{"id":4977,"date":"2020-11-11T19:53:56","date_gmt":"2020-11-11T19:53:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebra\/?post_type=chapter&#038;p=4977"},"modified":"2021-08-11T22:51:38","modified_gmt":"2021-08-11T22:51:38","slug":"analysis-of-quadratic-functions-custom-edit","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/chapter\/analysis-of-quadratic-functions-custom-edit\/","title":{"raw":"Analysis of Quadratic Functions","rendered":"Analysis of Quadratic Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul class=\"ul1\">\r\n \t<li class=\"li2\"><span class=\"s1\">Use the quadratic formula and factoring to find both real and complex roots ([latex]x[\/latex]-intercepts) of quadratic functions.<\/span><\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Use algebra to find the [latex]y[\/latex]-intercepts of a quadratic function.<\/span><\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Solve problems involving the roots and intercepts of a quadratic function.<\/span><\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Use the discriminant to determine the nature (real or complex) and quantity of solutions to quadratic equations.<\/span><\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Determine a quadratic function\u2019s minimum or maximum value.<\/span><\/li>\r\n \t<li class=\"li3\"><span class=\"s4\">Solve problems involving a quadratic function\u2019s minimum or maximum value.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\nIn this section we will investigate quadratic functions further, including solving\u00a0problems involving area and projectile motion. Working with quadratic functions can be less complex than working with higher degree polynomial functions, so they provide a good opportunity for a detailed study of function behavior.\r\n<h2>Intercepts of Quadratic Functions<\/h2>\r\nWe also need to find intercepts of quadratic equations for graphing parabolas. Recall that we find the <span class=\"s1\">[latex]y[\/latex]<\/span>-intercept of a quadratic by evaluating the function at an input of zero, and we find the <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts at locations where the output is zero. Notice\u00a0that the number of <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts can vary depending upon the location of the graph.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170357\/CNX_Precalc_Figure_03_02_0132.jpg\" alt=\"Three graphs where the first graph shows a parabola with no x-intercept, the second is a parabola with one \u2013intercept, and the third parabola is of two x-intercepts.\" width=\"975\" height=\"317\" \/> Number of <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts of a parabola[\/caption]Mathematicians also define <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts as roots of the quadratic function.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving a Quadratic Equation with the Quadratic Formula<\/h3>\r\nSolve [latex]{x}^{2}+x+2=0[\/latex].\r\n\r\n[reveal-answer q=\"757696\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"757696\"]\r\n\r\nLet\u2019s begin by writing the quadratic formula: [latex]x=\\dfrac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex].\r\n\r\nWhen applying the <strong>quadratic formula<\/strong>, we identify the coefficients [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex]. For the equation [latex]{x}^{2}+x+2=0[\/latex], we have [latex]a=1[\/latex], [latex]b=1[\/latex], and [latex]c=2[\/latex].\u00a0Substituting these values into the formula we have:\r\n<p style=\"text-align: center;\">[latex]\\begin{align}x&amp;=\\dfrac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a} \\\\[1.5mm] &amp;=\\dfrac{-1\\pm \\sqrt{{1}^{2}-4\\cdot 1\\cdot \\left(2\\right)}}{2\\cdot 1} \\\\[1.5mm] &amp;=\\dfrac{-1\\pm \\sqrt{1 - 8}}{2} \\\\[1.5mm] &amp;=\\dfrac{-1\\pm \\sqrt{-7}}{2} \\\\[1.5mm] &amp;=\\dfrac{-1\\pm i\\sqrt{7}}{2} \\end{align}[\/latex]<\/p>\r\nThe solutions to the equation are [latex]x=\\dfrac{-1+i\\sqrt{7}}{2}[\/latex] and [latex]x=\\dfrac{-1-i\\sqrt{7}}{2}[\/latex] or [latex]x=-\\dfrac{1}{2}+\\dfrac{i\\sqrt{7}}{2}[\/latex] and [latex]x=-\\dfrac{1}{2}-\\dfrac{i\\sqrt{7}}{2}[\/latex].\r\n<h4>Analysis of the Solution<\/h4>\r\nThis quadratic equation has only non-real solutions.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a quadratic function, find the <em>x<\/em>-intercepts by rewriting in standard form.<\/h3>\r\n<ol>\r\n \t<li>Substitute <span class=\"s1\">[latex]a[\/latex]<\/span>\u00a0and <span class=\"s1\">[latex]b[\/latex]<\/span>\u00a0into [latex]h=-\\dfrac{b}{2a}[\/latex].<\/li>\r\n \t<li>Substitute [latex]x=h[\/latex]\u00a0into the general form of the quadratic function to find <span class=\"s1\">[latex]k[\/latex]<\/span>.<\/li>\r\n \t<li>Rewrite the quadratic in standard form using <span class=\"s1\">[latex]h[\/latex]<\/span>\u00a0and <span class=\"s1\">[latex]k[\/latex]<\/span>.<\/li>\r\n \t<li>Solve for when the output of the function will be zero to find the <span class=\"s1\">[latex]x[\/latex]<\/span><em>-<\/em>intercepts.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Roots\u00a0of a Parabola<\/h3>\r\nFind the <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts of the quadratic function [latex]f\\left(x\\right)=2{x}^{2}+4x - 4[\/latex].\r\n\r\n[reveal-answer q=\"201989\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"201989\"]\r\n\r\nWe begin by solving for when the output will be zero.\r\n<p style=\"text-align: center;\">[latex]0=2{x}^{2}+4x - 4[\/latex]<\/p>\r\nBecause the quadratic is not easily factorable in this case, we solve for the intercepts by first rewriting the quadratic in standard form.\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/p>\r\nWe know that [latex]a=2[\/latex]. Then we solve for\u00a0<span class=\"s1\">[latex]h[\/latex]<\/span>\u00a0and <span class=\"s1\">[latex]k[\/latex].<\/span>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;h=-\\dfrac{b}{2a}=-\\dfrac{4}{2\\left(2\\right)}=-1\\\\[2mm]&amp;\\end{align}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}k&amp;=f\\left(h\\right)\\\\&amp;=f\\left(-1\\right)\\\\&amp;=2{\\left(-1\\right)}^{2}+4\\left(-1\\right)-4\\\\&amp;=-6\\end{align}[\/latex]<\/p>\r\nSo now we can rewrite in standard form.\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=2{\\left(x+1\\right)}^{2}-6[\/latex]<\/p>\r\nWe can now solve for when the output will be zero.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;0=2{\\left(x+1\\right)}^{2}-6 \\\\ &amp;6=2{\\left(x+1\\right)}^{2} \\\\ &amp;3={\\left(x+1\\right)}^{2} \\\\ &amp;x+1=\\pm \\sqrt{3} \\\\ &amp;x=-1\\pm \\sqrt{3} \\end{align}[\/latex]<\/p>\r\nThe graph has [latex]x[\/latex]<em>-<\/em>intercepts at [latex]\\left(-1-\\sqrt{3},0\\right)[\/latex] and [latex]\\left(-1+\\sqrt{3},0\\right)[\/latex].\r\n<h4>Analysis of the Solution<\/h4>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170402\/CNX_Precalc_Figure_03_02_0152.jpg\" alt=\"Graph of a parabola which has the following x-intercepts (-2.732, 0) and (0.732, 0).\" width=\"487\" height=\"517\" \/>\r\n\r\nWe can check our work by graphing the given function on a graphing utility and observing the roots.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=121416&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"150\" data-mce-fragment=\"1\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Applying the Vertex and <em>x<\/em>-Intercepts of a Parabola<\/h3>\r\nA ball is thrown upward from the top of a 40 foot high building at a speed of 80 feet per second. The ball\u2019s height above ground can be modeled by the equation [latex]H\\left(t\\right)=-16{t}^{2}+80t+40[\/latex].\r\n\r\na. When does the ball reach the maximum height?\r\n\r\nb. What is the maximum height of the ball?\r\n\r\nc. When does the ball hit the ground?\r\n\r\n[reveal-answer q=\"394530\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"394530\"]\r\n\r\na. The ball reaches the maximum height at the vertex of the parabola.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}h&amp;=-\\dfrac{80}{2\\left(-16\\right)} \\\\[1mm]&amp;=\\dfrac{80}{32} \\\\[1mm] &amp;=\\dfrac{5}{2} \\\\[1mm] &amp;=2.5 \\end{align}[\/latex]<\/p>\r\nThe ball reaches a maximum height after 2.5 seconds.\r\n\r\nb. To find the maximum height, find the [latex]y[\/latex]<em>\u00a0<\/em>coordinate of the vertex of the parabola.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}k&amp;=H\\left(2.5\\right) \\\\[1mm] &amp;=-16{\\left(2.5\\right)}^{2}+80\\left(2.5\\right)+40 \\\\[1mm] &amp;=140\\hfill \\end{align}[\/latex]<\/p>\r\nThe ball reaches a maximum height of 140 feet.\r\n\r\nc. To find when the ball hits the ground, we need to determine when the height is zero, [latex]H\\left(t\\right)=0[\/latex].\r\n\r\nWe use the quadratic formula.\r\n<p style=\"text-align: center;\">[latex]\\begin{align} t&amp;=\\dfrac{-80\\pm \\sqrt{{80}^{2}-4\\left(-16\\right)\\left(40\\right)}}{2\\left(-16\\right)} \\\\[1mm] &amp;=\\dfrac{-80\\pm \\sqrt{8960}}{-32} \\end{align}[\/latex]<\/p>\r\nBecause the square root does not simplify nicely, we can use a calculator to approximate the values of the solutions.\r\n<p style=\"text-align: center;\">[latex]t=\\dfrac{-80+\\sqrt{8960}}{-32}\\approx 5.458\\hspace{3mm}[\/latex] or [latex]\\hspace{3mm}t=\\dfrac{-80-\\sqrt{8960}}{-32}\\approx -0.458[\/latex]<\/p>\r\nSince the domain starts at [latex]t=0[\/latex] when the ball is thrown, the second answer is outside the reasonable domain of our model. The ball will hit the ground after about 5.46 seconds.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170404\/CNX_Precalc_Figure_03_02_0162.jpg\" alt=\"Graph of a negative parabola where x goes from -1 to 6.\" width=\"487\" height=\"254\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nA rock is thrown upward from the top of a 112-foot high cliff overlooking the ocean at a speed of 96 feet per second. The rock\u2019s height above ocean can be modeled by the equation [latex]H\\left(t\\right)=-16{t}^{2}+96t+112[\/latex].\r\n\r\na. When does the rock reach the maximum height?\r\n\r\nb. What is the maximum height of the rock?\r\n\r\nc. When does the rock hit the ocean?\r\n\r\n[reveal-answer q=\"174919\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"174919\"]\r\n\r\na.\u00a03 seconds \u00a0b.\u00a0256 feet \u00a0c.\u00a07 seconds\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=15809&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"375\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Complex\u00a0Roots<\/h2>\r\nConsider the following function: [latex]f(x)=x^2+2x+3[\/latex], and it's graph below:\r\n<p style=\"text-align: center;\"><img class=\"alignnone size-medium wp-image-4477\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18034032\/desmos-graph-300x300.png\" alt=\"Graph of quadratic function with the following points (-1,2), (-2,3), (0,3), (1,6), (-3,6). \" width=\"300\" height=\"300\" \/><\/p>\r\nDoes this function have roots? It's probably obvious that this function does not cross the [latex]x[\/latex]-axis, therefore it doesn't have any [latex]x[\/latex]-intercepts. Recall that the [latex]x[\/latex]-intercepts of a function are found by setting the function equal to zero:\r\n<p style=\"text-align: center;\">[latex]x^2+2x+3=0[\/latex]<\/p>\r\nIn the next example we will solve this equation. You will see that there are roots, but they are not [latex]x[\/latex]-intercepts because the function does not contain [latex](x,y)[\/latex] pairs that are on the [latex]x[\/latex]-axis. \u00a0We call these complex roots.\r\n\r\nBy setting the function equal to zero and using the quadratic formula to solve, you will see that the roots are complex numbers.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the [latex]x[\/latex]-intercepts of the quadratic function. [latex]f(x)=x^2+2x+3[\/latex]\r\n[reveal-answer q=\"698410\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"698410\"]\r\n\r\nThe [latex]x[\/latex]-intercepts of the function\u00a0[latex]f(x)=x^2+2x+3[\/latex] are found by setting it equal to zero, and solving for [latex]x[\/latex] since the [latex]y[\/latex] values of the [latex]x[\/latex]-intercepts are zero.\r\n\r\nFirst, identify [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex].\r\n<p style=\"text-align: center;\">[latex]x^2+2x+3=0[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]a=1,b=2,c=3[\/latex]<\/p>\r\nSubstitute these values into the quadratic formula.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}x&amp;=\\dfrac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}\\\\[1mm]&amp;=\\dfrac{-2\\pm \\sqrt{{2}^{2}-4(1)(3)}}{2(1)}\\\\[1mm]&amp;=\\dfrac{-2\\pm \\sqrt{4-12}}{2} \\\\[1mm]&amp;=\\dfrac{-2\\pm \\sqrt{-8}}{2}\\\\[1mm]&amp;=\\dfrac{-2\\pm 2i\\sqrt{2}}{2} \\\\[1mm]&amp;=-1\\pm i\\sqrt{2}\\\\[1mm]x&amp;=-1+i\\sqrt{2},-1-i\\sqrt{2}\\end{align}[\/latex]<\/p>\r\nThe solutions to this equation are complex, therefore there are no [latex]x[\/latex]-intercepts for the function\u00a0[latex]f(x)=x^2+2x+3[\/latex] in the set of real numbers that can be plotted on the Cartesian Coordinate plane. The graph of the function is plotted on the Cartesian Coordinate plane below:\r\n\r\n[caption id=\"attachment_4477\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-4477 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18034032\/desmos-graph-300x300.png\" alt=\"Graph of quadratic function with the following points (-1,2), (-2,3), (0,3), (1,6), (-3,6). \" width=\"300\" height=\"300\" \/> Graph of quadratic function with no [latex]x[\/latex]-intercepts in the real numbers.[\/caption]Note how the graph does not cross the [latex]x[\/latex]-axis, therefore there are no real [latex]x[\/latex]-intercepts for this function.[\/hidden-answer]<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=121401&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\nThe following video gives another example of how to use the quadratic formula to find complex solutions to a quadratic equation.\r\n\r\nhttps:\/\/youtu.be\/11EwTcRMPn8\r\n<h2>The Discriminant<\/h2>\r\nThe <strong>quadratic formula<\/strong> not only generates the solutions to a quadratic equation, it tells us about the nature of the solutions. When we consider the <strong>discriminant<\/strong>, or the expression under the radical, [latex]{b}^{2}-4ac[\/latex], it tells us whether the solutions are real numbers or complex numbers, and how many solutions of each type to expect. In turn, we can then determine whether a quadratic function has real or complex roots. The table below\u00a0relates the value of the discriminant to the solutions of a quadratic equation.\r\n<table summary=\"A table with 5 rows and 2 columns. The entries in the first row are: Value of Discriminant and Results. The entries in the second row are: b squared minus four times a times c equals zero and One rational solution (double solution). The entries in the third row are: b squared minus four times a times c is greater than zero, perfect square and Two rational solutions. The entries in the fourth row are: b squared minus four times a times c is greater than zero, not a perfect square and Two irrational solutions. The entries in the fifth row are: b squared minus four times a times c is less than zero and Two complex solutions.\">\r\n<thead>\r\n<tr>\r\n<th>Value of Discriminant<\/th>\r\n<th>Results<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]{b}^{2}-4ac=0[\/latex]<\/td>\r\n<td>One repeated rational solution<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{b}^{2}-4ac&gt;0[\/latex], perfect square<\/td>\r\n<td>Two rational solutions<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{b}^{2}-4ac&gt;0[\/latex], not a perfect square<\/td>\r\n<td>Two irrational solutions<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{b}^{2}-4ac&lt;0[\/latex]<\/td>\r\n<td>Two complex solutions<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox\">\r\n<h3>A General Note: The Discriminant<\/h3>\r\nFor [latex]a{x}^{2}+bx+c=0[\/latex], where [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are real numbers, the <strong>discriminant<\/strong> is the expression under the radical in the quadratic formula: [latex]{b}^{2}-4ac[\/latex]. It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse the discriminant to find the nature of the solutions to the following quadratic equations:\r\n<ol>\r\n \t<li>[latex]{x}^{2}+4x+4=0[\/latex]<\/li>\r\n \t<li>[latex]8{x}^{2}+14x+3=0[\/latex]<\/li>\r\n \t<li>[latex]3{x}^{2}-5x - 2=0[\/latex]<\/li>\r\n \t<li>[latex]3{x}^{2}-10x+15=0[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"497176\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"497176\"]\r\n\r\nCalculate the discriminant [latex]{b}^{2}-4ac[\/latex] for each equation and state the expected type of solutions.\r\n<ol>\r\n \t<li>[latex]\\begin{align}{x}^{2}+4x+4=0&amp;&amp;{b}^{2}-4ac={\\left(4\\right)}^{2}-4\\left(1\\right)\\left(4\\right)=0\\end{align}\\hspace{4mm}[\/latex]There will be one repeated rational solution.<\/li>\r\n \t<li>[latex]\\begin{align}8{x}^{2}+14x+3=0&amp;&amp;{b}^{2}-4ac={\\left(14\\right)}^{2}-4\\left(8\\right)\\left(3\\right)=100\\end{align}\\hspace{4mm}[\/latex]As [latex]100[\/latex] is a perfect square, there will be two rational solutions.<\/li>\r\n \t<li>[latex]\\begin{align}3{x}^{2}-5x - 2=0&amp;&amp;{b}^{2}-4ac={\\left(-5\\right)}^{2}-4\\left(3\\right)\\left(-2\\right)=49\\end{align}\\hspace{4mm}[\/latex]As [latex]49[\/latex] is a perfect square, there will be two rational solutions.<\/li>\r\n \t<li>[latex]\\begin{align}3{x}^{2}-10x+15=0&amp;&amp;{b}^{2}-4ac={\\left(-10\\right)}^{2}-4\\left(3\\right)\\left(15\\right)=-80\\end{align}\\hspace{4mm}[\/latex]There will be two complex solutions.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe have seen\u00a0that a quadratic equation may have two real solutions, one real solution, or two complex solutions.\r\n\r\nLet\u2019s summarize\u00a0how the discriminant affects the evaluation of [latex] \\sqrt{{{b}^{2}}-4ac}[\/latex], and how it helps to determine the solution set.\r\n<ul>\r\n \t<li>If [latex]b^{2}-4ac&gt;0[\/latex], then the number underneath the radical will be a positive value. You can always find the square root of a positive, so evaluating the Quadratic Formula will result in two real solutions (one by adding the square root, and one by subtracting it).<\/li>\r\n \t<li>If [latex]b^{2}-4ac=0[\/latex], then you will be taking the square root of 0, which is 0. Since adding and subtracting 0 both give the same result, the \"[latex]\\pm[\/latex]\" portion of the formula doesn't matter. There will be one real repeated solution.<\/li>\r\n \t<li>If [latex]b^{2}-4ac&lt;0[\/latex], then the number underneath the radical will be a negative value. Since you cannot find the square root of a negative number using real numbers, there are no real solutions. However, you can use imaginary numbers. You will then have two complex solutions, one by adding the imaginary square root and one by subtracting it.<\/li>\r\n<\/ul>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nUse the discriminant to determine how many and what kind of solutions the quadratic equation [latex]x^{2}-4x+10=0[\/latex]\u00a0has.\r\n\r\n[reveal-answer q=\"116245\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"116245\"]\r\n\r\nEvaluate [latex]b^{2}-4ac[\/latex]. First note that\u00a0[latex]a=1,b=\u22124[\/latex], and [latex]c=10[\/latex].\r\n<p style=\"text-align: center;\">[latex]{c}b^{2}-4ac[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\left(-4\\right)^{2}-4\\left(1\\right)\\left(10\\right)[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]16\u201340=\u221224[\/latex]<\/p>\r\nThe result is a negative number. The discriminant is negative, so [latex]x^{2}-4x+10=0[\/latex] has two complex solutions.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=35145&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Applications With Quadratic Functions<\/h2>\r\nThere are many real-world scenarios that involve finding the maximum or minimum value of a quadratic function, such as applications involving area and revenue.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170347\/CNX_Precalc_Figure_03_02_0092.jpg\" alt=\"Two graphs where the first graph shows the maximum value for f(x)=(x-2)^2+1 which occurs at (2, 1) and the second graph shows the minimum value for g(x)=-(x+3)^2+4 which occurs at (-3, 4).\" width=\"975\" height=\"558\" \/>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Maximum Value of a Quadratic Function<\/h3>\r\nA backyard farmer wants to enclose a rectangular space for a new garden within her fenced backyard. She has purchased 80 feet of wire fencing to enclose three sides, and she will use a section of the backyard fence as the fourth side.\r\n<ol>\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[reveal-answer q=\"704029\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"704029\"]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170349\/CNX_Precalc_Figure_03_02_0102.jpg\" alt=\"Diagram of the garden and the backyard.\" width=\"487\" height=\"310\" \/>\r\n\r\nLet\u2019s use a diagram such as the one above\u00a0to record the given information. It is also helpful to introduce a temporary variable, [latex]W[\/latex], to represent the width of the garden and the length of the fence section parallel to the backyard fence.\r\n<ol>\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[latex]W=80 - 2L[\/latex]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 [latex]A=LW=L\\left(80 - 2L\\right)=80L - 2{L}^{2}[\/latex]. This formula represents the area of the fence in terms of the variable length [latex]L[\/latex]. The function, written in general form, is[latex]A\\left(L\\right)=-2{L}^{2}+80L[\/latex].<\/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]\u00a0is the coefficient of the squared term, [latex]a=-2,b=80[\/latex], and [latex]c=0[\/latex].<\/li>\r\n<\/ol>\r\nTo find the vertex:\r\n<p style=\"text-align: center;\">[latex]h=-\\dfrac{80}{2\\left(-2\\right)}=20[\/latex]<\/p>\r\n<p style=\"text-align: center;\">and<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{align}k&amp;=A\\left(20\\right) \\\\&amp;=80\\left(20\\right)-2{\\left(20\\right)}^{2}\\\\&amp;=800 \\end{align}[\/latex]<\/p>\r\nThe maximum value of the function is an area of 800 square feet, which occurs when [latex]L=20[\/latex] feet. When the shorter sides are 20 feet, there is 40 feet of fencing left for the longer side. To maximize the area, she should enclose the garden so the two shorter sides have length 20 feet and the longer side parallel to the existing fence has length 40 feet.\r\n<h4>Analysis of the Solution<\/h4>\r\nThis problem also could be solved by graphing the quadratic function. We can see where the maximum area occurs on a graph of the quadratic function below.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170352\/CNX_Precalc_Figure_03_02_0112.jpg\" alt=\"Graph of the parabolic function A(L)=-2L^2+80L, which the x-axis is labeled Length (L) and the y-axis is labeled Area (A). The vertex is at (20, 800).\" width=\"487\" height=\"476\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe problem we solved above is called a constrained optimization problem. We can optimize our desired outcome given a constraint, which in this case was a limited amount of fencing materials. Try it yourself in the next problem.\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2451&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox\">\r\n<h3>How To: Given an application involving revenue, use a quadratic equation to find the maximum.<\/h3>\r\n<ol>\r\n \t<li>Write a quadratic equation for revenue.<\/li>\r\n \t<li>Find the vertex of the quadratic equation.<\/li>\r\n \t<li>Determine the [latex]y[\/latex]-value of the vertex.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding Maximum Revenue<\/h3>\r\nThe unit price of an item affects its supply and demand. That is, if the unit price goes up, the demand for the item will usually decrease. For example, a 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?\r\n\r\n[reveal-answer q=\"230015\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"230015\"]\r\n\r\nRevenue 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]\u00a0for price per subscription and [latex]Q[\/latex]\u00a0for quantity, giving us the equation [latex]\\text{Revenue}=pQ[\/latex].\r\n\r\nBecause 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\r\n<p style=\"text-align: center;\">[latex]\\begin{align}m&amp;=\\dfrac{79,000 - 84,000}{32 - 30} \\\\ &amp;=\\dfrac{-5,000}{2} \\\\ &amp;=-2,500 \\end{align}[\/latex]<\/p>\r\nThis tells us the paper will lose 2,500 subscribers for each dollar they raise the price. We can then solve for the [latex]y[\/latex]-intercept.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;Q=-2500p+b &amp;&amp;\\text{Substitute in the point }Q=84,000\\text{ and }p=30 \\\\ &amp;84,000=-2500\\left(30\\right)+b &amp;&amp;\\text{Solve for }b \\\\ &amp;b=159,000 \\end{align}[\/latex]<\/p>\r\nThis gives us the linear equation [latex]Q=-2,500p+159,000[\/latex] relating cost and subscribers. We now return to our revenue equation.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}&amp;\\text{Revenue}=pQ \\\\ &amp;\\text{Revenue}=p\\left(-2,500p+159,000\\right) \\\\ &amp;\\text{Revenue}=-2,500{p}^{2}+159,000p \\end{align}[\/latex]<\/p>\r\nWe now have a quadratic function for revenue as a function of the subscription charge. To find the price that will maximize revenue for the newspaper, we can find the vertex.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}h&amp;=-\\dfrac{159,000}{2\\left(-2,500\\right)} \\\\ &amp;=31.8 \\end{align}[\/latex]<\/p>\r\nThe 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.\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{maximum revenue}&amp;=-2,500{\\left(31.8\\right)}^{2}+159,000\\left(31.8\\right) \\\\ &amp;=\\$2,528,100\\hfill \\end{align}[\/latex]<\/p>\r\n\r\n<h4>Analysis of the Solution<\/h4>\r\nThis could also be solved by graphing the quadratic. We can see the maximum revenue on a graph of the quadratic function.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170354\/CNX_Precalc_Figure_03_02_0122.jpg\" alt=\"Graph of the parabolic function which the x-axis is labeled Price (p) and the y-axis is labeled Revenue ($). The vertex is at (31.80, 258100).\" width=\"487\" height=\"327\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the example above, we knew the number of subscribers to a newspaper and used that information to find the optimal price for each subscription. What if the price of subscriptions is affected by competition?\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nPreviously,\u00a0we found\u00a0a quadratic function that modeled revenue as a function of price.\r\n<p style=\"text-align: center;\">[latex]\\text{Revenue}-2,500{p}^{2}+159,000p[\/latex]<\/p>\r\n<p style=\"text-align: left;\">We found that selling the paper at [latex]\\$31.80[\/latex] per subscription would maximize revenue. \u00a0What if your closest competitor sells their paper for [latex]\\$25.00[\/latex] per subscription? What is the maximum revenue you can make you sell your paper for the same?<\/p>\r\n[reveal-answer q=\"410084\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"410084\"]\r\n\r\nEvaluating the function for [latex]p=25[\/latex] gives [latex]\\$2,412,500[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=15552&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\nThe quadratic formula [latex]x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex]\r\n\r\nThe discriminant is defined as [latex]b^2-4ac[\/latex]\r\n<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li>The zeros, or [latex]x[\/latex]-intercepts, are the points at which the parabola crosses the [latex]x[\/latex]-axis. The [latex]y[\/latex]-intercept is the point at which the parabola crosses the [latex]y[\/latex]<em>-<\/em>axis.<\/li>\r\n \t<li>The vertex can be found from an equation representing a quadratic function.<\/li>\r\n \t<li>A quadratic function\u2019s minimum or maximum value is given by the [latex]y[\/latex]-value of the vertex.<\/li>\r\n \t<li>The minium or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue.<\/li>\r\n \t<li>Some quadratic equations must be solved by using the quadratic formula.<\/li>\r\n \t<li>The vertex and the intercepts can be identified and interpreted to solve real-world problems.<\/li>\r\n \t<li>Some quadratic functions have complex roots.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165135623614\" class=\"definition\">\r\n \t<dt><strong>discriminant<\/strong><\/dt>\r\n \t<dd>the value under the radical in the quadratic formula, [latex]b^2-4ac[\/latex], which tells whether the quadratic has real or complex roots<\/dd>\r\n \t<dt><strong>vertex<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135623619\">the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135623624\" class=\"definition\">\r\n \t<dt><strong>vertex form of a quadratic function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135623630\">another name for the standard form of a quadratic function<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135623634\" class=\"definition\">\r\n \t<dt><strong>zeros<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135623639\">in a given function, the values of [latex]x[\/latex] at which [latex]y=0[\/latex], also called roots<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul class=\"ul1\">\n<li class=\"li2\"><span class=\"s1\">Use the quadratic formula and factoring to find both real and complex roots ([latex]x[\/latex]-intercepts) of quadratic functions.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Use algebra to find the [latex]y[\/latex]-intercepts of a quadratic function.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Solve problems involving the roots and intercepts of a quadratic function.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Use the discriminant to determine the nature (real or complex) and quantity of solutions to quadratic equations.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Determine a quadratic function\u2019s minimum or maximum value.<\/span><\/li>\n<li class=\"li3\"><span class=\"s4\">Solve problems involving a quadratic function\u2019s minimum or maximum value.<\/span><\/li>\n<\/ul>\n<\/div>\n<p>In this section we will investigate quadratic functions further, including solving\u00a0problems involving area and projectile motion. Working with quadratic functions can be less complex than working with higher degree polynomial functions, so they provide a good opportunity for a detailed study of function behavior.<\/p>\n<h2>Intercepts of Quadratic Functions<\/h2>\n<p>We also need to find intercepts of quadratic equations for graphing parabolas. Recall that we find the <span class=\"s1\">[latex]y[\/latex]<\/span>-intercept of a quadratic by evaluating the function at an input of zero, and we find the <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts at locations where the output is zero. Notice\u00a0that the number of <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts can vary depending upon the location of the graph.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170357\/CNX_Precalc_Figure_03_02_0132.jpg\" alt=\"Three graphs where the first graph shows a parabola with no x-intercept, the second is a parabola with one \u2013intercept, and the third parabola is of two x-intercepts.\" width=\"975\" height=\"317\" \/><\/p>\n<p class=\"wp-caption-text\">Number of <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts of a parabola<\/p>\n<\/div>\n<p>Mathematicians also define <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts as roots of the quadratic function.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Solving a Quadratic Equation with the Quadratic Formula<\/h3>\n<p>Solve [latex]{x}^{2}+x+2=0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q757696\">Show Solution<\/span><\/p>\n<div id=\"q757696\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let\u2019s begin by writing the quadratic formula: [latex]x=\\dfrac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex].<\/p>\n<p>When applying the <strong>quadratic formula<\/strong>, we identify the coefficients [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex]. For the equation [latex]{x}^{2}+x+2=0[\/latex], we have [latex]a=1[\/latex], [latex]b=1[\/latex], and [latex]c=2[\/latex].\u00a0Substituting these values into the formula we have:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}x&=\\dfrac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a} \\\\[1.5mm] &=\\dfrac{-1\\pm \\sqrt{{1}^{2}-4\\cdot 1\\cdot \\left(2\\right)}}{2\\cdot 1} \\\\[1.5mm] &=\\dfrac{-1\\pm \\sqrt{1 - 8}}{2} \\\\[1.5mm] &=\\dfrac{-1\\pm \\sqrt{-7}}{2} \\\\[1.5mm] &=\\dfrac{-1\\pm i\\sqrt{7}}{2} \\end{align}[\/latex]<\/p>\n<p>The solutions to the equation are [latex]x=\\dfrac{-1+i\\sqrt{7}}{2}[\/latex] and [latex]x=\\dfrac{-1-i\\sqrt{7}}{2}[\/latex] or [latex]x=-\\dfrac{1}{2}+\\dfrac{i\\sqrt{7}}{2}[\/latex] and [latex]x=-\\dfrac{1}{2}-\\dfrac{i\\sqrt{7}}{2}[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>This quadratic equation has only non-real solutions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a quadratic function, find the <em>x<\/em>-intercepts by rewriting in standard form.<\/h3>\n<ol>\n<li>Substitute <span class=\"s1\">[latex]a[\/latex]<\/span>\u00a0and <span class=\"s1\">[latex]b[\/latex]<\/span>\u00a0into [latex]h=-\\dfrac{b}{2a}[\/latex].<\/li>\n<li>Substitute [latex]x=h[\/latex]\u00a0into the general form of the quadratic function to find <span class=\"s1\">[latex]k[\/latex]<\/span>.<\/li>\n<li>Rewrite the quadratic in standard form using <span class=\"s1\">[latex]h[\/latex]<\/span>\u00a0and <span class=\"s1\">[latex]k[\/latex]<\/span>.<\/li>\n<li>Solve for when the output of the function will be zero to find the <span class=\"s1\">[latex]x[\/latex]<\/span><em>&#8211;<\/em>intercepts.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Roots\u00a0of a Parabola<\/h3>\n<p>Find the <span class=\"s1\">[latex]x[\/latex]<\/span>-intercepts of the quadratic function [latex]f\\left(x\\right)=2{x}^{2}+4x - 4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q201989\">Show Solution<\/span><\/p>\n<div id=\"q201989\" class=\"hidden-answer\" style=\"display: none\">\n<p>We begin by solving for when the output will be zero.<\/p>\n<p style=\"text-align: center;\">[latex]0=2{x}^{2}+4x - 4[\/latex]<\/p>\n<p>Because the quadratic is not easily factorable in this case, we solve for the intercepts by first rewriting the quadratic in standard form.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/p>\n<p>We know that [latex]a=2[\/latex]. Then we solve for\u00a0<span class=\"s1\">[latex]h[\/latex]<\/span>\u00a0and <span class=\"s1\">[latex]k[\/latex].<\/span><\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&h=-\\dfrac{b}{2a}=-\\dfrac{4}{2\\left(2\\right)}=-1\\\\[2mm]&\\end{align}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}k&=f\\left(h\\right)\\\\&=f\\left(-1\\right)\\\\&=2{\\left(-1\\right)}^{2}+4\\left(-1\\right)-4\\\\&=-6\\end{align}[\/latex]<\/p>\n<p>So now we can rewrite in standard form.<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=2{\\left(x+1\\right)}^{2}-6[\/latex]<\/p>\n<p>We can now solve for when the output will be zero.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&0=2{\\left(x+1\\right)}^{2}-6 \\\\ &6=2{\\left(x+1\\right)}^{2} \\\\ &3={\\left(x+1\\right)}^{2} \\\\ &x+1=\\pm \\sqrt{3} \\\\ &x=-1\\pm \\sqrt{3} \\end{align}[\/latex]<\/p>\n<p>The graph has [latex]x[\/latex]<em>&#8211;<\/em>intercepts at [latex]\\left(-1-\\sqrt{3},0\\right)[\/latex] and [latex]\\left(-1+\\sqrt{3},0\\right)[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170402\/CNX_Precalc_Figure_03_02_0152.jpg\" alt=\"Graph of a parabola which has the following x-intercepts (-2.732, 0) and (0.732, 0).\" width=\"487\" height=\"517\" \/><\/p>\n<p>We can check our work by graphing the given function on a graphing utility and observing the roots.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=121416&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"150\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Applying the Vertex and <em>x<\/em>-Intercepts of a Parabola<\/h3>\n<p>A ball is thrown upward from the top of a 40 foot high building at a speed of 80 feet per second. The ball\u2019s height above ground can be modeled by the equation [latex]H\\left(t\\right)=-16{t}^{2}+80t+40[\/latex].<\/p>\n<p>a. When does the ball reach the maximum height?<\/p>\n<p>b. What is the maximum height of the ball?<\/p>\n<p>c. When does the ball hit the ground?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q394530\">Show Solution<\/span><\/p>\n<div id=\"q394530\" class=\"hidden-answer\" style=\"display: none\">\n<p>a. The ball reaches the maximum height at the vertex of the parabola.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}h&=-\\dfrac{80}{2\\left(-16\\right)} \\\\[1mm]&=\\dfrac{80}{32} \\\\[1mm] &=\\dfrac{5}{2} \\\\[1mm] &=2.5 \\end{align}[\/latex]<\/p>\n<p>The ball reaches a maximum height after 2.5 seconds.<\/p>\n<p>b. To find the maximum height, find the [latex]y[\/latex]<em>\u00a0<\/em>coordinate of the vertex of the parabola.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}k&=H\\left(2.5\\right) \\\\[1mm] &=-16{\\left(2.5\\right)}^{2}+80\\left(2.5\\right)+40 \\\\[1mm] &=140\\hfill \\end{align}[\/latex]<\/p>\n<p>The ball reaches a maximum height of 140 feet.<\/p>\n<p>c. To find when the ball hits the ground, we need to determine when the height is zero, [latex]H\\left(t\\right)=0[\/latex].<\/p>\n<p>We use the quadratic formula.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align} t&=\\dfrac{-80\\pm \\sqrt{{80}^{2}-4\\left(-16\\right)\\left(40\\right)}}{2\\left(-16\\right)} \\\\[1mm] &=\\dfrac{-80\\pm \\sqrt{8960}}{-32} \\end{align}[\/latex]<\/p>\n<p>Because the square root does not simplify nicely, we can use a calculator to approximate the values of the solutions.<\/p>\n<p style=\"text-align: center;\">[latex]t=\\dfrac{-80+\\sqrt{8960}}{-32}\\approx 5.458\\hspace{3mm}[\/latex] or [latex]\\hspace{3mm}t=\\dfrac{-80-\\sqrt{8960}}{-32}\\approx -0.458[\/latex]<\/p>\n<p>Since the domain starts at [latex]t=0[\/latex] when the ball is thrown, the second answer is outside the reasonable domain of our model. The ball will hit the ground after about 5.46 seconds.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170404\/CNX_Precalc_Figure_03_02_0162.jpg\" alt=\"Graph of a negative parabola where x goes from -1 to 6.\" width=\"487\" height=\"254\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>A rock is thrown upward from the top of a 112-foot high cliff overlooking the ocean at a speed of 96 feet per second. The rock\u2019s height above ocean can be modeled by the equation [latex]H\\left(t\\right)=-16{t}^{2}+96t+112[\/latex].<\/p>\n<p>a. When does the rock reach the maximum height?<\/p>\n<p>b. What is the maximum height of the rock?<\/p>\n<p>c. When does the rock hit the ocean?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q174919\">Show Solution<\/span><\/p>\n<div id=\"q174919\" class=\"hidden-answer\" style=\"display: none\">\n<p>a.\u00a03 seconds \u00a0b.\u00a0256 feet \u00a0c.\u00a07 seconds<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=15809&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"375\"><\/iframe><\/p>\n<\/div>\n<h2>Complex\u00a0Roots<\/h2>\n<p>Consider the following function: [latex]f(x)=x^2+2x+3[\/latex], and it&#8217;s graph below:<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-4477\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18034032\/desmos-graph-300x300.png\" alt=\"Graph of quadratic function with the following points (-1,2), (-2,3), (0,3), (1,6), (-3,6).\" width=\"300\" height=\"300\" \/><\/p>\n<p>Does this function have roots? It&#8217;s probably obvious that this function does not cross the [latex]x[\/latex]-axis, therefore it doesn&#8217;t have any [latex]x[\/latex]-intercepts. Recall that the [latex]x[\/latex]-intercepts of a function are found by setting the function equal to zero:<\/p>\n<p style=\"text-align: center;\">[latex]x^2+2x+3=0[\/latex]<\/p>\n<p>In the next example we will solve this equation. You will see that there are roots, but they are not [latex]x[\/latex]-intercepts because the function does not contain [latex](x,y)[\/latex] pairs that are on the [latex]x[\/latex]-axis. \u00a0We call these complex roots.<\/p>\n<p>By setting the function equal to zero and using the quadratic formula to solve, you will see that the roots are complex numbers.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the [latex]x[\/latex]-intercepts of the quadratic function. [latex]f(x)=x^2+2x+3[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q698410\">Show Solution<\/span><\/p>\n<div id=\"q698410\" class=\"hidden-answer\" style=\"display: none\">\n<p>The [latex]x[\/latex]-intercepts of the function\u00a0[latex]f(x)=x^2+2x+3[\/latex] are found by setting it equal to zero, and solving for [latex]x[\/latex] since the [latex]y[\/latex] values of the [latex]x[\/latex]-intercepts are zero.<\/p>\n<p>First, identify [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]x^2+2x+3=0[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]a=1,b=2,c=3[\/latex]<\/p>\n<p>Substitute these values into the quadratic formula.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}x&=\\dfrac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}\\\\[1mm]&=\\dfrac{-2\\pm \\sqrt{{2}^{2}-4(1)(3)}}{2(1)}\\\\[1mm]&=\\dfrac{-2\\pm \\sqrt{4-12}}{2} \\\\[1mm]&=\\dfrac{-2\\pm \\sqrt{-8}}{2}\\\\[1mm]&=\\dfrac{-2\\pm 2i\\sqrt{2}}{2} \\\\[1mm]&=-1\\pm i\\sqrt{2}\\\\[1mm]x&=-1+i\\sqrt{2},-1-i\\sqrt{2}\\end{align}[\/latex]<\/p>\n<p>The solutions to this equation are complex, therefore there are no [latex]x[\/latex]-intercepts for the function\u00a0[latex]f(x)=x^2+2x+3[\/latex] in the set of real numbers that can be plotted on the Cartesian Coordinate plane. The graph of the function is plotted on the Cartesian Coordinate plane below:<\/p>\n<div id=\"attachment_4477\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4477\" class=\"wp-image-4477 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18034032\/desmos-graph-300x300.png\" alt=\"Graph of quadratic function with the following points (-1,2), (-2,3), (0,3), (1,6), (-3,6).\" width=\"300\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-4477\" class=\"wp-caption-text\">Graph of quadratic function with no [latex]x[\/latex]-intercepts in the real numbers.<\/p>\n<\/div>\n<p>Note how the graph does not cross the [latex]x[\/latex]-axis, therefore there are no real [latex]x[\/latex]-intercepts for this function.<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=121401&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<p>The following video gives another example of how to use the quadratic formula to find complex solutions to a quadratic equation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Quadratic Formula - Complex Solutions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/11EwTcRMPn8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>The Discriminant<\/h2>\n<p>The <strong>quadratic formula<\/strong> not only generates the solutions to a quadratic equation, it tells us about the nature of the solutions. When we consider the <strong>discriminant<\/strong>, or the expression under the radical, [latex]{b}^{2}-4ac[\/latex], it tells us whether the solutions are real numbers or complex numbers, and how many solutions of each type to expect. In turn, we can then determine whether a quadratic function has real or complex roots. The table below\u00a0relates the value of the discriminant to the solutions of a quadratic equation.<\/p>\n<table summary=\"A table with 5 rows and 2 columns. The entries in the first row are: Value of Discriminant and Results. The entries in the second row are: b squared minus four times a times c equals zero and One rational solution (double solution). The entries in the third row are: b squared minus four times a times c is greater than zero, perfect square and Two rational solutions. The entries in the fourth row are: b squared minus four times a times c is greater than zero, not a perfect square and Two irrational solutions. The entries in the fifth row are: b squared minus four times a times c is less than zero and Two complex solutions.\">\n<thead>\n<tr>\n<th>Value of Discriminant<\/th>\n<th>Results<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]{b}^{2}-4ac=0[\/latex]<\/td>\n<td>One repeated rational solution<\/td>\n<\/tr>\n<tr>\n<td>[latex]{b}^{2}-4ac>0[\/latex], perfect square<\/td>\n<td>Two rational solutions<\/td>\n<\/tr>\n<tr>\n<td>[latex]{b}^{2}-4ac>0[\/latex], not a perfect square<\/td>\n<td>Two irrational solutions<\/td>\n<\/tr>\n<tr>\n<td>[latex]{b}^{2}-4ac<0[\/latex]<\/td>\n<td>Two complex solutions<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox\">\n<h3>A General Note: The Discriminant<\/h3>\n<p>For [latex]a{x}^{2}+bx+c=0[\/latex], where [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are real numbers, the <strong>discriminant<\/strong> is the expression under the radical in the quadratic formula: [latex]{b}^{2}-4ac[\/latex]. It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use the discriminant to find the nature of the solutions to the following quadratic equations:<\/p>\n<ol>\n<li>[latex]{x}^{2}+4x+4=0[\/latex]<\/li>\n<li>[latex]8{x}^{2}+14x+3=0[\/latex]<\/li>\n<li>[latex]3{x}^{2}-5x - 2=0[\/latex]<\/li>\n<li>[latex]3{x}^{2}-10x+15=0[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q497176\">Show Solution<\/span><\/p>\n<div id=\"q497176\" class=\"hidden-answer\" style=\"display: none\">\n<p>Calculate the discriminant [latex]{b}^{2}-4ac[\/latex] for each equation and state the expected type of solutions.<\/p>\n<ol>\n<li>[latex]\\begin{align}{x}^{2}+4x+4=0&&{b}^{2}-4ac={\\left(4\\right)}^{2}-4\\left(1\\right)\\left(4\\right)=0\\end{align}\\hspace{4mm}[\/latex]There will be one repeated rational solution.<\/li>\n<li>[latex]\\begin{align}8{x}^{2}+14x+3=0&&{b}^{2}-4ac={\\left(14\\right)}^{2}-4\\left(8\\right)\\left(3\\right)=100\\end{align}\\hspace{4mm}[\/latex]As [latex]100[\/latex] is a perfect square, there will be two rational solutions.<\/li>\n<li>[latex]\\begin{align}3{x}^{2}-5x - 2=0&&{b}^{2}-4ac={\\left(-5\\right)}^{2}-4\\left(3\\right)\\left(-2\\right)=49\\end{align}\\hspace{4mm}[\/latex]As [latex]49[\/latex] is a perfect square, there will be two rational solutions.<\/li>\n<li>[latex]\\begin{align}3{x}^{2}-10x+15=0&&{b}^{2}-4ac={\\left(-10\\right)}^{2}-4\\left(3\\right)\\left(15\\right)=-80\\end{align}\\hspace{4mm}[\/latex]There will be two complex solutions.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>We have seen\u00a0that a quadratic equation may have two real solutions, one real solution, or two complex solutions.<\/p>\n<p>Let\u2019s summarize\u00a0how the discriminant affects the evaluation of [latex]\\sqrt{{{b}^{2}}-4ac}[\/latex], and how it helps to determine the solution set.<\/p>\n<ul>\n<li>If [latex]b^{2}-4ac>0[\/latex], then the number underneath the radical will be a positive value. You can always find the square root of a positive, so evaluating the Quadratic Formula will result in two real solutions (one by adding the square root, and one by subtracting it).<\/li>\n<li>If [latex]b^{2}-4ac=0[\/latex], then you will be taking the square root of 0, which is 0. Since adding and subtracting 0 both give the same result, the &#8220;[latex]\\pm[\/latex]&#8221; portion of the formula doesn&#8217;t matter. There will be one real repeated solution.<\/li>\n<li>If [latex]b^{2}-4ac<0[\/latex], then the number underneath the radical will be a negative value. Since you cannot find the square root of a negative number using real numbers, there are no real solutions. However, you can use imaginary numbers. You will then have two complex solutions, one by adding the imaginary square root and one by subtracting it.<\/li>\n<\/ul>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Use the discriminant to determine how many and what kind of solutions the quadratic equation [latex]x^{2}-4x+10=0[\/latex]\u00a0has.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q116245\">Show Solution<\/span><\/p>\n<div id=\"q116245\" class=\"hidden-answer\" style=\"display: none\">\n<p>Evaluate [latex]b^{2}-4ac[\/latex]. First note that\u00a0[latex]a=1,b=\u22124[\/latex], and [latex]c=10[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]{c}b^{2}-4ac[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\left(-4\\right)^{2}-4\\left(1\\right)\\left(10\\right)[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]16\u201340=\u221224[\/latex]<\/p>\n<p>The result is a negative number. The discriminant is negative, so [latex]x^{2}-4x+10=0[\/latex] has two complex solutions.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=35145&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h2>Applications With Quadratic Functions<\/h2>\n<p>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<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170347\/CNX_Precalc_Figure_03_02_0092.jpg\" alt=\"Two graphs where the first graph shows the maximum value for f(x)=(x-2)^2+1 which occurs at (2, 1) and the second graph shows the minimum value for g(x)=-(x+3)^2+4 which occurs at (-3, 4).\" width=\"975\" height=\"558\" \/><\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Maximum Value of a Quadratic Function<\/h3>\n<p>A backyard farmer wants to enclose a rectangular space for a new garden within her fenced backyard. She has purchased 80 feet of wire fencing to enclose three sides, and she will use a section of the backyard fence as the fourth side.<\/p>\n<ol>\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 class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q704029\">Show Solution<\/span><\/p>\n<div id=\"q704029\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170349\/CNX_Precalc_Figure_03_02_0102.jpg\" alt=\"Diagram of the garden and the backyard.\" width=\"487\" height=\"310\" \/><\/p>\n<p>Let\u2019s use a diagram such as the one above\u00a0to 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<ol>\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].<br \/>\n[latex]W=80 - 2L[\/latex]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 [latex]A=LW=L\\left(80 - 2L\\right)=80L - 2{L}^{2}[\/latex]. This formula represents the area of the fence in terms of the variable length [latex]L[\/latex]. The function, written in general form, is[latex]A\\left(L\\right)=-2{L}^{2}+80L[\/latex].<\/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]\u00a0is the coefficient of the squared term, [latex]a=-2,b=80[\/latex], and [latex]c=0[\/latex].<\/li>\n<\/ol>\n<p>To find the vertex:<\/p>\n<p style=\"text-align: center;\">[latex]h=-\\dfrac{80}{2\\left(-2\\right)}=20[\/latex]<\/p>\n<p style=\"text-align: center;\">and<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}k&=A\\left(20\\right) \\\\&=80\\left(20\\right)-2{\\left(20\\right)}^{2}\\\\&=800 \\end{align}[\/latex]<\/p>\n<p>The maximum value of the function is an area of 800 square feet, which occurs when [latex]L=20[\/latex] feet. When the shorter sides are 20 feet, there is 40 feet of fencing left for the longer side. To maximize the area, she should enclose the garden so the two shorter sides have length 20 feet and the longer side parallel to the existing fence has length 40 feet.<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>This problem also could be solved by graphing the quadratic function. We can see where the maximum area occurs on a graph of the quadratic function below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170352\/CNX_Precalc_Figure_03_02_0112.jpg\" alt=\"Graph of the parabolic function A(L)=-2L^2+80L, which the x-axis is labeled Length (L) and the y-axis is labeled Area (A). The vertex is at (20, 800).\" width=\"487\" height=\"476\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The problem we solved above is called a constrained optimization problem. We can optimize our desired outcome given a constraint, which in this case was a limited amount of fencing materials. Try it yourself in the next problem.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2451&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\">\n<h3>How To: Given an application involving revenue, use a quadratic equation to find the maximum.<\/h3>\n<ol>\n<li>Write a quadratic equation for revenue.<\/li>\n<li>Find the vertex of the quadratic equation.<\/li>\n<li>Determine the [latex]y[\/latex]-value of the vertex.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding Maximum Revenue<\/h3>\n<p>The unit price of an item affects its supply and demand. That is, if the unit price goes up, the demand for the item will usually decrease. For example, a local newspaper currently has 84,000 subscribers at a quarterly charge of $30. Market research has suggested that if the owners raise the price to $32, they would lose 5,000 subscribers. Assuming that subscriptions are linearly related to the price, what price should the newspaper charge for a quarterly subscription to maximize their revenue?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q230015\">Show Solution<\/span><\/p>\n<div id=\"q230015\" class=\"hidden-answer\" style=\"display: none\">\n<p>Revenue is the amount of money a company brings in. In this case, the revenue can be found by multiplying the price per subscription times the number of subscribers, or quantity. We can introduce variables, [latex]p[\/latex]\u00a0for price per subscription and [latex]Q[\/latex]\u00a0for quantity, giving us the equation [latex]\\text{Revenue}=pQ[\/latex].<\/p>\n<p>Because the number of subscribers changes with the price, we need to find a relationship between the variables. We know that currently [latex]p=30[\/latex] and [latex]Q=84,000[\/latex]. We also know that if the price rises to $32, the newspaper would lose 5,000 subscribers, giving a second pair of values, [latex]p=32[\/latex] and [latex]Q=79,000[\/latex]. From this we can find a linear equation relating the two quantities. The slope will be<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}m&=\\dfrac{79,000 - 84,000}{32 - 30} \\\\ &=\\dfrac{-5,000}{2} \\\\ &=-2,500 \\end{align}[\/latex]<\/p>\n<p>This tells us the paper will lose 2,500 subscribers for each dollar they raise the price. We can then solve for the [latex]y[\/latex]-intercept.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&Q=-2500p+b &&\\text{Substitute in the point }Q=84,000\\text{ and }p=30 \\\\ &84,000=-2500\\left(30\\right)+b &&\\text{Solve for }b \\\\ &b=159,000 \\end{align}[\/latex]<\/p>\n<p>This gives us the linear equation [latex]Q=-2,500p+159,000[\/latex] relating cost and subscribers. We now return to our revenue equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}&\\text{Revenue}=pQ \\\\ &\\text{Revenue}=p\\left(-2,500p+159,000\\right) \\\\ &\\text{Revenue}=-2,500{p}^{2}+159,000p \\end{align}[\/latex]<\/p>\n<p>We now have a quadratic function for revenue as a function of the subscription charge. To find the price that will maximize revenue for the newspaper, we can find the vertex.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}h&=-\\dfrac{159,000}{2\\left(-2,500\\right)} \\\\ &=31.8 \\end{align}[\/latex]<\/p>\n<p>The model tells us that the maximum revenue will occur if the newspaper charges $31.80 for a subscription. To find what the maximum revenue is, we evaluate the revenue function.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}\\text{maximum revenue}&=-2,500{\\left(31.8\\right)}^{2}+159,000\\left(31.8\\right) \\\\ &=\\$2,528,100\\hfill \\end{align}[\/latex]<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>This could also be solved by graphing the quadratic. We can see the maximum revenue on a graph of the quadratic function.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02170354\/CNX_Precalc_Figure_03_02_0122.jpg\" alt=\"Graph of the parabolic function which the x-axis is labeled Price (p) and the y-axis is labeled Revenue ($). The vertex is at (31.80, 258100).\" width=\"487\" height=\"327\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the example above, we knew the number of subscribers to a newspaper and used that information to find the optimal price for each subscription. What if the price of subscriptions is affected by competition?<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Previously,\u00a0we found\u00a0a quadratic function that modeled revenue as a function of price.<\/p>\n<p style=\"text-align: center;\">[latex]\\text{Revenue}-2,500{p}^{2}+159,000p[\/latex]<\/p>\n<p style=\"text-align: left;\">We found that selling the paper at [latex]\\$31.80[\/latex] per subscription would maximize revenue. \u00a0What if your closest competitor sells their paper for [latex]\\$25.00[\/latex] per subscription? What is the maximum revenue you can make you sell your paper for the same?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q410084\">Show Solution<\/span><\/p>\n<div id=\"q410084\" class=\"hidden-answer\" style=\"display: none\">\n<p>Evaluating the function for [latex]p=25[\/latex] gives [latex]\\$2,412,500[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=15552&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"350\"><\/iframe><\/p>\n<\/div>\n<h2>Key Equations<\/h2>\n<p>The quadratic formula [latex]x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex]<\/p>\n<p>The discriminant is defined as [latex]b^2-4ac[\/latex]<\/p>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>The zeros, or [latex]x[\/latex]-intercepts, are the points at which the parabola crosses the [latex]x[\/latex]-axis. The [latex]y[\/latex]-intercept is the point at which the parabola crosses the [latex]y[\/latex]<em>&#8211;<\/em>axis.<\/li>\n<li>The vertex can be found from an equation representing a quadratic function.<\/li>\n<li>A quadratic function\u2019s minimum or maximum value is given by the [latex]y[\/latex]-value of the vertex.<\/li>\n<li>The minium or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue.<\/li>\n<li>Some quadratic equations must be solved by using the quadratic formula.<\/li>\n<li>The vertex and the intercepts can be identified and interpreted to solve real-world problems.<\/li>\n<li>Some quadratic functions have complex roots.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165135623614\" class=\"definition\">\n<dt><strong>discriminant<\/strong><\/dt>\n<dd>the value under the radical in the quadratic formula, [latex]b^2-4ac[\/latex], which tells whether the quadratic has real or complex roots<\/dd>\n<dt><strong>vertex<\/strong><\/dt>\n<dd id=\"fs-id1165135623619\">the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135623624\" class=\"definition\">\n<dt><strong>vertex form of a quadratic function<\/strong><\/dt>\n<dd id=\"fs-id1165135623630\">another name for the standard form of a quadratic function<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135623634\" class=\"definition\">\n<dt><strong>zeros<\/strong><\/dt>\n<dd id=\"fs-id1165135623639\">in a given function, the values of [latex]x[\/latex] at which [latex]y=0[\/latex], also called roots<\/dd>\n<\/dl>\n","protected":false},"author":167848,"menu_order":3,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4977","chapter","type-chapter","status-publish","hentry"],"part":764,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters\/4977","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/users\/167848"}],"version-history":[{"count":7,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters\/4977\/revisions"}],"predecessor-version":[{"id":5268,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters\/4977\/revisions\/5268"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/parts\/764"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapters\/4977\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/media?parent=4977"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/pressbooks\/v2\/chapter-type?post=4977"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/contributor?post=4977"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-collegealgebrapclc\/wp-json\/wp\/v2\/license?post=4977"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}