{"id":3072,"date":"2017-01-13T21:40:19","date_gmt":"2017-01-13T21:40:19","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=3072"},"modified":"2017-04-19T16:24:58","modified_gmt":"2017-04-19T16:24:58","slug":"writing-formulas-for-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/writing-formulas-for-polynomial-functions\/","title":{"raw":"Writing Formulas for Polynomial Functions","rendered":"Writing Formulas for Polynomial Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Write the equation of a polynomial function given it's graph<\/li>\r\n<\/ul>\r\n<\/div>\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 horizontal intercepts 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 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 the 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\" data-media-type=\"image\/jpg\" \/>\r\n\r\n[reveal-answer q=\"338564\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"338564\"]\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 at the intercept, suggesting the corresponding factor of the polynomial will be second degree (quadratic). Together, this gives us\r\n\r\n[latex]f\\left(x\\right)=a\\left(x+3\\right){\\left(x - 2\\right)}^{2}\\left(x - 5\\right)[\/latex]\r\n\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 5<\/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\" data-media-type=\"image\/jpg\" \/>\r\n\r\n[reveal-answer q=\"427364\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"427364\"][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:\/\/www.myopenmath.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<div class=\"textbox key-takeaways\">\r\n<h3>Key Takeaways<\/h3>\r\nUse Desmos to help you write the equation of a degree 5 polynomial function with roots at [latex](-1,0),(0,2),\\text{and },(0,3)[\/latex] that passes through the point [latex](1,-32)[\/latex].\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/College+Algebra\/calculator.html\" target=\"_blank\"><img class=\"alignnone size-full wp-image-3370\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/13193222\/calculator.png\" alt=\"\" width=\"251\" height=\"46\" \/><\/a>\r\n[reveal-answer q=\"135031\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"135031\"]\r\n\r\n[latex]f(x)=-2(x-2)(x+1)^3(x-2)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2 data-type=\"title\"><\/h2>\r\n<h2 data-type=\"title\">Using Local and Global Extrema<\/h2>\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<div class=\"textbox shaded\">\r\n<h3>Local and Global Extrema<\/h3>\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\" data-media-type=\"image\/png\" \/>\r\n\r\n<\/div>\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\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"598415\"]\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\" data-media-type=\"image\/jpg\" \/>\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{ }=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\" data-media-type=\"image\/jpg\" \/>\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 the 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 Figure 24. 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\u2014values from 0 to 7.\r\n\r\nFrom 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\" data-media-type=\"image\/jpg\" \/>\r\n\r\nFrom this zoomed-in view, we can refine our estimate for the maximum volume to about 339 cubic cm, when the squares measure approximately 2.7 cm on each side.\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\nClick on the graph below\u00a0to find the maximum and minimum values on the interval [latex]\\left[-2,7\\right][\/latex] of the function [latex]f\\left(x\\right)=0.1{\\left(x - \\frac{5}{3}\\right)}^{3}{\\left(x+1\\right)}^{2}\\left(x - 7\\right)[\/latex].\r\n\r\nThen, use the slider to see how changing the value of a effects the end behavior and y-intercept of the graph.\r\n\r\nhttps:\/\/www.desmos.com\/calculator\/5oyyx0vbiv\r\n\r\n[reveal-answer q=\"43463\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"43463\"]The minimum occurs at approximately the point [latex]\\left(5.98,-398.8\\right)[\/latex], and the maximum occurs at approximately the point [latex]\\left(0.02,3.24\\right)[\/latex].[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Write the equation of a polynomial function given it&#8217;s graph<\/li>\n<\/ul>\n<\/div>\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 horizontal intercepts 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 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 the 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\" data-media-type=\"image\/jpg\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q338564\">Solution<\/span><\/p>\n<div id=\"q338564\" class=\"hidden-answer\" style=\"display: none\">\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 at the intercept, suggesting the corresponding factor of the polynomial will be second degree (quadratic). Together, this gives us<\/p>\n<p>[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 5<\/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\" data-media-type=\"image\/jpg\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q427364\">Solution<\/span><\/p>\n<div id=\"q427364\" class=\"hidden-answer\" style=\"display: none\">[latex]f\\left(x\\right)=-\\frac{1}{8}{\\left(x - 2\\right)}^{3}{\\left(x+1\\right)}^{2}\\left(x - 4\\right)[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=29478&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"450\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key Takeaways<\/h3>\n<p>Use Desmos to help you write the equation of a degree 5 polynomial function with roots at [latex](-1,0),(0,2),\\text{and },(0,3)[\/latex] that passes through the point [latex](1,-32)[\/latex].<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/oerfiles\/College+Algebra\/calculator.html\" target=\"_blank\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-3370\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/13193222\/calculator.png\" alt=\"\" width=\"251\" height=\"46\" \/><\/a><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q135031\">Show Answer<\/span><\/p>\n<div id=\"q135031\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f(x)=-2(x-2)(x+1)^3(x-2)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2 data-type=\"title\"><\/h2>\n<h2 data-type=\"title\">Using Local and Global Extrema<\/h2>\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<div class=\"textbox shaded\">\n<h3>Local and Global Extrema<\/h3>\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\" data-media-type=\"image\/png\" \/><\/p>\n<\/div>\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\">Solution<\/span><\/p>\n<div id=\"q598415\" class=\"hidden-answer\" style=\"display: none\">\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>.<\/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\" data-media-type=\"image\/jpg\" \/><\/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{ }=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\" data-media-type=\"image\/jpg\" \/><\/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 the 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 Figure 24. 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\u2014values from 0 to 7.<\/p>\n<p>From 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\" data-media-type=\"image\/jpg\" \/><\/p>\n<p>From this zoomed-in view, we can refine our estimate for the maximum volume to about 339 cubic cm, when the squares measure approximately 2.7 cm on each side.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Click on the graph below\u00a0to find the maximum and minimum values on the interval [latex]\\left[-2,7\\right][\/latex] of the function [latex]f\\left(x\\right)=0.1{\\left(x - \\frac{5}{3}\\right)}^{3}{\\left(x+1\\right)}^{2}\\left(x - 7\\right)[\/latex].<\/p>\n<p>Then, use the slider to see how changing the value of a effects the end behavior and y-intercept of the graph.<\/p>\n<p>https:\/\/www.desmos.com\/calculator\/5oyyx0vbiv<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q43463\">Solution<\/span><\/p>\n<div id=\"q43463\" class=\"hidden-answer\" style=\"display: none\">The minimum occurs at approximately the point [latex]\\left(5.98,-398.8\\right)[\/latex], and the maximum occurs at approximately the point [latex]\\left(0.02,3.24\\right)[\/latex].<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-3072\">\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: Zeros of a Polynomial. <strong>Authored by<\/strong>: Lumen Learning (With Desmos). <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div 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 29478. <strong>Authored by<\/strong>: McClure, Caren. <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 <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"},{\"type\":\"cc\",\"description\":\"Question ID 29478\",\"author\":\"McClure, Caren\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"original\",\"description\":\"Interactive: Zeros of a Polynomial\",\"author\":\"Lumen Learning (With Desmos)\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"93854fb6-2394-411d-80cb-326ecb8224fd","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3072","chapter","type-chapter","status-publish","hentry"],"part":1700,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/3072","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":9,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/3072\/revisions"}],"predecessor-version":[{"id":4373,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/3072\/revisions\/4373"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1700"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/3072\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/media?parent=3072"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=3072"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/contributor?post=3072"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/wp-json\/wp\/v2\/license?post=3072"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}