class=\"no-emphasis\">quadratic function<\/span> is a parabola.<\/p>\r\n<p id=\"eip-103\">The general form of a quadratic function is[latex]\\,f\\left(x\\right)=a{x}^{2}+bx+c\\,[\/latex]where[latex]\\,a,b,\\,[\/latex]and[latex]\\,c\\,[\/latex]are real numbers and[latex]\\,a\\ne 0.[\/latex]<\/p>\r\n<p id=\"fs-id1165137666538\">The standard form of a quadratic function is[latex]\\,f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\,[\/latex]where[latex]\\,a\\ne 0.[\/latex]<\/p>\r\n<p id=\"fs-id1165137762385\">The vertex[latex]\\,\\left(h,k\\right)\\,[\/latex]is located at<\/p>\r\n\r\n<div id=\"eip-301\" class=\"unnumbered aligncenter\">[latex]h=\u2013\\frac{b}{2a},\\text{ }k=f\\left(h\\right)=f\\left(\\frac{-b}{2a}\\right)[\/latex]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165131886746\" class=\"precalculus howto textbox tryit\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137650986\"><strong>Given a graph of a quadratic function, write the equation of the function in general form.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165134223276\" type=\"1\">\r\n \t<li>Identify the horizontal shift of the parabola; this value is[latex]\\,h.\\,[\/latex]Identify the vertical shift of the parabola; this value is[latex]\\,k.[\/latex]<\/li>\r\n \t<li>Substitute the values of the horizontal and vertical shift for[latex]\\,h\\,[\/latex]and[latex]\\,k.\\,[\/latex]in the function[latex]\\,f\\left(x\\right)=a{\\left(x\u2013h\\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[latex]\\,x\\,[\/latex]and[latex]\\,f\\left(x\\right).[\/latex]<\/li>\r\n \t<li>Solve for the stretch factor,[latex]\\,|a|.[\/latex]<\/li>\r\n \t<li>Expand and simplify to write in general form.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_02_02\" class=\"textbox examples\">\r\n<div id=\"fs-id1165135460939\">\r\n<div id=\"fs-id1165135460941\">\r\n<h3>Writing the Equation of a Quadratic Function from the Graph<\/h3>\r\n<p id=\"fs-id1165135532321\">Write an equation for the quadratic function[latex]\\,g\\,[\/latex]in <a class=\"autogenerated-content\" href=\"#Figure_03_02_007\">(Figure)<\/a> as 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.<\/p>\r\n\r\n<div id=\"Figure_03_02_007\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135457\/CNX_Precalc_Figure_03_02_007.jpg\" alt=\"Graph of a parabola with its vertex at (-2, -3).\" width=\"487\" height=\"443\" \/> <strong>Figure 7.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134211341\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134211341\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134211341\"]\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-\\left(-2\\right)\\right)}^{2}-3=a{\\left(x+2\\right)}^{2}\u20133.[\/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\r\n<div id=\"eip-id1165134221671\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill -1&amp; =&amp; a{\\left(0+2\\right)}^{2}-3\\hfill \\\\ \\hfill 2&amp; =&amp; 4a\\hfill \\\\ \\hfill a&amp; =&amp; \\frac{1}{2}\\hfill \\end{array}[\/latex]<\/div>\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}\u20133.[\/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\r\n<div id=\"eip-id1165137463836\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill g\\left(x\\right)&amp; =&amp; \\frac{1}{2}{\\left(x+2\\right)}^{2}-3\\hfill \\\\ &amp; =&amp; \\frac{1}{2}\\left(x+2\\right)\\left(x+2\\right)-3\\hfill \\\\ &amp; =&amp; \\frac{1}{2}\\left({x}^{2}+4x+4\\right)-3\\hfill \\\\ &amp; =&amp; \\frac{1}{2}{x}^{2}+2x+2-3\\hfill \\\\ &amp; =&amp; \\frac{1}{2}{x}^{2}+2x-1\\hfill \\end{array}[\/latex]<\/div>\r\nNotice that the horizontal and vertical shifts of the basic graph of the quadratic function determine the location of the vertex of the parabola; the vertex is unaffected by stretches and compressions.[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137838619\">\r\n<h4>Analysis<\/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}=\u20136\\,[\/latex]and[latex]\\,\\Delta \\text{Tbl = 2,}\\,[\/latex]and select[latex]\\,\\text{TABLE}\\text{.}\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Table_03_02_01\">(Figure)<\/a>.<\/p>\r\n\r\n<table id=\"Table_03_02_01\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/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><strong>[latex]y[\/latex]<\/strong><\/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\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137527658\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"ti_03_02_01\">\r\n<div id=\"fs-id1165137933940\">\r\n<p id=\"fs-id1165137933941\">A coordinate grid has been superimposed over the quadratic path of a basketball in <a class=\"autogenerated-content\" href=\"#Figure_03_02_008\">(Figure)<\/a>. Find an equation for the path of the ball. Does the shooter make the basket?<\/p>\r\n\r\n<div id=\"Figure_03_02_008\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135459\/CNX_Precalc_Figure_03_02_008.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\" \/> <strong>Figure 8. <\/strong>(credit: modification of work by Dan Meyer)[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135414238\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135414238\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135414238\"]\r\n<p id=\"fs-id1165135414239\">The path passes through the origin and has vertex at[latex]\\,\\left(-4,\\text{ }7\\right),\\,[\/latex]so[latex]\\,\\left(h\\right)x=\u2013\\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(\u20137.5\\right)\\approx 1.64;\\,[\/latex]he doesn\u2019t make it.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135168275\" class=\"precalculus howto textbox tryit\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165135574310\"><strong>Given a quadratic function in general form, find the vertex of the parabola.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165134108459\" type=\"1\">\r\n \t<li>Identify[latex]\\,a, b, \\text{and} c.[\/latex]<\/li>\r\n \t<li>Find[latex]\\,h,\\,[\/latex]the <em>x<\/em>-coordinate of the vertex, by substituting[latex]\\,a\\,[\/latex]and[latex]\\,b\\,[\/latex]into[latex]\\,h=\u2013\\frac{b}{2a}.[\/latex]<\/li>\r\n \t<li>Find[latex]\\,k,\\,[\/latex]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=\"textbox examples\">\r\n<div id=\"fs-id1165137658566\">\r\n<div id=\"fs-id1165137771901\">\r\n<h3>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}\u20136x+7.\\,[\/latex]Rewrite the quadratic in standard form (vertex form).<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137596321\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137596321\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137596321\"]\r\n<p id=\"fs-id1165137596323\">[latex]\\begin{array}{c}\\text{The horizontal coordinate of the vertex will be at}\\hfill \\\\ &amp; \\hfill h&amp; =&amp; -\\frac{b}{2a}\\hfill \\\\ &amp; &amp; =&amp; -\\frac{-6}{2\\left(2\\right)}\\hfill \\\\ &amp; &amp; =&amp; \\frac{6}{4}\\hfill \\\\ &amp; &amp; =&amp; \\frac{3}{2}\\hfill \\\\ \\text{The vertical coordinate of the vertex will be at}\\hfill \\\\ &amp; \\hfill k&amp; =&amp; f\\left(h\\right)\\hfill \\\\ &amp; &amp; =&amp; f\\left(\\frac{3}{2}\\right)\\hfill \\\\ &amp; &amp; =&amp; 2{\\left(\\frac{3}{2}\\right)}^{2}-6\\left(\\frac{3}{2}\\right)+7\\hfill \\\\ &amp; &amp; =&amp; \\frac{5}{2}\\hfill \\end{array}[\/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. First, find the horizontal coordinate of the vertex. Then find the vertical coordinate of the vertex. Substitute the values into standard form, using the \"[latex]a[\/latex]\" from the general form.<\/p>\r\n\r\n<div id=\"eip-id1165135499318\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill f\\left(x\\right)&amp; =&amp; a{x}^{2}+bx+c\\hfill \\\\ \\hfill f\\left(x\\right)&amp; =&amp; 2{x}^{2}-6x+7\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137653186\">The standard form of a quadratic function prior to writing the function then becomes the following:<\/p>\r\n\r\n<div id=\"eip-id1165134389821\" class=\"unnumbered\">[latex]f\\left(x\\right)=2{\\left(x\u2013\\frac{3}{2}\\right)}^{2}+\\frac{5}{2}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137591920\">\r\n<h4>Analysis<\/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,[latex]\\,k,[\/latex]and where it occurs,[latex]\\,x.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135362470\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"ti_03_02_02\">\r\n<div id=\"fs-id1165135193261\">\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\r\n<\/div>\r\n<div id=\"fs-id1165137638479\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137638479\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137638479\"]\r\n<p id=\"fs-id1165137638480\">[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>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133436210\" class=\"bc-section section\">\r\n<h3>Finding the Domain and Range of a Quadratic Function<\/h3>\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=\"textbox key-takeaways\">\r\n<h3>Domain and Range of a Quadratic Function<\/h3>\r\n<p id=\"fs-id1165135502927\">The domain of any <span class=\"no-emphasis\">quadratic function<\/span> is all real numbers unless the context of the function presents some restrictions.<\/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[latex]\\,a\\,[\/latex]value 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[latex]\\,a\\,[\/latex]value 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[latex]\\,a\\,[\/latex]value is[latex]\\,f\\left(x\\right)\\ge k;\\,[\/latex]the range of a quadratic function written in standard form with a negative[latex]\\,a\\,[\/latex]value is[latex]\\,f\\left(x\\right)\\le k.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135205144\" class=\"precalculus howto textbox tryit\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137409165\"><strong>Given a quadratic function, find the domain and range.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165137843779\" type=\"1\">\r\n \t<li>Identify the domain of any quadratic function as all real numbers.<\/li>\r\n \t<li>Determine whether[latex]\\,a\\,[\/latex]is positive or negative. If[latex]\\,a\\,[\/latex]is positive, the parabola has a minimum. If[latex]\\,a\\,[\/latex]is negative, the parabola has a maximum.<\/li>\r\n \t<li>Determine the maximum or minimum value of the parabola,[latex]\\,k.[\/latex]<\/li>\r\n \t<li>If the parabola has a minimum, the range is given by[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=\"textbox examples\">\r\n<div id=\"fs-id1165134257627\">\r\n<div id=\"fs-id1165134257629\">\r\n<h3>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\r\n<\/div>\r\n<div id=\"fs-id1165137837922\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137837922\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137837922\"]\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[latex]\\,a\\,[\/latex]is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding the[latex]\\,x\\text{-}[\/latex]value of the vertex.<\/p>\r\n\r\n<div id=\"eip-id1165132986104\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill h&amp; =&amp; -\\frac{b}{2a}\\hfill \\\\ \\hfill &amp; =&amp; -\\frac{9}{2\\left(-5\\right)}\\hfill \\\\ \\hfill &amp; =&amp; \\frac{9}{10}\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137736576\">The maximum value is given by[latex]\\,f\\left(h\\right).[\/latex]<\/p>\r\n\r\n<div id=\"eip-id1165135687688\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill f\\left(\\frac{9}{10}\\right)&amp; =&amp; -5{\\left(\\frac{9}{10}\\right)}^{2}+9\\left(\\frac{9}{10}\\right)-1\\hfill \\\\ &amp; =&amp; \\frac{61}{20}\\hfill \\end{array}[\/latex]<\/div>\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][\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133227752\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"ti_03_02_03\">\r\n<div id=\"fs-id1165135424648\">\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\r\n<\/div>\r\n<div id=\"fs-id1165137592393\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137592393\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137592393\"]\r\n<p id=\"fs-id1165137812604\">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>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137736541\" class=\"bc-section section\">\r\n<h3>Determining the Maximum and Minimum Values of Quadratic Functions<\/h3>\r\n<p id=\"fs-id1165137442167\">The output of the quadratic function at the vertex is the maximum or minimum value of the function, depending on the orientation of the <span class=\"no-emphasis\">parabola<\/span>. We can see the maximum and minimum values in <a class=\"autogenerated-content\" href=\"#Figure_03_02_009\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_03_02_009\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135506\/CNX_Precalc_Figure_03_02_009.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\" \/> <strong>Figure 9.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137431411\">There are many real-world scenarios that involve finding the maximum or minimum value of a quadratic function, such as applications involving area and revenue.<\/p>\r\n\r\n<div id=\"Example_03_02_05\" class=\"textbox examples\">\r\n<div id=\"fs-id1165134378616\">\r\n<div id=\"fs-id1165134378618\">\r\n<h3>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\" type=\"a\">\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[latex]\\,L.[\/latex]<\/li>\r\n \t<li>What dimensions should she make her garden to maximize the enclosed area?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1165137836806\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137836806\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137836806\"]\r\n<p id=\"fs-id1165137836808\">Let\u2019s use a diagram such as <a class=\"autogenerated-content\" href=\"#Figure_03_02_010\">(Figure)<\/a> to record the given information. It is also helpful to introduce a temporary variable,[latex]\\,W,\\,[\/latex]to represent the width of the garden and the length of the fence section parallel to the backyard fence.<\/p>\r\n\r\n<div id=\"Figure_03_02_010\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135510\/CNX_Precalc_Figure_03_02_010.jpg\" alt=\"Diagram of the garden and the backyard.\" width=\"487\" height=\"310\" \/> <strong>Figure 10.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<ol id=\"fs-id1165134363440\" type=\"a\">\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,[latex]\\,W,\\,[\/latex]in terms of[latex]\\,L.[\/latex]\r\n<div id=\"eip-id1165135697866\" class=\"unnumbered\">[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\r\n<div id=\"eip-624\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill A&amp; =&amp; LW=L\\left(80-2L\\right)\\hfill \\\\ \\hfill A\\left(L\\right)&amp; =&amp; 80L-2{L}^{2}\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165135258914\">This formula represents the area of the fence in terms of the variable length[latex]\\,L.\\,[\/latex]The function, written in general form, is<\/p>\r\n\r\n<div id=\"eip-382\" class=\"unnumbered aligncenter\">[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[latex]\\,a\\,[\/latex]is 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\r\n<div id=\"eip-id1165135202446\" class=\"unnumbered\">[latex]\\begin{array}{ccccccc}\\hfill h&amp; =&amp; -\\frac{b}{2a}\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}k&amp; =&amp; A\\left(20\\right)\\hfill \\\\ &amp; =&amp; -\\frac{80}{2\\left(-2\\right)}\\hfill &amp; \\phantom{\\rule{1em}{0ex}}\\text{and}&amp; &amp; =&amp; 80\\left(20\\right)-2{\\left(20\\right)}^{2}\\hfill \\\\ &amp; =&amp; 20\\hfill &amp; &amp; &amp; =&amp; 800\\hfill \\end{array}[\/latex]<\/div>\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[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135582226\">\r\n<h4>Analysis<\/h4>\r\n<p id=\"fs-id1165135582232\">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 <a class=\"autogenerated-content\" href=\"#Figure_03_02_011\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_03_02_011\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135518\/CNX_Precalc_Figure_03_02_011.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\" \/> <strong>Figure 11.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133340409\" class=\"precalculus howto textbox tryit\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165137803708\"><strong>Given an application involving revenue, use a quadratic equation to find the maximum.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165135436584\" type=\"1\">\r\n \t<li>Write a quadratic equation for a revenue function.<\/li>\r\n \t<li>Find the vertex of the quadratic equation.<\/li>\r\n \t<li>Determine the <em>y<\/em>-value of the vertex.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_02_06\" class=\"textbox examples\">\r\n<div id=\"fs-id1165134278696\">\r\n<div id=\"fs-id1165137473136\">\r\n<h3>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\r\n<\/div>\r\n<div id=\"fs-id1165135389886\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135389886\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135389886\"]\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,[latex]\\,p\\,[\/latex]for price per subscription and[latex]\\,Q\\,[\/latex]for 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\r\n<div id=\"eip-id1165135246622\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill m&amp; =&amp; \\frac{79,000-84,000}{32-30}\\hfill \\\\ &amp; =&amp; \\frac{-5,000}{2}\\hfill \\\\ &amp; =&amp; -2,500\\hfill \\end{array}[\/latex]<\/div>\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\r\n<div id=\"eip-id1165131968004\" class=\"unnumbered\">[latex]\\begin{array}{cccc}\\hfill Q&amp; =&amp; -2500p+b\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Substitute in the point}Q=84,000\\text{ and }p=30\\hfill \\\\ \\hfill 84,000&amp; =&amp; -2500\\left(30\\right)+b\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Solve for}b\\hfill \\\\ \\hfill b&amp; =&amp; 159,000\\hfill &amp; \\end{array}[\/latex]<\/div>\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\r\n<div id=\"eip-id1165132337192\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{Revenue}&amp; =&amp; pQ\\hfill \\\\ \\hfill \\mathrm{Revenue}&amp; =&amp; p\\left(-2,500p+159,000\\right)\\hfill \\\\ \\hfill \\mathrm{Revenue}&amp; =&amp; -2,500{p}^{2}+159,000p\\hfill \\end{array}[\/latex]<\/div>\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\r\n<div id=\"eip-id1165135170999\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill h&amp; =&amp; -\\frac{159,000}{2\\left(-2,500\\right)}\\hfill \\\\ &amp; =&amp; 31.8\\hfill \\end{array}[\/latex]<\/div>\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\r\n<div id=\"eip-id1165134323486\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill \\text{maximum revenue}&amp; =&amp; -2,500{\\left(31.8\\right)}^{2}+159,000\\left(31.8\\right)\\hfill \\\\ &amp; =&amp; 2,528,100\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135538766\">\r\n<h4>Analysis<\/h4>\r\n<p id=\"fs-id1165135538771\">This could also be solved by graphing the quadratic as in <a class=\"autogenerated-content\" href=\"#Figure_03_02_012\">(Figure)<\/a>. We can see the maximum revenue on a graph of the quadratic function.<\/p>\r\n\r\n<div id=\"Figure_03_02_012\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135523\/CNX_Precalc_Figure_03_02_012.jpg\" alt=\"Graph of the parabolic function which the x-axis is labeled Price (p) and the y-axis is labeled Revenue (\ud83d\udcb2). The vertex is at (31.80, 258100).\" width=\"487\" height=\"327\" \/> <strong>Figure 12.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135693703\" class=\"bc-section section\">\r\n<h4>Finding the <em>x<\/em>- and <em>y<\/em>-Intercepts of a Quadratic Function<\/h4>\r\n<p id=\"fs-id1165134569121\">Much as we did in the application problems above, we also need to find intercepts of quadratic equations for graphing parabolas. Recall that we find the[latex]\\,y\\text{-}[\/latex]intercept of a quadratic by evaluating the function at an input of zero, and we find the[latex]\\,x\\text{-}[\/latex]intercepts at locations where the output is zero. Notice in <a class=\"autogenerated-content\" href=\"#Figure_03_02_013\">(Figure)<\/a> that the number of[latex]\\,x\\text{-}[\/latex]intercepts can vary depending upon the location of the graph.<\/p>\r\n\r\n<div id=\"Figure_03_02_013\" class=\"wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135526\/CNX_Precalc_Figure_03_02_013.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\" \/> <strong>Figure 13. <\/strong>Number of x-intercepts of a parabola[\/caption]\r\n\r\n<div id=\"fs-id1165137464602\" class=\"precalculus howto textbox tryit\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165135638554\"><strong>Given a quadratic function[latex]\\,f\\left(x\\right),\\,[\/latex]find the[latex]\\,y\\text{-}[\/latex] and <em>x<\/em>-intercepts.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1165135378765\" type=\"1\">\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=\"textbox examples\">\r\n<div id=\"fs-id1165134129944\">\r\n<div id=\"fs-id1165134129946\">\r\n<h3>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\r\n<\/div>\r\n<div id=\"fs-id1165137901093\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137901093\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137901093\"]\r\n<p id=\"fs-id1165137901096\">We find the <em>y<\/em>-intercept by evaluating[latex]\\,f\\left(0\\right).[\/latex]<\/p>\r\n\r\n<div id=\"eip-id1165133349374\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill f\\left(0\\right)&amp; =&amp; 3{\\left(0\\right)}^{2}+5\\left(0\\right)-2\\hfill \\\\ &amp; =&amp; -2\\hfill \\end{array}[\/latex]<\/div>\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\r\n<div id=\"eip-id1165132926414\" class=\"unnumbered\">[latex]0=3{x}^{2}+5x-2[\/latex]<\/div>\r\n<p id=\"fs-id1165135690677\">In this case, the quadratic can be factored easily, providing the simplest method for solution.<\/p>\r\n\r\n<div id=\"eip-id1165135321232\" class=\"unnumbered\">[latex]0=\\left(3x-1\\right)\\left(x+2\\right)[\/latex]<\/div>\r\n<div id=\"eip-id1165134586903\" class=\"unnumbered\">[latex]\\begin{array}{cccccc}\\hfill h&amp; =&amp; -\\frac{b}{2a}&amp; \\hfill \\phantom{\\rule{2em}{0ex}}k&amp; =&amp; f\\left(-1\\right)\\hfill \\\\ &amp; =&amp; -\\frac{4}{2\\left(2\\right)}\\hfill &amp; &amp; =&amp; \\hfill 2{\\left(-1\\right)}^{2}+4\\left(-1\\right)-4\\\\ &amp; =&amp; -1\\hfill &amp; &amp; =&amp; -6\\hfill \\end{array}[\/latex]<\/div>\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[\/hidden-answer]\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137911614\">\r\n<h4>Analysis<\/h4>\r\n<p id=\"fs-id1165134170695\">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]See <a class=\"autogenerated-content\" href=\"#Figure_03_02_014\">(Figure)<\/a><\/p>\r\n\r\n<div id=\"Figure_03_02_014\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135532\/CNX_Precalc_Figure_03_02_014.jpg\" alt=\"Graph of a parabola which has the following intercepts (-2, 0), (1\/3, 0), and (0, -2).\" width=\"487\" height=\"480\" \/> <strong>Figure 14.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135381309\" class=\"bc-section section\">\r\n<h4>Rewriting Quadratics in Standard Form<\/h4>\r\n<p id=\"fs-id1165135381314\">In <a class=\"autogenerated-content\" href=\"#Example_03_02_07\">(Figure)<\/a>, 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=\"precalculus howto textbox tryit\">\r\n<h3>How To<\/h3>\r\n<p id=\"fs-id1165135694488\"><strong>Given a quadratic function, find the[latex]\\,x\\text{-}[\/latex]intercepts by rewriting in standard form<\/strong>.<\/p>\r\n\r\n<ol id=\"fs-id1165134113976\" type=\"1\">\r\n \t<li>Substitute[latex]\\,a\\,[\/latex]and[latex]\\,b\\,[\/latex]into[latex]\\,h=-\\frac{b}{2a}.[\/latex]<\/li>\r\n \t<li>Substitute[latex]\\,x=h\\,[\/latex]into the general form of the quadratic function to find[latex]\\,k.[\/latex]<\/li>\r\n \t<li>Rewrite the quadratic in standard form using[latex]\\,h\\,[\/latex]and[latex]\\,k.[\/latex]<\/li>\r\n \t<li>Solve for when the output of the function will be zero to find the[latex]\\,x\\text{-}[\/latex]intercepts.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_03_02_08\" class=\"textbox examples\">\r\n<div id=\"fs-id1165134060458\">\r\n<div id=\"fs-id1165134224010\">\r\n<h3>Finding the <em>x<\/em>-Intercepts of a Parabola<\/h3>\r\n<p id=\"fs-id1165134224020\">Find the[latex]\\,x\\text{-}[\/latex]intercepts of the quadratic function[latex]\\,f\\left(x\\right)=2{x}^{2}+4x-4.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135524476\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135524476\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135524476\"]\r\n<p id=\"fs-id1165135524478\">We begin by solving for when the output will be zero.<\/p>\r\n\r\n<div id=\"eip-id1165135701748\" class=\"unnumbered\">[latex]0=2{x}^{2}+4x-4[\/latex]<\/div>\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\r\n<div id=\"eip-id1165132111123\" class=\"unnumbered\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/div>\r\n<p id=\"fs-id1165137925165\">We know that[latex]\\,a=2.\\,[\/latex]Then we solve for[latex]\\,h\\,[\/latex]and[latex]\\,k.[\/latex]<\/p>\r\n\r\n<div id=\"eip-id1165137883696\" class=\"unnumbered\">[latex]\\begin{array}{cccccc}\\hfill h&amp; =&amp; -\\frac{b}{2a}\\hfill &amp; \\hfill \\phantom{\\rule{2em}{0ex}}k&amp; =&amp; f\\left(-1\\right)\\hfill \\\\ &amp; =&amp; -\\frac{4}{2\\left(2\\right)}\\hfill &amp; &amp; =&amp; 2{\\left(-1\\right)}^{2}+4\\left(-1\\right)-4\\hfill \\\\ &amp; =&amp; -1\\hfill &amp; &amp; =&amp; -6\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165134031332\">So now we can rewrite in standard form.<\/p>\r\n\r\n<div id=\"eip-id1165135480971\" class=\"unnumbered\">[latex]f\\left(x\\right)=2{\\left(x+1\\right)}^{2}-6[\/latex]<\/div>\r\n<p id=\"fs-id1165135381286\">We can now solve for when the output will be zero.<\/p>\r\n\r\n<div id=\"eip-id1165131840672\" class=\"unnumbered\">[latex]\\begin{array}{l}0=2{\\left(x+1\\right)}^{2}-6\\hfill \\\\ 6=2{\\left(x+1\\right)}^{2}\\hfill \\\\ 3={\\left(x+1\\right)}^{2}\\hfill \\\\ x+1=\u00b1\\sqrt{3}\\hfill \\\\ x=-1\u00b1\\sqrt{3}\\hfill \\end{array}[\/latex]<\/div>\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<p id=\"fs-id1165137843092\">We can check our work by graphing the given function on a graphing utility and observing the[latex]\\,x\\text{-}[\/latex]intercepts. See <a class=\"autogenerated-content\" href=\"#Figure_03_02_015\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_03_02_015\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135535\/CNX_Precalc_Figure_03_02_015.jpg\" alt=\"Graph of a parabola which has the following x-intercepts (-2.732, 0) and (0.732, 0).\" width=\"487\" height=\"517\" \/> <strong>Figure 15.<\/strong>[\/caption]\r\n\r\n[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137843086\">\r\n<h4>Analysis<\/h4>\r\n<p id=\"fs-id1709228\">We could have achieved the same results using the quadratic formula. Identify[latex]\\,a=2,b=4\\,[\/latex]and[latex]\\,c=-4.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1709335\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill x&amp; =&amp; \\frac{-b\u00b1\\sqrt{{b}^{2}-4ac}}{2a}\\hfill \\\\ &amp; =&amp; \\frac{-4\u00b1\\sqrt{{4}^{2}-4\\left(2\\right)\\left(-4\\right)}}{2\\left(2\\right)}\\hfill \\\\ &amp; =&amp; \\frac{-4\u00b1\\sqrt{48}}{4}\\hfill \\\\ &amp; =&amp; \\frac{-4\u00b1\\sqrt{3\\left(16\\right)}}{4}\\hfill \\\\ &amp; =&amp; -1\u00b1\\sqrt{3}\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1728602\">So the <em>x<\/em>-intercepts occur at[latex]\\,\\left(-1-\\sqrt{3},0\\right)\\,[\/latex]and[latex]\\,\\left(-1+\\sqrt{3},0\\right).[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134109655\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"ti_03_02_04\">\r\n<div id=\"fs-id1165134534212\">\r\n<p id=\"fs-id1165134534213\">In a <a href=\"#fs-id1165135362470\">Try It<\/a>, 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\r\n<\/div>\r\n<div id=\"fs-id1165135240997\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135240997\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135240997\"]\r\n<p id=\"fs-id1165135240998\"><em>y<\/em>-intercept at (0, 13), No[latex]\\,x\\text{-}[\/latex]intercepts<\/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=\"textbox examples\">\r\n<div id=\"fs-id1165134085786\">\r\n<div id=\"fs-id1165134085788\">\r\n<h3>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\r\n<ol id=\"fs-id1165137726798\" type=\"a\">\r\n \t<li>When does the ball reach the maximum height?<\/li>\r\n \t<li>What is the maximum height of the ball?<\/li>\r\n \t<li>When does the ball hit the ground?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1165135264685\" class=\"solution textbox shaded\">\r\n\r\n[reveal-answer q=\"85834\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"85834\"]\r\n<ol id=\"fs-id1165135264687\" type=\"a\">\r\n \t<li>The ball reaches the maximum height at the vertex of the parabola.\r\n<div id=\"eip-id1165135364049\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill h&amp; =&amp; -\\frac{80}{2\\left(-16\\right)}\\hfill \\\\ &amp; =&amp; \\frac{80}{32}\\hfill \\\\ &amp; =&amp; \\frac{5}{2}\\hfill \\\\ &amp; =&amp; 2.5\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165135528870\">The ball reaches a maximum height after 2.5 seconds.<\/p>\r\n<\/li>\r\n \t<li>To find the maximum height, find the[latex]\\,y\\text{-}[\/latex]coordinate of the vertex of the parabola.\r\n<div id=\"eip-id1165134188974\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill k&amp; =&amp; H\\left(-\\frac{b}{2a}\\right)\\hfill \\\\ &amp; =&amp; H\\left(2.5\\right)\\hfill \\\\ &amp; =&amp; -16{\\left(2.5\\right)}^{2}+80\\left(2.5\\right)+40\\hfill \\\\ &amp; =&amp; 140\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165135409750\">The ball reaches a maximum height of 140 feet.<\/p>\r\n<\/li>\r\n \t<li>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<p id=\"fs-id1165135181381\">We use the quadratic formula.<\/p>\r\n\r\n<div id=\"eip-id1165133166054\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill t&amp; =&amp; \\frac{-80\u00b1\\sqrt{{80}^{2}-4\\left(-16\\right)\\left(40\\right)}}{2\\left(-16\\right)}\\hfill \\\\ &amp; =&amp; \\frac{-80\u00b1\\sqrt{8960}}{-32}\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165137783269\">Because the square root does not simplify nicely, we can use a calculator to approximate the values of the solutions.<\/p>\r\n\r\n<div id=\"eip-id1165134299806\" class=\"unnumbered\">[latex]\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ \\begin{array}{lll}t=\\frac{-80-\\sqrt{8960}}{-32}\\approx 5.458\\hfill &amp; \\text{or}\\hfill &amp; t=\\frac{-80+\\sqrt{8960}}{-32}\\approx -0.458\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165135347508\">The second answer is outside the reasonable domain of our model, so we conclude the ball will hit the ground after about 5.458 seconds. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2319&amp;action=edit#Figure_03_02_016\">(Figure)<\/a>.<\/p>\r\n\r\n<div id=\"Figure_03_02_016\" class=\"small wp-caption aligncenter\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135536\/CNX_Precalc_Figure_03_02_016.jpg\" alt=\"Graph of a negative parabola where x goes from -1 to 6.\" width=\"487\" height=\"254\" \/> <strong>Figure 16.<\/strong>[\/caption]\r\n\r\n<\/div>\r\n<p id=\"eip-id1827109\">Note that the graph does not represent the physical path of the ball upward and downward. Keep the quantities on each axis in mind while interpreting the graph.<\/p>\r\n<\/li>\r\n<\/ol>\r\n<p id=\"eip-id1827109\">[\/hidden-answer]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134081290\" class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\n<div id=\"ti_03_02_05\">\r\n<div id=\"fs-id1165134081299\">\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\r\n<ol id=\"fs-id1165135152195\" type=\"a\">\r\n \t<li>When does the rock reach the maximum height?<\/li>\r\n \t<li>What is the maximum height of the rock?<\/li>\r\n \t<li>When does the rock hit the ocean?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1165135152213\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135152213\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135152213\"]\r\n<p id=\"fs-id1165137843187\">3 seconds256 feet7 seconds<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137475325\" class=\"precalculus media\">\r\n<p id=\"fs-id1165137475332\">Access these online resources for additional instruction and practice with quadratic equations.<\/p>\r\n\r\n<ul id=\"fs-id1165137475336\">\r\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphquadgen\">Graphing Quadratic Functions in General Form<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphquadstan\">Graphing Quadratic Functions in Standard Form<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/quadfuncrev\">Quadratic Function Review<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/characterquad\">Characteristics of a Quadratic Function<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134205927\" class=\"key-equations\">\r\n<h3>Key Equations<\/h3>\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>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<\/div>\r\n<div id=\"fs-id1165135426424\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\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[latex]\\,x\\text{-}[\/latex]intercepts, are the points at which the parabola crosses the[latex]\\,x\\text{-}[\/latex]axis. The[latex]\\,y\\text{-}[\/latex]intercept is the point at which the parabola crosses the[latex]\\,y\\text{-}[\/latex]axis. See <a class=\"autogenerated-content\" href=\"#Example_03_02_01\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_03_02_07\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_03_02_08\">(Figure)<\/a>.<\/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. See <a class=\"autogenerated-content\" href=\"#Example_03_02_02\">(Figure)<\/a>.<\/li>\r\n \t<li>The vertex can be found from an equation representing a quadratic function. See <a class=\"autogenerated-content\" href=\"#Example_03_02_03\">(Figure)<\/a><strong>.<\/strong><\/li>\r\n \t<li>The domain of a quadratic function is all real numbers. The range varies with the function. See <a class=\"autogenerated-content\" href=\"#Example_03_02_04\">(Figure)<\/a>.<\/li>\r\n \t<li>A quadratic function\u2019s minimum or maximum value is given by the[latex]\\,y\\text{-}[\/latex]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. See <a class=\"autogenerated-content\" href=\"#Example_03_02_05\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_03_02_06\">(Figure)<\/a>.<\/li>\r\n \t<li>The vertex and the intercepts can be identified and interpreted to solve real-world problems. See <a class=\"autogenerated-content\" href=\"#Example_03_02_10\">(Figure)<\/a>.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165135361324\" class=\"textbox exercises\">\r\n<h3>Section Exercises<\/h3>\r\n<div id=\"fs-id1165135361327\" class=\"bc-section section\">\r\n<h4>Verbal<\/h4>\r\n<div id=\"fs-id1165135361332\">\r\n<div id=\"fs-id1165135361334\">\r\n<p id=\"fs-id1165135361336\">Explain the advantage of writing a quadratic function in standard form.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135361339\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135361339\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135361339\"]\r\n<p id=\"fs-id1165135361340\">When written in that form, the vertex can be easily identified.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134367971\">\r\n<div id=\"fs-id1165134367972\">\r\n<p id=\"fs-id1165134367973\">How can the vertex of a parabola be used in solving real-world problems?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134367976\">\r\n<div id=\"fs-id1165134367977\">\r\n<p id=\"fs-id1165134367978\">Explain why the condition of[latex]\\,a\\ne 0\\,[\/latex]is imposed in the definition of the quadratic function.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135453265\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135453265\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135453265\"]\r\n<p id=\"fs-id1165135453266\">If[latex]\\,a=0\\,[\/latex]then the function becomes a linear function.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134094470\">\r\n<div id=\"fs-id1165134094472\">\r\n<p id=\"fs-id1165134094473\">What is another name for the standard form of a quadratic function?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134094476\">\r\n<div id=\"fs-id1165134094477\">\r\n<p id=\"fs-id1165134094478\">What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165133276250\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133276250\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133276250\"]\r\n<p id=\"fs-id1165133276252\">If possible, we can use factoring. Otherwise, we can use the quadratic formula.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133276257\" class=\"bc-section section\">\r\n<h4>Algebraic<\/h4>\r\n<p id=\"fs-id1165133276262\">For the following exercises, rewrite the quadratic functions in standard form and give the vertex.<\/p>\r\n\r\n<div id=\"fs-id1165133276266\">\r\n<div id=\"fs-id1165133276268\">\r\n<p id=\"fs-id1165133276271\">[latex]f\\left(x\\right)={x}^{2}-12x+32[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135336013\">\r\n<div id=\"fs-id1165133065125\">\r\n<p id=\"fs-id1165133065126\">[latex]g\\left(x\\right)={x}^{2}+2x-3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165132963686\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165132963686\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165132963686\"]\r\n<p id=\"fs-id1165132963687\">[latex]f\\left(x\\right)={\\left(x+1\\right)}^{2}-2,\\,[\/latex]Vertex[latex]\\,\\left(-1,-4\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134199469\">\r\n<div id=\"fs-id1165134199470\">\r\n<p id=\"fs-id1165134199472\">[latex]f\\left(x\\right)={x}^{2}-x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135536371\">\r\n<div id=\"fs-id1165135536373\">\r\n<p id=\"fs-id1165135536374\">[latex]f\\left(x\\right)={x}^{2}+5x-2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135584090\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135584090\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135584090\"]\r\n<p id=\"fs-id1165135584091\">[latex]f\\left(x\\right)={\\left(x+\\frac{5}{2}\\right)}^{2}-\\frac{33}{4},\\,[\/latex]Vertex[latex]\\,\\left(-\\frac{5}{2},-\\frac{33}{4}\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134476611\">\r\n<div id=\"fs-id1165134476612\">\r\n<p id=\"fs-id1165134476613\">[latex]h\\left(x\\right)=2{x}^{2}+8x-10[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133328085\">\r\n<div id=\"fs-id1165133328087\">\r\n<p id=\"fs-id1165133328088\">[latex]k\\left(x\\right)=3{x}^{2}-6x-9[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135382110\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135382110\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135382110\"]\r\n<p id=\"fs-id1165135382112\">[latex]f\\left(x\\right)=3{\\left(x-1\\right)}^{2}-12,\\,[\/latex]Vertex[latex]\\,\\left(1,-12\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134422869\">\r\n<div id=\"fs-id1165134422870\">\r\n<p id=\"fs-id1165134422871\">[latex]f\\left(x\\right)=2{x}^{2}-6x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135409784\">\r\n<div id=\"fs-id1165135409786\">\r\n<p id=\"fs-id1165135409787\">[latex]f\\left(x\\right)=3{x}^{2}-5x-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137749740\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137749740\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137749740\"]\r\n<p id=\"fs-id1165137749741\">[latex]f\\left(x\\right)=3{\\left(x-\\frac{5}{6}\\right)}^{2}-\\frac{37}{12},\\,[\/latex]Vertex[latex]\\,\\left(\\frac{5}{6},-\\frac{37}{12}\\right)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165135258891\">For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.<\/p>\r\n\r\n<div id=\"fs-id1165135258896\">\r\n<div id=\"fs-id1165135258897\">\r\n<p id=\"fs-id1165135258898\">[latex]y\\left(x\\right)=2{x}^{2}+10x+12[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133344089\">\r\n<div id=\"fs-id1165133344091\">\r\n<p id=\"fs-id1165133344092\">[latex]f\\left(x\\right)=2{x}^{2}-10x+4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135254602\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135254602\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135254602\"]\r\n<p id=\"fs-id1165135254604\">Minimum is[latex]\\,-\\frac{17}{2}\\,[\/latex]and occurs at[latex]\\,\\frac{5}{2}.\\,[\/latex]Axis of symmetry is[latex]\\,x=\\frac{5}{2}.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134091244\">\r\n<div id=\"fs-id1165134091247\">\r\n<p id=\"fs-id1165134091248\">[latex]f\\left(x\\right)=-{x}^{2}+4x+3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134278554\">\r\n<div id=\"fs-id1165134278556\">\r\n<p id=\"fs-id1165134278557\">[latex]f\\left(x\\right)=4{x}^{2}+x-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137852807\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137852807\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137852807\"]\r\n<p id=\"fs-id1165137852808\">Minimum is[latex]\\,-\\frac{17}{16}\\,[\/latex]and occurs at[latex]\\,-\\frac{1}{8}.\\,[\/latex]Axis of symmetry is[latex]\\,x=-\\frac{1}{8}.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135193269\">\r\n<div id=\"fs-id1165135193271\">\r\n<p id=\"fs-id1165135193272\">[latex]h\\left(t\\right)=-4{t}^{2}+6t-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135380708\">\r\n<div id=\"fs-id1165135380710\">\r\n<p id=\"fs-id1165135506299\">[latex]f\\left(x\\right)=\\frac{1}{2}{x}^{2}+3x+1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134328315\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134328315\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134328315\"]\r\n<p id=\"fs-id1165134328316\">Minimum is[latex]\\,-\\frac{7}{2}\\,[\/latex]and occurs at[latex]\\,-3.\\,[\/latex] Axis of symmetry is[latex]\\,x=-3.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134362815\">\r\n<div id=\"fs-id1165134362817\">\r\n<p id=\"fs-id1165134362818\">[latex]f\\left(x\\right)=-\\frac{1}{3}{x}^{2}-2x+3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165135419821\">For the following exercises, determine the domain and range of the quadratic function.<\/p>\r\n\r\n<div id=\"fs-id1165135419824\">\r\n<div id=\"fs-id1165135419827\">\r\n<p id=\"fs-id1165135419828\">[latex]f\\left(x\\right)={\\left(x-3\\right)}^{2}+2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135353077\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135353077\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135353077\"]\r\n<p id=\"fs-id1165135353078\">Domain is[latex]\\,\\left(-\\infty ,\\infty \\right).\\,[\/latex]Range is[latex]\\,\\left[2,\\infty \\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137681943\">\r\n<div id=\"fs-id1165137681945\">\r\n<p id=\"fs-id1165137681946\">[latex]f\\left(x\\right)=-2{\\left(x+3\\right)}^{2}-6[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135528310\">\r\n<div id=\"fs-id1165135528312\">\r\n<p id=\"fs-id1165135528313\">[latex]f\\left(x\\right)={x}^{2}+6x+4[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135548952\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135548952\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135548952\"]\r\n<p id=\"fs-id1165135548954\">Domain is[latex]\\,\\left(-\\infty ,\\infty \\right).\\,[\/latex]Range is[latex]\\,\\left[-5,\\infty \\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134199532\">\r\n<div id=\"fs-id1165134199534\">\r\n<p id=\"fs-id1165134199535\">[latex]f\\left(x\\right)=2{x}^{2}-4x+2[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134138647\">\r\n<div id=\"fs-id1165134138649\">\r\n<p id=\"fs-id1165134138650\">[latex]k\\left(x\\right)=3{x}^{2}-6x-9[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135470085\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135470085\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135470085\"]\r\n<p id=\"fs-id1165135470086\">Domain is[latex]\\,\\left(-\\infty ,\\infty \\right).\\,[\/latex]Range is[latex]\\,\\left[-12,\\infty \\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165133252496\">For the following exercises, use the vertex[latex]\\,\\left(h,k\\right)\\,[\/latex]and a point on the graph[latex]\\,\\left(x,y\\right)\\,[\/latex]to find the general form of the equation of the quadratic function.<\/p>\r\n\r\n<div id=\"fs-id1165137898126\">\r\n<div id=\"fs-id1165137898128\">\r\n<p id=\"fs-id1165137898129\">[latex]\\left(h,k\\right)=\\left(2,0\\right),\\left(x,y\\right)=\\left(4,4\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135403325\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135403325\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135403325\"]\r\n<p id=\"fs-id1165135403326\">[latex]f\\left(x\\right)={x}^{2}-4x+4[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134079556\">\r\n<div id=\"fs-id1165134079558\">\r\n<p id=\"fs-id1165134079559\">[latex]\\left(h,k\\right)=\\left(-2,-1\\right),\\left(x,y\\right)=\\left(-4,3\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134485654\">\r\n<div id=\"fs-id1165134485656\">\r\n<p id=\"fs-id1165134485657\">[latex]\\left(h,k\\right)=\\left(0,1\\right),\\left(x,y\\right)=\\left(2,5\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135551841\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135551841\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135551841\"]\r\n<p id=\"fs-id1165135551842\">[latex]f\\left(x\\right)={x}^{2}+1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133077523\">\r\n<div id=\"fs-id1165133077525\">\r\n<p id=\"fs-id1165133077526\">[latex]\\left(h,k\\right)=\\left(2,3\\right),\\left(x,y\\right)=\\left(5,12\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165132912701\">\r\n<div id=\"fs-id1165132912704\">\r\n<p id=\"fs-id1165132912705\">[latex]\\left(h,k\\right)=\\left(-5,3\\right),\\left(x,y\\right)=\\left(2,9\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137843237\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137843237\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137843237\"]\r\n<p id=\"fs-id1165137843238\">[latex]f\\left(x\\right)=\\frac{6}{49}{x}^{2}+\\frac{60}{49}x+\\frac{297}{49}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133252567\">\r\n<div id=\"fs-id1165135536384\">\r\n<p id=\"fs-id1165135536385\">[latex]\\left(h,k\\right)=\\left(3,2\\right),\\left(x,y\\right)=\\left(10,1\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135412141\">\r\n<div id=\"fs-id1165135412143\">\r\n<p id=\"fs-id1165135412144\">[latex]\\left(h,k\\right)=\\left(0,1\\right),\\left(x,y\\right)=\\left(1,0\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135384995\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135384995\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135384995\"]\r\n<p id=\"fs-id1165135384996\">[latex]f\\left(x\\right)=-{x}^{2}+1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134479000\">\r\n<div id=\"fs-id1165134479002\">\r\n<p id=\"fs-id1165134479003\">[latex]\\left(h,k\\right)=\\left(1,0\\right),\\left(x,y\\right)=\\left(0,1\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133095093\" class=\"bc-section section\">\r\n<h4>Graphical<\/h4>\r\n<p id=\"fs-id1165133095098\">For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.<\/p>\r\n\r\n<div id=\"fs-id1165133095103\">\r\n<div id=\"fs-id1165133095105\">\r\n<p id=\"fs-id1165133095106\">[latex]f\\left(x\\right)={x}^{2}-2x[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134148456\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134148456\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134148456\"]<span id=\"fs-id1165133280629\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135539\/CNX_Precalc_Figure_03_02_201.jpg\" alt=\"Graph of f(x) = x^2-2x\" \/><\/span>\r\n<p id=\"fs-id1165133280642\">Vertex[latex]\\left(1,\\text{ }-1\\right),\\,[\/latex]Axis of symmetry is[latex]\\,x=1.\\,[\/latex]Intercepts are[latex]\\,\\left(0,0\\right), \\left(2,0\\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135646115\">\r\n<div id=\"fs-id1165135646117\">\r\n<p id=\"fs-id1165135646118\">[latex]f\\left(x\\right)={x}^{2}-6x-1[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135692195\">\r\n<div id=\"fs-id1165135692197\">\r\n<p id=\"fs-id1165135692198\">[latex]f\\left(x\\right)={x}^{2}-5x-6[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135697888\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135697888\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135697888\"]<span id=\"fs-id1165134149910\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135545\/CNX_Precalc_Figure_03_02_203.jpg\" alt=\"Graph of f(x)x^2-5x-6\" \/><\/span>\r\n<p id=\"fs-id1165134149922\">Vertex[latex]\\,\\left(\\frac{5}{2},\\frac{-49}{4}\\right),\\,[\/latex]Axis of symmetry is[latex]\\,\\left(0,-6\\right),\\left(-1,0\\right),\\left(6,0\\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135205053\">\r\n<div id=\"fs-id1165135205055\">\r\n<p id=\"fs-id1165135205056\">[latex]f\\left(x\\right)={x}^{2}-7x+3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135500665\">\r\n<div id=\"fs-id1165135500667\">\r\n<p id=\"fs-id1165135500668\">[latex]f\\left(x\\right)=-2{x}^{2}+5x-8[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135347648\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135347648\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135347648\"]<span id=\"fs-id1165135347655\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135549\/CNX_Precalc_Figure_03_02_205.jpg\" alt=\"Graph of f(x)=-2x^2+5x-8\" \/><\/span>\r\n<p id=\"fs-id1165135347667\">Vertex[latex]\\,\\left(\\frac{5}{4}, -\\frac{39}{8}\\right),\\,[\/latex]Axis of symmetry is[latex]\\,x=\\frac{5}{4}.\\,[\/latex]Intercepts are[latex]\\,\\left(0, -8\\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135639341\">\r\n<div id=\"fs-id1165135639343\">\r\n<p id=\"fs-id1165135639344\">[latex]f\\left(x\\right)=4{x}^{2}-12x-3[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div id=\"eip-id1165134560567\"><span id=\"fs-id1165134391611\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135556\/CNX_Precalc_Figure_03_02_206.jpg\" alt=\"Graph of f(x)=4x^2-12x-3\" \/><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1165134391625\">For the following exercises, write the equation for the graphed quadratic function.<\/p>\r\n\r\n<div id=\"fs-id1165134391628\">\r\n<div id=\"fs-id1165134391630\"><span id=\"fs-id1165134391635\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135558\/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).\" \/><\/span><\/div>\r\n<div id=\"fs-id1165134391648\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134391648\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134391648\"]\r\n<p id=\"fs-id1165134391649\">[latex]f\\left(x\\right)={x}^{2}-4x+1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135262595\">\r\n<div id=\"fs-id1165135262597\"><span id=\"fs-id1165135262603\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135603\/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)\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135262616\">\r\n<div id=\"fs-id1165135262618\"><span id=\"fs-id1165135262624\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135605\/CNX_Precalc_Figure_03_02_209.jpg\" alt=\"Graph of a negative parabola with a vertex at (2, 7).\" \/><\/span><\/div>\r\n<div id=\"fs-id1165135536430\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135536430\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135536430\"]\r\n<p id=\"fs-id1165135536431\">[latex]f\\left(x\\right)=-2{x}^{2}+8x-1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134162152\">\r\n<div id=\"fs-id1165134162154\"><span id=\"fs-id1165134162160\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135607\/CNX_Precalc_Figure_03_02_210.jpg\" alt=\"Graph of a negative parabola with a vertex at (-1, 2).\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134162173\">\r\n<div id=\"fs-id1165134162175\"><span id=\"fs-id1165134196136\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135609\/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).\" \/><\/span><\/div>\r\n<div id=\"fs-id1165134196148\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134196148\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134196148\"]\r\n<p id=\"fs-id1165134196149\">[latex]f\\left(x\\right)=\\frac{1}{2}{x}^{2}-3x+\\frac{7}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137940515\">\r\n<div id=\"fs-id1165137940518\"><span id=\"fs-id1165131959568\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135619\/CNX_Precalc_Figure_03_02_212.jpg\" alt=\"Graph of a negative parabola with a vertex at (-2, 3).\" \/><\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165131959583\" class=\"bc-section section\">\r\n<h4>Numeric<\/h4>\r\n<p id=\"fs-id1165131959588\">For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.<\/p>\r\n\r\n<div id=\"fs-id1165131959593\">\r\n<div id=\"fs-id1165131959596\">\r\n<table id=\"fs-id1165131959598\" class=\"unnumbered\" summary=\"..\"><caption>\u00a0<\/caption>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/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><strong>[latex]y[\/latex]<\/strong><\/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\n<\/div>\r\n<div id=\"fs-id1165133247929\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133247929\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165133247929\"]\r\n<p id=\"fs-id1165133247930\">[latex]f\\left(x\\right)={x}^{2}+1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135485986\">\r\n<div id=\"fs-id1165135485987\">\r\n<table id=\"fs-id1165135485989\" class=\"unnumbered\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/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><strong>[latex]y[\/latex]<\/strong><\/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\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133248546\">\r\n<div id=\"fs-id1165133248548\">\r\n<table id=\"fs-id1165133248550\" class=\"unnumbered\" summary=\"..\"><caption>\u00a0<\/caption>\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/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><strong>[latex]y[\/latex]<\/strong><\/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\n<\/div>\r\n<div id=\"fs-id1165137900973\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137900973\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137900973\"]\r\n<p id=\"fs-id1165137900974\">[latex]f\\left(x\\right)=2-{x}^{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134138600\">\r\n<div id=\"fs-id1165134138602\">\r\n<table id=\"fs-id1165134138604\" class=\"unnumbered\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/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><strong>[latex]y[\/latex]<\/strong><\/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\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134328259\">\r\n<div id=\"fs-id1165134328261\">\r\n<table id=\"fs-id1165134328263\" class=\"unnumbered\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td><strong>[latex]x[\/latex]<\/strong><\/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><strong>[latex]y[\/latex]<\/strong><\/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\n<\/div>\r\n<div id=\"fs-id1165135537351\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135537351\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135537351\"]\r\n<p id=\"fs-id1165135537352\">[latex]f\\left(x\\right)=2{x}^{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135513626\" class=\"bc-section section\">\r\n<h4>Technology<\/h4>\r\n<p id=\"fs-id1165135513632\">For the following exercises, use a calculator to find the answer.<\/p>\r\n\r\n<div id=\"fs-id1165135513635\">\r\n<div id=\"fs-id1165135513637\">\r\n<p id=\"fs-id1165135513639\">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]<\/p>\r\n<p id=\"fs-id1165133402104\">What appears to be the effect of changing the coefficient?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165133402109\">\r\n<div id=\"fs-id1165133402111\">\r\n<p id=\"fs-id1165133402112\">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>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137843141\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137843141\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137843141\"]\r\n<p id=\"fs-id1165137843142\">The graph is shifted up or down (a vertical shift).<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137843146\">\r\n<div id=\"fs-id1165137843148\">\r\n<p id=\"fs-id1165137843151\">Graph 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]<\/p>\r\n<p id=\"fs-id1165137697907\">What appears to be the effect of adding or subtracting those numbers?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137697912\">\r\n<div id=\"fs-id1165137697915\">\r\n<p id=\"fs-id1165137697916\">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[latex]\\,x\\,[\/latex]is 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>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135533787\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135533787\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135533787\"]\r\n<p id=\"fs-id1165135533788\">50 feet<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135533793\">\r\n<div id=\"fs-id1165135533795\">\r\n<p id=\"fs-id1165135533796\">A 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[latex]\\,|x|\\,[\/latex]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>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135629622\" class=\"bc-section section\">\r\n<h4>Extensions<\/h4>\r\n<p id=\"fs-id1165135523290\">For the following exercises, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function.<\/p>\r\n\r\n<div id=\"fs-id1165135523294\">\r\n<div id=\"fs-id1165135523296\">\r\n<p id=\"fs-id1165135523298\">Vertex[latex]\\,\\left(1,-2\\right),\\,[\/latex]opens up.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135170999\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135170999\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135170999\"]\r\n<p id=\"fs-id1165135171000\">Domain is[latex]\\,\\left(-\\infty ,\\infty \\right).\\,[\/latex]Range is[latex]\\,\\left[-2,\\infty \\right).[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137897840\">\r\n<div id=\"fs-id1165137897842\">\r\n<p id=\"fs-id1165137897843\">Vertex[latex]\\,\\left(-1,2\\right)\\,[\/latex]opens down.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135609192\">\r\n<div id=\"fs-id1165135609194\">\r\n<p id=\"fs-id1165135609195\">Vertex[latex]\\,\\left(-5,11\\right),\\,[\/latex]opens down.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135609234\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135609234\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135609234\"]\r\n<p id=\"fs-id1165135609235\">Domain is[latex]\\,\\left(-\\infty ,\\infty \\right)\\,[\/latex]Range is[latex]\\,\\left(-\\infty ,11\\right].[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135501919\">\r\n<div id=\"fs-id1165135501921\">\r\n<p id=\"fs-id1165135501922\">Vertex[latex]\\,\\left(-100,100\\right),\\,[\/latex]opens up.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165135501959\">For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function.<\/p>\r\n\r\n<div id=\"fs-id1165135501964\">\r\n<div id=\"fs-id1165135501966\">\r\n<p id=\"fs-id1165135501967\">Contains[latex]\\,\\left(1,1\\right)\\,[\/latex]and has shape of[latex]\\,f\\left(x\\right)=2{x}^{2}.\\,[\/latex]Vertex is on the[latex]\\,y\\text{-}[\/latex]axis.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137642675\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137642675\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137642675\"]\r\n<p id=\"fs-id1165137642676\">[latex]f\\left(x\\right)=2{x}^{2}-1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137642716\">\r\n<div id=\"fs-id1165133289615\">\r\n<p id=\"fs-id1165133289616\">Contains[latex]\\,\\left(-1,4\\right)\\,[\/latex]and has the shape of[latex]\\,f\\left(x\\right)=2{x}^{2}.\\,[\/latex]Vertex is on the[latex]\\,y\\text{-}[\/latex]axis.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135551155\">\r\n<div id=\"fs-id1165135551157\">\r\n<p id=\"fs-id1165135551158\">Contains[latex]\\,\\left(2,3\\right)\\,[\/latex]and has the shape of[latex]\\,f\\left(x\\right)=3{x}^{2}.\\,[\/latex]Vertex is on the[latex]\\,y\\text{-}[\/latex]axis.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135341250\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135341250\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135341250\"]\r\n<p id=\"fs-id1165135341251\">[latex]f\\left(x\\right)=3{x}^{2}-9[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135525847\">\r\n<div id=\"fs-id1165135525850\">\r\n<p id=\"fs-id1165135525851\">Contains[latex]\\,\\left(1,-3\\right)\\,[\/latex]and has the shape of[latex]\\,f\\left(x\\right)=-{x}^{2}.\\,[\/latex]Vertex is on the[latex]\\,y\\text{-}[\/latex]axis.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135442559\">\r\n<div id=\"fs-id1165135442561\">\r\n<p id=\"fs-id1165135442562\">Contains[latex]\\,\\left(4,3\\right)\\,[\/latex]and has the shape of[latex]\\,f\\left(x\\right)=5{x}^{2}.\\,[\/latex]Vertex is on the[latex]\\,y\\text{-}[\/latex]axis.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135646094\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135646094\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135646094\"]\r\n<p id=\"fs-id1165135646095\">[latex]f\\left(x\\right)=5{x}^{2}-77[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137901057\">\r\n<div id=\"fs-id1165137901059\">\r\n<p id=\"fs-id1165137901060\">Contains[latex]\\,\\left(1,-6\\right)\\,[\/latex]has the shape of[latex]\\,f\\left(x\\right)=3{x}^{2}.\\,[\/latex]Vertex has x-coordinate of[latex]\\,-1.[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165131857387\" class=\"bc-section section\">\r\n<h4>Real-World Applications<\/h4>\r\n<div id=\"fs-id1165131857392\">\r\n<div id=\"fs-id1165131857394\">\r\n<p id=\"fs-id1165131857395\">Find the dimensions of the rectangular corral producing the greatest enclosed area given 200 feet of fencing.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165131857400\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165131857400\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165131857400\"]\r\n<p id=\"fs-id1165131857401\">50 feet by 50 feet. Maximize[latex]\\,f\\left(x\\right)=-{x}^{2}+100x.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165131857448\">\r\n<div id=\"fs-id1165137940524\">\r\n<p id=\"fs-id1165137940525\">Find 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>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137940531\">\r\n<div id=\"fs-id1165137940533\">\r\n<p id=\"fs-id1165137940534\">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>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165137940538\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137940538\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165137940538\"]\r\n<p id=\"fs-id1165137940540\">125 feet by 62.5 feet. Maximize[latex]\\,f\\left(x\\right)=-2{x}^{2}+250x.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137940588\">\r\n<div id=\"fs-id1165134089391\">\r\n<p id=\"fs-id1165134089392\">Among all of the pairs of numbers whose sum is 6, find the pair with the largest product. What is the product?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165134089397\">\r\n<div id=\"fs-id1165134089399\">\r\n<p id=\"fs-id1165134089400\">Among all of the pairs of numbers whose difference is 12, find the pair with the smallest product. What is the product?<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165134089405\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134089405\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165134089405\"]\r\n<p id=\"fs-id1165134089406\">[latex]6\\,[\/latex]and[latex]\\,-6;\\,[\/latex]product is \u201336; maximize[latex]\\,f\\left(x\\right)={x}^{2}+12x.[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135364089\">\r\n<div id=\"fs-id1165135364091\">\r\n<p id=\"fs-id1165135364092\">Suppose that the price per unit in dollars of a cell phone production is modeled by[latex]\\,p=\\text{\\$}45-0.0125x,\\,[\/latex]where[latex]\\,x\\,[\/latex]is in thousands of phones produced, and the revenue represented by thousands of dollars is[latex]\\,R=x\\cdot p.\\,[\/latex]Find the production level that will maximize revenue.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135571732\">\r\n<div id=\"fs-id1165135571734\">\r\n<p id=\"fs-id1165135571735\">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>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135394030\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135394030\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135394030\"]\r\n<p id=\"fs-id1165135394031\">2909.56 meters<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135394035\">\r\n<div id=\"fs-id1165135394038\">\r\n<p id=\"fs-id1165135394039\">A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time, in seconds, is given by[latex]\\,h\\left(t\\right)=-4.9{t}^{2}+24t+8.\\,[\/latex]How long does it take to reach maximum height?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135449627\">\r\n<div id=\"fs-id1165135449629\">\r\n<p id=\"fs-id1165135449630\">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>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135449636\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135449636\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1165135449636\"]\r\n<p id=\"fs-id1165135449637\">$10.70<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165135449642\">\r\n<div id=\"fs-id1165135449644\">\r\n<p id=\"fs-id1165135449645\">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>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Glossary<\/h3>\r\n<dl id=\"fs-id1165135449657\">\r\n \t<dt>axis of symmetry<\/dt>\r\n \t<dd id=\"fs-id1165135449662\">a vertical line drawn through the vertex of a parabola, that opens up or down, 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\">\r\n \t<dt>general form of a quadratic function<\/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[latex]\\,a,b,\\,[\/latex]and[latex]\\,c\\,[\/latex]are real numbers and[latex]\\,a\\ne 0.[\/latex]<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137931270\">\r\n \t<dt>roots<\/dt>\r\n \t<dd id=\"fs-id1165137931276\">in a given function, the values of[latex]\\,x\\,[\/latex]at which[latex]\\,y=0[\/latex], also called zeros<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137931314\">\r\n \t<dt>standard form of a quadratic function<\/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\">\r\n \t<dt>vertex<\/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\">\r\n \t<dt>vertex form of a quadratic function<\/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\">\r\n \t<dt>zeros<\/dt>\r\n \t<dd id=\"fs-id1165135623639\">in a given function, the values of[latex]\\,x\\,[\/latex]at which[latex]\\,y=0[\/latex], also called roots<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Recognize characteristics of parabolas.<\/li>\n<li>Understand how the graph of a parabola is related to its quadratic function.<\/li>\n<li>Determine a quadratic function\u2019s minimum or maximum value.<\/li>\n<li>Solve problems involving a quadratic function\u2019s minimum or maximum value.<\/li>\n<\/ul>\n<\/div>\n<div 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\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135358\/CNX_Precalc_Figure_03_02_001.jpg\" alt=\"Satellite dishes.\" width=\"731\" height=\"368\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 1. <\/strong>An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr)<\/p>\n<\/div>\n<div class=\"wp-caption-text\"><\/div>\n<\/div>\n<p id=\"fs-id1165134339909\">Curved antennas, such as the ones shown in <a class=\"autogenerated-content\" href=\"#Figure_03_02_001\">(Figure)<\/a>, 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<div id=\"fs-id1165137762207\" class=\"bc-section section\">\n<h3>Recognizing Characteristics of Parabolas<\/h3>\n<p id=\"fs-id1165137727999\">The graph of a quadratic function is a U-shaped curve called a <span class=\"no-emphasis\">parabola<\/span>. One important feature of the graph is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph, or the <span class=\"no-emphasis\">minimum value<\/span> of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the <span class=\"no-emphasis\">maximum value<\/span>. In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the axis of symmetry. These features are illustrated in <a class=\"autogenerated-content\" href=\"#Figure_03_02_002\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_03_02_002\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135405\/CNX_Precalc_Figure_03_02_002.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\"><strong>Figure 2.<\/strong><\/p>\n<\/div>\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 zeros<strong>, <\/strong>or roots, of the quadratic function, the values of[latex]\\,x\\,[\/latex]at which[latex]\\,y=0.[\/latex]<\/p>\n<div id=\"Example_03_02_01\" class=\"textbox examples\">\n<div id=\"fs-id1165131959514\">\n<div id=\"fs-id1165135541748\">\n<h3>Identifying the Characteristics of a Parabola<\/h3>\n<p id=\"fs-id1165135366534\">Determine the vertex, axis of symmetry, zeros, and[latex]\\,y\\text{-}[\/latex]intercept of the parabola shown in <a class=\"autogenerated-content\" href=\"#Figure_03_02_003\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_03_02_003\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135407\/CNX_Precalc_Figure_03_02_003.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\"><strong>Figure 3.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137892270\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137892270\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137892270\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137695151\">The vertex is the turning point of the graph. We can see that the vertex is at[latex]\\,\\left(3,1\\right).\\,[\/latex]Because this parabola opens upward, the axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is[latex]\\,x=3.\\,[\/latex]This parabola does not cross the[latex]\\,x\\text{-}[\/latex]axis, so it has no zeros. It crosses the[latex]\\,y\\text{-}[\/latex]axis at[latex]\\,\\left(0,7\\right)\\,[\/latex]so this is the <em>y<\/em>-intercept.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137641326\" class=\"bc-section section\">\n<h3>Understanding How the Graphs of Parabolas are Related to Their Quadratic Functions<\/h3>\n<p id=\"fs-id1165137652877\">The general form <strong>of a quadratic function <\/strong>presents the function in the form<\/p>\n<div id=\"fs-id1165137422466\">[latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex]<\/div>\n<p id=\"fs-id1165137544673\">where[latex]\\,a,b,\\,[\/latex]and[latex]\\,c\\,[\/latex]are 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\u00b1\\sqrt{{b}^{2}-4ac}}{2a},\\,[\/latex]to solve[latex]\\,a{x}^{2}+bx+c=0\\,[\/latex]for the[latex]\\,x\\text{-}[\/latex]intercepts, or zeros, we find the value of[latex]\\,x\\,[\/latex]halfway between them is always[latex]\\,x=-\\frac{b}{2a},\\,[\/latex]the equation for the axis of symmetry.<\/p>\n<p id=\"fs-id1165135190920\"><a class=\"autogenerated-content\" href=\"#Figure_03_02_004\">(Figure)<\/a> represents the graph of the quadratic function written in general form as[latex]\\,y={x}^{2}+4x+3.\\,[\/latex]In this form,[latex]\\,a=1,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[latex]\\,x\\text{-}[\/latex]intercepts, those points where the parabola crosses the[latex]\\,x\\text{-}[\/latex]axis, occur at[latex]\\,\\left(-3,0\\right)\\,[\/latex]and[latex]\\,\\left(-1,0\\right).[\/latex]<\/p>\n<div id=\"Figure_03_02_004\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135417\/CNX_Precalc_Figure_03_02_004.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\"><strong>Figure 4.<\/strong><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137676320\">The <strong>standard form of a quadratic function<\/strong> presents the function in the form<\/p>\n<div id=\"fs-id1165135320100\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/div>\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 vertex form of a quadratic function.<\/p>\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. <a class=\"autogenerated-content\" href=\"#Figure_03_02_005\">(Figure)<\/a> represents the graph of the quadratic function written in standard form as[latex]\\,y=-3{\\left(x+2\\right)}^{2}+4.\\,[\/latex]Since[latex]\\,x\u2013h=x+2\\,[\/latex]in this example,[latex]\\,h=\u20132.\\,[\/latex]In this form,[latex]\\,a=-3,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]<\/p>\n<div id=\"Figure_03_02_005\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135423\/CNX_Precalc_Figure_03_02_005.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\"><strong>Figure 5.<\/strong><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137453489\">The standard form is useful for determining how the graph is transformed from the graph of[latex]\\,y={x}^{2}.\\,[\/latex]<a class=\"autogenerated-content\" href=\"#Figure_03_02_006\">(Figure)<\/a> is the graph of this basic function.<\/p>\n<div id=\"Figure_03_02_006\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135436\/CNX_Precalc_Figure_03_02_006.jpg\" alt=\"Graph of y=x^2.\" width=\"487\" height=\"480\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 6.<\/strong><\/p>\n<\/div>\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 <a class=\"autogenerated-content\" href=\"#Figure_03_02_005\">(Figure)<\/a>,[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 <a class=\"autogenerated-content\" href=\"#Figure_03_02_005\">(Figure)<\/a>,[latex]\\,h<0,\\,[\/latex]so the graph is shifted 2 units to the left. The magnitude of[latex]\\,a\\,[\/latex]indicates the stretch of the graph. If[latex]|a|>1,[\/latex] the point associated with a particular[latex]\\,x\\text{-}[\/latex]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[latex]\\,x\\text{-}[\/latex]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 <a class=\"autogenerated-content\" href=\"#Figure_03_02_005\">(Figure)<\/a>,[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=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill a{\\left(x-h\\right)}^{2}+k& =& a{x}^{2}+bx+c\\hfill \\\\ \\hfill a{x}^{2}-2ahx+\\left(a{h}^{2}+k\\right)& =& a{x}^{2}+bx+c\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137409211\">For the linear terms to be equal, the coefficients must be equal.<\/p>\n<div id=\"eip-144\" class=\"unnumbered aligncenter\">[latex]\u20132ah=b,\\text{ so }h=-\\frac{b}{2a}[\/latex]<\/div>\n<p id=\"fs-id1165134118295\">This is the <span class=\"no-emphasis\">axis of symmetry<\/span> we defined earlier. Setting the constant terms equal:<\/p>\n<div id=\"eip-313\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill a{h}^{2}+k& =& c\\hfill \\\\ \\hfill k& =& c-a{h}^{2}\\hfill \\\\ & =& c-a-{\\left(\\frac{b}{2a}\\right)}^{2}\\hfill \\\\ & =& c-\\frac{{b}^{2}}{4a}\\hfill \\end{array}[\/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[latex]\\,h,\\,[\/latex]so[latex]\\,f\\left(h\\right)=k.[\/latex]<\/p>\n<div id=\"fs-id1165137749882\" class=\"textbox key-takeaways\">\n<h3>Forms of Quadratic Functions<\/h3>\n<p id=\"fs-id1165135333154\">A quadratic function is a polynomial function of degree two. The graph of a <span class=\"no-emphasis\">quadratic function<\/span> is a parabola.<\/p>\n<p id=\"eip-103\">The general form of a quadratic function is[latex]\\,f\\left(x\\right)=a{x}^{2}+bx+c\\,[\/latex]where[latex]\\,a,b,\\,[\/latex]and[latex]\\,c\\,[\/latex]are real numbers and[latex]\\,a\\ne 0.[\/latex]<\/p>\n<p id=\"fs-id1165137666538\">The standard form of a quadratic function is[latex]\\,f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k\\,[\/latex]where[latex]\\,a\\ne 0.[\/latex]<\/p>\n<p id=\"fs-id1165137762385\">The vertex[latex]\\,\\left(h,k\\right)\\,[\/latex]is located at<\/p>\n<div id=\"eip-301\" class=\"unnumbered aligncenter\">[latex]h=\u2013\\frac{b}{2a},\\text{ }k=f\\left(h\\right)=f\\left(\\frac{-b}{2a}\\right)[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165131886746\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137650986\"><strong>Given a graph of a quadratic function, write the equation of the function in general form.<\/strong><\/p>\n<ol id=\"fs-id1165134223276\" type=\"1\">\n<li>Identify the horizontal shift of the parabola; this value is[latex]\\,h.\\,[\/latex]Identify the vertical shift of the parabola; this value is[latex]\\,k.[\/latex]<\/li>\n<li>Substitute the values of the horizontal and vertical shift for[latex]\\,h\\,[\/latex]and[latex]\\,k.\\,[\/latex]in the function[latex]\\,f\\left(x\\right)=a{\\left(x\u2013h\\right)}^{2}+k.[\/latex]<\/li>\n<li>Substitute the values of any point, other than the vertex, on the graph of the parabola for[latex]\\,x\\,[\/latex]and[latex]\\,f\\left(x\\right).[\/latex]<\/li>\n<li>Solve for the stretch factor,[latex]\\,|a|.[\/latex]<\/li>\n<li>Expand and simplify to write in general form.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_02_02\" class=\"textbox examples\">\n<div id=\"fs-id1165135460939\">\n<div id=\"fs-id1165135460941\">\n<h3>Writing the Equation of a Quadratic Function from the Graph<\/h3>\n<p id=\"fs-id1165135532321\">Write an equation for the quadratic function[latex]\\,g\\,[\/latex]in <a class=\"autogenerated-content\" href=\"#Figure_03_02_007\">(Figure)<\/a> as 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.<\/p>\n<div id=\"Figure_03_02_007\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135457\/CNX_Precalc_Figure_03_02_007.jpg\" alt=\"Graph of a parabola with its vertex at (-2, -3).\" width=\"487\" height=\"443\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 7.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134211341\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134211341\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134211341\" 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-\\left(-2\\right)\\right)}^{2}-3=a{\\left(x+2\\right)}^{2}\u20133.[\/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<div id=\"eip-id1165134221671\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill -1& =& a{\\left(0+2\\right)}^{2}-3\\hfill \\\\ \\hfill 2& =& 4a\\hfill \\\\ \\hfill a& =& \\frac{1}{2}\\hfill \\end{array}[\/latex]<\/div>\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}\u20133.[\/latex]<\/p>\n<p id=\"fs-id1165137844164\">To write this in general polynomial form, we can expand the formula and simplify terms.<\/p>\n<div id=\"eip-id1165137463836\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill g\\left(x\\right)& =& \\frac{1}{2}{\\left(x+2\\right)}^{2}-3\\hfill \\\\ & =& \\frac{1}{2}\\left(x+2\\right)\\left(x+2\\right)-3\\hfill \\\\ & =& \\frac{1}{2}\\left({x}^{2}+4x+4\\right)-3\\hfill \\\\ & =& \\frac{1}{2}{x}^{2}+2x+2-3\\hfill \\\\ & =& \\frac{1}{2}{x}^{2}+2x-1\\hfill \\end{array}[\/latex]<\/div>\n<p>Notice that the horizontal and vertical shifts of the basic graph of the quadratic function determine the location of the vertex of the parabola; the vertex is unaffected by stretches and compressions.<\/p><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137838619\">\n<h4>Analysis<\/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}=\u20136\\,[\/latex]and[latex]\\,\\Delta \\text{Tbl = 2,}\\,[\/latex]and select[latex]\\,\\text{TABLE}\\text{.}\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Table_03_02_01\">(Figure)<\/a>.<\/p>\n<table id=\"Table_03_02_01\" summary=\"..\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/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><strong>[latex]y[\/latex]<\/strong><\/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<\/div>\n<div id=\"fs-id1165137527658\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_02_01\">\n<div id=\"fs-id1165137933940\">\n<p id=\"fs-id1165137933941\">A coordinate grid has been superimposed over the quadratic path of a basketball in <a class=\"autogenerated-content\" href=\"#Figure_03_02_008\">(Figure)<\/a>. Find an equation for the path of the ball. Does the shooter make the basket?<\/p>\n<div id=\"Figure_03_02_008\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135459\/CNX_Precalc_Figure_03_02_008.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\"><strong>Figure 8. <\/strong>(credit: modification of work by Dan Meyer)<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135414238\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135414238\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135414238\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135414239\">The path passes through the origin and has vertex at[latex]\\,\\left(-4,\\text{ }7\\right),\\,[\/latex]so[latex]\\,\\left(h\\right)x=\u2013\\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(\u20137.5\\right)\\approx 1.64;\\,[\/latex]he doesn\u2019t make it.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135168275\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135574310\"><strong>Given a quadratic function in general form, find the vertex of the parabola.<\/strong><\/p>\n<ol id=\"fs-id1165134108459\" type=\"1\">\n<li>Identify[latex]\\,a, b, \\text{and} c.[\/latex]<\/li>\n<li>Find[latex]\\,h,\\,[\/latex]the <em>x<\/em>-coordinate of the vertex, by substituting[latex]\\,a\\,[\/latex]and[latex]\\,b\\,[\/latex]into[latex]\\,h=\u2013\\frac{b}{2a}.[\/latex]<\/li>\n<li>Find[latex]\\,k,\\,[\/latex]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=\"textbox examples\">\n<div id=\"fs-id1165137658566\">\n<div id=\"fs-id1165137771901\">\n<h3>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}\u20136x+7.\\,[\/latex]Rewrite the quadratic in standard form (vertex form).<\/p>\n<\/div>\n<div id=\"fs-id1165137596321\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137596321\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137596321\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137596323\">[latex]\\begin{array}{c}\\text{The horizontal coordinate of the vertex will be at}\\hfill \\\\ & \\hfill h& =& -\\frac{b}{2a}\\hfill \\\\ & & =& -\\frac{-6}{2\\left(2\\right)}\\hfill \\\\ & & =& \\frac{6}{4}\\hfill \\\\ & & =& \\frac{3}{2}\\hfill \\\\ \\text{The vertical coordinate of the vertex will be at}\\hfill \\\\ & \\hfill k& =& f\\left(h\\right)\\hfill \\\\ & & =& f\\left(\\frac{3}{2}\\right)\\hfill \\\\ & & =& 2{\\left(\\frac{3}{2}\\right)}^{2}-6\\left(\\frac{3}{2}\\right)+7\\hfill \\\\ & & =& \\frac{5}{2}\\hfill \\end{array}[\/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. First, find the horizontal coordinate of the vertex. Then find the vertical coordinate of the vertex. Substitute the values into standard form, using the &#8220;[latex]a[\/latex]&#8221; from the general form.<\/p>\n<div id=\"eip-id1165135499318\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill f\\left(x\\right)& =& a{x}^{2}+bx+c\\hfill \\\\ \\hfill f\\left(x\\right)& =& 2{x}^{2}-6x+7\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137653186\">The standard form of a quadratic function prior to writing the function then becomes the following:<\/p>\n<div id=\"eip-id1165134389821\" class=\"unnumbered\">[latex]f\\left(x\\right)=2{\\left(x\u2013\\frac{3}{2}\\right)}^{2}+\\frac{5}{2}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137591920\">\n<h4>Analysis<\/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,[latex]\\,k,[\/latex]and where it occurs,[latex]\\,x.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135362470\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_02_02\">\n<div id=\"fs-id1165135193261\">\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>\n<div id=\"fs-id1165137638479\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137638479\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137638479\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137638480\">[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<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133436210\" class=\"bc-section section\">\n<h3>Finding the Domain and Range of a Quadratic Function<\/h3>\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=\"textbox key-takeaways\">\n<h3>Domain and Range of a Quadratic Function<\/h3>\n<p id=\"fs-id1165135502927\">The domain of any <span class=\"no-emphasis\">quadratic function<\/span> is all real numbers unless the context of the function presents some restrictions.<\/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[latex]\\,a\\,[\/latex]value 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[latex]\\,a\\,[\/latex]value 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[latex]\\,a\\,[\/latex]value is[latex]\\,f\\left(x\\right)\\ge k;\\,[\/latex]the range of a quadratic function written in standard form with a negative[latex]\\,a\\,[\/latex]value is[latex]\\,f\\left(x\\right)\\le k.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135205144\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137409165\"><strong>Given a quadratic function, find the domain and range.<\/strong><\/p>\n<ol id=\"fs-id1165137843779\" type=\"1\">\n<li>Identify the domain of any quadratic function as all real numbers.<\/li>\n<li>Determine whether[latex]\\,a\\,[\/latex]is positive or negative. If[latex]\\,a\\,[\/latex]is positive, the parabola has a minimum. If[latex]\\,a\\,[\/latex]is negative, the parabola has a maximum.<\/li>\n<li>Determine the maximum or minimum value of the parabola,[latex]\\,k.[\/latex]<\/li>\n<li>If the parabola has a minimum, the range is given by[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=\"textbox examples\">\n<div id=\"fs-id1165134257627\">\n<div id=\"fs-id1165134257629\">\n<h3>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>\n<div id=\"fs-id1165137837922\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137837922\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137837922\" 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[latex]\\,a\\,[\/latex]is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding the[latex]\\,x\\text{-}[\/latex]value of the vertex.<\/p>\n<div id=\"eip-id1165132986104\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill h& =& -\\frac{b}{2a}\\hfill \\\\ \\hfill & =& -\\frac{9}{2\\left(-5\\right)}\\hfill \\\\ \\hfill & =& \\frac{9}{10}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137736576\">The maximum value is given by[latex]\\,f\\left(h\\right).[\/latex]<\/p>\n<div id=\"eip-id1165135687688\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill f\\left(\\frac{9}{10}\\right)& =& -5{\\left(\\frac{9}{10}\\right)}^{2}+9\\left(\\frac{9}{10}\\right)-1\\hfill \\\\ & =& \\frac{61}{20}\\hfill \\end{array}[\/latex]<\/div>\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]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133227752\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_02_03\">\n<div id=\"fs-id1165135424648\">\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>\n<div id=\"fs-id1165137592393\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137592393\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137592393\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137812604\">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<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736541\" class=\"bc-section section\">\n<h3>Determining the Maximum and Minimum Values of Quadratic Functions<\/h3>\n<p id=\"fs-id1165137442167\">The output of the quadratic function at the vertex is the maximum or minimum value of the function, depending on the orientation of the <span class=\"no-emphasis\">parabola<\/span>. We can see the maximum and minimum values in <a class=\"autogenerated-content\" href=\"#Figure_03_02_009\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_03_02_009\" class=\"wp-caption aligncenter\">\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135506\/CNX_Precalc_Figure_03_02_009.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\"><strong>Figure 9.<\/strong><\/p>\n<\/div>\n<\/div>\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 id=\"Example_03_02_05\" class=\"textbox examples\">\n<div id=\"fs-id1165134378616\">\n<div id=\"fs-id1165134378618\">\n<h3>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\" type=\"a\">\n<li>Find a formula for the area enclosed by the fence if the sides of fencing perpendicular to the existing fence have length[latex]\\,L.[\/latex]<\/li>\n<li>What dimensions should she make her garden to maximize the enclosed area?<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165137836806\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137836806\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137836806\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137836808\">Let\u2019s use a diagram such as <a class=\"autogenerated-content\" href=\"#Figure_03_02_010\">(Figure)<\/a> to record the given information. It is also helpful to introduce a temporary variable,[latex]\\,W,\\,[\/latex]to represent the width of the garden and the length of the fence section parallel to the backyard fence.<\/p>\n<div id=\"Figure_03_02_010\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135510\/CNX_Precalc_Figure_03_02_010.jpg\" alt=\"Diagram of the garden and the backyard.\" width=\"487\" height=\"310\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 10.<\/strong><\/p>\n<\/div>\n<\/div>\n<ol id=\"fs-id1165134363440\" type=\"a\">\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,[latex]\\,W,\\,[\/latex]in terms of[latex]\\,L.[\/latex]\n<div id=\"eip-id1165135697866\" class=\"unnumbered\">[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<div id=\"eip-624\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill A& =& LW=L\\left(80-2L\\right)\\hfill \\\\ \\hfill A\\left(L\\right)& =& 80L-2{L}^{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135258914\">This formula represents the area of the fence in terms of the variable length[latex]\\,L.\\,[\/latex]The function, written in general form, is<\/p>\n<div id=\"eip-382\" class=\"unnumbered aligncenter\">[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[latex]\\,a\\,[\/latex]is 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<div id=\"eip-id1165135202446\" class=\"unnumbered\">[latex]\\begin{array}{ccccccc}\\hfill h& =& -\\frac{b}{2a}\\hfill & & \\hfill \\phantom{\\rule{1em}{0ex}}k& =& A\\left(20\\right)\\hfill \\\\ & =& -\\frac{80}{2\\left(-2\\right)}\\hfill & \\phantom{\\rule{1em}{0ex}}\\text{and}& & =& 80\\left(20\\right)-2{\\left(20\\right)}^{2}\\hfill \\\\ & =& 20\\hfill & & & =& 800\\hfill \\end{array}[\/latex]<\/div>\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<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135582226\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135582232\">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 <a class=\"autogenerated-content\" href=\"#Figure_03_02_011\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_03_02_011\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135518\/CNX_Precalc_Figure_03_02_011.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\"><strong>Figure 11.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133340409\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137803708\"><strong>Given an application involving revenue, use a quadratic equation to find the maximum.<\/strong><\/p>\n<ol id=\"fs-id1165135436584\" type=\"1\">\n<li>Write a quadratic equation for a revenue function.<\/li>\n<li>Find the vertex of the quadratic equation.<\/li>\n<li>Determine the <em>y<\/em>-value of the vertex.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_02_06\" class=\"textbox examples\">\n<div id=\"fs-id1165134278696\">\n<div id=\"fs-id1165137473136\">\n<h3>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>\n<div id=\"fs-id1165135389886\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135389886\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135389886\" 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,[latex]\\,p\\,[\/latex]for price per subscription and[latex]\\,Q\\,[\/latex]for 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<div id=\"eip-id1165135246622\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill m& =& \\frac{79,000-84,000}{32-30}\\hfill \\\\ & =& \\frac{-5,000}{2}\\hfill \\\\ & =& -2,500\\hfill \\end{array}[\/latex]<\/div>\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<div id=\"eip-id1165131968004\" class=\"unnumbered\">[latex]\\begin{array}{cccc}\\hfill Q& =& -2500p+b\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Substitute in the point}Q=84,000\\text{ and }p=30\\hfill \\\\ \\hfill 84,000& =& -2500\\left(30\\right)+b\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Solve for}b\\hfill \\\\ \\hfill b& =& 159,000\\hfill & \\end{array}[\/latex]<\/div>\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<div id=\"eip-id1165132337192\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{Revenue}& =& pQ\\hfill \\\\ \\hfill \\mathrm{Revenue}& =& p\\left(-2,500p+159,000\\right)\\hfill \\\\ \\hfill \\mathrm{Revenue}& =& -2,500{p}^{2}+159,000p\\hfill \\end{array}[\/latex]<\/div>\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<div id=\"eip-id1165135170999\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill h& =& -\\frac{159,000}{2\\left(-2,500\\right)}\\hfill \\\\ & =& 31.8\\hfill \\end{array}[\/latex]<\/div>\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<div id=\"eip-id1165134323486\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill \\text{maximum revenue}& =& -2,500{\\left(31.8\\right)}^{2}+159,000\\left(31.8\\right)\\hfill \\\\ & =& 2,528,100\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135538766\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165135538771\">This could also be solved by graphing the quadratic as in <a class=\"autogenerated-content\" href=\"#Figure_03_02_012\">(Figure)<\/a>. We can see the maximum revenue on a graph of the quadratic function.<\/p>\n<div id=\"Figure_03_02_012\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135523\/CNX_Precalc_Figure_03_02_012.jpg\" alt=\"Graph of the parabolic function which the x-axis is labeled Price (p) and the y-axis is labeled Revenue (\ud83d\udcb2). The vertex is at (31.80, 258100).\" width=\"487\" height=\"327\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 12.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135693703\" class=\"bc-section section\">\n<h4>Finding the <em>x<\/em>&#8211; and <em>y<\/em>-Intercepts of a Quadratic Function<\/h4>\n<p id=\"fs-id1165134569121\">Much as we did in the application problems above, we also need to find intercepts of quadratic equations for graphing parabolas. Recall that we find the[latex]\\,y\\text{-}[\/latex]intercept of a quadratic by evaluating the function at an input of zero, and we find the[latex]\\,x\\text{-}[\/latex]intercepts at locations where the output is zero. Notice in <a class=\"autogenerated-content\" href=\"#Figure_03_02_013\">(Figure)<\/a> that the number of[latex]\\,x\\text{-}[\/latex]intercepts can vary depending upon the location of the graph.<\/p>\n<div id=\"Figure_03_02_013\" class=\"wp-caption aligncenter\">\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135526\/CNX_Precalc_Figure_03_02_013.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\"><strong>Figure 13. <\/strong>Number of x-intercepts of a parabola<\/p>\n<\/div>\n<div id=\"fs-id1165137464602\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135638554\"><strong>Given a quadratic function[latex]\\,f\\left(x\\right),\\,[\/latex]find the[latex]\\,y\\text{-}[\/latex] and <em>x<\/em>-intercepts.<\/strong><\/p>\n<ol id=\"fs-id1165135378765\" type=\"1\">\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=\"textbox examples\">\n<div id=\"fs-id1165134129944\">\n<div id=\"fs-id1165134129946\">\n<h3>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>\n<div id=\"fs-id1165137901093\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137901093\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137901093\" 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<div id=\"eip-id1165133349374\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill f\\left(0\\right)& =& 3{\\left(0\\right)}^{2}+5\\left(0\\right)-2\\hfill \\\\ & =& -2\\hfill \\end{array}[\/latex]<\/div>\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<div id=\"eip-id1165132926414\" class=\"unnumbered\">[latex]0=3{x}^{2}+5x-2[\/latex]<\/div>\n<p id=\"fs-id1165135690677\">In this case, the quadratic can be factored easily, providing the simplest method for solution.<\/p>\n<div id=\"eip-id1165135321232\" class=\"unnumbered\">[latex]0=\\left(3x-1\\right)\\left(x+2\\right)[\/latex]<\/div>\n<div id=\"eip-id1165134586903\" class=\"unnumbered\">[latex]\\begin{array}{cccccc}\\hfill h& =& -\\frac{b}{2a}& \\hfill \\phantom{\\rule{2em}{0ex}}k& =& f\\left(-1\\right)\\hfill \\\\ & =& -\\frac{4}{2\\left(2\\right)}\\hfill & & =& \\hfill 2{\\left(-1\\right)}^{2}+4\\left(-1\\right)-4\\\\ & =& -1\\hfill & & =& -6\\hfill \\end{array}[\/latex]<\/div>\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<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137911614\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165134170695\">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]See <a class=\"autogenerated-content\" href=\"#Figure_03_02_014\">(Figure)<\/a><\/p>\n<div id=\"Figure_03_02_014\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135532\/CNX_Precalc_Figure_03_02_014.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\"><strong>Figure 14.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135381309\" class=\"bc-section section\">\n<h4>Rewriting Quadratics in Standard Form<\/h4>\n<p id=\"fs-id1165135381314\">In <a class=\"autogenerated-content\" href=\"#Example_03_02_07\">(Figure)<\/a>, 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=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135694488\"><strong>Given a quadratic function, find the[latex]\\,x\\text{-}[\/latex]intercepts by rewriting in standard form<\/strong>.<\/p>\n<ol id=\"fs-id1165134113976\" type=\"1\">\n<li>Substitute[latex]\\,a\\,[\/latex]and[latex]\\,b\\,[\/latex]into[latex]\\,h=-\\frac{b}{2a}.[\/latex]<\/li>\n<li>Substitute[latex]\\,x=h\\,[\/latex]into the general form of the quadratic function to find[latex]\\,k.[\/latex]<\/li>\n<li>Rewrite the quadratic in standard form using[latex]\\,h\\,[\/latex]and[latex]\\,k.[\/latex]<\/li>\n<li>Solve for when the output of the function will be zero to find the[latex]\\,x\\text{-}[\/latex]intercepts.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_02_08\" class=\"textbox examples\">\n<div id=\"fs-id1165134060458\">\n<div id=\"fs-id1165134224010\">\n<h3>Finding the <em>x<\/em>-Intercepts of a Parabola<\/h3>\n<p id=\"fs-id1165134224020\">Find the[latex]\\,x\\text{-}[\/latex]intercepts of the quadratic function[latex]\\,f\\left(x\\right)=2{x}^{2}+4x-4.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135524476\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135524476\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135524476\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135524478\">We begin by solving for when the output will be zero.<\/p>\n<div id=\"eip-id1165135701748\" class=\"unnumbered\">[latex]0=2{x}^{2}+4x-4[\/latex]<\/div>\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<div id=\"eip-id1165132111123\" class=\"unnumbered\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/div>\n<p id=\"fs-id1165137925165\">We know that[latex]\\,a=2.\\,[\/latex]Then we solve for[latex]\\,h\\,[\/latex]and[latex]\\,k.[\/latex]<\/p>\n<div id=\"eip-id1165137883696\" class=\"unnumbered\">[latex]\\begin{array}{cccccc}\\hfill h& =& -\\frac{b}{2a}\\hfill & \\hfill \\phantom{\\rule{2em}{0ex}}k& =& f\\left(-1\\right)\\hfill \\\\ & =& -\\frac{4}{2\\left(2\\right)}\\hfill & & =& 2{\\left(-1\\right)}^{2}+4\\left(-1\\right)-4\\hfill \\\\ & =& -1\\hfill & & =& -6\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134031332\">So now we can rewrite in standard form.<\/p>\n<div id=\"eip-id1165135480971\" class=\"unnumbered\">[latex]f\\left(x\\right)=2{\\left(x+1\\right)}^{2}-6[\/latex]<\/div>\n<p id=\"fs-id1165135381286\">We can now solve for when the output will be zero.<\/p>\n<div id=\"eip-id1165131840672\" class=\"unnumbered\">[latex]\\begin{array}{l}0=2{\\left(x+1\\right)}^{2}-6\\hfill \\\\ 6=2{\\left(x+1\\right)}^{2}\\hfill \\\\ 3={\\left(x+1\\right)}^{2}\\hfill \\\\ x+1=\u00b1\\sqrt{3}\\hfill \\\\ x=-1\u00b1\\sqrt{3}\\hfill \\end{array}[\/latex]<\/div>\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<p id=\"fs-id1165137843092\">We can check our work by graphing the given function on a graphing utility and observing the[latex]\\,x\\text{-}[\/latex]intercepts. See <a class=\"autogenerated-content\" href=\"#Figure_03_02_015\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_03_02_015\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135535\/CNX_Precalc_Figure_03_02_015.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\"><strong>Figure 15.<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137843086\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1709228\">We could have achieved the same results using the quadratic formula. Identify[latex]\\,a=2,b=4\\,[\/latex]and[latex]\\,c=-4.[\/latex]<\/p>\n<div id=\"fs-id1709335\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill x& =& \\frac{-b\u00b1\\sqrt{{b}^{2}-4ac}}{2a}\\hfill \\\\ & =& \\frac{-4\u00b1\\sqrt{{4}^{2}-4\\left(2\\right)\\left(-4\\right)}}{2\\left(2\\right)}\\hfill \\\\ & =& \\frac{-4\u00b1\\sqrt{48}}{4}\\hfill \\\\ & =& \\frac{-4\u00b1\\sqrt{3\\left(16\\right)}}{4}\\hfill \\\\ & =& -1\u00b1\\sqrt{3}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1728602\">So the <em>x<\/em>-intercepts occur at[latex]\\,\\left(-1-\\sqrt{3},0\\right)\\,[\/latex]and[latex]\\,\\left(-1+\\sqrt{3},0\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134109655\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_02_04\">\n<div id=\"fs-id1165134534212\">\n<p id=\"fs-id1165134534213\">In a <a href=\"#fs-id1165135362470\">Try It<\/a>, 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>\n<div id=\"fs-id1165135240997\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135240997\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135240997\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135240998\"><em>y<\/em>-intercept at (0, 13), No[latex]\\,x\\text{-}[\/latex]intercepts<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_02_10\" class=\"textbox examples\">\n<div id=\"fs-id1165134085786\">\n<div id=\"fs-id1165134085788\">\n<h3>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<ol id=\"fs-id1165137726798\" type=\"a\">\n<li>When does the ball reach the maximum height?<\/li>\n<li>What is the maximum height of the ball?<\/li>\n<li>When does the ball hit the ground?<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165135264685\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q85834\">Show Solution<\/span><\/p>\n<div id=\"q85834\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1165135264687\" type=\"a\">\n<li>The ball reaches the maximum height at the vertex of the parabola.\n<div id=\"eip-id1165135364049\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill h& =& -\\frac{80}{2\\left(-16\\right)}\\hfill \\\\ & =& \\frac{80}{32}\\hfill \\\\ & =& \\frac{5}{2}\\hfill \\\\ & =& 2.5\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135528870\">The ball reaches a maximum height after 2.5 seconds.<\/p>\n<\/li>\n<li>To find the maximum height, find the[latex]\\,y\\text{-}[\/latex]coordinate of the vertex of the parabola.\n<div id=\"eip-id1165134188974\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill k& =& H\\left(-\\frac{b}{2a}\\right)\\hfill \\\\ & =& H\\left(2.5\\right)\\hfill \\\\ & =& -16{\\left(2.5\\right)}^{2}+80\\left(2.5\\right)+40\\hfill \\\\ & =& 140\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135409750\">The ball reaches a maximum height of 140 feet.<\/p>\n<\/li>\n<li>To find when the ball hits the ground, we need to determine when the height is zero,[latex]\\,H\\left(t\\right)=0.[\/latex]\n<p id=\"fs-id1165135181381\">We use the quadratic formula.<\/p>\n<div id=\"eip-id1165133166054\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill t& =& \\frac{-80\u00b1\\sqrt{{80}^{2}-4\\left(-16\\right)\\left(40\\right)}}{2\\left(-16\\right)}\\hfill \\\\ & =& \\frac{-80\u00b1\\sqrt{8960}}{-32}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137783269\">Because the square root does not simplify nicely, we can use a calculator to approximate the values of the solutions.<\/p>\n<div id=\"eip-id1165134299806\" class=\"unnumbered\">[latex]\\begin{array}{l}\\hfill \\\\ \\hfill \\\\ \\begin{array}{lll}t=\\frac{-80-\\sqrt{8960}}{-32}\\approx 5.458\\hfill & \\text{or}\\hfill & t=\\frac{-80+\\sqrt{8960}}{-32}\\approx -0.458\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135347508\">The second answer is outside the reasonable domain of our model, so we conclude the ball will hit the ground after about 5.458 seconds. See <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2319&amp;action=edit#Figure_03_02_016\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_03_02_016\" class=\"small wp-caption aligncenter\">\n<div style=\"width: 497px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135536\/CNX_Precalc_Figure_03_02_016.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\"><strong>Figure 16.<\/strong><\/p>\n<\/div>\n<\/div>\n<p id=\"eip-id1827109\">Note that the graph does not represent the physical path of the ball upward and downward. Keep the quantities on each axis in mind while interpreting the graph.<\/p>\n<\/li>\n<\/ol>\n<p id=\"eip-id1827109\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134081290\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_02_05\">\n<div id=\"fs-id1165134081299\">\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<ol id=\"fs-id1165135152195\" type=\"a\">\n<li>When does the rock reach the maximum height?<\/li>\n<li>What is the maximum height of the rock?<\/li>\n<li>When does the rock hit the ocean?<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1165135152213\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135152213\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135152213\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137843187\">3 seconds256 feet7 seconds<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137475325\" class=\"precalculus media\">\n<p id=\"fs-id1165137475332\">Access these online resources for additional instruction and practice with quadratic equations.<\/p>\n<ul id=\"fs-id1165137475336\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphquadgen\">Graphing Quadratic Functions in General Form<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphquadstan\">Graphing Quadratic Functions in Standard Form<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/quadfuncrev\">Quadratic Function Review<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/characterquad\">Characteristics of a Quadratic Function<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134205927\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\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>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<\/div>\n<div id=\"fs-id1165135426424\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\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[latex]\\,x\\text{-}[\/latex]intercepts, are the points at which the parabola crosses the[latex]\\,x\\text{-}[\/latex]axis. The[latex]\\,y\\text{-}[\/latex]intercept is the point at which the parabola crosses the[latex]\\,y\\text{-}[\/latex]axis. See <a class=\"autogenerated-content\" href=\"#Example_03_02_01\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_03_02_07\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_03_02_08\">(Figure)<\/a>.<\/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. See <a class=\"autogenerated-content\" href=\"#Example_03_02_02\">(Figure)<\/a>.<\/li>\n<li>The vertex can be found from an equation representing a quadratic function. See <a class=\"autogenerated-content\" href=\"#Example_03_02_03\">(Figure)<\/a><strong>.<\/strong><\/li>\n<li>The domain of a quadratic function is all real numbers. The range varies with the function. See <a class=\"autogenerated-content\" href=\"#Example_03_02_04\">(Figure)<\/a>.<\/li>\n<li>A quadratic function\u2019s minimum or maximum value is given by the[latex]\\,y\\text{-}[\/latex]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. See <a class=\"autogenerated-content\" href=\"#Example_03_02_05\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_03_02_06\">(Figure)<\/a>.<\/li>\n<li>The vertex and the intercepts can be identified and interpreted to solve real-world problems. See <a class=\"autogenerated-content\" href=\"#Example_03_02_10\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135361324\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165135361327\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165135361332\">\n<div id=\"fs-id1165135361334\">\n<p id=\"fs-id1165135361336\">Explain the advantage of writing a quadratic function in standard form.<\/p>\n<\/div>\n<div id=\"fs-id1165135361339\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135361339\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135361339\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135361340\">When written in that form, the vertex can be easily identified.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134367971\">\n<div id=\"fs-id1165134367972\">\n<p id=\"fs-id1165134367973\">How can the vertex of a parabola be used in solving real-world problems?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134367976\">\n<div id=\"fs-id1165134367977\">\n<p id=\"fs-id1165134367978\">Explain why the condition of[latex]\\,a\\ne 0\\,[\/latex]is imposed in the definition of the quadratic function.<\/p>\n<\/div>\n<div id=\"fs-id1165135453265\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135453265\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135453265\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135453266\">If[latex]\\,a=0\\,[\/latex]then the function becomes a linear function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134094470\">\n<div id=\"fs-id1165134094472\">\n<p id=\"fs-id1165134094473\">What is another name for the standard form of a quadratic function?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134094476\">\n<div id=\"fs-id1165134094477\">\n<p id=\"fs-id1165134094478\">What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?<\/p>\n<\/div>\n<div id=\"fs-id1165133276250\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165133276250\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165133276250\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165133276252\">If possible, we can use factoring. Otherwise, we can use the quadratic formula.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133276257\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165133276262\">For the following exercises, rewrite the quadratic functions in standard form and give the vertex.<\/p>\n<div id=\"fs-id1165133276266\">\n<div id=\"fs-id1165133276268\">\n<p id=\"fs-id1165133276271\">[latex]f\\left(x\\right)={x}^{2}-12x+32[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135336013\">\n<div id=\"fs-id1165133065125\">\n<p id=\"fs-id1165133065126\">[latex]g\\left(x\\right)={x}^{2}+2x-3[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165132963686\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165132963686\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165132963686\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165132963687\">[latex]f\\left(x\\right)={\\left(x+1\\right)}^{2}-2,\\,[\/latex]Vertex[latex]\\,\\left(-1,-4\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134199469\">\n<div id=\"fs-id1165134199470\">\n<p id=\"fs-id1165134199472\">[latex]f\\left(x\\right)={x}^{2}-x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135536371\">\n<div id=\"fs-id1165135536373\">\n<p id=\"fs-id1165135536374\">[latex]f\\left(x\\right)={x}^{2}+5x-2[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135584090\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135584090\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135584090\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135584091\">[latex]f\\left(x\\right)={\\left(x+\\frac{5}{2}\\right)}^{2}-\\frac{33}{4},\\,[\/latex]Vertex[latex]\\,\\left(-\\frac{5}{2},-\\frac{33}{4}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134476611\">\n<div id=\"fs-id1165134476612\">\n<p id=\"fs-id1165134476613\">[latex]h\\left(x\\right)=2{x}^{2}+8x-10[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133328085\">\n<div id=\"fs-id1165133328087\">\n<p id=\"fs-id1165133328088\">[latex]k\\left(x\\right)=3{x}^{2}-6x-9[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135382110\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135382110\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135382110\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135382112\">[latex]f\\left(x\\right)=3{\\left(x-1\\right)}^{2}-12,\\,[\/latex]Vertex[latex]\\,\\left(1,-12\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134422869\">\n<div id=\"fs-id1165134422870\">\n<p id=\"fs-id1165134422871\">[latex]f\\left(x\\right)=2{x}^{2}-6x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135409784\">\n<div id=\"fs-id1165135409786\">\n<p id=\"fs-id1165135409787\">[latex]f\\left(x\\right)=3{x}^{2}-5x-1[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137749740\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137749740\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137749740\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137749741\">[latex]f\\left(x\\right)=3{\\left(x-\\frac{5}{6}\\right)}^{2}-\\frac{37}{12},\\,[\/latex]Vertex[latex]\\,\\left(\\frac{5}{6},-\\frac{37}{12}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135258891\">For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.<\/p>\n<div id=\"fs-id1165135258896\">\n<div id=\"fs-id1165135258897\">\n<p id=\"fs-id1165135258898\">[latex]y\\left(x\\right)=2{x}^{2}+10x+12[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133344089\">\n<div id=\"fs-id1165133344091\">\n<p id=\"fs-id1165133344092\">[latex]f\\left(x\\right)=2{x}^{2}-10x+4[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135254602\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135254602\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135254602\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135254604\">Minimum is[latex]\\,-\\frac{17}{2}\\,[\/latex]and occurs at[latex]\\,\\frac{5}{2}.\\,[\/latex]Axis of symmetry is[latex]\\,x=\\frac{5}{2}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134091244\">\n<div id=\"fs-id1165134091247\">\n<p id=\"fs-id1165134091248\">[latex]f\\left(x\\right)=-{x}^{2}+4x+3[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134278554\">\n<div id=\"fs-id1165134278556\">\n<p id=\"fs-id1165134278557\">[latex]f\\left(x\\right)=4{x}^{2}+x-1[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137852807\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137852807\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137852807\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137852808\">Minimum is[latex]\\,-\\frac{17}{16}\\,[\/latex]and occurs at[latex]\\,-\\frac{1}{8}.\\,[\/latex]Axis of symmetry is[latex]\\,x=-\\frac{1}{8}.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135193269\">\n<div id=\"fs-id1165135193271\">\n<p id=\"fs-id1165135193272\">[latex]h\\left(t\\right)=-4{t}^{2}+6t-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135380708\">\n<div id=\"fs-id1165135380710\">\n<p id=\"fs-id1165135506299\">[latex]f\\left(x\\right)=\\frac{1}{2}{x}^{2}+3x+1[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134328315\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134328315\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134328315\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134328316\">Minimum is[latex]\\,-\\frac{7}{2}\\,[\/latex]and occurs at[latex]\\,-3.\\,[\/latex] Axis of symmetry is[latex]\\,x=-3.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134362815\">\n<div id=\"fs-id1165134362817\">\n<p id=\"fs-id1165134362818\">[latex]f\\left(x\\right)=-\\frac{1}{3}{x}^{2}-2x+3[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135419821\">For the following exercises, determine the domain and range of the quadratic function.<\/p>\n<div id=\"fs-id1165135419824\">\n<div id=\"fs-id1165135419827\">\n<p id=\"fs-id1165135419828\">[latex]f\\left(x\\right)={\\left(x-3\\right)}^{2}+2[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135353077\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135353077\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135353077\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135353078\">Domain is[latex]\\,\\left(-\\infty ,\\infty \\right).\\,[\/latex]Range is[latex]\\,\\left[2,\\infty \\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137681943\">\n<div id=\"fs-id1165137681945\">\n<p id=\"fs-id1165137681946\">[latex]f\\left(x\\right)=-2{\\left(x+3\\right)}^{2}-6[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135528310\">\n<div id=\"fs-id1165135528312\">\n<p id=\"fs-id1165135528313\">[latex]f\\left(x\\right)={x}^{2}+6x+4[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135548952\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135548952\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135548952\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135548954\">Domain is[latex]\\,\\left(-\\infty ,\\infty \\right).\\,[\/latex]Range is[latex]\\,\\left[-5,\\infty \\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134199532\">\n<div id=\"fs-id1165134199534\">\n<p id=\"fs-id1165134199535\">[latex]f\\left(x\\right)=2{x}^{2}-4x+2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134138647\">\n<div id=\"fs-id1165134138649\">\n<p id=\"fs-id1165134138650\">[latex]k\\left(x\\right)=3{x}^{2}-6x-9[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135470085\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135470085\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135470085\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135470086\">Domain is[latex]\\,\\left(-\\infty ,\\infty \\right).\\,[\/latex]Range is[latex]\\,\\left[-12,\\infty \\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165133252496\">For the following exercises, use the vertex[latex]\\,\\left(h,k\\right)\\,[\/latex]and a point on the graph[latex]\\,\\left(x,y\\right)\\,[\/latex]to find the general form of the equation of the quadratic function.<\/p>\n<div id=\"fs-id1165137898126\">\n<div id=\"fs-id1165137898128\">\n<p id=\"fs-id1165137898129\">[latex]\\left(h,k\\right)=\\left(2,0\\right),\\left(x,y\\right)=\\left(4,4\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135403325\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135403325\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135403325\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135403326\">[latex]f\\left(x\\right)={x}^{2}-4x+4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134079556\">\n<div id=\"fs-id1165134079558\">\n<p id=\"fs-id1165134079559\">[latex]\\left(h,k\\right)=\\left(-2,-1\\right),\\left(x,y\\right)=\\left(-4,3\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134485654\">\n<div id=\"fs-id1165134485656\">\n<p id=\"fs-id1165134485657\">[latex]\\left(h,k\\right)=\\left(0,1\\right),\\left(x,y\\right)=\\left(2,5\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135551841\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135551841\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135551841\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135551842\">[latex]f\\left(x\\right)={x}^{2}+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133077523\">\n<div id=\"fs-id1165133077525\">\n<p id=\"fs-id1165133077526\">[latex]\\left(h,k\\right)=\\left(2,3\\right),\\left(x,y\\right)=\\left(5,12\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132912701\">\n<div id=\"fs-id1165132912704\">\n<p id=\"fs-id1165132912705\">[latex]\\left(h,k\\right)=\\left(-5,3\\right),\\left(x,y\\right)=\\left(2,9\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137843237\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137843237\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137843237\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137843238\">[latex]f\\left(x\\right)=\\frac{6}{49}{x}^{2}+\\frac{60}{49}x+\\frac{297}{49}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133252567\">\n<div id=\"fs-id1165135536384\">\n<p id=\"fs-id1165135536385\">[latex]\\left(h,k\\right)=\\left(3,2\\right),\\left(x,y\\right)=\\left(10,1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135412141\">\n<div id=\"fs-id1165135412143\">\n<p id=\"fs-id1165135412144\">[latex]\\left(h,k\\right)=\\left(0,1\\right),\\left(x,y\\right)=\\left(1,0\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135384995\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135384995\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135384995\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135384996\">[latex]f\\left(x\\right)=-{x}^{2}+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134479000\">\n<div id=\"fs-id1165134479002\">\n<p id=\"fs-id1165134479003\">[latex]\\left(h,k\\right)=\\left(1,0\\right),\\left(x,y\\right)=\\left(0,1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133095093\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165133095098\">For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.<\/p>\n<div id=\"fs-id1165133095103\">\n<div id=\"fs-id1165133095105\">\n<p id=\"fs-id1165133095106\">[latex]f\\left(x\\right)={x}^{2}-2x[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134148456\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134148456\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134148456\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165133280629\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135539\/CNX_Precalc_Figure_03_02_201.jpg\" alt=\"Graph of f(x) = x^2-2x\" \/><\/span><\/p>\n<p id=\"fs-id1165133280642\">Vertex[latex]\\left(1,\\text{ }-1\\right),\\,[\/latex]Axis of symmetry is[latex]\\,x=1.\\,[\/latex]Intercepts are[latex]\\,\\left(0,0\\right), \\left(2,0\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135646115\">\n<div id=\"fs-id1165135646117\">\n<p id=\"fs-id1165135646118\">[latex]f\\left(x\\right)={x}^{2}-6x-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135692195\">\n<div id=\"fs-id1165135692197\">\n<p id=\"fs-id1165135692198\">[latex]f\\left(x\\right)={x}^{2}-5x-6[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135697888\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135697888\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135697888\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165134149910\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135545\/CNX_Precalc_Figure_03_02_203.jpg\" alt=\"Graph of f(x)x^2-5x-6\" \/><\/span><\/p>\n<p id=\"fs-id1165134149922\">Vertex[latex]\\,\\left(\\frac{5}{2},\\frac{-49}{4}\\right),\\,[\/latex]Axis of symmetry is[latex]\\,\\left(0,-6\\right),\\left(-1,0\\right),\\left(6,0\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135205053\">\n<div id=\"fs-id1165135205055\">\n<p id=\"fs-id1165135205056\">[latex]f\\left(x\\right)={x}^{2}-7x+3[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135500665\">\n<div id=\"fs-id1165135500667\">\n<p id=\"fs-id1165135500668\">[latex]f\\left(x\\right)=-2{x}^{2}+5x-8[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135347648\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135347648\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135347648\" class=\"hidden-answer\" style=\"display: none\"><span id=\"fs-id1165135347655\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135549\/CNX_Precalc_Figure_03_02_205.jpg\" alt=\"Graph of f(x)=-2x^2+5x-8\" \/><\/span><\/p>\n<p id=\"fs-id1165135347667\">Vertex[latex]\\,\\left(\\frac{5}{4}, -\\frac{39}{8}\\right),\\,[\/latex]Axis of symmetry is[latex]\\,x=\\frac{5}{4}.\\,[\/latex]Intercepts are[latex]\\,\\left(0, -8\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135639341\">\n<div id=\"fs-id1165135639343\">\n<p id=\"fs-id1165135639344\">[latex]f\\left(x\\right)=4{x}^{2}-12x-3[\/latex]<\/p>\n<\/div>\n<div id=\"eip-id1165134560567\"><span id=\"fs-id1165134391611\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135556\/CNX_Precalc_Figure_03_02_206.jpg\" alt=\"Graph of f(x)=4x^2-12x-3\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1165134391625\">For the following exercises, write the equation for the graphed quadratic function.<\/p>\n<div id=\"fs-id1165134391628\">\n<div id=\"fs-id1165134391630\"><span id=\"fs-id1165134391635\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135558\/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).\" \/><\/span><\/div>\n<div id=\"fs-id1165134391648\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134391648\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134391648\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134391649\">[latex]f\\left(x\\right)={x}^{2}-4x+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135262595\">\n<div id=\"fs-id1165135262597\"><span id=\"fs-id1165135262603\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135603\/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)\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165135262616\">\n<div id=\"fs-id1165135262618\"><span id=\"fs-id1165135262624\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135605\/CNX_Precalc_Figure_03_02_209.jpg\" alt=\"Graph of a negative parabola with a vertex at (2, 7).\" \/><\/span><\/div>\n<div id=\"fs-id1165135536430\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135536430\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135536430\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135536431\">[latex]f\\left(x\\right)=-2{x}^{2}+8x-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134162152\">\n<div id=\"fs-id1165134162154\"><span id=\"fs-id1165134162160\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135607\/CNX_Precalc_Figure_03_02_210.jpg\" alt=\"Graph of a negative parabola with a vertex at (-1, 2).\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165134162173\">\n<div id=\"fs-id1165134162175\"><span id=\"fs-id1165134196136\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135609\/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).\" \/><\/span><\/div>\n<div id=\"fs-id1165134196148\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134196148\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134196148\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134196149\">[latex]f\\left(x\\right)=\\frac{1}{2}{x}^{2}-3x+\\frac{7}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137940515\">\n<div id=\"fs-id1165137940518\"><span id=\"fs-id1165131959568\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135619\/CNX_Precalc_Figure_03_02_212.jpg\" alt=\"Graph of a negative parabola with a vertex at (-2, 3).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165131959583\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<p id=\"fs-id1165131959588\">For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.<\/p>\n<div id=\"fs-id1165131959593\">\n<div id=\"fs-id1165131959596\">\n<table id=\"fs-id1165131959598\" class=\"unnumbered\" summary=\"..\">\n<caption>\u00a0<\/caption>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/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><strong>[latex]y[\/latex]<\/strong><\/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<\/div>\n<div id=\"fs-id1165133247929\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165133247929\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165133247929\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165133247930\">[latex]f\\left(x\\right)={x}^{2}+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135485986\">\n<div id=\"fs-id1165135485987\">\n<table id=\"fs-id1165135485989\" class=\"unnumbered\" summary=\"..\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/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><strong>[latex]y[\/latex]<\/strong><\/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<\/div>\n<\/div>\n<div id=\"fs-id1165133248546\">\n<div id=\"fs-id1165133248548\">\n<table id=\"fs-id1165133248550\" class=\"unnumbered\" summary=\"..\">\n<caption>\u00a0<\/caption>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/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><strong>[latex]y[\/latex]<\/strong><\/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<\/div>\n<div id=\"fs-id1165137900973\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137900973\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137900973\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137900974\">[latex]f\\left(x\\right)=2-{x}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134138600\">\n<div id=\"fs-id1165134138602\">\n<table id=\"fs-id1165134138604\" class=\"unnumbered\" summary=\"..\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/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><strong>[latex]y[\/latex]<\/strong><\/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<\/div>\n<\/div>\n<div id=\"fs-id1165134328259\">\n<div id=\"fs-id1165134328261\">\n<table id=\"fs-id1165134328263\" class=\"unnumbered\" summary=\"..\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/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><strong>[latex]y[\/latex]<\/strong><\/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<\/div>\n<div id=\"fs-id1165135537351\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135537351\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135537351\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135537352\">[latex]f\\left(x\\right)=2{x}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135513626\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1165135513632\">For the following exercises, use a calculator to find the answer.<\/p>\n<div id=\"fs-id1165135513635\">\n<div id=\"fs-id1165135513637\">\n<p id=\"fs-id1165135513639\">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]<\/p>\n<p id=\"fs-id1165133402104\">What appears to be the effect of changing the coefficient?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133402109\">\n<div id=\"fs-id1165133402111\">\n<p id=\"fs-id1165133402112\">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<\/div>\n<div id=\"fs-id1165137843141\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137843141\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137843141\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137843142\">The graph is shifted up or down (a vertical shift).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137843146\">\n<div id=\"fs-id1165137843148\">\n<p id=\"fs-id1165137843151\">Graph 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]<\/p>\n<p id=\"fs-id1165137697907\">What appears to be the effect of adding or subtracting those numbers?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137697912\">\n<div id=\"fs-id1165137697915\">\n<p id=\"fs-id1165137697916\">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[latex]\\,x\\,[\/latex]is 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<\/div>\n<div id=\"fs-id1165135533787\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135533787\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135533787\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135533788\">50 feet<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135533793\">\n<div id=\"fs-id1165135533795\">\n<p id=\"fs-id1165135533796\">A 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[latex]\\,|x|\\,[\/latex]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<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135629622\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id1165135523290\">For the following exercises, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function.<\/p>\n<div id=\"fs-id1165135523294\">\n<div id=\"fs-id1165135523296\">\n<p id=\"fs-id1165135523298\">Vertex[latex]\\,\\left(1,-2\\right),\\,[\/latex]opens up.<\/p>\n<\/div>\n<div id=\"fs-id1165135170999\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135170999\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135170999\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135171000\">Domain is[latex]\\,\\left(-\\infty ,\\infty \\right).\\,[\/latex]Range is[latex]\\,\\left[-2,\\infty \\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137897840\">\n<div id=\"fs-id1165137897842\">\n<p id=\"fs-id1165137897843\">Vertex[latex]\\,\\left(-1,2\\right)\\,[\/latex]opens down.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135609192\">\n<div id=\"fs-id1165135609194\">\n<p id=\"fs-id1165135609195\">Vertex[latex]\\,\\left(-5,11\\right),\\,[\/latex]opens down.<\/p>\n<\/div>\n<div id=\"fs-id1165135609234\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135609234\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135609234\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135609235\">Domain is[latex]\\,\\left(-\\infty ,\\infty \\right)\\,[\/latex]Range is[latex]\\,\\left(-\\infty ,11\\right].[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135501919\">\n<div id=\"fs-id1165135501921\">\n<p id=\"fs-id1165135501922\">Vertex[latex]\\,\\left(-100,100\\right),\\,[\/latex]opens up.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135501959\">For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function.<\/p>\n<div id=\"fs-id1165135501964\">\n<div id=\"fs-id1165135501966\">\n<p id=\"fs-id1165135501967\">Contains[latex]\\,\\left(1,1\\right)\\,[\/latex]and has shape of[latex]\\,f\\left(x\\right)=2{x}^{2}.\\,[\/latex]Vertex is on the[latex]\\,y\\text{-}[\/latex]axis.<\/p>\n<\/div>\n<div id=\"fs-id1165137642675\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137642675\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137642675\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137642676\">[latex]f\\left(x\\right)=2{x}^{2}-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137642716\">\n<div id=\"fs-id1165133289615\">\n<p id=\"fs-id1165133289616\">Contains[latex]\\,\\left(-1,4\\right)\\,[\/latex]and has the shape of[latex]\\,f\\left(x\\right)=2{x}^{2}.\\,[\/latex]Vertex is on the[latex]\\,y\\text{-}[\/latex]axis.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135551155\">\n<div id=\"fs-id1165135551157\">\n<p id=\"fs-id1165135551158\">Contains[latex]\\,\\left(2,3\\right)\\,[\/latex]and has the shape of[latex]\\,f\\left(x\\right)=3{x}^{2}.\\,[\/latex]Vertex is on the[latex]\\,y\\text{-}[\/latex]axis.<\/p>\n<\/div>\n<div id=\"fs-id1165135341250\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135341250\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135341250\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135341251\">[latex]f\\left(x\\right)=3{x}^{2}-9[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135525847\">\n<div id=\"fs-id1165135525850\">\n<p id=\"fs-id1165135525851\">Contains[latex]\\,\\left(1,-3\\right)\\,[\/latex]and has the shape of[latex]\\,f\\left(x\\right)=-{x}^{2}.\\,[\/latex]Vertex is on the[latex]\\,y\\text{-}[\/latex]axis.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135442559\">\n<div id=\"fs-id1165135442561\">\n<p id=\"fs-id1165135442562\">Contains[latex]\\,\\left(4,3\\right)\\,[\/latex]and has the shape of[latex]\\,f\\left(x\\right)=5{x}^{2}.\\,[\/latex]Vertex is on the[latex]\\,y\\text{-}[\/latex]axis.<\/p>\n<\/div>\n<div id=\"fs-id1165135646094\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135646094\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135646094\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135646095\">[latex]f\\left(x\\right)=5{x}^{2}-77[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137901057\">\n<div id=\"fs-id1165137901059\">\n<p id=\"fs-id1165137901060\">Contains[latex]\\,\\left(1,-6\\right)\\,[\/latex]has the shape of[latex]\\,f\\left(x\\right)=3{x}^{2}.\\,[\/latex]Vertex has x-coordinate of[latex]\\,-1.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165131857387\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<div id=\"fs-id1165131857392\">\n<div id=\"fs-id1165131857394\">\n<p id=\"fs-id1165131857395\">Find the dimensions of the rectangular corral producing the greatest enclosed area given 200 feet of fencing.<\/p>\n<\/div>\n<div id=\"fs-id1165131857400\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165131857400\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165131857400\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165131857401\">50 feet by 50 feet. Maximize[latex]\\,f\\left(x\\right)=-{x}^{2}+100x.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165131857448\">\n<div id=\"fs-id1165137940524\">\n<p id=\"fs-id1165137940525\">Find 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<\/div>\n<\/div>\n<div id=\"fs-id1165137940531\">\n<div id=\"fs-id1165137940533\">\n<p id=\"fs-id1165137940534\">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<\/div>\n<div id=\"fs-id1165137940538\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165137940538\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165137940538\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165137940540\">125 feet by 62.5 feet. Maximize[latex]\\,f\\left(x\\right)=-2{x}^{2}+250x.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137940588\">\n<div id=\"fs-id1165134089391\">\n<p id=\"fs-id1165134089392\">Among all of the pairs of numbers whose sum is 6, find the pair with the largest product. What is the product?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134089397\">\n<div id=\"fs-id1165134089399\">\n<p id=\"fs-id1165134089400\">Among all of the pairs of numbers whose difference is 12, find the pair with the smallest product. What is the product?<\/p>\n<\/div>\n<div id=\"fs-id1165134089405\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165134089405\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165134089405\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134089406\">[latex]6\\,[\/latex]and[latex]\\,-6;\\,[\/latex]product is \u201336; maximize[latex]\\,f\\left(x\\right)={x}^{2}+12x.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135364089\">\n<div id=\"fs-id1165135364091\">\n<p id=\"fs-id1165135364092\">Suppose that the price per unit in dollars of a cell phone production is modeled by[latex]\\,p=\\text{\\$}45-0.0125x,\\,[\/latex]where[latex]\\,x\\,[\/latex]is in thousands of phones produced, and the revenue represented by thousands of dollars is[latex]\\,R=x\\cdot p.\\,[\/latex]Find the production level that will maximize revenue.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135571732\">\n<div id=\"fs-id1165135571734\">\n<p id=\"fs-id1165135571735\">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<\/div>\n<div id=\"fs-id1165135394030\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135394030\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135394030\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135394031\">2909.56 meters<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135394035\">\n<div id=\"fs-id1165135394038\">\n<p id=\"fs-id1165135394039\">A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time, in seconds, is given by[latex]\\,h\\left(t\\right)=-4.9{t}^{2}+24t+8.\\,[\/latex]How long does it take to reach maximum height?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135449627\">\n<div id=\"fs-id1165135449629\">\n<p id=\"fs-id1165135449630\">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<\/div>\n<div id=\"fs-id1165135449636\" class=\"solution textbox shaded\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1165135449636\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1165135449636\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165135449637\">$10.70<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135449642\">\n<div id=\"fs-id1165135449644\">\n<p id=\"fs-id1165135449645\">A farmer finds that if she plants 75 trees per acre, each tree will yield 20 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165135449657\">\n<dt>axis of symmetry<\/dt>\n<dd id=\"fs-id1165135449662\">a vertical line drawn through the vertex of a parabola, that opens up or down, around which the parabola is symmetric; it is defined by[latex]\\,x=-\\frac{b}{2a}.[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135502777\">\n<dt>general form of a quadratic function<\/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[latex]\\,a,b,\\,[\/latex]and[latex]\\,c\\,[\/latex]are real numbers and[latex]\\,a\\ne 0.[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137931270\">\n<dt>roots<\/dt>\n<dd id=\"fs-id1165137931276\">in a given function, the values of[latex]\\,x\\,[\/latex]at which[latex]\\,y=0[\/latex], also called zeros<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137931314\">\n<dt>standard form of a quadratic function<\/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\">\n<dt>vertex<\/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\">\n<dt>vertex form of a quadratic function<\/dt>\n<dd id=\"fs-id1165135623630\">another name for the standard form of a quadratic function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135623634\">\n<dt>zeros<\/dt>\n<dd id=\"fs-id1165135623639\">in a given function, the values of[latex]\\,x\\,[\/latex]at which[latex]\\,y=0[\/latex], also called roots<\/dd>\n<\/dl>\n<\/div>\n<\/div>\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-2319\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Algebra and Trigonometry. <strong>Authored by<\/strong>: Jay Abramson, et. al. <strong>Provided by<\/strong>: OpenStax CNX. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1\">http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1<\/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":53384,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Algebra and Trigonometry\",\"author\":\"Jay Abramson, et. al\",\"organization\":\"OpenStax CNX\",\"url\":\"http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/13ac107a-f15f-49d2-97e8-60ab2e3b519c@11.1\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2319","chapter","type-chapter","status-publish","hentry"],"part":2278,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters\/2319","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/users\/53384"}],"version-history":[{"count":2,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters\/2319\/revisions"}],"predecessor-version":[{"id":3625,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters\/2319\/revisions\/3625"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/parts\/2278"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapters\/2319\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/media?parent=2319"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/pressbooks\/v2\/chapter-type?post=2319"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/contributor?post=2319"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-json\/wp\/v2\/license?post=2319"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}