{"id":3953,"date":"2017-04-24T16:23:03","date_gmt":"2017-04-24T16:23:03","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/collegealgebra2017\/?post_type=chapter&#038;p=3953"},"modified":"2019-02-18T03:05:25","modified_gmt":"2019-02-18T03:05:25","slug":"graphs-of-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/wmopen-collegealgebra\/chapter\/graphs-of-polynomial-functions\/","title":{"raw":"Graphs of Polynomial Functions","rendered":"Graphs of Polynomial Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li class=\"li2\"><span class=\"s1\">Identify zeros of polynomial functions with even and odd multiplicity.<\/span><\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Draw the graph of a polynomial function using end behavior, turning points, intercepts, and the Intermediate Value Theorem.<\/span><\/li>\r\n \t<li class=\"li2\"><span class=\"s1\">Write the equation of a polynomial function given its graph.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\nThe revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in the table below<b>.<\/b>\r\n<table summary=\"Two rows and nine columns. The first row is labeled,\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Year<\/strong><\/td>\r\n<td>2006<\/td>\r\n<td>2007<\/td>\r\n<td>2008<\/td>\r\n<td>2009<\/td>\r\n<td>2010<\/td>\r\n<td>2011<\/td>\r\n<td>2012<\/td>\r\n<td>2013<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Revenue<\/strong><\/td>\r\n<td>52.4<\/td>\r\n<td>52.8<\/td>\r\n<td>51.2<\/td>\r\n<td>49.5<\/td>\r\n<td>48.6<\/td>\r\n<td>48.6<\/td>\r\n<td>48.7<\/td>\r\n<td>47.1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe revenue can be modeled by the polynomial function\r\n<p style=\"text-align: center;\">[latex]R\\left(t\\right)=-0.037{t}^{4}+1.414{t}^{3}-19.777{t}^{2}+118.696t - 205.332[\/latex]<\/p>\r\nwhere <em>R<\/em>\u00a0represents the revenue in millions of dollars and <em>t<\/em>\u00a0represents the year, with <em>t<\/em> = 6\u00a0corresponding to 2006. Over which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company decreasing? These questions, along with many others, can be answered by examining the graph of the polynomial function. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general.\r\n<h2>Multiplicity and Turning Points<\/h2>\r\nGraphs behave differently at various <em>x<\/em>-intercepts. Sometimes the graph will cross over the x-axis at an intercept. Other times the graph will touch the x-axis and bounce off.\r\n\r\nSuppose, for example, we graph the function [latex]f\\left(x\\right)=\\left(x+3\\right){\\left(x - 2\\right)}^{2}{\\left(x+1\\right)}^{3}[\/latex].\r\n<p style=\"text-align: left;\">Notice in the figure below\u00a0that the behavior of the function at each of the <em>x<\/em>-intercepts is different.<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201554\/CNX_Precalc_Figure_03_04_0072.jpg\" alt=\"Graph of h(x)=x^3+4x^2+x-6.\" width=\"487\" height=\"329\" \/> The behavior of a graph at an x-intercept can be determined by examining the multiplicity of the zero.[\/caption]\r\n\r\nThe <em>x<\/em>-intercept [latex]x=-3[\/latex]\u00a0is the solution to the equation [latex]\\left(x+3\\right)=0[\/latex]. The graph passes directly through the <em>x<\/em>-intercept at [latex]x=-3[\/latex]. The factor is linear (has a degree of 1), so the behavior near the intercept is like that of a line; it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.\r\n\r\nThe <em>x<\/em>-intercept [latex]x=2[\/latex] is the repeated solution to the equation [latex]{\\left(x - 2\\right)}^{2}=0[\/latex]. The graph touches the axis at the intercept and changes direction. The factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic\u2014it bounces off of the horizontal axis at the intercept.\r\n<p style=\"text-align: center;\">[latex]{\\left(x - 2\\right)}^{2}=\\left(x - 2\\right)\\left(x - 2\\right)[\/latex]<\/p>\r\nThe factor is repeated, that is, the factor [latex]\\left(x - 2\\right)[\/latex] appears twice. The number of times a given factor appears in the factored form of the equation of a polynomial is called the <strong>multiplicity<\/strong>. The zero associated with this factor, [latex]x=2[\/latex], has multiplicity 2 because the factor [latex]\\left(x - 2\\right)[\/latex] occurs twice.\r\n\r\nThe <em>x-<\/em>intercept [latex]x=-1[\/latex] is the repeated solution of factor [latex]{\\left(x+1\\right)}^{3}=0[\/latex]. The graph passes through the axis at the intercept but flattens out a bit first. This factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic with the same S-shape near the intercept as the function [latex]f\\left(x\\right)={x}^{3}[\/latex]. We call this a triple zero, or a zero with multiplicity 3.\r\n\r\nFor <strong>zeros<\/strong> with even multiplicities, the graphs\u00a0<em>touch<\/em> or are tangent to the <em>x<\/em>-axis at these x-values. For zeros with odd multiplicities, the graphs <em>cross<\/em> or intersect the <em>x<\/em>-axis at these x-values. See the graphs below\u00a0for examples of graphs of polynomial functions with multiplicity 1, 2, and 3.\r\n\r\n<img class=\"small aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201556\/CNX_Precalc_Figure_03_04_0082.jpg\" alt=\"Three graphs. The first is a single zero graph, where p equals 1. The graph is of a line with a slight curve. The second graph is a zero with multiplicity 2 graph where p equals 2. The graph is u-shaped, with both positive and negative ends pointed upwards (positive). The third graph is a zero with multiplicity 3 graph, where p equals 3. The graph is shaped somewhat like an s.\" width=\"975\" height=\"325\" \/>\r\n\r\nFor higher even powers, such as 4, 6, and 8, the graph will still touch and bounce off of the x-axis, but for each increasing even power the graph will appear flatter as it approaches and leaves the <em>x<\/em>-axis.\r\n\r\nFor higher odd powers, such as 5, 7, and 9, the graph will still cross through the x-axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the <em>x<\/em>-axis.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Graphical Behavior of Polynomials at <em>x<\/em>-Intercepts<\/h3>\r\nIf a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], the behavior near the <em>x<\/em>-intercept <em>h\u00a0<\/em>is determined by the power <em>p<\/em>. We say that [latex]x=h[\/latex] is a zero of <strong>multiplicity<\/strong> <em>p<\/em>.\r\n\r\nThe graph of a polynomial function will touch the <em>x<\/em>-axis at zeros with even multiplicities. The graph will cross the <em>x<\/em>-axis at zeros with odd multiplicities.\r\n\r\nThe sum of the multiplicities is the degree of the polynomial function.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a graph of a polynomial function of degree [latex]n[\/latex], identify the zeros and their multiplicities.<\/h3>\r\n<ol>\r\n \t<li>If the graph crosses the <em>x<\/em>-axis and appears almost linear at the intercept, it is a single zero.<\/li>\r\n \t<li>If the graph touches the <em>x<\/em>-axis and bounces off of the axis, it is a zero with even multiplicity.<\/li>\r\n \t<li>If the graph crosses the <em>x<\/em>-axis at a zero, it is a zero with odd multiplicity.<\/li>\r\n \t<li>The sum of the multiplicities is the degree\u00a0<em>n<\/em>.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying Zeros and Their Multiplicities<\/h3>\r\nUse the graph of the function of degree 6 to identify the zeros of the function and their possible multiplicities.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201558\/CNX_Precalc_Figure_03_04_0092.jpg\" alt=\"Three graphs showing three different polynomial functions with multiplicity 1, 2, and 3.\" width=\"487\" height=\"628\" \/>\r\n[reveal-answer q=\"583908\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"583908\"]\r\n\r\nThe polynomial function is of degree <em>n<\/em> which is 6. The sum of the multiplicities must be\u00a06.\r\n\r\nStarting from the left, the first zero occurs at [latex]x=-3[\/latex]. The graph touches the <em>x<\/em>-axis, so the multiplicity of the zero must be even. The zero of \u20133 has multiplicity 2.\r\n\r\nThe next zero occurs at [latex]x=-1[\/latex]. The graph looks almost linear at this point. This is a single zero of multiplicity 1.\r\n\r\nThe last zero occurs at [latex]x=4[\/latex]. The graph crosses the<em> x<\/em>-axis, so the multiplicity of the zero must be odd. We know that the multiplicity is 3 and that the sum of the multiplicities must be 6.\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\nUse the graph of the function of degree 7 to identify the zeros of the function and their multiplicities.\r\n\r\n<img class=\"small aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201600\/CNX_Precalc_Figure_03_04_0102.jpg\" alt=\"Graph of an even-degree polynomial with degree 6.\" width=\"487\" height=\"253\" \/>\r\n[reveal-answer q=\"874458\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"874458\"]\r\n\r\nThe graph has a zero of \u20135 with multiplicity 3, a zero of \u20131 with multiplicity 2, and a zero of 3 with multiplicity 2.[\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=121830&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"450\"><\/iframe>\r\n\r\n<\/div>\r\n\r\n<h2>Determining End Behavior<\/h2>\r\nAs we have already learned, the behavior of a graph of a <strong>polynomial function<\/strong> of the form\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+...+{a}_{1}x+{a}_{0}[\/latex]<\/p>\r\nwill either ultimately rise or fall as <em>x<\/em>\u00a0increases without bound and will either rise or fall as <em>x\u00a0<\/em>decreases without bound. This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. The same is true for very small inputs, say \u2013100 or \u20131,000.\r\n\r\nRecall that we call this behavior the <em>end behavior<\/em> of a function. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function, [latex]{a}_{n}{x}^{n}[\/latex], is an even power function, as <em>x<\/em>\u00a0increases or decreases without bound, [latex]f\\left(x\\right)[\/latex] increases without bound. When the leading term is an odd power function, as\u00a0<em>x<\/em>\u00a0decreases without bound, [latex]f\\left(x\\right)[\/latex] also decreases without bound; as <em>x<\/em>\u00a0increases without bound, [latex]f\\left(x\\right)[\/latex] also increases without bound. If the leading term is negative, it will change the direction of the end behavior. The table below\u00a0summarizes all four cases.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\">Even Degree<\/th>\r\n<th style=\"text-align: center;\">Odd Degree<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/11.png\"><img class=\"alignnone size-full wp-image-12504\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201602\/11.png\" alt=\"11\" width=\"423\" height=\"559\" \/><\/a><\/td>\r\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/12.png\"><img class=\"alignnone size-full wp-image-12505\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201605\/12.png\" alt=\"12\" width=\"397\" height=\"560\" \/><\/a><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/13.png\"><img class=\"alignnone size-full wp-image-12506\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201607\/13.png\" alt=\"13\" width=\"387\" height=\"574\" \/><\/a><\/td>\r\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/14.png\"><img class=\"alignnone size-full wp-image-12507\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201609\/14.png\" alt=\"14\" width=\"404\" height=\"564\" \/><\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h3><\/h3>\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=29473&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<h2>Graphing Polynomial Functions<\/h2>\r\nWe can use what we have learned about multiplicities, end behavior, and turning points to sketch graphs of polynomial functions. Let us put this all together and look at the steps required to graph polynomial functions.\r\n<div class=\"textbox\">\r\n<h3>How To: Given a polynomial function, sketch the graph<\/h3>\r\n<ol>\r\n \t<li>Find the intercepts.<\/li>\r\n \t<li>Check for symmetry. If the function is an even function, its graph is symmetric with respect to the\u00a0<em>y<\/em>-axis, that is,\u00a0<em>f<\/em>(\u2013<em>x<\/em>) = <em>f<\/em>(<em>x<\/em>).\r\nIf a function is an odd function, its graph is symmetric with respect to the origin, that is,\u00a0<em>f<\/em>(\u2013<em>x<\/em>) = <em>\u2013<\/em><em>f<\/em>(<em>x<\/em>).<\/li>\r\n \t<li>Use the multiplicities of the zeros to determine the behavior of the polynomial at the <em>x<\/em>-intercepts.<\/li>\r\n \t<li>Determine the end behavior by examining the leading term.<\/li>\r\n \t<li>Use the end behavior and the behavior at the intercepts to sketch the graph.<\/li>\r\n \t<li>Ensure that the number of turning points does not exceed one less than the degree of the polynomial.<\/li>\r\n \t<li>Optionally, use technology to check the graph.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Sketching the Graph of a Polynomial Function<\/h3>\r\nSketch a possible graph for [latex]f\\left(x\\right)=-2{\\left(x+3\\right)}^{2}\\left(x - 5\\right)[\/latex].\r\n\r\n[reveal-answer q=\"707233\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"707233\"]\r\n\r\nThis graph has two <em>x-<\/em>intercepts. At <em>x\u00a0<\/em>= \u20133, the factor is squared, indicating a multiplicity of 2. The graph will bounce off the\u00a0<em>x<\/em>-intercept at this value. At <em>x\u00a0<\/em>= 5, the function has a multiplicity of one, indicating the graph will cross through the axis at this intercept.\r\n\r\nThe <em>y<\/em>-intercept is found by evaluating <em>f<\/em>(0).\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\hfill \\\\ f\\left(0\\right)=-2{\\left(0+3\\right)}^{2}\\left(0 - 5\\right)\\hfill \\\\ \\text{}f\\left(0\\right)=-2\\cdot 9\\cdot \\left(-5\\right)\\hfill \\\\ \\text{}f\\left(0\\right)=90\\hfill \\end{array}[\/latex]<\/p>\r\nThe <em>y<\/em>-intercept is (0, 90).\r\n\r\nAdditionally, we can see the leading term, if this polynomial were multiplied out, would be [latex]-2{x}^{3}[\/latex], so the end behavior, as seen in the following graph, is that of a vertically reflected cubic with the outputs decreasing as the inputs approach infinity and the outputs increasing as the inputs approach negative infinity.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201614\/CNX_Precalc_Figure_03_04_0172.jpg\" alt=\"Showing the distribution for the leading term.\" width=\"487\" height=\"362\" \/>\r\n\r\nTo sketch the graph, we consider the following:\r\n<ul id=\"fs-id1165134374741\">\r\n \t<li>As [latex]x\\to -\\infty [\/latex] the function [latex]f\\left(x\\right)\\to \\infty [\/latex], so we know the graph starts in the second quadrant and is decreasing toward the <em>x<\/em>-axis.<\/li>\r\n \t<li>Since [latex]f\\left(-x\\right)=-2{\\left(-x+3\\right)}^{2}\\left(-x - 5\\right)[\/latex] is not equal to <em>f<\/em>(<em>x<\/em>), the graph does not have any symmetry.<\/li>\r\n \t<li>At [latex]\\left(-3,0\\right)[\/latex] the graph bounces off of the <em>x<\/em>-axis, so the function must start increasing after this point.<\/li>\r\n \t<li>At (0, 90), the graph crosses the <em>y<\/em>-axis.<\/li>\r\n<\/ul>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201617\/CNX_Precalc_Figure_03_04_0182.jpg\" alt=\"Graph of the end behavior and intercepts, (-3, 0) and (0, 90), for the function f(x)=-2(x+3)^2(x-5).\" width=\"487\" height=\"362\" \/>\r\n\r\n&nbsp;\r\n\r\nSomewhere after this point, the graph must turn back down or start decreasing toward the horizontal axis because the graph passes through the next intercept at (5, 0).\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201618\/CNX_Precalc_Figure_03_04_0192.jpg\" alt=\"Graph of the end behavior and intercepts, (-3, 0), (0, 90) and (5, 0), for the function f(x)=-2(x+3)^2(x-5).\" width=\"487\" height=\"362\" \/>\r\n\r\nAs [latex]x\\to \\infty [\/latex] the function [latex]f\\left(x\\right)\\to \\mathrm{-\\infty }[\/latex], so we know the graph continues to decrease, and we can stop drawing the graph in the fourth quadrant.\r\n\r\nThe complete graph of the polynomial function [latex]f\\left(x\\right)=-2{\\left(x+3\\right)}^{2}\\left(x - 5\\right)[\/latex] is as follows:\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201620\/CNX_Precalc_Figure_03_04_0202.jpg\" alt=\"Graph of f(x)=-2(x+3)^2(x-5).\" width=\"487\" height=\"366\" \/>\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\nSketch a possible graph for [latex]f\\left(x\\right)=\\frac{1}{4}x{\\left(x - 1\\right)}^{4}{\\left(x+3\\right)}^{3}[\/latex].\r\n\r\n[reveal-answer q=\"812296\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"812296\"]\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/30233150\/CNX_Precalc_Figure_03_04_0212.jpg\"><img class=\"aligncenter size-full wp-image-2911\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/30233150\/CNX_Precalc_Figure_03_04_0212.jpg\" alt=\"Graph of f(x)=(1\/4)x(x-1)^4(x+3)^3.\" width=\"487\" height=\"366\" \/><\/a>\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=113372&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"450\"><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n<h3>The Intermediate Value Theorem<\/h3>\r\nIn some situations, we may know two points on a graph but not the zeros. If those two points are on opposite sides of the <em>x<\/em>-axis, we can confirm that there is a zero between them. Consider a polynomial function <em>f<\/em>\u00a0whose graph is smooth and continuous. The <strong>Intermediate Value Theorem<\/strong> states that for two numbers <em>a<\/em>\u00a0and <em>b<\/em>\u00a0in the domain of <em>f<\/em>,\u00a0if <em>a\u00a0<\/em>&lt; <em>b<\/em>\u00a0and [latex]f\\left(a\\right)\\ne f\\left(b\\right)[\/latex], then the function <em>f<\/em>\u00a0takes on every value between [latex]f\\left(a\\right)[\/latex] and [latex]f\\left(b\\right)[\/latex].\r\n\r\nWe can apply this theorem to a special case that is useful for graphing polynomial functions. If a point on the graph of a continuous function <em>f<\/em>\u00a0at [latex]x=a[\/latex] lies above the <em>x<\/em>-axis and another point at [latex]x=b[\/latex] lies below the <em>x<\/em>-axis, there must exist a third point between [latex]x=a[\/latex] and [latex]x=b[\/latex] where the graph crosses the <em>x<\/em>-axis. Call this point [latex]\\left(c,\\text{ }f\\left(c\\right)\\right)[\/latex]. This means that we are assured there is a value\u00a0<em>c<\/em>\u00a0where [latex]f\\left(c\\right)=0[\/latex].\r\n\r\nIn other words, the Intermediate Value Theorem tells us that when a polynomial function changes from a negative value to a positive value, the function must cross the <em>x<\/em>-axis. The figure below\u00a0shows that there is a zero between <em>a<\/em>\u00a0and <em>b<\/em>.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201623\/CNX_Precalc_Figure_03_04_0222.jpg\" alt=\"Graph of an odd-degree polynomial function that shows a point f(a) that\u2019s negative, f(b) that\u2019s positive, and f(c) that\u2019s 0.\" width=\"487\" height=\"368\" \/> The Intermediate Value Theorem can be used to show there exists a zero.[\/caption]\r\n\r\n<div class=\"textbox\">\r\n<h3>A General Note: Intermediate Value Theorem<\/h3>\r\nLet <em>f<\/em>\u00a0be a polynomial function. The <strong>Intermediate Value Theorem<\/strong> states that if [latex]f\\left(a\\right)[\/latex]\u00a0and [latex]f\\left(b\\right)[\/latex]\u00a0have opposite signs, then there exists at least one value <em>c<\/em>\u00a0between <em>a<\/em>\u00a0and <em>b<\/em>\u00a0for which [latex]f\\left(c\\right)=0[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using the Intermediate Value Theorem<\/h3>\r\nShow that the function [latex]f\\left(x\\right)={x}^{3}-5{x}^{2}+3x+6[\/latex]\u00a0has at least two real zeros between [latex]x=1[\/latex]\u00a0and [latex]x=4[\/latex].\r\n\r\n[reveal-answer q=\"33137\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"33137\"]\r\n\r\nTo start, evaluate [latex]f\\left(x\\right)[\/latex]\u00a0at the integer values [latex]x=1,2,3,\\text{ and }4[\/latex].\r\n<table><colgroup> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<tbody>\r\n<tr>\r\n<td><em><strong>x<\/strong><\/em><\/td>\r\n<td>1<\/td>\r\n<td>2<\/td>\r\n<td>3<\/td>\r\n<td>4<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em><strong>f<\/strong><\/em><strong>(<em>x<\/em>)<\/strong><\/td>\r\n<td>5<\/td>\r\n<td>0<\/td>\r\n<td>\u20133<\/td>\r\n<td>2<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe see that one zero occurs at [latex]x=2[\/latex]. Also, since [latex]f\\left(3\\right)[\/latex] is negative and [latex]f\\left(4\\right)[\/latex] is positive, by the Intermediate Value Theorem, there must be at least one real zero between 3 and 4.\r\n\r\nWe have shown that there are at least two real zeros between [latex]x=1[\/latex]\u00a0and [latex]x=4[\/latex].\r\n<h4>Analysis of the Solution<\/h4>\r\nWe can also graphically see that there are two real zeros between [latex]x=1[\/latex]\u00a0and [latex]x=4[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201625\/CNX_Precalc_Figure_03_04_0232.jpg\" alt=\"Graph of f(x)=x^3-5x^2+3x+6 and shows, by the Intermediate Value Theorem, that there exists two zeros since f(1)=5 and f(4)=2 are positive and f(3) = -3 is negative.\" width=\"487\" height=\"591\" \/>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nShow that the function [latex]f\\left(x\\right)=7{x}^{5}-9{x}^{4}-{x}^{2}[\/latex] has at least one real zero between [latex]x=1[\/latex] and [latex]x=2[\/latex].\r\n\r\n[reveal-answer q=\"778313\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"778313\"]\r\n\r\nBecause <em>f<\/em>\u00a0is a polynomial function and since [latex]f\\left(1\\right)[\/latex] is negative and [latex]f\\left(2\\right)[\/latex] is positive, there is at least one real zero between [latex]x=1[\/latex] and [latex]x=2[\/latex].[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Writing Formulas for Polynomial Functions<\/h3>\r\nNow that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. Because a <strong>polynomial function<\/strong> written in factored form will have an <em>x<\/em>-intercept where each factor is equal to zero, we can form a function that will pass through a set of <em>x<\/em>-intercepts by introducing a corresponding set of factors.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Factored Form of Polynomials<\/h3>\r\nIf a polynomial of lowest degree <em>p<\/em>\u00a0has zeros at [latex]x={x}_{1},{x}_{2},\\dots ,{x}_{n}[\/latex],\u00a0then the polynomial can be written in the factored form: [latex]f\\left(x\\right)=a{\\left(x-{x}_{1}\\right)}^{{p}_{1}}{\\left(x-{x}_{2}\\right)}^{{p}_{2}}\\cdots {\\left(x-{x}_{n}\\right)}^{{p}_{n}}[\/latex]\u00a0where the powers [latex]{p}_{i}[\/latex]\u00a0on each factor can be determined by the behavior of the graph at the corresponding intercept, and the stretch factor <em>a<\/em>\u00a0can be determined given a value of the function other than the <em>x<\/em>-intercept.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a graph of a polynomial function, write a formula for the function<\/h3>\r\n<ol>\r\n \t<li>Identify the <em>x<\/em>-intercepts of the graph to find the factors of the polynomial.<\/li>\r\n \t<li>Examine the behavior of the graph at the <em>x<\/em>-intercepts to determine the multiplicity of each factor.<\/li>\r\n \t<li>Find the polynomial of least degree containing all of the factors found in the previous step.<\/li>\r\n \t<li>Use any other point on the graph (the <em>y<\/em>-intercept may be easiest) to determine the stretch factor.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Writing a Formula for a Polynomial Function from Its Graph<\/h3>\r\nWrite a formula for the polynomial 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\/02201627\/CNX_Precalc_Figure_03_04_0242.jpg\" alt=\"Graph of a positive even-degree polynomial with zeros at x=-3, 2, 5 and y=-2.\" width=\"487\" height=\"366\" \/>\r\n\r\n[reveal-answer q=\"338564\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"338564\"]\r\n\r\nThis graph has three <em>x<\/em>-intercepts: <em>x\u00a0<\/em>= \u20133, 2, and 5. The <em>y<\/em>-intercept is located at (0, 2). At <em>x\u00a0<\/em>= \u20133 and <em>x\u00a0<\/em>= 5,\u00a0the graph passes through the axis linearly, suggesting the corresponding factors of the polynomial will be linear. At <em>x\u00a0<\/em>= 2, the graph bounces off the x-axis at the intercept suggesting the corresponding factor of the polynomial will be second degree (quadratic). Together, this gives us\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a\\left(x+3\\right){\\left(x - 2\\right)}^{2}\\left(x - 5\\right)[\/latex]<\/p>\r\nTo determine the stretch factor, we utilize another point on the graph. We will use the <em>y<\/em>-intercept (0, \u20132), to solve for <em>a<\/em>.\r\n\r\n[latex]\\begin{array}{l}f\\left(0\\right)=a\\left(0+3\\right){\\left(0 - 2\\right)}^{2}\\left(0 - 5\\right)\\hfill \\\\ \\text{ }-2=a\\left(0+3\\right){\\left(0 - 2\\right)}^{2}\\left(0 - 5\\right)\\hfill \\\\ \\text{ }-2=-60a\\hfill \\\\ \\text{ }a=\\frac{1}{30}\\hfill \\end{array}[\/latex]\r\n\r\nThe graphed polynomial appears to represent the function [latex]f\\left(x\\right)=\\frac{1}{30}\\left(x+3\\right){\\left(x - 2\\right)}^{2}\\left(x - 5\\right)[\/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\nGiven the graph below, write a formula for the function shown.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201629\/CNX_Precalc_Figure_03_04_0252.jpg\" alt=\"Graph of a negative even-degree polynomial with zeros at x=-1, 2, 4 and y=-4.\" width=\"487\" height=\"291\" \/>\r\n\r\n[reveal-answer q=\"427364\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"427364\"]\r\n\r\n[latex]f\\left(x\\right)=-\\frac{1}{8}{\\left(x - 2\\right)}^{3}{\\left(x+1\\right)}^{2}\\left(x - 4\\right)[\/latex][\/hidden-answer]\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=29478&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"450\"><\/iframe>\r\n\r\n<\/div>\r\n\r\n<h3>Local and Global Extrema<\/h3>\r\nWith quadratics, we were able to algebraically find the maximum or minimum value of the function by finding the vertex. For general polynomials, finding these turning points is not possible without more advanced techniques from calculus. Even then, finding where extrema occur can still be algebraically challenging. For now, we will estimate the locations of turning points using technology to generate a graph.\r\n\r\nEach turning point represents a local minimum or maximum. Sometimes, a turning point is the highest or lowest point on the entire graph. In these cases, we say that the turning point is a <strong>global maximum <\/strong>or a <strong>global minimum<\/strong>. These are also referred to as the absolute maximum and absolute minimum values of the function.\r\n\r\nA <strong>local maximum<\/strong> or <strong>local minimum<\/strong> at <em>x\u00a0<\/em>= <em>a<\/em>\u00a0(sometimes called the relative maximum or minimum, respectively) is the output at the highest or lowest point on the graph in an open interval around <em>x\u00a0<\/em>= <em>a<\/em>. If a function has a local maximum at <em>a<\/em>, then [latex]f\\left(a\\right)\\ge f\\left(x\\right)[\/latex] for all <em>x<\/em>\u00a0in an open interval around <em>x<\/em> =\u00a0<em>a<\/em>. If a function has a local minimum at <em>a<\/em>, then [latex]f\\left(a\\right)\\le f\\left(x\\right)[\/latex] for all <em>x<\/em>\u00a0in an open interval around <em>x\u00a0<\/em>= <em>a<\/em>.\r\n\r\nA <strong>global maximum<\/strong> or <strong>global minimum<\/strong> is the output at the highest or lowest point of the function. If a function has a global maximum at <em>a<\/em>, then [latex]f\\left(a\\right)\\ge f\\left(x\\right)[\/latex] for all <em>x<\/em>. If a function has a global minimum at <em>a<\/em>, then [latex]f\\left(a\\right)\\le f\\left(x\\right)[\/latex] for all <em>x<\/em>.\r\n\r\nWe can see the difference between local and global extrema below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201631\/CNX_Precalc_Figure_03_04_026n2.jpg\" alt=\"Graph of an even-degree polynomial that denotes the local maximum and minimum and the global maximum.\" width=\"487\" height=\"475\" \/>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<strong>Do all polynomial functions have a global minimum or maximum?<\/strong>\r\n\r\n<em>No. Only polynomial functions of even degree have a global minimum or maximum. For example, [latex]f\\left(x\\right)=x[\/latex] has neither a global maximum nor a global minimum.<\/em>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using Local Extrema to Solve Applications<\/h3>\r\nAn open-top box is to be constructed by cutting out squares from each corner of a 14 cm by 20 cm sheet of plastic then folding up the sides. Find the size of squares that should be cut out to maximize the volume enclosed by the box.\r\n\r\n[reveal-answer q=\"598415\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"598415\"]\r\n\r\nWe will start this problem by drawing a picture like the one below, labeling the width of the cut-out squares with a variable, <em>w<\/em>.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201633\/CNX_Precalc_Figure_03_04_0272.jpg\" alt=\"Diagram of a rectangle with four squares at the corners.\" width=\"487\" height=\"298\" \/>\r\n\r\nNotice that after a square is cut out from each end, it leaves a [latex]\\left(14 - 2w\\right)[\/latex] cm by [latex]\\left(20 - 2w\\right)[\/latex] cm rectangle for the base of the box, and the box will be <em>w<\/em>\u00a0cm tall. This gives the volume\r\n\r\n[latex]\\begin{array}{l}V\\left(w\\right)=\\left(20 - 2w\\right)\\left(14 - 2w\\right)w\\hfill \\\\ \\text{}V\\left(w\\right)=280w - 68{w}^{2}+4{w}^{3}\\hfill \\end{array}[\/latex]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201635\/CNX_Precalc_Figure_03_04_0282.jpg\" alt=\"Graph of V(w)=(20-2w)(14-2w)w where the x-axis is labeled w and the y-axis is labeled V(w).\" width=\"487\" height=\"406\" \/>\r\n\r\nNotice, since the factors are <em>w<\/em>, [latex]20 - 2w[\/latex] and [latex]14 - 2w[\/latex], the three zeros are 10, 7, and 0, respectively. Because a height of 0 cm is not reasonable, we consider only the zeros 10 and 7. The shortest side is 14 and we are cutting off two squares, so values <em>w<\/em>\u00a0may take on are greater than zero or less than 7. This means we will restrict the domain of this function to [latex]0&lt;w&lt;7[\/latex]. Using technology to sketch the graph of [latex]V\\left(w\\right)[\/latex] on this reasonable domain, we get a graph like the one above. We can use this graph to estimate the maximum value for the volume, restricted to values for <em>w<\/em>\u00a0that are reasonable for this problem, values from 0 to 7.\r\n\r\nOn this graph, we turn our focus to only the portion on the reasonable domain, [latex]\\left[0,\\text{ }7\\right][\/latex]. We can estimate the maximum value to be around 340 cubic cm, which occurs when the squares are about 2.75 cm on each side. To improve this estimate, we could use advanced features of our technology, if available, or simply change our window to zoom in on our graph to produce the graph below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201637\/CNX_Precalc_Figure_03_04_0292.jpg\" alt=\"Graph of V(w)=(20-2w)(14-2w)w where the x-axis is labeled w and the y-axis is labeled V(w) on the domain [2.4, 3].\" width=\"487\" height=\"444\" \/>\r\n\r\nFrom this zoomed-in view, we can refine our estimate for the maximum volume to about 339 cubic cm which occurs when the squares measure approximately 2.7 cm on each side.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li>Polynomial functions of degree 2 or more are smooth, continuous functions.<\/li>\r\n \t<li>To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero.<\/li>\r\n \t<li>Another way to find the <em>x-<\/em>intercepts of a polynomial function is to graph the function and identify the points where the graph crosses the <em>x<\/em>-axis.<\/li>\r\n \t<li>The multiplicity of a zero determines how the graph behaves at the <em>x<\/em>-intercept.<\/li>\r\n \t<li>The graph of a polynomial will cross the x-axis at a zero with odd multiplicity.<\/li>\r\n \t<li>The graph of a polynomial will touch and bounce off the x-axis at a zero with even multiplicity.<\/li>\r\n \t<li>The end behavior of a polynomial function depends on the leading term.<\/li>\r\n \t<li>The graph of a polynomial function changes direction at its turning points.<\/li>\r\n \t<li>A polynomial function of degree <em>n<\/em>\u00a0has at most\u00a0<em>n <\/em>\u2013\u00a01 turning points.<\/li>\r\n \t<li>To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most<em>\u00a0n <\/em>\u2013\u00a01 turning points.<\/li>\r\n \t<li>Graphing a polynomial function helps to estimate local and global extremas.<\/li>\r\n \t<li>The Intermediate Value Theorem tells us that if [latex]f\\left(a\\right) \\text{and} f\\left(b\\right)[\/latex]\u00a0have opposite signs, then there exists at least one value <em>c<\/em>\u00a0between <em>a<\/em>\u00a0and <em>b<\/em>\u00a0for which [latex]f\\left(c\\right)=0[\/latex].<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165132943522\" class=\"definition\">\r\n \t<dt><strong>global maximum<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165132943525\">highest turning point on a graph; [latex]f\\left(a\\right)[\/latex]\u00a0where [latex]f\\left(a\\right)\\ge f\\left(x\\right)[\/latex]\u00a0for all <em>x<\/em>.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943522\" class=\"definition\">\r\n \t<dt><b>global minimum<\/b><\/dt>\r\n \t<dd id=\"fs-id1165132943525\">lowest turning point on a graph; [latex]f\\left(a\\right)[\/latex]\u00a0where [latex]f\\left(a\\right)\\le f\\left(x\\right)[\/latex] for all <em>x<\/em>.<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943522\" class=\"definition\">\r\n \t<dt><strong>Intermediate Value Theorem<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165132943525\">for two numbers <em>a<\/em>\u00a0and <em>b<\/em>\u00a0in the domain of <em>f<\/em>,\u00a0if [latex]a&lt;b[\/latex]\u00a0and [latex]f\\left(a\\right)\\ne f\\left(b\\right)[\/latex],\u00a0then the function <em>f<\/em>\u00a0takes on every value between [latex]f\\left(a\\right)[\/latex]\u00a0and [latex]f\\left(b\\right)[\/latex];\u00a0specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the <em>x<\/em>-axis<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165132943522\" class=\"definition\">\r\n \t<dt><strong>multiplicity<\/strong><\/dt>\r\n \t<dd id=\"fs-id1165132943525\">the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], [latex]x=h[\/latex]\u00a0is a zero of multiplicity <em>p<\/em>.<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li class=\"li2\"><span class=\"s1\">Identify zeros of polynomial functions with even and odd multiplicity.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Draw the graph of a polynomial function using end behavior, turning points, intercepts, and the Intermediate Value Theorem.<\/span><\/li>\n<li class=\"li2\"><span class=\"s1\">Write the equation of a polynomial function given its graph.<\/span><\/li>\n<\/ul>\n<\/div>\n<p>The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in the table below<b>.<\/b><\/p>\n<table summary=\"Two rows and nine columns. The first row is labeled,\">\n<tbody>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td>2006<\/td>\n<td>2007<\/td>\n<td>2008<\/td>\n<td>2009<\/td>\n<td>2010<\/td>\n<td>2011<\/td>\n<td>2012<\/td>\n<td>2013<\/td>\n<\/tr>\n<tr>\n<td><strong>Revenue<\/strong><\/td>\n<td>52.4<\/td>\n<td>52.8<\/td>\n<td>51.2<\/td>\n<td>49.5<\/td>\n<td>48.6<\/td>\n<td>48.6<\/td>\n<td>48.7<\/td>\n<td>47.1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The revenue can be modeled by the polynomial function<\/p>\n<p style=\"text-align: center;\">[latex]R\\left(t\\right)=-0.037{t}^{4}+1.414{t}^{3}-19.777{t}^{2}+118.696t - 205.332[\/latex]<\/p>\n<p>where <em>R<\/em>\u00a0represents the revenue in millions of dollars and <em>t<\/em>\u00a0represents the year, with <em>t<\/em> = 6\u00a0corresponding to 2006. Over which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company decreasing? These questions, along with many others, can be answered by examining the graph of the polynomial function. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general.<\/p>\n<h2>Multiplicity and Turning Points<\/h2>\n<p>Graphs behave differently at various <em>x<\/em>-intercepts. Sometimes the graph will cross over the x-axis at an intercept. Other times the graph will touch the x-axis and bounce off.<\/p>\n<p>Suppose, for example, we graph the function [latex]f\\left(x\\right)=\\left(x+3\\right){\\left(x - 2\\right)}^{2}{\\left(x+1\\right)}^{3}[\/latex].<\/p>\n<p style=\"text-align: left;\">Notice in the figure below\u00a0that the behavior of the function at each of the <em>x<\/em>-intercepts is different.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201554\/CNX_Precalc_Figure_03_04_0072.jpg\" alt=\"Graph of h(x)=x^3+4x^2+x-6.\" width=\"487\" height=\"329\" \/><\/p>\n<p class=\"wp-caption-text\">The behavior of a graph at an x-intercept can be determined by examining the multiplicity of the zero.<\/p>\n<\/div>\n<p>The <em>x<\/em>-intercept [latex]x=-3[\/latex]\u00a0is the solution to the equation [latex]\\left(x+3\\right)=0[\/latex]. The graph passes directly through the <em>x<\/em>-intercept at [latex]x=-3[\/latex]. The factor is linear (has a degree of 1), so the behavior near the intercept is like that of a line; it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.<\/p>\n<p>The <em>x<\/em>-intercept [latex]x=2[\/latex] is the repeated solution to the equation [latex]{\\left(x - 2\\right)}^{2}=0[\/latex]. The graph touches the axis at the intercept and changes direction. The factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic\u2014it bounces off of the horizontal axis at the intercept.<\/p>\n<p style=\"text-align: center;\">[latex]{\\left(x - 2\\right)}^{2}=\\left(x - 2\\right)\\left(x - 2\\right)[\/latex]<\/p>\n<p>The factor is repeated, that is, the factor [latex]\\left(x - 2\\right)[\/latex] appears twice. The number of times a given factor appears in the factored form of the equation of a polynomial is called the <strong>multiplicity<\/strong>. The zero associated with this factor, [latex]x=2[\/latex], has multiplicity 2 because the factor [latex]\\left(x - 2\\right)[\/latex] occurs twice.<\/p>\n<p>The <em>x-<\/em>intercept [latex]x=-1[\/latex] is the repeated solution of factor [latex]{\\left(x+1\\right)}^{3}=0[\/latex]. The graph passes through the axis at the intercept but flattens out a bit first. This factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic with the same S-shape near the intercept as the function [latex]f\\left(x\\right)={x}^{3}[\/latex]. We call this a triple zero, or a zero with multiplicity 3.<\/p>\n<p>For <strong>zeros<\/strong> with even multiplicities, the graphs\u00a0<em>touch<\/em> or are tangent to the <em>x<\/em>-axis at these x-values. For zeros with odd multiplicities, the graphs <em>cross<\/em> or intersect the <em>x<\/em>-axis at these x-values. See the graphs below\u00a0for examples of graphs of polynomial functions with multiplicity 1, 2, and 3.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"small aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201556\/CNX_Precalc_Figure_03_04_0082.jpg\" alt=\"Three graphs. The first is a single zero graph, where p equals 1. The graph is of a line with a slight curve. The second graph is a zero with multiplicity 2 graph where p equals 2. The graph is u-shaped, with both positive and negative ends pointed upwards (positive). The third graph is a zero with multiplicity 3 graph, where p equals 3. The graph is shaped somewhat like an s.\" width=\"975\" height=\"325\" \/><\/p>\n<p>For higher even powers, such as 4, 6, and 8, the graph will still touch and bounce off of the x-axis, but for each increasing even power the graph will appear flatter as it approaches and leaves the <em>x<\/em>-axis.<\/p>\n<p>For higher odd powers, such as 5, 7, and 9, the graph will still cross through the x-axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the <em>x<\/em>-axis.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Graphical Behavior of Polynomials at <em>x<\/em>-Intercepts<\/h3>\n<p>If a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], the behavior near the <em>x<\/em>-intercept <em>h\u00a0<\/em>is determined by the power <em>p<\/em>. We say that [latex]x=h[\/latex] is a zero of <strong>multiplicity<\/strong> <em>p<\/em>.<\/p>\n<p>The graph of a polynomial function will touch the <em>x<\/em>-axis at zeros with even multiplicities. The graph will cross the <em>x<\/em>-axis at zeros with odd multiplicities.<\/p>\n<p>The sum of the multiplicities is the degree of the polynomial function.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a graph of a polynomial function of degree [latex]n[\/latex], identify the zeros and their multiplicities.<\/h3>\n<ol>\n<li>If the graph crosses the <em>x<\/em>-axis and appears almost linear at the intercept, it is a single zero.<\/li>\n<li>If the graph touches the <em>x<\/em>-axis and bounces off of the axis, it is a zero with even multiplicity.<\/li>\n<li>If the graph crosses the <em>x<\/em>-axis at a zero, it is a zero with odd multiplicity.<\/li>\n<li>The sum of the multiplicities is the degree\u00a0<em>n<\/em>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying Zeros and Their Multiplicities<\/h3>\n<p>Use the graph of the function of degree 6 to identify the zeros of the function and their possible multiplicities.<\/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\/02201558\/CNX_Precalc_Figure_03_04_0092.jpg\" alt=\"Three graphs showing three different polynomial functions with multiplicity 1, 2, and 3.\" width=\"487\" height=\"628\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q583908\">Show Solution<\/span><\/p>\n<div id=\"q583908\" class=\"hidden-answer\" style=\"display: none\">\n<p>The polynomial function is of degree <em>n<\/em> which is 6. The sum of the multiplicities must be\u00a06.<\/p>\n<p>Starting from the left, the first zero occurs at [latex]x=-3[\/latex]. The graph touches the <em>x<\/em>-axis, so the multiplicity of the zero must be even. The zero of \u20133 has multiplicity 2.<\/p>\n<p>The next zero occurs at [latex]x=-1[\/latex]. The graph looks almost linear at this point. This is a single zero of multiplicity 1.<\/p>\n<p>The last zero occurs at [latex]x=4[\/latex]. The graph crosses the<em> x<\/em>-axis, so the multiplicity of the zero must be odd. We know that the multiplicity is 3 and that the sum of the multiplicities must be 6.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use the graph of the function of degree 7 to identify the zeros of the function and their multiplicities.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"small aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201600\/CNX_Precalc_Figure_03_04_0102.jpg\" alt=\"Graph of an even-degree polynomial with degree 6.\" width=\"487\" height=\"253\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q874458\">Show Solution<\/span><\/p>\n<div id=\"q874458\" class=\"hidden-answer\" style=\"display: none\">\n<p>The graph has a zero of \u20135 with multiplicity 3, a zero of \u20131 with multiplicity 2, and a zero of 3 with multiplicity 2.<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=121830&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"450\"><\/iframe><\/p>\n<\/div>\n<h2>Determining End Behavior<\/h2>\n<p>As we have already learned, the behavior of a graph of a <strong>polynomial function<\/strong> of the form<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+...+{a}_{1}x+{a}_{0}[\/latex]<\/p>\n<p>will either ultimately rise or fall as <em>x<\/em>\u00a0increases without bound and will either rise or fall as <em>x\u00a0<\/em>decreases without bound. This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. The same is true for very small inputs, say \u2013100 or \u20131,000.<\/p>\n<p>Recall that we call this behavior the <em>end behavior<\/em> of a function. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function, [latex]{a}_{n}{x}^{n}[\/latex], is an even power function, as <em>x<\/em>\u00a0increases or decreases without bound, [latex]f\\left(x\\right)[\/latex] increases without bound. When the leading term is an odd power function, as\u00a0<em>x<\/em>\u00a0decreases without bound, [latex]f\\left(x\\right)[\/latex] also decreases without bound; as <em>x<\/em>\u00a0increases without bound, [latex]f\\left(x\\right)[\/latex] also increases without bound. If the leading term is negative, it will change the direction of the end behavior. The table below\u00a0summarizes all four cases.<\/p>\n<table>\n<thead>\n<tr>\n<th style=\"text-align: center;\">Even Degree<\/th>\n<th style=\"text-align: center;\">Odd Degree<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/11.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12504\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201602\/11.png\" alt=\"11\" width=\"423\" height=\"559\" \/><\/a><\/td>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/12.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12505\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201605\/12.png\" alt=\"12\" width=\"397\" height=\"560\" \/><\/a><\/td>\n<\/tr>\n<tr>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/13.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12506\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201607\/13.png\" alt=\"13\" width=\"387\" height=\"574\" \/><\/a><\/td>\n<td><a href=\"https:\/\/courses.candelalearning.com\/precalcone1xmommaster\/wp-content\/uploads\/sites\/1226\/2015\/08\/14.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12507\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201609\/14.png\" alt=\"14\" width=\"404\" height=\"564\" \/><\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3><\/h3>\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=29473&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h2>Graphing Polynomial Functions<\/h2>\n<p>We can use what we have learned about multiplicities, end behavior, and turning points to sketch graphs of polynomial functions. Let us put this all together and look at the steps required to graph polynomial functions.<\/p>\n<div class=\"textbox\">\n<h3>How To: Given a polynomial function, sketch the graph<\/h3>\n<ol>\n<li>Find the intercepts.<\/li>\n<li>Check for symmetry. If the function is an even function, its graph is symmetric with respect to the\u00a0<em>y<\/em>-axis, that is,\u00a0<em>f<\/em>(\u2013<em>x<\/em>) = <em>f<\/em>(<em>x<\/em>).<br \/>\nIf a function is an odd function, its graph is symmetric with respect to the origin, that is,\u00a0<em>f<\/em>(\u2013<em>x<\/em>) = <em>\u2013<\/em><em>f<\/em>(<em>x<\/em>).<\/li>\n<li>Use the multiplicities of the zeros to determine the behavior of the polynomial at the <em>x<\/em>-intercepts.<\/li>\n<li>Determine the end behavior by examining the leading term.<\/li>\n<li>Use the end behavior and the behavior at the intercepts to sketch the graph.<\/li>\n<li>Ensure that the number of turning points does not exceed one less than the degree of the polynomial.<\/li>\n<li>Optionally, use technology to check the graph.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Sketching the Graph of a Polynomial Function<\/h3>\n<p>Sketch a possible graph for [latex]f\\left(x\\right)=-2{\\left(x+3\\right)}^{2}\\left(x - 5\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q707233\">Show Solution<\/span><\/p>\n<div id=\"q707233\" class=\"hidden-answer\" style=\"display: none\">\n<p>This graph has two <em>x-<\/em>intercepts. At <em>x\u00a0<\/em>= \u20133, the factor is squared, indicating a multiplicity of 2. The graph will bounce off the\u00a0<em>x<\/em>-intercept at this value. At <em>x\u00a0<\/em>= 5, the function has a multiplicity of one, indicating the graph will cross through the axis at this intercept.<\/p>\n<p>The <em>y<\/em>-intercept is found by evaluating <em>f<\/em>(0).<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\hfill \\\\ f\\left(0\\right)=-2{\\left(0+3\\right)}^{2}\\left(0 - 5\\right)\\hfill \\\\ \\text{}f\\left(0\\right)=-2\\cdot 9\\cdot \\left(-5\\right)\\hfill \\\\ \\text{}f\\left(0\\right)=90\\hfill \\end{array}[\/latex]<\/p>\n<p>The <em>y<\/em>-intercept is (0, 90).<\/p>\n<p>Additionally, we can see the leading term, if this polynomial were multiplied out, would be [latex]-2{x}^{3}[\/latex], so the end behavior, as seen in the following graph, is that of a vertically reflected cubic with the outputs decreasing as the inputs approach infinity and the outputs increasing as the inputs approach negative infinity.<\/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\/02201614\/CNX_Precalc_Figure_03_04_0172.jpg\" alt=\"Showing the distribution for the leading term.\" width=\"487\" height=\"362\" \/><\/p>\n<p>To sketch the graph, we consider the following:<\/p>\n<ul id=\"fs-id1165134374741\">\n<li>As [latex]x\\to -\\infty[\/latex] the function [latex]f\\left(x\\right)\\to \\infty[\/latex], so we know the graph starts in the second quadrant and is decreasing toward the <em>x<\/em>-axis.<\/li>\n<li>Since [latex]f\\left(-x\\right)=-2{\\left(-x+3\\right)}^{2}\\left(-x - 5\\right)[\/latex] is not equal to <em>f<\/em>(<em>x<\/em>), the graph does not have any symmetry.<\/li>\n<li>At [latex]\\left(-3,0\\right)[\/latex] the graph bounces off of the <em>x<\/em>-axis, so the function must start increasing after this point.<\/li>\n<li>At (0, 90), the graph crosses the <em>y<\/em>-axis.<\/li>\n<\/ul>\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\/02201617\/CNX_Precalc_Figure_03_04_0182.jpg\" alt=\"Graph of the end behavior and intercepts, (-3, 0) and (0, 90), for the function f(x)=-2(x+3)^2(x-5).\" width=\"487\" height=\"362\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>Somewhere after this point, the graph must turn back down or start decreasing toward the horizontal axis because the graph passes through the next intercept at (5, 0).<\/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\/02201618\/CNX_Precalc_Figure_03_04_0192.jpg\" alt=\"Graph of the end behavior and intercepts, (-3, 0), (0, 90) and (5, 0), for the function f(x)=-2(x+3)^2(x-5).\" width=\"487\" height=\"362\" \/><\/p>\n<p>As [latex]x\\to \\infty[\/latex] the function [latex]f\\left(x\\right)\\to \\mathrm{-\\infty }[\/latex], so we know the graph continues to decrease, and we can stop drawing the graph in the fourth quadrant.<\/p>\n<p>The complete graph of the polynomial function [latex]f\\left(x\\right)=-2{\\left(x+3\\right)}^{2}\\left(x - 5\\right)[\/latex] is as follows:<\/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\/02201620\/CNX_Precalc_Figure_03_04_0202.jpg\" alt=\"Graph of f(x)=-2(x+3)^2(x-5).\" width=\"487\" height=\"366\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Sketch a possible graph for [latex]f\\left(x\\right)=\\frac{1}{4}x{\\left(x - 1\\right)}^{4}{\\left(x+3\\right)}^{3}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q812296\">Show Solution<\/span><\/p>\n<div id=\"q812296\" class=\"hidden-answer\" style=\"display: none\">\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/30233150\/CNX_Precalc_Figure_03_04_0212.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-2911\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/30233150\/CNX_Precalc_Figure_03_04_0212.jpg\" alt=\"Graph of f(x)=(1\/4)x(x-1)^4(x+3)^3.\" width=\"487\" height=\"366\" \/><\/a><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=113372&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"450\"><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<h3>The Intermediate Value Theorem<\/h3>\n<p>In some situations, we may know two points on a graph but not the zeros. If those two points are on opposite sides of the <em>x<\/em>-axis, we can confirm that there is a zero between them. Consider a polynomial function <em>f<\/em>\u00a0whose graph is smooth and continuous. The <strong>Intermediate Value Theorem<\/strong> states that for two numbers <em>a<\/em>\u00a0and <em>b<\/em>\u00a0in the domain of <em>f<\/em>,\u00a0if <em>a\u00a0<\/em>&lt; <em>b<\/em>\u00a0and [latex]f\\left(a\\right)\\ne f\\left(b\\right)[\/latex], then the function <em>f<\/em>\u00a0takes on every value between [latex]f\\left(a\\right)[\/latex] and [latex]f\\left(b\\right)[\/latex].<\/p>\n<p>We can apply this theorem to a special case that is useful for graphing polynomial functions. If a point on the graph of a continuous function <em>f<\/em>\u00a0at [latex]x=a[\/latex] lies above the <em>x<\/em>-axis and another point at [latex]x=b[\/latex] lies below the <em>x<\/em>-axis, there must exist a third point between [latex]x=a[\/latex] and [latex]x=b[\/latex] where the graph crosses the <em>x<\/em>-axis. Call this point [latex]\\left(c,\\text{ }f\\left(c\\right)\\right)[\/latex]. This means that we are assured there is a value\u00a0<em>c<\/em>\u00a0where [latex]f\\left(c\\right)=0[\/latex].<\/p>\n<p>In other words, the Intermediate Value Theorem tells us that when a polynomial function changes from a negative value to a positive value, the function must cross the <em>x<\/em>-axis. The figure below\u00a0shows that there is a zero between <em>a<\/em>\u00a0and <em>b<\/em>.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02201623\/CNX_Precalc_Figure_03_04_0222.jpg\" alt=\"Graph of an odd-degree polynomial function that shows a point f(a) that\u2019s negative, f(b) that\u2019s positive, and f(c) that\u2019s 0.\" width=\"487\" height=\"368\" \/><\/p>\n<p class=\"wp-caption-text\">The Intermediate Value Theorem can be used to show there exists a zero.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>A General Note: Intermediate Value Theorem<\/h3>\n<p>Let <em>f<\/em>\u00a0be a polynomial function. The <strong>Intermediate Value Theorem<\/strong> states that if [latex]f\\left(a\\right)[\/latex]\u00a0and [latex]f\\left(b\\right)[\/latex]\u00a0have opposite signs, then there exists at least one value <em>c<\/em>\u00a0between <em>a<\/em>\u00a0and <em>b<\/em>\u00a0for which [latex]f\\left(c\\right)=0[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using the Intermediate Value Theorem<\/h3>\n<p>Show that the function [latex]f\\left(x\\right)={x}^{3}-5{x}^{2}+3x+6[\/latex]\u00a0has at least two real zeros between [latex]x=1[\/latex]\u00a0and [latex]x=4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q33137\">Show Solution<\/span><\/p>\n<div id=\"q33137\" class=\"hidden-answer\" style=\"display: none\">\n<p>To start, evaluate [latex]f\\left(x\\right)[\/latex]\u00a0at the integer values [latex]x=1,2,3,\\text{ and }4[\/latex].<\/p>\n<table>\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><em><strong>x<\/strong><\/em><\/td>\n<td>1<\/td>\n<td>2<\/td>\n<td>3<\/td>\n<td>4<\/td>\n<\/tr>\n<tr>\n<td><em><strong>f<\/strong><\/em><strong>(<em>x<\/em>)<\/strong><\/td>\n<td>5<\/td>\n<td>0<\/td>\n<td>\u20133<\/td>\n<td>2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We see that one zero occurs at [latex]x=2[\/latex]. Also, since [latex]f\\left(3\\right)[\/latex] is negative and [latex]f\\left(4\\right)[\/latex] is positive, by the Intermediate Value Theorem, there must be at least one real zero between 3 and 4.<\/p>\n<p>We have shown that there are at least two real zeros between [latex]x=1[\/latex]\u00a0and [latex]x=4[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>We can also graphically see that there are two real zeros between [latex]x=1[\/latex]\u00a0and [latex]x=4[\/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\/896\/2016\/11\/02201625\/CNX_Precalc_Figure_03_04_0232.jpg\" alt=\"Graph of f(x)=x^3-5x^2+3x+6 and shows, by the Intermediate Value Theorem, that there exists two zeros since f(1)=5 and f(4)=2 are positive and f(3) = -3 is negative.\" width=\"487\" height=\"591\" \/>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Show that the function [latex]f\\left(x\\right)=7{x}^{5}-9{x}^{4}-{x}^{2}[\/latex] has at least one real zero between [latex]x=1[\/latex] and [latex]x=2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q778313\">Show Solution<\/span><\/p>\n<div id=\"q778313\" class=\"hidden-answer\" style=\"display: none\">\n<p>Because <em>f<\/em>\u00a0is a polynomial function and since [latex]f\\left(1\\right)[\/latex] is negative and [latex]f\\left(2\\right)[\/latex] is positive, there is at least one real zero between [latex]x=1[\/latex] and [latex]x=2[\/latex].<\/div>\n<\/div>\n<\/div>\n<h3>Writing Formulas for Polynomial Functions<\/h3>\n<p>Now that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. Because a <strong>polynomial function<\/strong> written in factored form will have an <em>x<\/em>-intercept where each factor is equal to zero, we can form a function that will pass through a set of <em>x<\/em>-intercepts by introducing a corresponding set of factors.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Factored Form of Polynomials<\/h3>\n<p>If a polynomial of lowest degree <em>p<\/em>\u00a0has zeros at [latex]x={x}_{1},{x}_{2},\\dots ,{x}_{n}[\/latex],\u00a0then the polynomial can be written in the factored form: [latex]f\\left(x\\right)=a{\\left(x-{x}_{1}\\right)}^{{p}_{1}}{\\left(x-{x}_{2}\\right)}^{{p}_{2}}\\cdots {\\left(x-{x}_{n}\\right)}^{{p}_{n}}[\/latex]\u00a0where the powers [latex]{p}_{i}[\/latex]\u00a0on each factor can be determined by the behavior of the graph at the corresponding intercept, and the stretch factor <em>a<\/em>\u00a0can be determined given a value of the function other than the <em>x<\/em>-intercept.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>How To: Given a graph of a polynomial function, write a formula for the function<\/h3>\n<ol>\n<li>Identify the <em>x<\/em>-intercepts of the graph to find the factors of the polynomial.<\/li>\n<li>Examine the behavior of the graph at the <em>x<\/em>-intercepts to determine the multiplicity of each factor.<\/li>\n<li>Find the polynomial of least degree containing all of the factors found in the previous step.<\/li>\n<li>Use any other point on the graph (the <em>y<\/em>-intercept may be easiest) to determine the stretch factor.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing a Formula for a Polynomial Function from Its Graph<\/h3>\n<p>Write a formula for the polynomial 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\/02201627\/CNX_Precalc_Figure_03_04_0242.jpg\" alt=\"Graph of a positive even-degree polynomial with zeros at x=-3, 2, 5 and y=-2.\" width=\"487\" height=\"366\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q338564\">Show Solution<\/span><\/p>\n<div id=\"q338564\" class=\"hidden-answer\" style=\"display: none\">\n<p>This graph has three <em>x<\/em>-intercepts: <em>x\u00a0<\/em>= \u20133, 2, and 5. The <em>y<\/em>-intercept is located at (0, 2). At <em>x\u00a0<\/em>= \u20133 and <em>x\u00a0<\/em>= 5,\u00a0the graph passes through the axis linearly, suggesting the corresponding factors of the polynomial will be linear. At <em>x\u00a0<\/em>= 2, the graph bounces off the x-axis at the intercept suggesting the corresponding factor of the polynomial will be second degree (quadratic). Together, this gives us<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a\\left(x+3\\right){\\left(x - 2\\right)}^{2}\\left(x - 5\\right)[\/latex]<\/p>\n<p>To determine the stretch factor, we utilize another point on the graph. We will use the <em>y<\/em>-intercept (0, \u20132), to solve for <em>a<\/em>.<\/p>\n<p>[latex]\\begin{array}{l}f\\left(0\\right)=a\\left(0+3\\right){\\left(0 - 2\\right)}^{2}\\left(0 - 5\\right)\\hfill \\\\ \\text{ }-2=a\\left(0+3\\right){\\left(0 - 2\\right)}^{2}\\left(0 - 5\\right)\\hfill \\\\ \\text{ }-2=-60a\\hfill \\\\ \\text{ }a=\\frac{1}{30}\\hfill \\end{array}[\/latex]<\/p>\n<p>The graphed polynomial appears to represent the function [latex]f\\left(x\\right)=\\frac{1}{30}\\left(x+3\\right){\\left(x - 2\\right)}^{2}\\left(x - 5\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Given the graph below, write a formula for the function shown.<\/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\/02201629\/CNX_Precalc_Figure_03_04_0252.jpg\" alt=\"Graph of a negative even-degree polynomial with zeros at x=-1, 2, 4 and y=-4.\" width=\"487\" height=\"291\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q427364\">Show Solution<\/span><\/p>\n<div id=\"q427364\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f\\left(x\\right)=-\\frac{1}{8}{\\left(x - 2\\right)}^{3}{\\left(x+1\\right)}^{2}\\left(x - 4\\right)[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=29478&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"450\"><\/iframe><\/p>\n<\/div>\n<h3>Local and Global Extrema<\/h3>\n<p>With quadratics, we were able to algebraically find the maximum or minimum value of the function by finding the vertex. For general polynomials, finding these turning points is not possible without more advanced techniques from calculus. Even then, finding where extrema occur can still be algebraically challenging. For now, we will estimate the locations of turning points using technology to generate a graph.<\/p>\n<p>Each turning point represents a local minimum or maximum. Sometimes, a turning point is the highest or lowest point on the entire graph. In these cases, we say that the turning point is a <strong>global maximum <\/strong>or a <strong>global minimum<\/strong>. These are also referred to as the absolute maximum and absolute minimum values of the function.<\/p>\n<p>A <strong>local maximum<\/strong> or <strong>local minimum<\/strong> at <em>x\u00a0<\/em>= <em>a<\/em>\u00a0(sometimes called the relative maximum or minimum, respectively) is the output at the highest or lowest point on the graph in an open interval around <em>x\u00a0<\/em>= <em>a<\/em>. If a function has a local maximum at <em>a<\/em>, then [latex]f\\left(a\\right)\\ge f\\left(x\\right)[\/latex] for all <em>x<\/em>\u00a0in an open interval around <em>x<\/em> =\u00a0<em>a<\/em>. If a function has a local minimum at <em>a<\/em>, then [latex]f\\left(a\\right)\\le f\\left(x\\right)[\/latex] for all <em>x<\/em>\u00a0in an open interval around <em>x\u00a0<\/em>= <em>a<\/em>.<\/p>\n<p>A <strong>global maximum<\/strong> or <strong>global minimum<\/strong> is the output at the highest or lowest point of the function. If a function has a global maximum at <em>a<\/em>, then [latex]f\\left(a\\right)\\ge f\\left(x\\right)[\/latex] for all <em>x<\/em>. If a function has a global minimum at <em>a<\/em>, then [latex]f\\left(a\\right)\\le f\\left(x\\right)[\/latex] for all <em>x<\/em>.<\/p>\n<p>We can see the difference between local and global extrema 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\/896\/2016\/11\/02201631\/CNX_Precalc_Figure_03_04_026n2.jpg\" alt=\"Graph of an even-degree polynomial that denotes the local maximum and minimum and the global maximum.\" width=\"487\" height=\"475\" \/><\/p>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Do all polynomial functions have a global minimum or maximum?<\/strong><\/p>\n<p><em>No. Only polynomial functions of even degree have a global minimum or maximum. For example, [latex]f\\left(x\\right)=x[\/latex] has neither a global maximum nor a global minimum.<\/em><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using Local Extrema to Solve Applications<\/h3>\n<p>An open-top box is to be constructed by cutting out squares from each corner of a 14 cm by 20 cm sheet of plastic then folding up the sides. Find the size of squares that should be cut out to maximize the volume enclosed by the box.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q598415\">Show Solution<\/span><\/p>\n<div id=\"q598415\" class=\"hidden-answer\" style=\"display: none\">\n<p>We will start this problem by drawing a picture like the one below, labeling the width of the cut-out squares with a variable, <em>w<\/em>.<\/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\/02201633\/CNX_Precalc_Figure_03_04_0272.jpg\" alt=\"Diagram of a rectangle with four squares at the corners.\" width=\"487\" height=\"298\" \/><\/p>\n<p>Notice that after a square is cut out from each end, it leaves a [latex]\\left(14 - 2w\\right)[\/latex] cm by [latex]\\left(20 - 2w\\right)[\/latex] cm rectangle for the base of the box, and the box will be <em>w<\/em>\u00a0cm tall. This gives the volume<\/p>\n<p>[latex]\\begin{array}{l}V\\left(w\\right)=\\left(20 - 2w\\right)\\left(14 - 2w\\right)w\\hfill \\\\ \\text{}V\\left(w\\right)=280w - 68{w}^{2}+4{w}^{3}\\hfill \\end{array}[\/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\/896\/2016\/11\/02201635\/CNX_Precalc_Figure_03_04_0282.jpg\" alt=\"Graph of V(w)=(20-2w)(14-2w)w where the x-axis is labeled w and the y-axis is labeled V(w).\" width=\"487\" height=\"406\" \/><\/p>\n<p>Notice, since the factors are <em>w<\/em>, [latex]20 - 2w[\/latex] and [latex]14 - 2w[\/latex], the three zeros are 10, 7, and 0, respectively. Because a height of 0 cm is not reasonable, we consider only the zeros 10 and 7. The shortest side is 14 and we are cutting off two squares, so values <em>w<\/em>\u00a0may take on are greater than zero or less than 7. This means we will restrict the domain of this function to [latex]0<w<7[\/latex]. Using technology to sketch the graph of [latex]V\\left(w\\right)[\/latex] on this reasonable domain, we get a graph like the one above. We can use this graph to estimate the maximum value for the volume, restricted to values for <em>w<\/em>\u00a0that are reasonable for this problem, values from 0 to 7.<\/p>\n<p>On this graph, we turn our focus to only the portion on the reasonable domain, [latex]\\left[0,\\text{ }7\\right][\/latex]. We can estimate the maximum value to be around 340 cubic cm, which occurs when the squares are about 2.75 cm on each side. To improve this estimate, we could use advanced features of our technology, if available, or simply change our window to zoom in on our graph to produce the graph 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\/896\/2016\/11\/02201637\/CNX_Precalc_Figure_03_04_0292.jpg\" alt=\"Graph of V(w)=(20-2w)(14-2w)w where the x-axis is labeled w and the y-axis is labeled V(w) on the domain &#091;2.4, 3&#093;.\" width=\"487\" height=\"444\" \/><\/p>\n<p>From this zoomed-in view, we can refine our estimate for the maximum volume to about 339 cubic cm which occurs when the squares measure approximately 2.7 cm on each side.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Key Concepts<\/h2>\n<ul>\n<li>Polynomial functions of degree 2 or more are smooth, continuous functions.<\/li>\n<li>To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero.<\/li>\n<li>Another way to find the <em>x-<\/em>intercepts of a polynomial function is to graph the function and identify the points where the graph crosses the <em>x<\/em>-axis.<\/li>\n<li>The multiplicity of a zero determines how the graph behaves at the <em>x<\/em>-intercept.<\/li>\n<li>The graph of a polynomial will cross the x-axis at a zero with odd multiplicity.<\/li>\n<li>The graph of a polynomial will touch and bounce off the x-axis at a zero with even multiplicity.<\/li>\n<li>The end behavior of a polynomial function depends on the leading term.<\/li>\n<li>The graph of a polynomial function changes direction at its turning points.<\/li>\n<li>A polynomial function of degree <em>n<\/em>\u00a0has at most\u00a0<em>n <\/em>\u2013\u00a01 turning points.<\/li>\n<li>To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most<em>\u00a0n <\/em>\u2013\u00a01 turning points.<\/li>\n<li>Graphing a polynomial function helps to estimate local and global extremas.<\/li>\n<li>The Intermediate Value Theorem tells us that if [latex]f\\left(a\\right) \\text{and} f\\left(b\\right)[\/latex]\u00a0have opposite signs, then there exists at least one value <em>c<\/em>\u00a0between <em>a<\/em>\u00a0and <em>b<\/em>\u00a0for which [latex]f\\left(c\\right)=0[\/latex].<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n<dt><strong>global maximum<\/strong><\/dt>\n<dd id=\"fs-id1165132943525\">highest turning point on a graph; [latex]f\\left(a\\right)[\/latex]\u00a0where [latex]f\\left(a\\right)\\ge f\\left(x\\right)[\/latex]\u00a0for all <em>x<\/em>.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n<dt><b>global minimum<\/b><\/dt>\n<dd id=\"fs-id1165132943525\">lowest turning point on a graph; [latex]f\\left(a\\right)[\/latex]\u00a0where [latex]f\\left(a\\right)\\le f\\left(x\\right)[\/latex] for all <em>x<\/em>.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n<dt><strong>Intermediate Value Theorem<\/strong><\/dt>\n<dd id=\"fs-id1165132943525\">for two numbers <em>a<\/em>\u00a0and <em>b<\/em>\u00a0in the domain of <em>f<\/em>,\u00a0if [latex]a<b[\/latex]\u00a0and [latex]f\\left(a\\right)\\ne f\\left(b\\right)[\/latex],\u00a0then the function <em>f<\/em>\u00a0takes on every value between [latex]f\\left(a\\right)[\/latex]\u00a0and [latex]f\\left(b\\right)[\/latex];\u00a0specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the <em>x<\/em>-axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\" class=\"definition\">\n<dt><strong>multiplicity<\/strong><\/dt>\n<dd id=\"fs-id1165132943525\">the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], [latex]x=h[\/latex]\u00a0is a zero of multiplicity <em>p<\/em>.<\/dd>\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-3953\">\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>Question ID 121830. <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><\/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 29470, 29478. <strong>Authored by<\/strong>: Caren McClure. <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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