{"id":4679,"date":"2017-12-26T16:54:48","date_gmt":"2017-12-26T16:54:48","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/chapter\/graphs-of-quadratic-functions\/"},"modified":"2023-11-08T13:20:34","modified_gmt":"2023-11-08T13:20:34","slug":"graphs-of-quadratic-functions","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/cuny-hunter-collegealgebra\/chapter\/graphs-of-quadratic-functions\/","title":{"raw":"10.2 - Quadratic Functions and their Graphs","rendered":"10.2 &#8211; Quadratic Functions and their Graphs"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>(10.2.1) - Identify characteristics of a parabola\r\n<ul>\r\n \t<li>vertex<\/li>\r\n \t<li style=\"list-style-type: none\"><\/li>\r\n \t<li>axis of symmetry<\/li>\r\n \t<li>[latex]x[\/latex]\/[latex]y[\/latex]-intercepts<\/li>\r\n \t<li>General and standard forms of quadratic functions<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>(10.2.2) - Classifying Solutions to Quadratic Equations<\/li>\r\n<\/ul>\r\n<\/div>\r\nCurved antennas, such as the ones shown in the photo, are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function.\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\/2862\/2017\/12\/26165435\/CNX_Precalc_Figure_03_02_0012.jpg\" alt=\"Satellite dishes.\" width=\"731\" height=\"368\" \/> An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr)[\/caption]\r\n<h1>(10.2.1) - Identify characteristics of a parabola<\/h1>\r\nThe graph of a quadratic function is a U-shaped curve called a <strong>parabola<\/strong>. One important feature of the graph is that it has an extreme point, called the <strong>vertex<\/strong>. If the parabola opens up, the vertex represents the lowest point on the graph, or the <strong>minimum value<\/strong> of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the <strong>maximum value<\/strong>. In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the <strong>axis of symmetry<\/strong>.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165437\/CNX_Precalc_Figure_03_02_0022.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are.\" width=\"487\" height=\"480\" \/>\r\n\r\nThe [latex]y[\/latex]-intercept is the point at which the parabola crosses the [latex]y[\/latex]-axis. The [latex]x[\/latex]-intercepts are the points at which the parabola crosses the [latex]x[\/latex]-axis. If they exist, the [latex]x[\/latex]-intercepts represent the <strong>zeros<\/strong>, or <strong>roots<\/strong>, of the quadratic function, the values of [latex]x[\/latex]\u00a0at which [latex]y=0[\/latex].\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying the Characteristics of a Parabola<\/h3>\r\nDetermine the vertex, axis of symmetry, zeros, and <em>y<\/em>-intercept of the parabola shown below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165440\/CNX_Precalc_Figure_03_02_0032.jpg\" alt=\"Graph of a parabola with a vertex at (3, 1) and a y-intercept at (0, 7).\" width=\"487\" height=\"517\" \/>\r\n\r\n[reveal-answer q=\"366804\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"366804\"]\r\n\r\nThe vertex is the turning point of the graph. We can see that the vertex is at [latex](3,1)[\/latex]. Because this parabola opens upward, the axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is [latex]x=3[\/latex]. This parabola does not cross the [latex]x[\/latex]-axis, so it has no zeros. It crosses the [latex]y[\/latex]-axis at [latex](0, 7)[\/latex] so this is the [latex]y[\/latex]-intercept.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>TRY IT<\/h3>\r\n[ohm_question]147099[\/ohm_question]\r\n\r\n<\/div>\r\n<h3>General and Standard Forms of Quadratic Functions<\/h3>\r\nThe <strong>general form of a quadratic function<\/strong> presents the function in the form\r\n<p style=\"text-align: center\">[latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex]<\/p>\r\nwhere [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex]\u00a0are real numbers and [latex]a\\ne 0[\/latex]. If [latex]a&gt;0[\/latex], the parabola opens upward. If [latex]a&lt;0[\/latex], the parabola opens downward. We can use the general form of a parabola to find the equation for the axis of symmetry.\r\n\r\nThe axis of symmetry is defined by [latex]\\displaystyle x=-\\frac{b}{2a}[\/latex]. If we use the quadratic formula, [latex]\\displaystyle x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex], to solve [latex]a{x}^{2}+bx+c=0[\/latex] for the [latex]x[\/latex]-intercepts, or zeros, we find the value of [latex]x[\/latex]\u00a0halfway between them is always [latex]\\displaystyle x=-\\frac{b}{2a}[\/latex], the equation for the axis of symmetry.\r\n\r\nThe figure below shows\u00a0the graph of the quadratic function written in general form as [latex]y={x}^{2}+4x+3[\/latex]. In this form, [latex]a=1,\\text{ }b=4[\/latex], and [latex]c=3[\/latex]. Because [latex]a&gt;0[\/latex], the parabola opens upward. The axis of symmetry is [latex]\\displaystyle 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[\/latex]-intercepts, those points where the parabola crosses the [latex]x[\/latex]-axis, occur at [latex]\\left(-3,0\\right)[\/latex] and [latex]\\left(-1,0\\right)[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165442\/CNX_Precalc_Figure_03_02_0042.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are for the function y=x^2+4x+3.\" width=\"487\" height=\"555\" \/>\r\n\r\nThe <strong>standard form of a quadratic function<\/strong> presents the function in the form\r\n<p style=\"text-align: center\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/p>\r\nwhere [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the <strong>vertex form of a quadratic function<\/strong>.\r\n\r\n<h3>Given a quadratic function in general form, find the vertex of the parabola.<\/h3>\r\nOne reason we may want to identify the vertex of the parabola is that this point will inform us where the maximum or minimum value of the output occurs, ([latex]k[\/latex]), and where it occurs, ([latex]x[\/latex]). If we are given the general form of a quadratic function:\r\n<p style=\"text-align: center\">[latex]f(x)=ax^2+bx+c[\/latex]<\/p>\r\nWe can define the vertex, [latex](h,k)[\/latex], by doing the following:\r\n<ul>\r\n \t<li>Identify [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex].<\/li>\r\n \t<li>Find [latex]h[\/latex], the [latex]x[\/latex]-coordinate of the vertex, by substituting [latex]a[\/latex] and [latex]b[\/latex]\u00a0into [latex]\\displaystyle h=-\\frac{b}{2a}[\/latex].<\/li>\r\n \t<li>Find [latex]k[\/latex], the [latex]y[\/latex]-coordinate of the vertex, by evaluating [latex]\\displaystyle k=f\\left(h\\right)=f\\left(-\\frac{b}{2a}\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Vertex of a Quadratic Function<\/h3>\r\nFind the vertex of the quadratic function [latex]f\\left(x\\right)=2{x}^{2}-6x+7[\/latex]. Rewrite the quadratic in standard form (vertex form).\r\n\r\n[reveal-answer q=\"466886\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"466886\"]\r\n\r\nThe horizontal coordinate of the vertex will be at\r\n<p style=\"text-align: center\">[latex]\\large \\begin{array}{c}h=-\\frac{b}{2a}\\hfill \\\\ \\text{ }=-\\frac{-6}{2\\left(2\\right)}\\hfill \\\\ \\text{ }=\\frac{6}{4}\\hfill \\\\ \\text{ }=\\frac{3}{2}\\hfill \\end{array}[\/latex]<\/p>\r\nThe vertical coordinate of the vertex will be at\r\n<p style=\"text-align: center\">[latex]\\large \\begin{array}{c}k=f\\left(h\\right)\\hfill \\\\ \\text{ }=f\\left(\\frac{3}{2}\\right)\\hfill \\\\ \\text{ }=2{\\left(\\frac{3}{2}\\right)}^{2}-6\\left(\\frac{3}{2}\\right)+7\\hfill \\\\ \\text{ }=\\frac{5}{2}\\hfill \\end{array}[\/latex]<\/p>\r\nRewriting into standard form, the stretch factor will be the same as the [latex]a[\/latex] in the original quadratic.\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}f\\left(x\\right)=a{x}^{2}+bx+c\\hfill \\\\ f\\left(x\\right)=2{x}^{2}-6x+7\\hfill \\end{array}[\/latex]<\/p>\r\nUsing the vertex to determine the shifts,\r\n<p style=\"text-align: center\">[latex]\\displaystyle f\\left(x\\right)=2{\\left(x-\\frac{3}{2}\\right)}^{2}+\\frac{5}{2}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>EXAMPLE<\/h3>\r\nGiven the equation [latex]g\\left(x\\right)=13+{x}^{2}-6x[\/latex], write the equation in general form and then in standard form.\r\n\r\n[reveal-answer q=\"713769\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"713769\"][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[\/hidden-answer]\r\n\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<h1>(10.2.2) - Classifying Solutions to Quadratic Equations<\/h1>\r\nMuch as we did in the application problems above, we also need to find intercepts of quadratic equations for graphing parabolas. Recall that we find the <em>y<\/em>-intercept of a quadratic by evaluating the function at an input of zero, and we find the <em>x<\/em>-intercepts at locations where the output is zero. Notice\u00a0that the number of <em>x<\/em>-intercepts can vary depending upon the location of the graph.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165449\/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 x-intercepts of a parabola[\/caption]\r\n\r\nMathematicians also define <em>x<\/em>-intercepts as roots of the quadratic function.\r\n<div class=\"textbox\">\r\n<h3>How To: Given a quadratic function [latex]f\\left(x\\right)[\/latex], find the [latex]y[\/latex]-\u00a0and [latex]x[\/latex]-intercepts.<\/h3>\r\n<ol>\r\n \t<li>Evaluate [latex]f\\left(0\\right)[\/latex] to find the [latex]y[\/latex]-intercept.<\/li>\r\n \t<li>Solve the quadratic equation [latex]f\\left(x\\right)=0[\/latex] to find the [latex]x[\/latex]-intercepts.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the [latex]y[\/latex]- and [latex]x[\/latex]-Intercepts of a Parabola<\/h3>\r\nFind the [latex]y[\/latex]- and [latex]x[\/latex]-intercepts of the quadratic [latex]f\\left(x\\right)=3{x}^{2}+5x - 2[\/latex].\r\n\r\n[reveal-answer q=\"14680\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"14680\"]\r\nWe find the [latex]y[\/latex]-intercept by evaluating [latex]f\\left(0\\right)[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}{c}f\\left(0\\right)=3{\\left(0\\right)}^{2}+5\\left(0\\right)-2\\hfill \\\\ \\text{ }=-2\\hfill \\end{array}[\/latex]<\/p>\r\nSo the [latex]y[\/latex]-intercept is at [latex]\\left(0,-2\\right)[\/latex].\r\n\r\nFor the [latex]x[\/latex]-intercepts, or roots, we find all solutions of [latex]f\\left(x\\right)=0[\/latex].\r\n<p style=\"text-align: center\">[latex]0=3{x}^{2}+5x - 2[\/latex]<\/p>\r\nIn this case, the quadratic can be factored easily, providing the simplest method for solution.\r\n<p style=\"text-align: center\">[latex]0=\\left(3x - 1\\right)\\left(x+2\\right)[\/latex]\r\n[latex]\\begin{array}{c}0=3x - 1\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill &amp; 0=x+2\\hfill \\\\ x=\\frac{1}{3}\\hfill &amp; \\hfill &amp; \\text{or}\\hfill &amp; \\hfill &amp; x=-2\\hfill \\end{array}[\/latex]<\/p>\r\nSo the <em>roots<\/em>\u00a0are at [latex]\\left(\\frac{1}{3},0\\right)[\/latex] and [latex]\\left(-2,0\\right)[\/latex].\r\n<h4>Analysis of the Solution<\/h4>\r\nBy graphing the function, we can confirm that the graph crosses the <em>y<\/em>-axis at [latex]\\left(0,-2\\right)[\/latex]. We can also confirm that the graph crosses the [latex]x[\/latex]-axis at [latex]\\left(\\frac{1}{3},0\\right)[\/latex] and [latex]\\left(-2,0\\right)[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165452\/CNX_Precalc_Figure_03_02_0142.jpg\" alt=\"Graph of a parabola which has the following intercepts (-2, 0), (1\/3, 0), and (0, -2).\" width=\"487\" height=\"480\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the previous example, finding the [latex]y[\/latex]- and [latex]x[\/latex]-Intercepts of a parabola, 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.\r\n<div class=\"textbox\">\r\n<h3>How To: Given a quadratic function, find the [latex]x[\/latex]-intercepts by rewriting in standard form.<\/h3>\r\n<ol>\r\n \t<li>Substitute [latex]a[\/latex]\u00a0and [latex]b[\/latex]\u00a0into [latex]\\displaystyle h=-\\frac{b}{2a}[\/latex].<\/li>\r\n \t<li>Substitute [latex]x=h[\/latex]\u00a0into 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]\u00a0and [latex]k[\/latex].<\/li>\r\n \t<li>Solve for when the output of the function will be zero to find the [latex]x[\/latex]<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 [latex]x[\/latex]-intercepts of the quadratic function [latex]f\\left(x\\right)=2{x}^{2}+4x - 4[\/latex].\r\n\r\n[reveal-answer q=\"201989\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"201989\"]\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 [latex]h[\/latex]\u00a0and [latex]k[\/latex].\r\n<p style=\"text-align: center\">[latex]\\large \\begin{array}{c}h=-\\frac{b}{2a}\\hfill &amp; \\hfill &amp; \\hfill &amp; k=f\\left(-1\\right)\\hfill \\\\ \\text{ }=-\\frac{4}{2\\left(2\\right)}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{ }=2{\\left(-1\\right)}^{2}+4\\left(-1\\right)-4\\hfill \\\\ \\text{ }=-1\\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{ }=-6\\hfill \\end{array}[\/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{array}{c}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=\\pm \\sqrt{3}\\hfill \\\\ x=-1\\pm \\sqrt{3}\\hfill \\end{array}[\/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\/2862\/2017\/12\/26165454\/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>EXAMPLE<\/h3>\r\nFind the [latex]y[\/latex]-intercept for the function [latex]g\\left(x\\right)=13+{x}^{2}-6x[\/latex].\r\n\r\n[reveal-answer q=\"275171\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"275171\"][latex]y[\/latex]-intercept at [latex](0,13)[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<h4>Complex\u00a0Roots<\/h4>\r\nNow you will hopefully begin to understand why we introduced complex numbers at the beginning of this module. Consider the following function: [latex]f(x)=x^2+2x+3[\/latex], and it's graph below:\r\n\r\n<img class=\"wp-image-3475 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165458\/Screen-Shot-2016-08-04-at-11.34.19-AM.png\" alt=\"Graph of quadratic function with the following points (-1,2), (-2,3), (0,3), (1,6), (-3,6).\" width=\"241\" height=\"249\" \/>\r\n\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. \u00a0You 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 contain complex numbers:\r\n<p style=\"text-align: center\">[latex]x^2+2x+3=0[\/latex]<\/p>\r\n\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 Answer[\/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], [latex]c[\/latex].\r\n<p style=\"text-align: center\">[latex]\\begin{array}{ccc}x^2+2x+3=0\\\\a=1,b=2,c=3\\end{array}[\/latex]<\/p>\r\nSubstitute these values into the quadratic formula.\r\n<p style=\"text-align: center\">[latex]\\large \\begin{array}{c}x&amp;=&amp;\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}\\\\&amp;=&amp;\\frac{-2\\pm \\sqrt{{2}^{2}-4(1)(3)}}{2(1)}\\\\&amp;=&amp;\\frac{-2\\pm \\sqrt{4-12}}{2} \\\\&amp;=&amp;\\frac{-2\\pm \\sqrt{-8}}{2}\\\\&amp;=&amp;\\frac{-2\\pm 2i\\sqrt{2}}{2} \\\\-1\\pm i\\sqrt{2}&amp;=&amp;-1+\\sqrt{2},-1-\\sqrt{2}\\end{array}[\/latex]<\/p>\r\nThe solutions to this equations 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_3475\" align=\"aligncenter\" width=\"241\"]<img class=\"wp-image-3475\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165458\/Screen-Shot-2016-08-04-at-11.34.19-AM.png\" alt=\"Graph of quadratic function with the following points (-1,2), (-2,3), (0,3), (1,6), (-3,6).\" width=\"241\" height=\"249\" \/> Graph of quadratic function with no x-intercepts in the real numbers.[\/caption]\r\n\r\nNote 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]\r\n\r\n<\/div>\r\n<h4>Important Terms<\/h4>\r\n<dl id=\"fs-id1165135449657\" class=\"definition\">\r\n \t<dt><strong>axis of symmetry<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135449662\">a vertical line drawn through the vertex of a parabola around which the parabola is symmetric; it is defined by [latex]\\displaystyle x=-\\frac{b}{2a}[\/latex].<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135502777\" class=\"definition\">\r\n \t<dt><strong>general form of a quadratic function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165135502783\">the function that describes a parabola, written in the form [latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex], where [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex]\u00a0are real numbers and [latex]a\\ne 0[\/latex].<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137931314\" class=\"definition\">\r\n \t<dt><strong>standard form of a quadratic function<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165137931319\">the function that describes a parabola, written in the form [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex], where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex.<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dd><\/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>zeros<\/strong><\/dt>\r\n<\/dl>\r\n<dl id=\"fs-id1165135623634\" class=\"definition\">\r\n \t<dd id=\"fs-id1165135623639\">in a given function, the values of [latex]x[\/latex]\u00a0at which [latex]y = 0[\/latex], also called roots<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135623614\" class=\"definition\">\r\n \t<dt><\/dt>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>(10.2.1) &#8211; Identify characteristics of a parabola\n<ul>\n<li>vertex<\/li>\n<li style=\"list-style-type: none\"><\/li>\n<li>axis of symmetry<\/li>\n<li>[latex]x[\/latex]\/[latex]y[\/latex]-intercepts<\/li>\n<li>General and standard forms of quadratic functions<\/li>\n<\/ul>\n<\/li>\n<li>(10.2.2) &#8211; Classifying Solutions to Quadratic Equations<\/li>\n<\/ul>\n<\/div>\n<p>Curved antennas, such as the ones shown in the photo, are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function.<\/p>\n<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\/2862\/2017\/12\/26165435\/CNX_Precalc_Figure_03_02_0012.jpg\" alt=\"Satellite dishes.\" width=\"731\" height=\"368\" \/><\/p>\n<p class=\"wp-caption-text\">An array of satellite dishes. (credit: Matthew Colvin de Valle, Flickr)<\/p>\n<\/div>\n<h1>(10.2.1) &#8211; Identify characteristics of a parabola<\/h1>\n<p>The graph of a quadratic function is a U-shaped curve called a <strong>parabola<\/strong>. One important feature of the graph is that it has an extreme point, called the <strong>vertex<\/strong>. If the parabola opens up, the vertex represents the lowest point on the graph, or the <strong>minimum value<\/strong> of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the <strong>maximum value<\/strong>. In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the <strong>axis of symmetry<\/strong>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165437\/CNX_Precalc_Figure_03_02_0022.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are.\" width=\"487\" height=\"480\" \/><\/p>\n<p>The [latex]y[\/latex]-intercept is the point at which the parabola crosses the [latex]y[\/latex]-axis. The [latex]x[\/latex]-intercepts are the points at which the parabola crosses the [latex]x[\/latex]-axis. If they exist, the [latex]x[\/latex]-intercepts represent the <strong>zeros<\/strong>, or <strong>roots<\/strong>, of the quadratic function, the values of [latex]x[\/latex]\u00a0at which [latex]y=0[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying the Characteristics of a Parabola<\/h3>\n<p>Determine the vertex, axis of symmetry, zeros, and <em>y<\/em>-intercept of the parabola shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165440\/CNX_Precalc_Figure_03_02_0032.jpg\" alt=\"Graph of a parabola with a vertex at (3, 1) and a y-intercept at (0, 7).\" width=\"487\" height=\"517\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q366804\">Solution<\/span><\/p>\n<div id=\"q366804\" class=\"hidden-answer\" style=\"display: none\">\n<p>The vertex is the turning point of the graph. We can see that the vertex is at [latex](3,1)[\/latex]. Because this parabola opens upward, the axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is [latex]x=3[\/latex]. This parabola does not cross the [latex]x[\/latex]-axis, so it has no zeros. It crosses the [latex]y[\/latex]-axis at [latex](0, 7)[\/latex] so this is the [latex]y[\/latex]-intercept.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<h3>TRY IT<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm147099\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=147099&theme=oea&iframe_resize_id=ohm147099&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h3>General and Standard Forms of Quadratic Functions<\/h3>\n<p>The <strong>general form of a quadratic function<\/strong> presents the function in the form<\/p>\n<p style=\"text-align: center\">[latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex]<\/p>\n<p>where [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex]\u00a0are real numbers and [latex]a\\ne 0[\/latex]. If [latex]a>0[\/latex], the parabola opens upward. If [latex]a<0[\/latex], the parabola opens downward. We can use the general form of a parabola to find the equation for the axis of symmetry.\n\nThe axis of symmetry is defined by [latex]\\displaystyle x=-\\frac{b}{2a}[\/latex]. If we use the quadratic formula, [latex]\\displaystyle x=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}[\/latex], to solve [latex]a{x}^{2}+bx+c=0[\/latex] for the [latex]x[\/latex]-intercepts, or zeros, we find the value of [latex]x[\/latex]\u00a0halfway between them is always [latex]\\displaystyle x=-\\frac{b}{2a}[\/latex], the equation for the axis of symmetry.\n\nThe figure below shows\u00a0the graph of the quadratic function written in general form as [latex]y={x}^{2}+4x+3[\/latex]. In this form, [latex]a=1,\\text{ }b=4[\/latex], and [latex]c=3[\/latex]. Because [latex]a>0[\/latex], the parabola opens upward. The axis of symmetry is [latex]\\displaystyle 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[\/latex]-intercepts, those points where the parabola crosses the [latex]x[\/latex]-axis, occur at [latex]\\left(-3,0\\right)[\/latex] and [latex]\\left(-1,0\\right)[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165442\/CNX_Precalc_Figure_03_02_0042.jpg\" alt=\"Graph of a parabola showing where the x and y intercepts, vertex, and axis of symmetry are for the function y=x^2+4x+3.\" width=\"487\" height=\"555\" \/><\/p>\n<p>The <strong>standard form of a quadratic function<\/strong> presents the function in the form<\/p>\n<p style=\"text-align: center\">[latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex]<\/p>\n<p>where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the <strong>vertex form of a quadratic function<\/strong>.<\/p>\n<h3>Given a quadratic function in general form, find the vertex of the parabola.<\/h3>\n<p>One reason we may want to identify the vertex of the parabola is that this point will inform us where the maximum or minimum value of the output occurs, ([latex]k[\/latex]), and where it occurs, ([latex]x[\/latex]). If we are given the general form of a quadratic function:<\/p>\n<p style=\"text-align: center\">[latex]f(x)=ax^2+bx+c[\/latex]<\/p>\n<p>We can define the vertex, [latex](h,k)[\/latex], by doing the following:<\/p>\n<ul>\n<li>Identify [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex].<\/li>\n<li>Find [latex]h[\/latex], the [latex]x[\/latex]-coordinate of the vertex, by substituting [latex]a[\/latex] and [latex]b[\/latex]\u00a0into [latex]\\displaystyle h=-\\frac{b}{2a}[\/latex].<\/li>\n<li>Find [latex]k[\/latex], the [latex]y[\/latex]-coordinate of the vertex, by evaluating [latex]\\displaystyle k=f\\left(h\\right)=f\\left(-\\frac{b}{2a}\\right)[\/latex]<\/li>\n<\/ul>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Vertex of a Quadratic Function<\/h3>\n<p>Find the vertex of the quadratic function [latex]f\\left(x\\right)=2{x}^{2}-6x+7[\/latex]. Rewrite the quadratic in standard form (vertex form).<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q466886\">Solution<\/span><\/p>\n<div id=\"q466886\" class=\"hidden-answer\" style=\"display: none\">\n<p>The horizontal coordinate of the vertex will be at<\/p>\n<p style=\"text-align: center\">[latex]\\large \\begin{array}{c}h=-\\frac{b}{2a}\\hfill \\\\ \\text{ }=-\\frac{-6}{2\\left(2\\right)}\\hfill \\\\ \\text{ }=\\frac{6}{4}\\hfill \\\\ \\text{ }=\\frac{3}{2}\\hfill \\end{array}[\/latex]<\/p>\n<p>The vertical coordinate of the vertex will be at<\/p>\n<p style=\"text-align: center\">[latex]\\large \\begin{array}{c}k=f\\left(h\\right)\\hfill \\\\ \\text{ }=f\\left(\\frac{3}{2}\\right)\\hfill \\\\ \\text{ }=2{\\left(\\frac{3}{2}\\right)}^{2}-6\\left(\\frac{3}{2}\\right)+7\\hfill \\\\ \\text{ }=\\frac{5}{2}\\hfill \\end{array}[\/latex]<\/p>\n<p>Rewriting into standard form, the stretch factor will be the same as the [latex]a[\/latex] in the original quadratic.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}f\\left(x\\right)=a{x}^{2}+bx+c\\hfill \\\\ f\\left(x\\right)=2{x}^{2}-6x+7\\hfill \\end{array}[\/latex]<\/p>\n<p>Using the vertex to determine the shifts,<\/p>\n<p style=\"text-align: center\">[latex]\\displaystyle f\\left(x\\right)=2{\\left(x-\\frac{3}{2}\\right)}^{2}+\\frac{5}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>EXAMPLE<\/h3>\n<p>Given the equation [latex]g\\left(x\\right)=13+{x}^{2}-6x[\/latex], write the equation in general form and then in standard form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q713769\">Solution<\/span><\/p>\n<div id=\"q713769\" class=\"hidden-answer\" style=\"display: none\">[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<\/div>\n<\/div>\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<h1>(10.2.2) &#8211; Classifying Solutions to Quadratic Equations<\/h1>\n<p>Much as we did in the application problems above, we also need to find intercepts of quadratic equations for graphing parabolas. Recall that we find the <em>y<\/em>-intercept of a quadratic by evaluating the function at an input of zero, and we find the <em>x<\/em>-intercepts at locations where the output is zero. Notice\u00a0that the number of <em>x<\/em>-intercepts can vary depending upon the location of the graph.<\/p>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165449\/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 x-intercepts of a parabola<\/p>\n<\/div>\n<p>Mathematicians also define <em>x<\/em>-intercepts as roots of the quadratic function.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a quadratic function [latex]f\\left(x\\right)[\/latex], find the [latex]y[\/latex]&#8211;\u00a0and [latex]x[\/latex]-intercepts.<\/h3>\n<ol>\n<li>Evaluate [latex]f\\left(0\\right)[\/latex] to find the [latex]y[\/latex]-intercept.<\/li>\n<li>Solve the quadratic equation [latex]f\\left(x\\right)=0[\/latex] to find the [latex]x[\/latex]-intercepts.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the [latex]y[\/latex]&#8211; and [latex]x[\/latex]-Intercepts of a Parabola<\/h3>\n<p>Find the [latex]y[\/latex]&#8211; and [latex]x[\/latex]-intercepts of the quadratic [latex]f\\left(x\\right)=3{x}^{2}+5x - 2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q14680\">Solution<\/span><\/p>\n<div id=\"q14680\" class=\"hidden-answer\" style=\"display: none\">\nWe find the [latex]y[\/latex]-intercept by evaluating [latex]f\\left(0\\right)[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{c}f\\left(0\\right)=3{\\left(0\\right)}^{2}+5\\left(0\\right)-2\\hfill \\\\ \\text{ }=-2\\hfill \\end{array}[\/latex]<\/p>\n<p>So the [latex]y[\/latex]-intercept is at [latex]\\left(0,-2\\right)[\/latex].<\/p>\n<p>For the [latex]x[\/latex]-intercepts, or roots, we find all solutions of [latex]f\\left(x\\right)=0[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]0=3{x}^{2}+5x - 2[\/latex]<\/p>\n<p>In this case, the quadratic can be factored easily, providing the simplest method for solution.<\/p>\n<p style=\"text-align: center\">[latex]0=\\left(3x - 1\\right)\\left(x+2\\right)[\/latex]<br \/>\n[latex]\\begin{array}{c}0=3x - 1\\hfill & \\hfill & \\hfill & \\hfill & 0=x+2\\hfill \\\\ x=\\frac{1}{3}\\hfill & \\hfill & \\text{or}\\hfill & \\hfill & x=-2\\hfill \\end{array}[\/latex]<\/p>\n<p>So the <em>roots<\/em>\u00a0are at [latex]\\left(\\frac{1}{3},0\\right)[\/latex] and [latex]\\left(-2,0\\right)[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>By graphing the function, we can confirm that the graph crosses the <em>y<\/em>-axis at [latex]\\left(0,-2\\right)[\/latex]. We can also confirm that the graph crosses the [latex]x[\/latex]-axis at [latex]\\left(\\frac{1}{3},0\\right)[\/latex] and [latex]\\left(-2,0\\right)[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165452\/CNX_Precalc_Figure_03_02_0142.jpg\" alt=\"Graph of a parabola which has the following intercepts (-2, 0), (1\/3, 0), and (0, -2).\" width=\"487\" height=\"480\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the previous example, finding the [latex]y[\/latex]&#8211; and [latex]x[\/latex]-Intercepts of a parabola, the quadratic was easily solved by factoring. However, there are many quadratics that cannot be factored. We can solve these quadratics by first rewriting them in standard form.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a quadratic function, find the [latex]x[\/latex]-intercepts by rewriting in standard form.<\/h3>\n<ol>\n<li>Substitute [latex]a[\/latex]\u00a0and [latex]b[\/latex]\u00a0into [latex]\\displaystyle h=-\\frac{b}{2a}[\/latex].<\/li>\n<li>Substitute [latex]x=h[\/latex]\u00a0into 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]\u00a0and [latex]k[\/latex].<\/li>\n<li>Solve for when the output of the function will be zero to find the [latex]x[\/latex]<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 [latex]x[\/latex]-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\">Solution<\/span><\/p>\n<div id=\"q201989\" class=\"hidden-answer\" style=\"display: none\">\nWe 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 [latex]h[\/latex]\u00a0and [latex]k[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\large \\begin{array}{c}h=-\\frac{b}{2a}\\hfill & \\hfill & \\hfill & k=f\\left(-1\\right)\\hfill \\\\ \\text{ }=-\\frac{4}{2\\left(2\\right)}\\hfill & \\hfill & \\hfill & \\text{ }=2{\\left(-1\\right)}^{2}+4\\left(-1\\right)-4\\hfill \\\\ \\text{ }=-1\\hfill & \\hfill & \\hfill & \\text{ }=-6\\hfill \\end{array}[\/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{array}{c}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=\\pm \\sqrt{3}\\hfill \\\\ x=-1\\pm \\sqrt{3}\\hfill \\end{array}[\/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\/2862\/2017\/12\/26165454\/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>EXAMPLE<\/h3>\n<p>Find the [latex]y[\/latex]-intercept for the function [latex]g\\left(x\\right)=13+{x}^{2}-6x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q275171\">Solution<\/span><\/p>\n<div id=\"q275171\" class=\"hidden-answer\" style=\"display: none\">[latex]y[\/latex]-intercept at [latex](0,13)[\/latex]<\/div>\n<\/div>\n<\/div>\n<h4>Complex\u00a0Roots<\/h4>\n<p>Now you will hopefully begin to understand why we introduced complex numbers at the beginning of this module. Consider the following function: [latex]f(x)=x^2+2x+3[\/latex], and it&#8217;s graph below:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3475 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165458\/Screen-Shot-2016-08-04-at-11.34.19-AM.png\" alt=\"Graph of quadratic function with the following points (-1,2), (-2,3), (0,3), (1,6), (-3,6).\" width=\"241\" height=\"249\" \/><\/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. \u00a0You 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 contain complex numbers:<\/p>\n<p style=\"text-align: center\">[latex]x^2+2x+3=0[\/latex]<\/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 Answer<\/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], [latex]c[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\begin{array}{ccc}x^2+2x+3=0\\\\a=1,b=2,c=3\\end{array}[\/latex]<\/p>\n<p>Substitute these values into the quadratic formula.<\/p>\n<p style=\"text-align: center\">[latex]\\large \\begin{array}{c}x&=&\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}\\\\&=&\\frac{-2\\pm \\sqrt{{2}^{2}-4(1)(3)}}{2(1)}\\\\&=&\\frac{-2\\pm \\sqrt{4-12}}{2} \\\\&=&\\frac{-2\\pm \\sqrt{-8}}{2}\\\\&=&\\frac{-2\\pm 2i\\sqrt{2}}{2} \\\\-1\\pm i\\sqrt{2}&=&-1+\\sqrt{2},-1-\\sqrt{2}\\end{array}[\/latex]<\/p>\n<p>The solutions to this equations 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_3475\" style=\"width: 251px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3475\" class=\"wp-image-3475\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2862\/2017\/12\/26165458\/Screen-Shot-2016-08-04-at-11.34.19-AM.png\" alt=\"Graph of quadratic function with the following points (-1,2), (-2,3), (0,3), (1,6), (-3,6).\" width=\"241\" height=\"249\" \/><\/p>\n<p id=\"caption-attachment-3475\" class=\"wp-caption-text\">Graph of quadratic function with no x-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<h4>Important Terms<\/h4>\n<dl id=\"fs-id1165135449657\" class=\"definition\">\n<dt><strong>axis of symmetry<\/strong><\/dt>\n<dd id=\"fs-id1165135449662\">a vertical line drawn through the vertex of a parabola around which the parabola is symmetric; it is defined by [latex]\\displaystyle x=-\\frac{b}{2a}[\/latex].<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135502777\" class=\"definition\">\n<dt><strong>general form of a quadratic function<\/strong><\/dt>\n<dd id=\"fs-id1165135502783\">the function that describes a parabola, written in the form [latex]f\\left(x\\right)=a{x}^{2}+bx+c[\/latex], where [latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex]\u00a0are real numbers and [latex]a\\ne 0[\/latex].<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137931314\" class=\"definition\">\n<dt><strong>standard form of a quadratic function<\/strong><\/dt>\n<dd id=\"fs-id1165137931319\">the function that describes a parabola, written in the form [latex]f\\left(x\\right)=a{\\left(x-h\\right)}^{2}+k[\/latex], where [latex]\\left(h,\\text{ }k\\right)[\/latex] is the vertex.<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dd><\/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>zeros<\/strong><\/dt>\n<\/dl>\n<dl id=\"fs-id1165135623634\" class=\"definition\">\n<dd id=\"fs-id1165135623639\">in a given function, the values of [latex]x[\/latex]\u00a0at which [latex]y = 0[\/latex], also called roots<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135623614\" class=\"definition\">\n<dt><\/dt>\n<\/dl>\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-4679\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Interactive: Transform Quadratic 1. <strong>Provided by<\/strong>: Lumen Learning (with Desmos). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/fpatj6tbcn\">https:\/\/www.desmos.com\/calculator\/fpatj6tbcn<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Interactive: Transform Quadratic 2. <strong>Provided by<\/strong>: Lumen Learning (with Desmos). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/5g3xfhkklq\">https:\/\/www.desmos.com\/calculator\/5g3xfhkklq<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Interactive: Transform Quadratic 3. <strong>Provided by<\/strong>: Lumen Learning (with Desmos). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/ha6gh59rq7\">https:\/\/www.desmos.com\/calculator\/ha6gh59rq7<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Interactive: Tranform Quadratic 4. <strong>Provided by<\/strong>: Lumen Learning (with Desmos). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/pimelalx4i\">https:\/\/www.desmos.com\/calculator\/pimelalx4i<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Vertex and Axis of Symmetry Interactive. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/q3e3ymnpnn\">https:\/\/www.desmos.com\/calculator\/q3e3ymnpnn<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/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\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 120303, 120300. <strong>Authored by<\/strong>: Lumen Learning. <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>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID# 147099. <strong>Authored by<\/strong>: Day,Alyson. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 35145. <strong>Authored by<\/strong>: Jim Smart. <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>: IMathAS Community License CC-BY + GPL<\/li><\/ul><\/div>\n\t\t\t\t\t\t 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