{"id":81,"date":"2019-08-20T17:02:11","date_gmt":"2019-08-20T21:02:11","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/power-functions-and-polynomial-functions\/"},"modified":"2022-06-01T10:39:26","modified_gmt":"2022-06-01T14:39:26","slug":"power-functions-and-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/power-functions-and-polynomial-functions\/","title":{"raw":"Power Functions and Polynomial Functions","rendered":"Power Functions and Polynomial Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section, you will:\n<ul>\n \t<li>Identify power functions.<\/li>\n \t<li>Identify end behavior of power functions.<\/li>\n \t<li>Identify polynomial functions.<\/li>\n \t<li>Identify the degree and leading coefficient of polynomial functions.<\/li>\n<\/ul>\n<\/div>\n<div id=\"CNX_Precalc_Figure_03_03_001.jpg\" class=\"small aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135627\/CNX_Precalc_Figure_03_03_001.jpg\" alt=\"Three birds on a cliff with the sun rising in the background.\" width=\"488\" height=\"366\"> <strong>Figure 1. <\/strong>(credit: Jason Bay, Flickr)[\/caption]\n\n<\/div>\n<p id=\"fs-id1165134540133\">Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown in <a class=\"autogenerated-content\" href=\"#Table_03_03_01\">(Figure)<\/a>.<\/p>\n\n<table id=\"Table_03_03_01\" summary=\"..\">\n<tbody>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td>[latex]2009[\/latex]<\/td>\n<td>[latex]2010[\/latex]<\/td>\n<td>[latex]2011[\/latex]<\/td>\n<td>[latex]2012[\/latex]<\/td>\n<td>[latex]2013[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Bird Population<\/strong><\/td>\n<td>[latex]800[\/latex]<\/td>\n<td>[latex]897[\/latex]<\/td>\n<td>[latex]992[\/latex]<\/td>\n<td>[latex]1,083[\/latex]<\/td>\n<td>[latex]1,169[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137442798\">The population can be estimated using the function[latex]\\,P\\left(t\\right)=-0.3{t}^{3}+97t+800,\\,[\/latex]where[latex]\\,P\\left(t\\right)\\,[\/latex]represents the bird population on the island[latex]\\,t\\,[\/latex]years after 2009. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island. In this section, we will examine functions that we can use to estimate and predict these types of changes.<\/p>\n\n<div id=\"fs-id1165137540446\" class=\"bc-section section\">\n<h3>Identifying Power Functions<\/h3>\n<p id=\"fs-id1165137570394\">Before we can understand the bird problem, it will be helpful to understand a different type of function. A <strong>power function <\/strong>is a function with a single term that is the product of a real number, a <strong>coefficient,<\/strong> and a variable raised to a fixed real number.<\/p>\n<p id=\"fs-id1165135320417\">As an example, consider functions for area or volume. The function for the <span class=\"no-emphasis\">area of a circle<\/span> with radius[latex]\\,r\\,[\/latex]\nis<\/p>\n\n<div id=\"eip-544\" class=\"unnumbered aligncenter\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/div>\n<p id=\"fs-id1165135191346\">and the function for the <span class=\"no-emphasis\">volume of a sphere<\/span> with radius[latex]\\,r\\,[\/latex]\nis<\/p>\n\n<div id=\"eip-640\" class=\"unnumbered aligncenter\">[latex]V\\left(r\\right)=\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/div>\n<p id=\"fs-id1165137579058\">Both of these are examples of power functions because they consist of a coefficient,[latex]\\,\\pi \\,[\/latex]or[latex]\\,\\frac{4}{3}\\pi ,\\,[\/latex]multiplied by a variable[latex]\\,r\\,[\/latex]raised to a power.<\/p>\n\n<div id=\"fs-id1165135356525\" class=\"textbox key-takeaways\">\n<h3>Power Function<\/h3>\n<p id=\"fs-id1165137771947\">A power function is a function that can be represented in the form<\/p>\n\n<div id=\"eip-826\" class=\"unnumbered aligncenter\">[latex]f\\left(x\\right)=k{x}^{p}[\/latex]<\/div>\n<p id=\"eip-id1165135584093\">where[latex]\\,k\\,[\/latex]\nand[latex]\\,p\\,[\/latex]are real numbers, and[latex]\\,k\\,[\/latex]\nis known as the coefficient.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137661479\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165137582131\"><strong>Is[latex]\\,f\\left(x\\right)={2}^{x}\\,[\/latex]a power function?<\/strong><\/p>\n<p id=\"fs-id1165137598469\"><em>No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function.<\/em><\/p>\n\n<\/div>\n<div id=\"Example_03_03_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137745179\">\n<div id=\"fs-id1165137742710\">\n<h3>Identifying Power Functions<\/h3>\n<p id=\"fs-id1165137824370\">Which of the following functions are power functions?<\/p>\n[latex]\\begin{array}{cccc}\\hfill f\\left(x\\right)&amp; =&amp; 1\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Constant function}\\hfill \\\\ \\hfill f\\left(x\\right)&amp; =&amp; x\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Identify function}\\hfill \\\\ \\hfill f\\left(x\\right)&amp; =&amp; {x}^{2}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Quadratic function}\\hfill \\\\ \\hfill f\\left(x\\right)&amp; =&amp; {x}^{3}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Cubic function}\\hfill \\\\ \\hfill f\\left(x\\right)&amp; =&amp; \\frac{1}{x}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Reciprocal function}\\hfill \\\\ \\hfill f\\left(x\\right)&amp; =&amp; \\frac{1}{{x}^{2}}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Reciprocal squared function}\\hfill \\\\ \\hfill f\\left(x\\right)&amp; =&amp; \\sqrt{x}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Square root function}\\hfill \\\\ \\hfill f\\left(x\\right)&amp; =&amp; \\sqrt[3]{x}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Cube root function}\\hfill \\end{array}[\/latex]\n\n<\/div>\n<div id=\"fs-id1165137422823\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137422823\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137422823\"]\n<p id=\"fs-id1165137843987\">All of the listed functions are power functions.<\/p>\n<p id=\"fs-id1165135533093\">The constant and identity functions are power functions because they can be written as[latex]\\,f\\left(x\\right)={x}^{0}\\,[\/latex]and[latex]\\,f\\left(x\\right)={x}^{1}\\,[\/latex]respectively.<\/p>\n<p id=\"fs-id1165137411464\">The quadratic and cubic functions are power functions with whole number powers[latex]\\,f\\left(x\\right)={x}^{2}\\,[\/latex]and[latex]\\,f\\left(x\\right)={x}^{3}.[\/latex]<\/p>\n<p id=\"fs-id1165137475956\">The <span class=\"no-emphasis\">reciprocal<\/span> and reciprocal squared functions are power functions with negative whole number powers because they can be written as[latex]\\,f\\left(x\\right)={x}^{-1}\\,[\/latex]and[latex]\\,f\\left(x\\right)={x}^{-2}.[\/latex]<\/p>\n<p id=\"fs-id1165135704907\">The square and <span class=\"no-emphasis\">cube root<\/span> functions are power functions with fractional powers because they can be written as[latex]\\,f\\left(x\\right)={x}^{\\frac{1}{2}}\\,[\/latex]or[latex]\\,f\\left(x\\right)={x}^{\\frac{1}{3}}.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137660222\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_03_01\">\n<div id=\"fs-id1165137475224\">\n<p id=\"fs-id1165137475225\">Which functions are power functions?<\/p>\n\n<div><\/div>\n<p id=\"fs-id1165137824385\">[latex]\\begin{array}{ccc}\\hfill f\\left(x\\right)&amp; =&amp; 2x\\cdot 4{x}^{3}\\hfill \\\\ \\hfill g\\left(x\\right)&amp; =&amp; -{x}^{5}+5{x}^{3}\\hfill \\\\ \\hfill h\\left(x\\right)&amp; =&amp; \\frac{2{x}^{5}-1}{3{x}^{2}+4}\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134312227\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134312227\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134312227\"]\n<p id=\"fs-id1165134094624\">[latex]f\\left(x\\right)\\,[\/latex]\nis a power function because it can be written as[latex]\\,f\\left(x\\right)=8{x}^{5}.\\,[\/latex]\nThe other functions are not power functions.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134269023\" class=\"bc-section section\">\n<h3>Identifying End Behavior of Power Functions<\/h3>\n<p id=\"fs-id1165135436540\"><a class=\"autogenerated-content\" href=\"#Figure_03_03_002\">(Figure)<\/a> shows the graphs of[latex]\\,f\\left(x\\right)={x}^{2},\\,g\\left(x\\right)={x}^{4}\\,[\/latex]and[latex]\\,h\\left(x\\right)={x}^{6},\\,[\/latex]which are all power functions with even, whole-number powers. Notice that these graphs have similar shapes, very much like that of the quadratic function in the toolkit. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.<\/p>\n\n<div id=\"Figure_03_03_002\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135645\/CNX_Precalc_Figure_03_03_002.jpg\" alt=\"Graph of three functions, h(x)=x^2 in green, g(x)=x^4 in orange, and f(x)=x^6 in blue.\" width=\"487\" height=\"253\"> <strong>Figure 2. <\/strong>Even-power functions[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137911555\">To describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbol[latex]\\,\\infty \\,[\/latex]for positive infinity and[latex]\\,\\mathrm{-\\infty }\\,[\/latex]for negative infinity. When we say that \u201c[latex]x\\,[\/latex]approaches infinity,\u201d which can be symbolically written as[latex]\\,x\\to \\infty ,\\,[\/latex]we are describing a behavior; we are saying that[latex]\\,x\\,[\/latex]is increasing without bound.<\/p>\n<p id=\"fs-id1165137658268\">With the positive even-power function, as the input increases or decreases without bound, the output values become very large, positive numbers. Equivalently, we could describe this behavior by saying that as[latex]\\,x\\,[\/latex]approaches positive or negative infinity, the[latex]\\,f\\left(x\\right)\\,[\/latex]values increase without bound. In symbolic form, we could write<\/p>\n\n<div id=\"eip-742\" class=\"unnumbered aligncenter\">[latex]\\text{as }x\\to \u00b1\\infty , f\\left(x\\right)\\to \\infty [\/latex]<\/div>\n<p id=\"fs-id1165137533222\"><a class=\"autogenerated-content\" href=\"#Figure_03_03_003\">(Figure)<\/a> shows the graphs of[latex]\\,f\\left(x\\right)={x}^{3},\\,g\\left(x\\right)={x}^{5},[\/latex]and[latex]\\,h\\left(x\\right)={x}^{7},[\/latex]which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function in the toolkit. Again, as the power increases, the graphs flatten near the origin and become steeper away from the origin.<\/p>\n\n<div id=\"Figure_03_03_003\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135711\/CNX_Precalc_Figure_03_03_003.jpg\" alt=\"Graph of three functions, f(x)=x^3 in green, g(x)=x^5 in orange, and h(x)=x^7 in blue.\" width=\"487\" height=\"366\"> <strong> Figure 3.<\/strong> Odd-power functions[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137730237\">These examples illustrate that functions of the form[latex]\\,f\\left(x\\right)={x}^{n}\\,[\/latex]reveal symmetry of one kind or another. First, in <a class=\"autogenerated-content\" href=\"#Figure_03_03_002\">(Figure)<\/a> we see that even functions of the form[latex]\\,f\\left(x\\right)={x}^{n}\\text{, }n\\,[\/latex]even, are symmetric about the[latex]\\,y\\text{-}[\/latex]axis. In <a class=\"autogenerated-content\" href=\"#Figure_03_03_003\">(Figure)<\/a> we see that odd functions of the form[latex]\\,f\\left(x\\right)={x}^{n}\\text{, }n\\,[\/latex] odd, are symmetric about the origin.<\/p>\n<p id=\"fs-id1165137812578\">For these odd power functions, as[latex]\\,x\\,[\/latex] approaches negative infinity,[latex]\\,f\\left(x\\right)\\,[\/latex] decreases without bound. As[latex]\\,x\\,[\/latex] approaches positive infinity,[latex]\\,f\\left(x\\right)\\,[\/latex] increases without bound. In symbolic form we write<\/p>\n\n<div id=\"eip-77\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{as} x\\to -\\infty , f\\left(x\\right)\\to -\\infty \\\\ \\text{as} x\\to \\infty , f\\left(x\\right)\\to \\infty \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137425284\">The behavior of the graph of a function as the input values get very small ([latex]\\,x\\to -\\infty \\,[\/latex]) and get very large ([latex]\\,x\\to \\infty \\,[\/latex]) is referred to as the end behavior of the function. We can use words or symbols to describe end behavior.<\/p>\n<p id=\"fs-id1165137433212\"><a class=\"autogenerated-content\" href=\"#Figure_03_03_004abcd\">(Figure)<\/a> shows the end behavior of power functions in the form[latex]\\,f\\left(x\\right)=k{x}^{n}\\,[\/latex]where[latex]\\,n\\,[\/latex]is a non-negative integer depending on the power and the constant.<\/p>\n\n<div id=\"Figure_03_03_004abcd\" class=\"wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135713\/CNX_Precalc_Figure_03_03_004abcd.jpg\" alt=\"Graph of an even-powered function with a positive constant. As x goes to negative infinity, the function goes to positive infinity; as x goes to positive infinity, the function goes to positive infinity. Graph of an odd-powered function with a positive constant. As x goes to negative infinity, the function goes to positive infinity; as x goes to positive infinity, the function goes to negative infinity. Graph of an even-powered function with a negative constant. As x goes to negative infinity, the function goes to negative infinity; as x goes to positive infinity, the function goes to negative infinity. Graph of an odd-powered function with a negative constant. As x goes to negative infinity, the function goes to negative infinity; as x goes to positive infinity, the function goes to negative infinity.\" width=\"731\" height=\"734\"> <strong>Figure 4.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"fs-id1165135161436\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137415258\"><strong>Given a power function[latex]\\,f\\left(x\\right)=k{x}^{n}\\,[\/latex]where<\/strong>[latex]\\,n\\,[\/latex]<strong>is a non-negative integer, identify the end behavior.<\/strong><\/p>\n\n<ol id=\"fs-id1165137409522\" type=\"1\">\n \t<li>Determine whether the power is even or odd.<\/li>\n \t<li>Determine whether the constant is positive or negative.<\/li>\n \t<li>Use <a class=\"autogenerated-content\" href=\"#Figure_03_03_004abcd\">(Figure)<\/a> to identify the end behavior.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_03_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137923491\">\n<div id=\"fs-id1165137599768\">\n<h3>Identifying the End Behavior of a Power Function<\/h3>\n<p id=\"fs-id1165137644554\">Describe the end behavior of the graph of[latex]\\,f\\left(x\\right)={x}^{8}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135169237\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135169237\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135169237\"]\n<p id=\"fs-id1165137502379\">The coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). As[latex]\\,x\\,[\/latex]approaches infinity, the output (value of[latex]\\,f\\left(x\\right)\\,[\/latex]) increases without bound. We write as[latex]\\,x\\to \\infty ,f\\left(x\\right)\\to \\infty .\\,[\/latex]As[latex]\\,x\\,[\/latex]approaches negative infinity, the output increases without bound. In symbolic form, as[latex]x\\to -\\infty , f\\left(x\\right)\\to \\infty .[\/latex] We can graphically represent the function as shown in <a class=\"autogenerated-content\" href=\"#Figure_03_03_008\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_03_03_008\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135720\/CNX_Precalc_Figure_03_03_008.jpg\" alt=\"Graph of f(x)=x^8.\" width=\"487\" height=\"330\"> <strong>Figure 5.<\/strong>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137535914\">\n<div id=\"fs-id1165137811997\">\n<h3>Identifying the End Behavior of a Power Function.<\/h3>\n<p id=\"fs-id1165137453217\">Describe the end behavior of the graph of[latex]\\,f\\left(x\\right)=-{x}^{9}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137722696\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137722696\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137722696\"]\n<p id=\"fs-id1165137409513\">The exponent of the power function is 9 (an odd number). Because the coefficient is[latex]\\,\u20131\\,[\/latex](negative), the graph is the reflection about the[latex]\\,x\\text{-}[\/latex]axis of the graph of[latex]\\,f\\left(x\\right)={x}^{9}.\\,[\/latex]<a class=\"autogenerated-content\" href=\"#Figure_03_03_009\">(Figure)<\/a> shows that as[latex]\\,x\\,[\/latex]approaches infinity, the output decreases without bound. As[latex]\\,x\\,[\/latex]approaches negative infinity, the output increases without bound. In symbolic form, we would write<\/p>\n\n<div id=\"eip-id1165134384400\" class=\"unnumbered\">[latex]\\begin{array}{l}\\text{as} x\\to -\\infty , f\\left(x\\right)\\to \\infty \\\\ \\text{as} x\\to \\infty , f\\left(x\\right)\\to -\\infty \\end{array}[\/latex]<\/div>\n<div id=\"Figure_03_03_009\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135741\/CNX_Precalc_Figure_03_03_009.jpg\" alt=\"Graph of f(x)=-x^9.\" width=\"487\" height=\"667\"> <strong>Figure 6.<\/strong>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135259565\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137548471\">We can check our work by using the table feature on a graphing utility.<\/p>\n\n<table id=\"Table_03_03_03\" summary=\"..\">\n<thead>\n<tr>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]f\\left(x\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\u201310<\/td>\n<td>1,000,000,000<\/td>\n<\/tr>\n<tr>\n<td>\u20135<\/td>\n<td>1,953,125<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>\u20131,953,125<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>\u20131,000,000,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137644426\">We can see from <a class=\"autogenerated-content\" href=\"#Table_03_03_03\">(Figure)<\/a> that, when we substitute very small values for[latex]\\,x,\\,[\/latex]the output is very large, and when we substitute very large values for[latex]\\,x,\\,[\/latex]the output is very small (meaning that it is a very large negative value).<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137626838\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165137734867\">\n<p id=\"fs-id1165137734868\">Describe in words and symbols the end behavior of[latex]\\,f\\left(x\\right)=-5{x}^{4}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137647550\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137647550\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137647550\"]\n<p id=\"fs-id1165137647551\">As[latex]\\,x\\,[\/latex]approaches positive or negative infinity,[latex]\\,f\\left(x\\right)\\,[\/latex]decreases without bound: as[latex]\\,x\\to \u00b1\\infty , f\\left(x\\right)\\to -\\infty \\,[\/latex]because of the negative coefficient.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134069294\" class=\"bc-section section\">\n<h3>Identifying Polynomial Functions<\/h3>\n<p id=\"fs-id1165135689465\">An oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius[latex]\\,r\\,[\/latex]\nof the spill depends on the number of weeks[latex]\\,w\\,[\/latex]\nthat have passed. This relationship is linear.<\/p>\n\n<div class=\"unnumbered\">[latex]r\\left(w\\right)=24+8w[\/latex]<\/div>\n<p id=\"fs-id1165133432974\">We can combine this with the formula for the area[latex]\\,A\\,[\/latex]\nof a circle.<\/p>\n\n<div id=\"eip-731\" class=\"unnumbered aligncenter\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/div>\n<p id=\"fs-id1165137704887\">Composing these functions gives a formula for the area in terms of weeks.<\/p>\n\n<div id=\"eip-645\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill A\\left(w\\right)&amp; =&amp; A\\left(r\\left(w\\right)\\right)\\hfill \\\\ &amp; =&amp; A\\left(24+8w\\right)\\hfill \\\\ &amp; =&amp; \\pi {\\left(24+8w\\right)}^{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137835475\">Multiplying gives the formula.<\/p>\n\n<div id=\"eip-290\" class=\"unnumbered aligncenter\">[latex]A\\left(w\\right)=576\\pi +384\\pi w+64\\pi {w}^{2}[\/latex]<\/div>\n<p id=\"fs-id1165135205726\">This formula is an example of a <strong>polynomial function<\/strong>. A polynomial function consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.<\/p>\n\n<div id=\"fs-id1165137715427\" class=\"textbox key-takeaways\">\n<h3>Polynomial Functions<\/h3>\n<p id=\"fs-id1165137823247\">Let[latex]\\,n\\,[\/latex]\nbe a non-negative integer. A polynomial function is a function that can be written in the form<\/p>\n\n<div id=\"fs-id1165131937978\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/div>\n<p id=\"eip-id1165137832690\">This is called the general form of a polynomial function. Each[latex]\\,{a}_{i}\\,[\/latex]\nis a coefficient and can be any real number, but\n[latex]\\,{a}_{n}\\,[\/latex]cannot = 0. Each expression[latex]\\,{a}_{i}{x}^{i}\\,[\/latex]\nis a term of a polynomial function.<\/p>\n\n<\/div>\n<div id=\"Example_03_03_04\" class=\"textbox examples\">\n<div id=\"fs-id1165137817691\">\n<div id=\"fs-id1165137817693\">\n<h3>Identifying Polynomial Functions<\/h3>\n<p id=\"fs-id1165135262000\">Which of the following are polynomial functions?<\/p>\n\n<div><\/div>\n<div id=\"eip-id1165134474011\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill f\\left(x\\right)&amp; =&amp; 2{x}^{3}\\cdot 3x+4\\hfill \\\\ \\hfill g\\left(x\\right)&amp; =&amp; -x\\left({x}^{2}-4\\right)\\hfill \\\\ \\hfill h\\left(x\\right)&amp; =&amp; 5\\sqrt{x+2}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165134221783\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134221783\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134221783\"]\n<p id=\"fs-id1165134094645\">The first two functions are examples of polynomial functions because they can be written in the form[latex]\\,f\\left(x\\right)={a}_{n}{x}^{n}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0},\\,[\/latex]\nwhere the powers are non-negative integers and the coefficients are real numbers.<\/p>\n\n<ul id=\"fs-id1165137864157\">\n \t<li>[latex]f\\left(x\\right)\\,[\/latex]\ncan be written as[latex]\\,f\\left(x\\right)=6{x}^{4}+4.[\/latex]<\/li>\n \t<li>[latex]g\\left(x\\right)\\,[\/latex]\ncan be written as[latex]\\,g\\left(x\\right)=-{x}^{3}+4x.[\/latex]<\/li>\n \t<li>[latex]h\\left(x\\right)\\,[\/latex]\ncannot be written in this form and is therefore not a polynomial function.<\/li>\n<\/ul>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135508329\" class=\"bc-section section\">\n<h3>Identifying the Degree and Leading Coefficient of a Polynomial Function<\/h3>\n<p id=\"fs-id1165137831216\">Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order of power, or in general form. The degree of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form. The leading term is the term containing the highest power of the variable, or the term with the highest degree. The leading coefficient is the coefficient of the leading term.<\/p>\n\n<div id=\"fs-id1165135193124\" class=\"textbox key-takeaways\">\n<h3>Terminology of Polynomial Functions<\/h3>\n<p id=\"fs-id1165137921667\">We often rearrange polynomials so that the powers are descending.<\/p>\n<span id=\"fs-id1165137406148\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135744\/CNX_Precalc_Figure_03_03_010n.jpg\" alt=\"Diagram to show what the components of the leading term in a function are. The leading coefficient is a_n and the degree of the variable is the exponent in x^n. Both the leading coefficient and highest degree variable make up the leading term. So the function looks like f(x)=a_nx^n +\u2026+a_2x^2+a_1x+a_0.\"><\/span>\n<p id=\"fs-id1165137482568\">When a polynomial is written in this way, we say that it is in general form.<\/p>\n\n<\/div>\n<div id=\"fs-id1165134031372\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137803898\"><strong>Given a polynomial function, identify the degree and leading coefficient.<\/strong><\/p>\n\n<ol id=\"fs-id1165135587816\" type=\"1\">\n \t<li>Find the highest power of[latex]\\,x\\,[\/latex]\nto determine the degree function.<\/li>\n \t<li>Identify the term containing the highest power of[latex]\\,x\\,[\/latex]\nto find the leading term.<\/li>\n \t<li>Identify the coefficient of the leading term.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_03_05\" class=\"textbox examples\">\n<div id=\"fs-id1165137401820\">\n<div id=\"fs-id1165137862379\">\n<h3>Identifying the Degree and Leading Coefficient of a Polynomial Function<\/h3>\n<p id=\"fs-id1165137435372\">Identify the degree, leading term, and leading coefficient of the following polynomial functions.<\/p>\n\n<div id=\"eip-id1165134242117\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill f\\left(x\\right)&amp; =&amp; 3+2{x}^{2}-4{x}^{3}\\hfill \\\\ \\hfill g\\left(t\\right)&amp; =&amp; 5{t}^{5}-2{t}^{3}+7t\\hfill \\\\ h\\left(p\\right)\\hfill &amp; =&amp; 6p-{p}^{3}-2\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135527012\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135527012\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135527012\"]\n<p id=\"fs-id1165137722510\">For the function[latex]\\,f\\left(x\\right),\\,[\/latex]the highest power of[latex]\\,x\\,[\/latex]is 3, so the degree is 3. The leading term is the term containing that degree,[latex]\\,-4{x}^{3}.\\,[\/latex]The leading coefficient is the coefficient of that term,[latex]\\,-4.[\/latex]<\/p>\n<p id=\"fs-id1165135457771\">For the function[latex]\\,g\\left(t\\right),\\,[\/latex]the highest power of[latex]\\,t\\,[\/latex]is[latex]\\,5,\\,[\/latex]so the degree is[latex]\\,5.\\,[\/latex]The leading term is the term containing that degree,[latex]\\,5{t}^{5}.\\,[\/latex]The leading coefficient is the coefficient of that term,[latex]\\,5.[\/latex]<\/p>\n<p id=\"fs-id1165135503949\">For the function[latex]\\,h\\left(p\\right),\\,[\/latex]the highest power of[latex]\\,p\\,[\/latex]is[latex]\\,3,\\,[\/latex]so the degree is[latex]\\,3.\\,[\/latex]The leading term is the term containing that degree,[latex]\\,-{p}^{3}.\\,[\/latex]The leading coefficient is the coefficient of that term,[latex]\\,-1.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137534935\" class=\"textbox tryit\">\n<div id=\"ti_03_03_03\">\n<div id=\"fs-id1165137424483\">\n<p id=\"fs-id1165137424484\">Identify the degree, leading term, and leading coefficient of the polynomial[latex]\\,f\\left(x\\right)=4{x}^{2}-{x}^{6}+2x-6.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135701674\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135701674\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135701674\"]\n<p id=\"fs-id1165135701675\">The degree is 6. The leading term is[latex]\\,-{x}^{6}.\\,[\/latex]The leading coefficient is[latex]\\,-1.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137702213\" class=\"bc-section section\">\n<h4>Identifying End Behavior of Polynomial Functions<\/h4>\n<p id=\"fs-id1165137601421\">Knowing the degree of a polynomial function is useful in helping us predict its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as[latex]\\,x\\,[\/latex] gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the power function consisting of the leading term. See <a class=\"autogenerated-content\" href=\"#Table_03_03_04\">(Figure)<\/a>.<\/p>\n\n<table id=\"Table_03_03_04\" summary=\"..\"><colgroup> <col> <col> <col><\/colgroup>\n<thead>\n<tr>\n<th>Polynomial Function<\/th>\n<th>Leading Term<\/th>\n<th>Graph of Polynomial Function<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]f\\left(x\\right)=5{x}^{4}+2{x}^{3}-x-4[\/latex]<\/td>\n<td>[latex]5{x}^{4}[\/latex]<\/td>\n<td><span id=\"fs-id1165137768814\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135746\/CNX_Precalc_Figure_03_03_011.jpg\" alt=\"Graph of f(x)=5x^4+2x^3-x-4.\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>[latex]f\\left(x\\right)=-2{x}^{6}-{x}^{5}+3{x}^{4}+{x}^{3}[\/latex]<\/td>\n<td>[latex]-2{x}^{6}[\/latex]<\/td>\n<td><span id=\"fs-id1165137714206\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135752\/CNX_Precalc_Figure_03_03_012.jpg\" alt=\"Graph of f(x)=-2x^6-x^5+3x^4+x^3.\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>[latex]f\\left(x\\right)=3{x}^{5}-4{x}^{4}+2{x}^{2}+1[\/latex]<\/td>\n<td>[latex]3{x}^{5}[\/latex]<\/td>\n<td><span id=\"fs-id1165137540879\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135755\/CNX_Precalc_Figure_03_03_013.jpg\" alt=\"Graph of f(x)=3x^5-4x^4+2x^2+1.\"><\/span><\/td>\n<\/tr>\n<tr>\n<td>[latex]f\\left(x\\right)=-6{x}^{3}+7{x}^{2}+3x+1[\/latex]<\/td>\n<td>[latex]-6{x}^{3}[\/latex]<\/td>\n<td><span id=\"fs-id1165137600670\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135801\/CNX_Precalc_Figure_03_03_014.jpg\" alt=\"Graph of f(x)=-6x^3+7x^2+3x+1.\"><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"Example_03_03_06\" class=\"textbox examples\">\n<div id=\"fs-id1165137452413\">\n<div id=\"fs-id1165137452415\">\n<h3>Identifying End Behavior and Degree of a Polynomial Function<\/h3>\n<p id=\"fs-id1165137831279\">Describe the end behavior and determine a possible degree of the polynomial function in <a class=\"autogenerated-content\" href=\"#Figure_03_03_015\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_03_03_015\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135812\/CNX_Precalc_Figure_03_03_015.jpg\" alt=\"Graph of an odd-degree polynomial.\" width=\"487\" height=\"443\"> <strong>Figure 7.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135251309\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135251309\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135251309\"]\n<p id=\"fs-id1165135251312\">As the input values[latex]\\,x\\,[\/latex]\nget very large, the output values[latex]\\,f\\left(x\\right)\\,[\/latex]increase without bound. As the input values[latex]\\,x\\,[\/latex]\nget very small, the output values[latex]\\,f\\left(x\\right)\\,[\/latex]decrease without bound. We can describe the end behavior symbolically by writing<\/p>\n\n<div id=\"eip-id1165137778911\" class=\"unnumbered\">[latex]\\begin{array}{l}\\text{as} x\\to -\\infty , f\\left(x\\right)\\to -\\infty \\\\ \\text{as} x\\to \\infty , f\\left(x\\right)\\to \\infty \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137454991\">In words, we could say that as[latex]\\,x\\,[\/latex]values approach infinity, the function values approach infinity, and as[latex]\\,x\\,[\/latex]values approach negative infinity, the function values approach negative infinity.<\/p>\n<p id=\"fs-id1165134113949\">We can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polynomial creating this graph must be odd and the leading coefficient must be positive.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137470875\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_03_04\">\n<div id=\"fs-id1165137732301\">\n<p id=\"fs-id1165135460938\">Describe the end behavior, and determine a possible degree of the polynomial function in <a class=\"autogenerated-content\" href=\"#Figure_03_03_016\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_03_03_016\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135827\/CNX_Precalc_Figure_03_03_016n.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"487\" height=\"440\"> <strong>Figure 8.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134047710\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134047710\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134047710\"]\n<p id=\"fs-id1165134047711\">As[latex]\\,x\\to \\infty , f\\left(x\\right)\\to -\\infty ; as x\\to -\\infty , f\\left(x\\right)\\to -\\infty .\\,[\/latex]It has the shape of an even degree power function with a negative coefficient.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_07\" class=\"textbox examples\">\n<div id=\"fs-id1165137470361\">\n<div id=\"fs-id1165137470363\">\n<h3>Identifying End Behavior and Degree of a Polynomial Function<\/h3>\n<p id=\"fs-id1165132011287\">Given the function[latex]\\,f\\left(x\\right)=-3{x}^{2}\\left(x-1\\right)\\left(x+4\\right),\\,[\/latex]express the function as a polynomial in general form, and determine the leading term, degree, and end behavior of the function.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137401107\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137401107\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137401107\"]\n<p id=\"fs-id1165137401109\">Obtain the general form by expanding the given expression for[latex]\\,f\\left(x\\right).[\/latex]<\/p>\n\n<div id=\"eip-id1165132051075\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill f\\left(x\\right)&amp; =&amp; -3{x}^{2}\\left(x-1\\right)\\left(x+4\\right)\\hfill \\\\ &amp; =&amp; -3{x}^{2}\\left({x}^{2}+3x-4\\right)\\hfill \\\\ &amp; =&amp; -3{x}^{4}-9{x}^{3}+12{x}^{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137634030\">The general form is[latex]\\,f\\left(x\\right)=-3{x}^{4}-9{x}^{3}+12{x}^{2}.\\,[\/latex]\nThe leading term is[latex]\\,-3{x}^{4};\\,[\/latex]\ntherefore, the degree of the polynomial is 4. The degree is even (4) and the leading coefficient is negative (\u20133), so the end behavior is<\/p>\n\n<div id=\"eip-id1165133007607\" class=\"unnumbered\">[latex]\\begin{array}{l}\\text{as} x\\to -\\infty , f\\left(x\\right)\\to -\\infty \\\\ \\text{as} x\\to \\infty , f\\left(x\\right)\\to -\\infty \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_03_05\">\n<div id=\"fs-id1165137722131\">\n<p id=\"fs-id1165137416652\">Given the function[latex]\\,f\\left(x\\right)=0.2\\left(x-2\\right)\\left(x+1\\right)\\left(x-5\\right),\\,[\/latex]express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135409431\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135409431\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135409431\"]\n<p id=\"fs-id1165137749856\">The leading term is[latex]\\,0.2{x}^{3},\\,[\/latex]so it is a degree 3 polynomial. As[latex]\\,x\\,[\/latex]approaches positive infinity,[latex]\\,f\\left(x\\right)\\,[\/latex]increases without bound; as[latex]\\,x\\,[\/latex]approaches negative infinity,[latex]\\,f\\left(x\\right)\\,[\/latex]decreases without bound.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137735781\" class=\"bc-section section\">\n<h4>Identifying Local Behavior of Polynomial Functions<\/h4>\n<p id=\"fs-id1165134054039\">In addition to the end behavior of polynomial functions, we are also interested in what happens in the \u201cmiddle\u201d of the function. In particular, we are interested in locations where graph behavior changes. A <strong>turning point <\/strong>is a point at which the function values change from increasing to decreasing or decreasing to increasing.<\/p>\n<p id=\"fs-id1165137417044\">We are also interested in the intercepts. As with all functions, the <em>y-<\/em>intercept is the point at which the graph intersects the vertical axis. The point corresponds to the coordinate pair in which the input value is zero. Because a polynomial is a function, only one output value corresponds to each input value so there can be only one <em>y-<\/em>intercept[latex]\\,\\left(0,{a}_{0}\\right).\\,[\/latex]The <em>x-<\/em>intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one <em>x-<\/em>intercept. See <a class=\"autogenerated-content\" href=\"#Figure_03_03_017\">(Figure)<\/a><strong>.<\/strong><\/p>\n\n<div id=\"Figure_03_03_017\" class=\"wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135837\/CNX_Precalc_Figure_03_03_017.jpg\" alt=\"\" width=\"731\" height=\"629\"> <strong>Figure 9.<\/strong>[\/caption]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Intercepts and Turning Points of Polynomial Functions<\/h3>\n<p id=\"fs-id1165137638552\">A turning point of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. The <em>y-<\/em>intercept is the point at which the function has an input value of zero. The <em>x<\/em>-intercepts are the points at which the output value is zero.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137766902\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137645233\"><strong>Given a polynomial function, determine the intercepts.<\/strong><\/p>\n\n<ol id=\"fs-id1165137571388\" type=\"1\">\n \t<li>Determine the <em>y-<\/em>intercept by setting [latex]\\,x=0\\,[\/latex] and finding the corresponding output value.<\/li>\n \t<li>Determine the <em>x<\/em>-intercepts by solving for the input values that yield an output value of zero.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_03_08\" class=\"textbox examples\">\n<div id=\"fs-id1165137435581\">\n<div id=\"fs-id1165137803210\">\n<h3>Determining the Intercepts of a Polynomial Function<\/h3>\n<p id=\"fs-id1165137441767\">Given the polynomial function[latex]\\,f\\left(x\\right)=\\left(x-2\\right)\\left(x+1\\right)\\left(x-4\\right),\\,[\/latex]written in factored form for your convenience, determine the <em>y<\/em>- and <em>x<\/em>-intercepts.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135251466\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135251466\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135251466\"]\n<p id=\"fs-id1165135251468\">The <em>y-<\/em>intercept occurs when the input is zero so substitute 0 for[latex]\\,x.[\/latex]<\/p>\n\n<div id=\"eip-id1165133032876\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill f\\left(0\\right)&amp; =&amp; {\\left(0\\right)}^{4}-4{\\left(0\\right)}^{2}-45\\hfill \\\\ &amp; =&amp; -45\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135689436\">The <em>y-<\/em>intercept is (0, 8).<\/p>\n<p id=\"fs-id1165137863224\">The <em>x<\/em>-intercepts occur when the output is zero.<\/p>\n\n<div id=\"eip-id1165134380311\" class=\"unnumbered\">[latex]0=\\left(x-2\\right)\\left(x+1\\right)\\left(x-4\\right)[\/latex]<\/div>\n<div id=\"fs-id2036938\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccccccccccc}\\hfill x-2&amp; =&amp; 0\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{or}\\phantom{\\rule{2em}{0ex}}&amp; \\hfill x+1&amp; =&amp; 0\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{or}\\phantom{\\rule{2em}{0ex}}&amp; \\hfill x-4&amp; =&amp; 0\\hfill \\\\ \\hfill x&amp; =&amp; 2\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{or}\\phantom{\\rule{2em}{0ex}}&amp; \\hfill x&amp; =&amp; -1\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{or}\\phantom{\\rule{2em}{0ex}}&amp; \\hfill x&amp; =&amp; 4\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135316178\">The <em>x<\/em>-intercepts are[latex]\\,\\left(2,0\\right),\\left(\u20131,0\\right),\\,[\/latex]and[latex]\\,\\left(4,0\\right).[\/latex]<\/p>\n<p id=\"fs-id1165134380385\">We can see these intercepts on the graph of the function shown in <a class=\"autogenerated-content\" href=\"#Figure_03_03_018\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_03_03_018\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135844\/CNX_Precalc_Figure_03_03_018.jpg\" alt=\"Graph of f(x)=(x-2)(x+1)(x-4), which labels all the intercepts.\" width=\"487\" height=\"630\"> <strong>Figure 10.<\/strong>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_09\" class=\"textbox examples\">\n<div id=\"fs-id1165137834894\">\n<div id=\"fs-id1165137834896\">\n<h3>Determining the Intercepts of a Polynomial Function with Factoring<\/h3>\nGiven the polynomial function[latex]\\,f\\left(x\\right)={x}^{4}-4{x}^{2}-45,\\,[\/latex]determine the <em>y<\/em>- and <em>x<\/em>-intercepts.\n\n<\/div>\n<div id=\"fs-id1165137634473\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137634473\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137634473\"]\n<p id=\"fs-id1165137634475\">The <em>y-<\/em>intercept occurs when the input is zero.<\/p>\n\n<div id=\"eip-id1165132943488\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill f\\left(0\\right)&amp; =&amp; {\\left(0\\right)}^{4}-4{\\left(0\\right)}^{2}-45\\hfill \\\\ &amp; =&amp; -45\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135653967\">The <em>y-<\/em>intercept is[latex]\\,\\left(0,-45\\right).[\/latex]<\/p>\n<p id=\"fs-id1165135152099\">The <em>x<\/em>-intercepts occur when the output is zero. To determine when the output is zero, we will need to factor the polynomial.<\/p>\n[latex]\\begin{array}{ccc}\\hfill f\\left(x\\right)&amp; =&amp; {x}^{4}-4{x}^{2}-45\\hfill \\\\ &amp; =&amp; \\left({x}^{2}-9\\right)\\left({x}^{2}+5\\right)\\hfill \\\\ &amp; =&amp; \\left(x-3\\right)\\left(x+3\\right)\\left({x}^{2}+5\\right)\\hfill \\end{array}[\/latex]\n[latex]\\phantom{\\rule{2em}{0ex}}0=\\left(x-3\\right)\\left(x+3\\right)\\left({x}^{2}+5\\right)[\/latex]\n\n[latex]\\begin{array}{ccccccccc}\\hfill x-3&amp; =&amp; 0\\hfill &amp; \\text{or}&amp; \\hfill x+3&amp; =&amp; 0\\hfill &amp;\\text{or}&amp; {x}^{2}+5=0\\\\ \\hfill x&amp; =&amp; 3\\hfill &amp; \\text{or} &amp; \\hfill x&amp; =&amp; -3\\hfill &amp; \\text{or}&amp; (\\text{no real solution)}\\end{array}[\/latex]\n<p id=\"fs-id1165135436471\">The <em>x<\/em>-intercepts are[latex]\\,\\left(3,0\\right)\\,[\/latex]and[latex]\\,\\left(\u20133,0\\right).[\/latex]<\/p>\nWe can see these intercepts on the graph of the function shown in <a class=\"autogenerated-content\" href=\"#Figure_03_03_019\">(Figure)<\/a>. We can see that the function is even because[latex]\\,f\\left(x\\right)=f\\left(-x\\right).[\/latex]\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135849\/CNX_Precalc_Figure_03_03_019.jpg\" alt=\"Graph of f(x)=x^4-4x^2-45, which labels all the intercepts at (-3, 0), (3, 0), and (0, -45).\" width=\"487\" height=\"426\"> <strong>Figure 11.<\/strong>[\/caption]\n\n[\/hidden-answer]<span id=\"fs-id1165137803348\"><\/span>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137749604\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_03_06\">\n<div id=\"fs-id1165137405243\">\n<p id=\"fs-id1165137405244\">Given the polynomial function[latex]\\,f\\left(x\\right)=2{x}^{3}-6{x}^{2}-20x,\\,[\/latex]determine the <em>y<\/em>- and <em>x<\/em>-intercepts.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137762370\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137762370\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137762370\"]\n<p id=\"fs-id1165137762371\"><em>y<\/em>-intercept[latex]\\,\\left(0,0\\right);\\,[\/latex]<em>x<\/em>-intercepts[latex]\\,\\left(0,0\\right),\\left(\u20132,0\\right),\\,[\/latex]and[latex]\\left(5,0\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134080932\" class=\"bc-section section\">\n<h4>Comparing Smooth and Continuous Graphs<\/h4>\n<p id=\"fs-id1165137692509\">The degree of a polynomial function helps us to determine the number of <em>x<\/em>-intercepts and the number of turning points. A polynomial function of[latex]\\,n\\text{th}\\,[\/latex]degree is the product of[latex]\\,n\\,[\/latex]factors, so it will have at most[latex]\\,n\\,[\/latex]roots or zeros, or <em>x<\/em>-intercepts. The graph of the polynomial function of degree[latex]\\,n\\,[\/latex]must have at most[latex]\\,n\u20131\\,[\/latex]turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.<\/p>\n<p id=\"fs-id1165137657937\">A continuous function has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A smooth curve is a graph that has no sharp corners. The turning points of a smooth graph must always occur at rounded curves. The graphs of polynomial functions are both continuous and smooth.<\/p>\n\n<div id=\"fs-id1165137847104\" class=\"textbox key-takeaways\">\n<h3>Intercepts and Turning Points of Polynomials<\/h3>\n<p id=\"fs-id1165137405499\">A polynomial of degree[latex]\\,n\\,[\/latex]will have, at most,[latex]\\,n\\,[\/latex]<em>x<\/em>-intercepts and[latex]\\,n-1\\,[\/latex]turning points.<\/p>\n\n<\/div>\n<div id=\"Example_03_03_10\" class=\"textbox examples\">\n<div id=\"fs-id1165135237034\">\n<div id=\"fs-id1165135237036\">\n<h3>Determining the Number of Intercepts and Turning Points of a Polynomial<\/h3>\n<p id=\"fs-id1165134152759\">Without graphing the function, determine the local behavior of the function by finding the maximum number of <em>x<\/em>-intercepts and turning points for[latex]\\,f\\left(x\\right)=-3{x}^{10}+4{x}^{7}-{x}^{4}+2{x}^{3}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135414339\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135414339\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135414339\"]\n<p id=\"fs-id1165135414341\">The polynomial has a degree of[latex]\\,10,\\,[\/latex]so there are at most 10 <em>x<\/em>-intercepts and at most 9 turning points.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137628834\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_03_07\">\n<div id=\"fs-id1165135188273\">\n<p id=\"fs-id1165135188274\">Without graphing the function, determine the maximum number of <em>x<\/em>-intercepts and turning points for[latex]\\,f\\left(x\\right)=108-13{x}^{9}-8{x}^{4}+14{x}^{12}+2{x}^{3}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137660801\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137660801\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137660801\"]\n<p id=\"fs-id1165137660802\">There are at most 12[latex]\\,x\\text{-}[\/latex]intercepts and at most 11 turning points.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_11\" class=\"textbox examples\">\n<div id=\"fs-id1165137435064\">\n<div id=\"fs-id1165137435066\">\n<h3>Drawing Conclusions about a Polynomial Function from the Graph<\/h3>\n<p id=\"fs-id1165137843783\">What can we conclude about the polynomial represented by the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_03_03_020\">(Figure)<\/a> based on its intercepts and turning points?<\/p>\n\n<div id=\"Figure_03_03_020\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135851\/CNX_Precalc_Figure_03_03_020.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"487\" height=\"367\"> <strong>Figure 12.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137737264\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137737264\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137737264\"]\n<p id=\"fs-id1165131926327\">The end behavior of the graph tells us this is the graph of an even-degree polynomial. See <a class=\"autogenerated-content\" href=\"#Figure_03_03_021\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_03_03_021\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135900\/CNX_Precalc_Figure_03_03_021.jpg\" alt=\"Graph of an even-degree polynomial that denotes the turning points and intercepts.\" width=\"487\" height=\"368\"> <strong>Figure 13.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165135670389\">The graph has 2 <em>x<\/em>-intercepts, suggesting a degree of 2 or greater, and 3 turning points, suggesting a degree of 4 or greater. Based on this, it would be reasonable to conclude that the degree is even and at least 4.[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137871106\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_03_08\">\n<div id=\"fs-id1165137834183\">\n<p id=\"fs-id1165137454180\">What can we conclude about the polynomial represented by the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_03_03_022\">(Figure)<\/a> based on its intercepts and turning points?<\/p>\n\n<div id=\"Figure_03_03_022\" class=\"wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135910\/CNX_Precalc_Figure_03_03_022.jpg\" alt=\"Graph of an odd-degree polynomial.\" width=\"487\" height=\"442\"> <strong>Figure 14.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137666790\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137666790\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137666790\"]\n<p id=\"fs-id1165137666791\">The end behavior indicates an odd-degree polynomial function; there are 3[latex]\\,x\\text{-}[\/latex]intercepts and 2 turning points, so the degree is odd and at least 3. Because of the end behavior, we know that the lead coefficient must be negative.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_12\" class=\"textbox examples\">\n<div id=\"fs-id1165135184013\">\n<div id=\"fs-id1165137725458\">\n<h3>Drawing Conclusions about a Polynomial Function from the Factors<\/h3>\n<p id=\"fs-id1165135435639\">Given the function[latex]\\,f\\left(x\\right)=-4x\\left(x+3\\right)\\left(x-4\\right),\\,[\/latex]\ndetermine the local behavior.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135457721\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135457721\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135457721\"]\n<p id=\"fs-id1165135457723\">The <em>y<\/em>-intercept is found by evaluating[latex]\\,f\\left(0\\right).[\/latex]<\/p>\n\n<div id=\"eip-id1165134587897\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill f\\left(0\\right)&amp; =&amp; -4\\left(0\\right)\\left(0+3\\right)\\left(0-4\\\\ &amp; =&amp; 0\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135245749\">The <em>y<\/em>-intercept is[latex]\\,\\left(0,0\\right).[\/latex]<\/p>\n<p id=\"fs-id1165135203755\">The <em>x<\/em>-intercepts are found by determining the zeros of the function.<\/p>\n\n<div id=\"eip-id1165135401630\" class=\"unnumbered\">[latex]0=-4x\\left(x+3\\right)\\left(x-4\\right)[\/latex]<\/div>\n<div id=\"fs-id1178321\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccccccccccc}\\hfill x&amp; =&amp; 0\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{or}\\phantom{\\rule{2em}{0ex}}&amp; \\hfill x+3&amp; =&amp; 0\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{or}\\phantom{\\rule{2em}{0ex}}&amp; \\hfill x-4&amp; =&amp; 0\\hfill \\\\ x&amp; =&amp; 0&amp; \\phantom{\\rule{2em}{0ex}}\\text{or}\\phantom{\\rule{2em}{0ex}}&amp; x&amp; =&amp; -3&amp; \\phantom{\\rule{2em}{0ex}}\\text{or}\\phantom{\\rule{2em}{0ex}}&amp; x&amp; =&amp; 4\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135431016\">The <em>x<\/em>-intercepts are[latex]\\,\\left(0,0\\right),\\left(\u20133,0\\right),\\,[\/latex]and[latex]\\,\\left(4,0\\right).[\/latex]<\/p>\n<p id=\"fs-id1165137472984\">The degree is 3 so the graph has at most 2 turning points.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137661075\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_03_09\">\n<div id=\"fs-id1165137575430\">\n<p id=\"fs-id1165137575431\">Given the function[latex]\\,f\\left(x\\right)=0.2\\left(x-2\\right)\\left(x+1\\right)\\left(x-5\\right),\\,[\/latex]determine the local behavior.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137833005\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137833005\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137833005\"]\n<p id=\"fs-id1165137833006\">The[latex]\\,x\\text{-}[\/latex]intercepts are[latex]\\,\\left(2,0\\right),\\left(-1,0\\right),[\/latex]and[latex]\\,\\left(5,0\\right),\\,[\/latex]the <em>y-<\/em>intercept is[latex]\\,\\left(0,\\text{2}\\right),\\,[\/latex]and the graph has at most 2 turning points.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137653058\" class=\"precalculus media\">\n<p id=\"fs-id1165135456729\">Access these online resources for additional instruction and practice with power and polinomial functions.<\/p>\n\n<ul id=\"fs-id1165137410802\">\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/keyinfopoly\">Find Key Information about a Given Polynomial Function<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/endbehavior\">End Behavior of a Polynomial Function<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/turningpoints\">Turning Points and[latex]\\,x\\text{-}[\/latex]intercepts of Polynomial Functions<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/leastposdegree\">Least Possible Degree of a Polynomial Function<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165134063974\" summary=\"..\">\n<tbody>\n<tr>\n<td>general form of a polynomial function<\/td>\n<td>[latex]f\\left(x\\right)={a}_{n}{x}^{n}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137731646\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165135438864\">\n \t<li>A power function is a variable base raised to a number power. See <a class=\"autogenerated-content\" href=\"#Example_03_03_01\">(Figure)<\/a>.<\/li>\n \t<li>The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.<\/li>\n \t<li>The end behavior depends on whether the power is even or odd. See <a class=\"autogenerated-content\" href=\"#Example_03_03_02\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_03_03_03\">(Figure)<\/a>.<\/li>\n \t<li>A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power. See <a class=\"autogenerated-content\" href=\"#Example_03_03_04\">(Figure)<\/a>.<\/li>\n \t<li>The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient. See <a class=\"autogenerated-content\" href=\"#Example_03_03_05\">(Figure)<\/a>.<\/li>\n \t<li>The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function. See <a class=\"autogenerated-content\" href=\"#Example_03_03_06\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_03_03_07\">(Figure)<\/a>.<\/li>\n \t<li>A polynomial of degree[latex]\\,n\\,[\/latex]\nwill have at most[latex]\\,n\\,[\/latex]\n<em>x-<\/em>intercepts and at most[latex]\\,n-1\\,[\/latex]\nturning points. See <a class=\"autogenerated-content\" href=\"#Example_03_03_08\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_03_03_09\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_03_03_10\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_03_03_11\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_03_03_12\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137553381\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165137761271\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165137651029\">\n<div id=\"fs-id1165137651030\">\n<p id=\"fs-id1165137651031\">Explain the difference between the coefficient of a power function and its degree.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137423709\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137423709\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137423709\"]\n<p id=\"fs-id1165137423710\">The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137784642\">\n<div id=\"fs-id1165137784643\">\n\nIf a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137794232\">\n<div id=\"fs-id1165137794233\">\n<p id=\"fs-id1165137794234\">In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135205853\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135205853\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135205853\"]\n<p id=\"fs-id1165135205854\">As[latex]\\,x\\,[\/latex]\ndecreases without bound, so does[latex]\\,f\\left(x\\right).\\,[\/latex]\nAs[latex]\\,x\\,[\/latex]\nincreases without bound, so does[latex]\\,f\\left(x\\right).[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137409144\">\n<div id=\"fs-id1165137409146\">\n<p id=\"fs-id1165137628457\">What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137628462\">\n<div id=\"fs-id1165137628463\">\n\nWhat can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As[latex]\\,x\\to -\\infty ,\\,f\\left(x\\right)\\to -\\infty \\,[\/latex]\nand as[latex]\\,x\\to \\infty ,\\,f\\left(x\\right)\\to -\\infty .\\,[\/latex]\n\n<\/div>\n<div id=\"fs-id1165137679007\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137679007\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137679007\"]\n<p id=\"fs-id1165137679008\">The polynomial function is of even degree and leading coefficient is negative.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135444041\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165137807114\">For the following exercises, identify the function as a power function, a polynomial function, or neither.<\/p>\n\n<div id=\"fs-id1165137807118\">\n<div id=\"fs-id1165137807119\">\n<p id=\"fs-id1165137431407\">[latex]f\\left(x\\right)={x}^{5}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137585053\">\n<div id=\"fs-id1165137585054\">\n<p id=\"fs-id1165137585055\">[latex]f\\left(x\\right)={\\left({x}^{2}\\right)}^{3}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134043753\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134043753\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134043753\"]\n<p id=\"fs-id1165137771535\">Power function<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137771538\">\n<div id=\"fs-id1165137771539\">\n<p id=\"fs-id1165137771540\">[latex]f\\left(x\\right)=x-{x}^{4}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135339598\">\n<div id=\"fs-id1165135339599\">\n<p id=\"fs-id1165134475621\">[latex]f\\left(x\\right)=\\frac{{x}^{2}}{{x}^{2}-1}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137723151\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137723151\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137723151\"]\n<p id=\"fs-id1165137723152\">Neither<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137723156\">\n<div id=\"fs-id1165137723157\">\n<p id=\"fs-id1165134043631\">[latex]f\\left(x\\right)=2x\\left(x+2\\right){\\left(x-1\\right)}^{2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134149794\">\n<div id=\"fs-id1165134149795\">\n<p id=\"fs-id1165134149796\">[latex]f\\left(x\\right)={3}^{x+1}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137619578\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137619578\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137619578\"]\n<p id=\"fs-id1165137619580\">Neither<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137619583\">For the following exercises, find the degree and leading coefficient for the given polynomial.<\/p>\n\n<div id=\"fs-id1165137464314\">\n<div id=\"fs-id1165137464315\">\n<p id=\"fs-id1165137464316\">[latex]-3x{}^{4}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134380913\">\n<div id=\"fs-id1165134380914\">\n<p id=\"fs-id1165134380915\">[latex]7-2{x}^{2}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137424474\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137424474\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137424474\"]\n<p id=\"fs-id1165137424475\">Degree = 2, Coefficient = \u20132<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137425870\">\n<p id=\"fs-id1165137425871\">[latex]-2{x}^{2}-3{x}^{5}+x-6 [\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137742433\">\n<div id=\"fs-id1165137742434\">\n<p id=\"fs-id1165137742435\">[latex]x\\left(4-{x}^{2}\\right)\\left(2x+1\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137842483\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137842483\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137842483\"]\n<p id=\"fs-id1165132943532\">Degree =4, Coefficient = \u20132<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165132943536\">\n<div id=\"fs-id1165132943537\">\n<p id=\"fs-id1165132943538\">[latex]{x}^{2}{\\left(2x-3\\right)}^{2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137455116\">For the following exercises, determine the end behavior of the functions.<\/p>\n\n<div id=\"fs-id1165137455119\">\n<div id=\"fs-id1165133078101\">\n<p id=\"fs-id1165133078102\">[latex]f\\left(x\\right)={x}^{4}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137445856\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137445856\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137445856\"]\n<p id=\"fs-id1165137445857\">[latex]\\text{As}\\,x\\to \\infty ,\\,\\,f\\left(x\\right)\\to \\infty ,\\,\\text{as}\\,x\\to -\\infty ,\\,f\\left(x\\right)\\to \\infty [\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137423948\">\n<div id=\"fs-id1165137423949\">\n<p id=\"fs-id1165137423950\">[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137678592\">\n<div id=\"fs-id1165137678593\">\n<p id=\"fs-id1165137678594\">[latex]f\\left(x\\right)=-{x}^{4}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137643430\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137643430\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137643430\"]\n<p id=\"fs-id1165137643431\">[latex]\\text{As}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to -\\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to -\\infty [\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135557775\">\n<div id=\"fs-id1165135457738\">\n<p id=\"fs-id1165135457739\">[latex]f\\left(x\\right)=-{x}^{9}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137526614\">\n<div id=\"fs-id1165137526615\">\n<p id=\"fs-id1165137526616\">[latex]f\\left(x\\right)=-2{x}^{4}-3{x}^{2}+x-1 [\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137656888\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137656888\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137656888\"]\n<p id=\"fs-id1165137656889\">[latex]\\text{As}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to -\\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to -\\infty [\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135696746\">\n<div id=\"fs-id1165135696747\">\n<p id=\"fs-id1165135696748\">[latex]f\\left(x\\right)=3{x}^{2}+x-2[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137627936\">\n<div id=\"fs-id1165137627937\">\n<p id=\"fs-id1165137806389\">[latex]f\\left(x\\right)={x}^{2}\\left(2{x}^{3}-x+1\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137794123\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137794123\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137794123\"]\n<p id=\"fs-id1165135333271\">[latex]\\text{As}\\,x\\to \\infty ,\\,\\,f\\left(x\\right)\\to \\infty ,\\,\\text{as}\\,x\\to -\\infty ,\\,f\\left(x\\right)\\to -\\infty [\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137761673\">\n<div id=\"fs-id1165137761674\">\n<p id=\"fs-id1165137761675\">[latex]f\\left(x\\right)={\\left(2-x\\right)}^{7}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135650636\">For the following exercises, find the intercepts of the functions.<\/p>\n\n<div id=\"fs-id1165135650639\">\n<div id=\"fs-id1165135650640\">\n<p id=\"fs-id1165135436503\">[latex]f\\left(t\\right)=2\\left(t-1\\right)\\left(t+2\\right)\\left(t-3\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135697918\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135697918\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135697918\"]\n<p id=\"fs-id1165135697919\"><em>y<\/em>-intercept is[latex]\\,\\left(0,12\\right),\\,[\/latex]<em>t<\/em>-intercepts are[latex]\\,\\left(1,0\\right);\\left(\u20132,0\\right);\\text{and }\\left(3,0\\right).[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137634317\">\n<div id=\"fs-id1165137634318\">\n<p id=\"fs-id1165137634319\">[latex]g\\left(n\\right)=-2\\left(3n-1\\right)\\left(2n+1\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134113935\">\n<div id=\"fs-id1165137645540\">\n<p id=\"fs-id1165137645541\">[latex]f\\left(x\\right)={x}^{4}-16[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137728286\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137728286\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137728286\"]\n<p id=\"fs-id1165137728287\"><em>y<\/em>-intercept is[latex]\\,\\left(0,-16\\right).\\,[\/latex]<em>x<\/em>-intercepts are[latex]\\,\\left(2,0\\right)\\,[\/latex]and[latex]\\,\\left(-2,0\\right).[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137783610\">\n<div id=\"fs-id1165137783611\">\n<p id=\"fs-id1165137783612\">[latex]f\\left(x\\right)={x}^{3}+27[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137823331\">\n<div id=\"fs-id1165137734444\">\n<p id=\"fs-id1165137734445\">[latex]f\\left(x\\right)=x\\left({x}^{2}-2x-8\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137896961\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137896961\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137896961\"]\n<p id=\"fs-id1165137896962\"><em>y<\/em>-intercept is[latex]\\,\\left(0,0\\right).\\,[\/latex]<em>x<\/em>-intercepts are[latex]\\,\\left(0,0\\right),\\left(4,0\\right),\\,[\/latex]and[latex]\\,\\left(-2, 0\\right).[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134223322\">\n<div id=\"fs-id1165134223324\">\n<p id=\"fs-id1165137456270\">[latex]f\\left(x\\right)=\\left(x+3\\right)\\left(4{x}^{2}-1\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135416569\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165134486753\">For the following exercises, determine the least possible degree of the polynomial function shown.<\/p>\n\n<div id=\"fs-id1165134486757\">\n<div id=\"fs-id1165134486758\"><span id=\"fs-id1165137942457\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135918\/CNX_Precalc_Figure_03_03_201.jpg\" alt=\"Graph of an odd-degree polynomial.\"><\/span><\/div>\n<div id=\"fs-id1165135388469\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135388469\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135388469\"]\n<p id=\"fs-id1165135388470\">3<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137823453\">\n<div id=\"fs-id1165137823454\"><span id=\"fs-id1165135530394\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135921\/CNX_Precalc_Figure_03_03_202.jpg\" alt=\"Graph of an even-degree polynomial.\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165135436619\">\n<div id=\"fs-id1165135436620\"><span id=\"fs-id1165135543346\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135936\/CNX_Precalc_Figure_03_03_203.jpg\" alt=\"Graph of an odd-degree polynomial.\"><\/span><\/div>\n<div id=\"fs-id1165135181683\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135181683\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135181683\"]\n<p id=\"fs-id1165135181684\">5<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137781545\">\n<div id=\"fs-id1165137781546\"><span id=\"fs-id1165137433119\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135938\/CNX_Precalc_Figure_03_03_204.jpg\" alt=\"Graph of an odd-degree polynomial.\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165135457734\">\n<div id=\"fs-id1165137833940\"><span id=\"fs-id1165137833945\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135940\/CNX_Precalc_Figure_03_03_205.jpg\" alt=\"Graph of an odd-degree polynomial.\"><\/span><\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165135255943\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135255943\"]\n<p id=\"fs-id1165135255943\">3<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137410264\">\n<div id=\"fs-id1165137410266\"><span id=\"fs-id1165134323766\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135944\/CNX_Precalc_Figure_03_03_206.jpg\" alt=\"Graph of an even-degree polynomial.\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165134129751\">\n<div id=\"fs-id1165134129752\"><span id=\"fs-id1165137405932\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135955\/CNX_Precalc_Figure_03_03_207.jpg\" alt=\"Graph of an odd-degree polynomial.\"><\/span><\/div>\n<div id=\"fs-id1165135191388\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135191388\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135191388\"]\n<p id=\"fs-id1165135191389\">5<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135191392\">\n<div id=\"fs-id1165135191393\"><span id=\"fs-id1165137653349\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135957\/CNX_Precalc_Figure_03_03_208.jpg\" alt=\"Graph of an even-degree polynomial.\"><\/span><\/div>\n<\/div>\n<p id=\"fs-id1165134262467\">For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.<\/p>\n\n<div id=\"fs-id1165135332833\">\n<div id=\"fs-id1165135332834\"><span id=\"fs-id1165135678606\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140002\/CNX_Precalc_Figure_03_03_209.jpg\" alt=\"Graph of an odd-degree polynomial.\"><\/span><\/div>\n<div id=\"fs-id1165137444866\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137444866\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137444866\"]\n<p id=\"fs-id1165137444867\">Yes. Number of turning points is 2. Least possible degree is 3.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137444870\">\n<div id=\"fs-id1165137444871\"><span id=\"fs-id1165135264631\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140011\/CNX_Precalc_Figure_03_03_210.jpg\" alt=\"Graph of an equation.\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137704867\">\n<div id=\"fs-id1165137704868\"><span id=\"fs-id1165134089461\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140017\/CNX_Precalc_Figure_03_03_211.jpg\" alt=\"Graph of an even-degree polynomial.\"><\/span><\/div>\n<div id=\"fs-id1165137570516\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137570516\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137570516\"]\n<p id=\"fs-id1165137570517\">Yes. Number of turning points is 1. Least possible degree is 2.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135191925\">\n<div id=\"fs-id1165135191926\"><span id=\"fs-id1165135591035\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140023\/CNX_Precalc_Figure_03_03_212.jpg\" alt=\"Graph of an odd-degree polynomial.\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165134313927\">\n<div id=\"fs-id1165137476631\"><span id=\"fs-id1165135319603\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140025\/CNX_Precalc_Figure_03_03_213.jpg\" alt=\"Graph of an odd-degree polynomial.\"><\/span><\/div>\n<div id=\"fs-id1165134042451\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134042451\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134042451\"]\n<p id=\"fs-id1165134042452\">Yes. Number of turning points is 0. Least possible degree is 1.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137436429\">\n<div id=\"fs-id1165137436430\"><span id=\"fs-id1165137772222\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140031\/CNX_Precalc_Figure_03_03_214.jpg\" alt=\"Graph of an equation.\"><\/span><\/div>\n<div id=\"fs-id1165133310452\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133310452\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133310452\"]\n<p id=\"fs-id1165133310453\">No.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133310456\">\n<div><span id=\"fs-id1165137642864\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140035\/CNX_Precalc_Figure_03_03_215.jpg\" alt=\"Graph of an odd-degree polynomial.\"><\/span><\/div>\n<div id=\"fs-id1165137569703\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137569703\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137569703\"]\n<p id=\"fs-id1165137569704\">Yes. Number of turning points is 0. Least possible degree is 1.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134130094\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<p id=\"fs-id1165137828131\">For the following exercises, make a table to confirm the end behavior of the function.<\/p>\n\n<div id=\"fs-id1165137828134\">\n<div id=\"fs-id1165137416865\">\n<p id=\"fs-id1165137416866\">[latex]f\\left(x\\right)=-{x}^{3}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137734861\">\n<div id=\"fs-id1165137734862\">\n<p id=\"fs-id1165137734863\">[latex]f\\left(x\\right)={x}^{4}-5{x}^{2}[\/latex]<\/p>\n\n<div class=\"textbox shaded\">[reveal-answer q=\"983153\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"983153\"]\n<table id=\"fs-id1165137654655\" class=\"unnumbered\" summary=\"..\"><caption>&nbsp;<\/caption>\n<thead>\n<tr>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]f\\left(x\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>10<\/td>\n<td>9,500<\/td>\n<\/tr>\n<tr>\n<td>100<\/td>\n<td>99,950,000<\/td>\n<\/tr>\n<tr>\n<td>\u201310<\/td>\n<td>9,500<\/td>\n<\/tr>\n<tr>\n<td>\u2013100<\/td>\n<td>99,950,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137654654\">[latex]\\text{as}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to \\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty [\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133045350\">\n<div id=\"fs-id1165133045351\">\n<p id=\"fs-id1165133045352\">[latex]f\\left(x\\right)={x}^{2}{\\left(1-x\\right)}^{2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135456684\">\n<div id=\"fs-id1165135456685\">\n<p id=\"fs-id1165135456686\">[latex]f\\left(x\\right)=\\left(x-1\\right)\\left(x-2\\right)\\left(3-x\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134122928\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134122928\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134122928\"]\n<table id=\"fs-id1165134122930\" class=\"unnumbered\" summary=\"..\"><caption>&nbsp;<\/caption>\n<thead>\n<tr>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]f\\left(x\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>10<\/td>\n<td>\u2013504<\/td>\n<\/tr>\n<tr>\n<td>100<\/td>\n<td>\u2013941,094<\/td>\n<\/tr>\n<tr>\n<td>\u201310<\/td>\n<td>1,716<\/td>\n<\/tr>\n<tr>\n<td>\u2013100<\/td>\n<td>1,061,106<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165134122929\">[latex]\\text{as}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to \\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to -\\infty [\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134122886\">\n<div id=\"fs-id1165134122887\">\n<p id=\"fs-id1165134122888\">[latex]f\\left(x\\right)=\\frac{{x}^{5}}{10}-{x}^{4}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137755588\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1165133238486\">For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.<\/p>\n\n<div id=\"fs-id1165135317475\">\n<div>\n<p id=\"fs-id1165135317477\">[latex]f\\left(x\\right)={x}^{3}\\left(x-2\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137629480\" class=\"solution textbox shaded\">\n\n[reveal-answer q=\"41150\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"41150\"]\n\n<span id=\"fs-id1165135168133\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140037\/CNX_Precalc_Figure_03_03_216.jpg\" alt=\"Graph of f(x)=x^3(x-2).\"><\/span>\n<p id=\"fs-id1165135168128\">The[latex]\\,y\\text{-}[\/latex]intercept is[latex]\\,\\left(0, 0\\right).\\,[\/latex]\nThe[latex]\\,x\\text{-}[\/latex]intercepts are[latex]\\,\\left(0, 0\\right),\\text{ }\\left(2, 0\\right).\\,[\/latex]\n[latex]\\text{As}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to \\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty [\/latex]<\/p>\n<p id=\"fs-id1165135168128\">[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135388458\">\n<div id=\"fs-id1165135388459\">\n<p id=\"fs-id1165135388460\">[latex]f\\left(x\\right)=x\\left(x-3\\right)\\left(x+3\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137920714\">\n<div id=\"fs-id1165137920716\">\n<p id=\"fs-id1165137920717\">[latex]f\\left(x\\right)=x\\left(14-2x\\right)\\left(10-2x\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135436598\" class=\"solution textbox shaded\">\n\n[reveal-answer q=\"891902\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"891902\"]\n\n<span id=\"fs-id1165137438048\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140052\/CNX_Precalc_Figure_03_03_218.jpg\" alt=\"Graph of f(x)=x(14-2x)(10-2x).\"><\/span>\n<p id=\"fs-id1165135436600\">The[latex]\\,y\\text{-}[\/latex]intercept is[latex]\\,\\left(0,0\\right)[\/latex]\n. The[latex]\\,x\\text{-}[\/latex]intercepts are[latex]\\,\\left(0, 0\\right),\\text{ }\\left(5, 0\\right),\\text{ }\\left(7, 0\\right).\\,[\/latex]\n[latex]\\text{As}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to -\\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty [\/latex]<\/p>\n<p id=\"fs-id1165135436600\">[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133448049\">\n<div>\n<p id=\"fs-id1165133448052\">[latex]f\\left(x\\right)=x\\left(14-2x\\right){\\left(10-2x\\right)}^{2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134417902\">\n<div id=\"fs-id1165134417903\">\n<p id=\"fs-id1165134417904\">[latex]f\\left(x\\right)={x}^{3}-16x[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137676496\" class=\"solution textbox shaded\">\n\n[reveal-answer q=\"218879\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"218879\"]\n\n<span id=\"fs-id1165137676502\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140059\/CNX_Precalc_Figure_03_03_220.jpg\" alt=\"\"><\/span>\n<p id=\"fs-id1165137676497\">The[latex]\\,y\\text{-}[\/latex]intercept is[latex]\\,\\left(0, 0\\right).\\,[\/latex] The[latex]\\,x\\text{-}[\/latex]intercept is[latex]\\,\\left(-4, 0\\right),\\text{ }\\left(0, 0\\right),\\text{ }\\left(4, 0\\right).\\,[\/latex]\n[latex]As\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to -\\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty [\/latex]<\/p>\n<p id=\"fs-id1165137676497\">[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137837875\">\n<div id=\"fs-id1165137837876\">\n<p id=\"fs-id1165137837878\">[latex]f\\left(x\\right)={x}^{3}-27[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135160696\">\n<div id=\"fs-id1165135160697\">\n<p id=\"fs-id1165131891794\">[latex]f\\left(x\\right)={x}^{4}-81[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165131891798\" class=\"solution textbox shaded\">\n\n[reveal-answer q=\"110405\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"110405\"]\n\n<span id=\"fs-id1165135555475\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140101\/CNX_Precalc_Figure_03_03_222.jpg\" alt=\"Graph of f(x)=x^3-27.\"><\/span>\n<p id=\"fs-id1165131891799\">The[latex]\\,y\\text{-}[\/latex]intercept is[latex]\\,\\left(0, -81\\right).\\,[\/latex]\nThe[latex]\\,x\\text{-}[\/latex]intercept are[latex]\\,\\left(3, 0\\right),\\text{ }\\left(-3, 0\\right).\\,[\/latex]\n[latex]\\text{As}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to \\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty [\/latex]<\/p>\n<p id=\"fs-id1165131891799\">[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135484565\">\n<div id=\"fs-id1165135484566\">\n<p id=\"fs-id1165135484567\">[latex]f\\left(x\\right)=-{x}^{3}+{x}^{2}+2x[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133092744\">\n<div id=\"fs-id1165133092745\">\n<p id=\"fs-id1165133092746\">[latex]f\\left(x\\right)={x}^{3}-2{x}^{2}-15x[\/latex]<\/p>\n\n<div class=\"textbox shaded\">[reveal-answer q=\"678775\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"678775\"]\n<div id=\"fs-id1165133092745\">\n<p id=\"fs-id1165133092746\"><span id=\"fs-id1165137401080\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140107\/CNX_Precalc_Figure_03_03_224.jpg\" alt=\"Graph of f(x)=-x^3+x^2+2x.\">The[latex]\\,y\\text{-}[\/latex]intercept is[latex]\\,\\left(0, 0\\right).\\,[\/latex] The[latex]\\,x\\text{-}[\/latex]intercepts are[latex]\\,\\left(-3, 0\\right),\\text{ }\\left(0, 0\\right),\\text{ }\\left(5, 0\\right).\\,[\/latex]\n[latex]\\text{As}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to -\\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty [\/latex][\/hidden-answer]<\/span><\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134378662\">\n<div id=\"fs-id1165134378663\">\n<p id=\"fs-id1165134378664\">[latex]f\\left(x\\right)={x}^{3}-0.01x[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137811752\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id1165133111641\">For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or \u20131. There may be more than one correct answer.<\/p>\n\n<div id=\"fs-id1165133448009\">\n<div id=\"fs-id1165133448010\">\n<p id=\"fs-id1165133448012\">The[latex]\\,y-[\/latex]intercept is[latex]\\,\\left(0,-4\\right).\\,[\/latex]The[latex]\\,x-[\/latex]intercepts are[latex]\\,\\left(-2,0\\right),\\,\\left(2,0\\right).\\,[\/latex]Degree is 2.<\/p>\n<p id=\"eip-id1165134540124\">End behavior:[latex]\\,\\text{as}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to \\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty .[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135160156\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135160156\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135160156\"]\n<p id=\"fs-id1165135160157\">[latex]f\\left(x\\right)={x}^{2}-4[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137894256\">\n<div id=\"fs-id1165137894257\">\n<p id=\"fs-id1165137894258\">The[latex]\\,y-[\/latex]intercept is[latex]\\,\\left(0,9\\right).\\,[\/latex]The[latex]\\,x\\text{-}[\/latex]intercepts are[latex]\\,\\left(-3,0\\right),\\,\\left(3,0\\right).\\,[\/latex]Degree is 2.<\/p>\n<p id=\"eip-id1165134566570\">End behavior:[latex]\\,\\text{as}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to -\\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to -\\infty .[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137897907\">\n<div id=\"fs-id1165137897908\">\n<p id=\"fs-id1165137897909\">The[latex]\\,y-[\/latex]intercept is[latex]\\,\\left(0,0\\right).\\,[\/latex]The[latex]\\,x-[\/latex]intercepts are[latex]\\,\\left(0,0\\right),\\,\\left(2,0\\right).\\,[\/latex]Degree is 3.<\/p>\n<p id=\"eip-id1165137749966\">End behavior:[latex]\\,\\text{as}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to -\\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty .[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134389974\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134389974\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134389974\"]\n<p id=\"fs-id1165134389976\">[latex]f\\left(x\\right)={x}^{3}-4{x}^{2}+4x[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137921541\">\n<div id=\"fs-id1165137921542\">\n<p id=\"fs-id1165137921544\">The[latex]\\,y-[\/latex]intercept is[latex]\\,\\left(0,1\\right).\\,[\/latex]The[latex]\\,x-[\/latex]intercept is[latex]\\,\\left(1,0\\right).\\,[\/latex]Degree is 3.<\/p>\n<p id=\"eip-id1165137895228\">End behavior:[latex]\\,\\text{as}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to \\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to -\\infty .[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135536216\">\n<div id=\"fs-id1165135536217\">\n<p id=\"fs-id1165135536218\">The[latex]\\,y-[\/latex]intercept is[latex]\\,\\left(0,1\\right).\\,[\/latex]There is no[latex]\\,x-[\/latex]intercept. Degree is 4.<\/p>\n<p id=\"eip-id1165135434911\">End behavior:[latex]\\,\\text{as}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to \\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty .[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137922537\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137922537\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137922537\"]\n<p id=\"fs-id1165135386380\">[latex]f\\left(x\\right)={x}^{4}+1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134085635\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<p id=\"fs-id1165134085641\">For the following exercises, use the written statements to construct a polynomial function that represents the required information.<\/p>\n\n<div id=\"fs-id1165135581116\">\n<div id=\"fs-id1165135581117\">\n<p id=\"fs-id1165135581118\">An oil slick is expanding as a circle. The radius of the circle is increasing at the rate of 20 meters per day. Express the area of the circle as a function of[latex]\\,d,\\,[\/latex]the number of days elapsed.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137892519\">\n<div id=\"fs-id1165137892520\">\n<p id=\"fs-id1165137892521\">A cube has an edge of 3 feet. The edge is increasing at the rate of 2 feet per minute. Express the volume of the cube as a function of[latex]\\,m,\\,[\/latex]the number of minutes elapsed.<\/p>\n\n<\/div>\n<div id=\"fs-id1165134032299\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134032299\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134032299\"]\n<p id=\"fs-id1165137704612\">[latex]V\\left(m\\right)=8{m}^{3}+36{m}^{2}+54m+27[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137848042\">\n<div id=\"fs-id1165137848043\">\n<p id=\"fs-id1165137848044\">A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased by[latex]\\,x\\,[\/latex]inches and the width increased by twice that amount, express the area of the rectangle as a function of[latex]\\,x.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137891257\">\n<div id=\"fs-id1165137891258\">\n<p id=\"fs-id1165137891259\">An open box is to be constructed by cutting out square corners of [latex]\\,x-[\/latex]inch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function of[latex]\\,x.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135571779\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135571779\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135571779\"]\n<p id=\"fs-id1165135571780\">[latex]V\\left(x\\right)=4{x}^{3}-32{x}^{2}+64x[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135190825\">\n<div id=\"fs-id1165135190826\">\n<p id=\"fs-id1165135190827\">A rectangle is twice as long as it is wide. Squares of side 2 feet are cut out from each corner. Then the sides are folded up to make an open box. Express the volume of the box as a function of the width ([latex]x[\/latex]).<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165137668266\">\n \t<dt>coefficient<\/dt>\n \t<dd id=\"fs-id1165135194915\">a nonzero real number multiplied by a variable raised to an exponent<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135194918\">\n \t<dt>continuous function<\/dt>\n \t<dd id=\"fs-id1165135194921\">a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137832108\">\n \t<dt>degree<\/dt>\n \t<dd id=\"fs-id1165137832112\">the highest power of the variable that occurs in a polynomial<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137832115\">\n \t<dt>end behavior<\/dt>\n \t<dd>the behavior of the graph of a function as the input decreases without bound and increases without bound<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131990658\">\n \t<dt>leading coefficient<\/dt>\n \t<dd id=\"fs-id1165131990661\">the coefficient of the leading term<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\">\n \t<dt>leading term<\/dt>\n \t<dd id=\"fs-id1165132943525\">the term containing the highest power of the variable<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943528\">\n \t<dt>polynomial function<\/dt>\n \t<dd id=\"fs-id1165134297639\">a function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134297646\">\n \t<dt>power function<\/dt>\n \t<dd id=\"fs-id1165135486042\">a function that can be represented in the form[latex]\\,f\\left(x\\right)=k{x}^{p}\\,[\/latex]where[latex]\\,k\\,[\/latex]is a constant, the base is a variable, and the exponent,[latex]\\,p[\/latex], is a constant<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137833929\">\n \t<dt>smooth curve<\/dt>\n \t<dd>a graph with no sharp corners<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\">\n \t<dt>term of a polynomial function<\/dt>\n \t<dd id=\"fs-id1165137644990\">any[latex]\\,{a}_{i}{x}^{i}\\,[\/latex]of a polynomial function in the form[latex]\\,f\\left(x\\right)={a}_{n}{x}^{n}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\">\n \t<dt>turning point<\/dt>\n \t<dd id=\"fs-id1165133085665\">the location at which the graph of a function changes direction<\/dd>\n<\/dl>\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Identify power functions.<\/li>\n<li>Identify end behavior of power functions.<\/li>\n<li>Identify polynomial functions.<\/li>\n<li>Identify the degree and leading coefficient of polynomial functions.<\/li>\n<\/ul>\n<\/div>\n<div id=\"CNX_Precalc_Figure_03_03_001.jpg\" class=\"small aligncenter\">\n<figure style=\"width: 488px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135627\/CNX_Precalc_Figure_03_03_001.jpg\" alt=\"Three birds on a cliff with the sun rising in the background.\" width=\"488\" height=\"366\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1. <\/strong>(credit: Jason Bay, Flickr)<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165134540133\">Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown in <a class=\"autogenerated-content\" href=\"#Table_03_03_01\">(Figure)<\/a>.<\/p>\n<table id=\"Table_03_03_01\" summary=\"..\">\n<tbody>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td>[latex]2009[\/latex]<\/td>\n<td>[latex]2010[\/latex]<\/td>\n<td>[latex]2011[\/latex]<\/td>\n<td>[latex]2012[\/latex]<\/td>\n<td>[latex]2013[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>Bird Population<\/strong><\/td>\n<td>[latex]800[\/latex]<\/td>\n<td>[latex]897[\/latex]<\/td>\n<td>[latex]992[\/latex]<\/td>\n<td>[latex]1,083[\/latex]<\/td>\n<td>[latex]1,169[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137442798\">The population can be estimated using the function[latex]\\,P\\left(t\\right)=-0.3{t}^{3}+97t+800,\\,[\/latex]where[latex]\\,P\\left(t\\right)\\,[\/latex]represents the bird population on the island[latex]\\,t\\,[\/latex]years after 2009. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island. In this section, we will examine functions that we can use to estimate and predict these types of changes.<\/p>\n<div id=\"fs-id1165137540446\" class=\"bc-section section\">\n<h3>Identifying Power Functions<\/h3>\n<p id=\"fs-id1165137570394\">Before we can understand the bird problem, it will be helpful to understand a different type of function. A <strong>power function <\/strong>is a function with a single term that is the product of a real number, a <strong>coefficient,<\/strong> and a variable raised to a fixed real number.<\/p>\n<p id=\"fs-id1165135320417\">As an example, consider functions for area or volume. The function for the <span class=\"no-emphasis\">area of a circle<\/span> with radius[latex]\\,r\\,[\/latex]<br \/>\nis<\/p>\n<div id=\"eip-544\" class=\"unnumbered aligncenter\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/div>\n<p id=\"fs-id1165135191346\">and the function for the <span class=\"no-emphasis\">volume of a sphere<\/span> with radius[latex]\\,r\\,[\/latex]<br \/>\nis<\/p>\n<div id=\"eip-640\" class=\"unnumbered aligncenter\">[latex]V\\left(r\\right)=\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/div>\n<p id=\"fs-id1165137579058\">Both of these are examples of power functions because they consist of a coefficient,[latex]\\,\\pi \\,[\/latex]or[latex]\\,\\frac{4}{3}\\pi ,\\,[\/latex]multiplied by a variable[latex]\\,r\\,[\/latex]raised to a power.<\/p>\n<div id=\"fs-id1165135356525\" class=\"textbox key-takeaways\">\n<h3>Power Function<\/h3>\n<p id=\"fs-id1165137771947\">A power function is a function that can be represented in the form<\/p>\n<div id=\"eip-826\" class=\"unnumbered aligncenter\">[latex]f\\left(x\\right)=k{x}^{p}[\/latex]<\/div>\n<p id=\"eip-id1165135584093\">where[latex]\\,k\\,[\/latex]<br \/>\nand[latex]\\,p\\,[\/latex]are real numbers, and[latex]\\,k\\,[\/latex]<br \/>\nis known as the coefficient.<\/p>\n<\/div>\n<div id=\"fs-id1165137661479\" class=\"precalculus qa textbox shaded\">\n<p id=\"fs-id1165137582131\"><strong>Is[latex]\\,f\\left(x\\right)={2}^{x}\\,[\/latex]a power function?<\/strong><\/p>\n<p id=\"fs-id1165137598469\"><em>No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function.<\/em><\/p>\n<\/div>\n<div id=\"Example_03_03_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137745179\">\n<div id=\"fs-id1165137742710\">\n<h3>Identifying Power Functions<\/h3>\n<p id=\"fs-id1165137824370\">Which of the following functions are power functions?<\/p>\n<p>[latex]\\begin{array}{cccc}\\hfill f\\left(x\\right)& =& 1\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Constant function}\\hfill \\\\ \\hfill f\\left(x\\right)& =& x\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Identify function}\\hfill \\\\ \\hfill f\\left(x\\right)& =& {x}^{2}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Quadratic function}\\hfill \\\\ \\hfill f\\left(x\\right)& =& {x}^{3}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Cubic function}\\hfill \\\\ \\hfill f\\left(x\\right)& =& \\frac{1}{x}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Reciprocal function}\\hfill \\\\ \\hfill f\\left(x\\right)& =& \\frac{1}{{x}^{2}}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Reciprocal squared function}\\hfill \\\\ \\hfill f\\left(x\\right)& =& \\sqrt{x}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Square root function}\\hfill \\\\ \\hfill f\\left(x\\right)& =& \\sqrt[3]{x}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Cube root function}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137422823\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137843987\">All of the listed functions are power functions.<\/p>\n<p id=\"fs-id1165135533093\">The constant and identity functions are power functions because they can be written as[latex]\\,f\\left(x\\right)={x}^{0}\\,[\/latex]and[latex]\\,f\\left(x\\right)={x}^{1}\\,[\/latex]respectively.<\/p>\n<p id=\"fs-id1165137411464\">The quadratic and cubic functions are power functions with whole number powers[latex]\\,f\\left(x\\right)={x}^{2}\\,[\/latex]and[latex]\\,f\\left(x\\right)={x}^{3}.[\/latex]<\/p>\n<p id=\"fs-id1165137475956\">The <span class=\"no-emphasis\">reciprocal<\/span> and reciprocal squared functions are power functions with negative whole number powers because they can be written as[latex]\\,f\\left(x\\right)={x}^{-1}\\,[\/latex]and[latex]\\,f\\left(x\\right)={x}^{-2}.[\/latex]<\/p>\n<p id=\"fs-id1165135704907\">The square and <span class=\"no-emphasis\">cube root<\/span> functions are power functions with fractional powers because they can be written as[latex]\\,f\\left(x\\right)={x}^{\\frac{1}{2}}\\,[\/latex]or[latex]\\,f\\left(x\\right)={x}^{\\frac{1}{3}}.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137660222\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_03_01\">\n<div id=\"fs-id1165137475224\">\n<p id=\"fs-id1165137475225\">Which functions are power functions?<\/p>\n<div><\/div>\n<p id=\"fs-id1165137824385\">[latex]\\begin{array}{ccc}\\hfill f\\left(x\\right)& =& 2x\\cdot 4{x}^{3}\\hfill \\\\ \\hfill g\\left(x\\right)& =& -{x}^{5}+5{x}^{3}\\hfill \\\\ \\hfill h\\left(x\\right)& =& \\frac{2{x}^{5}-1}{3{x}^{2}+4}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134312227\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134094624\">[latex]f\\left(x\\right)\\,[\/latex]<br \/>\nis a power function because it can be written as[latex]\\,f\\left(x\\right)=8{x}^{5}.\\,[\/latex]<br \/>\nThe other functions are not power functions.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134269023\" class=\"bc-section section\">\n<h3>Identifying End Behavior of Power Functions<\/h3>\n<p id=\"fs-id1165135436540\"><a class=\"autogenerated-content\" href=\"#Figure_03_03_002\">(Figure)<\/a> shows the graphs of[latex]\\,f\\left(x\\right)={x}^{2},\\,g\\left(x\\right)={x}^{4}\\,[\/latex]and[latex]\\,h\\left(x\\right)={x}^{6},\\,[\/latex]which are all power functions with even, whole-number powers. Notice that these graphs have similar shapes, very much like that of the quadratic function in the toolkit. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.<\/p>\n<div id=\"Figure_03_03_002\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135645\/CNX_Precalc_Figure_03_03_002.jpg\" alt=\"Graph of three functions, h(x)=x^2 in green, g(x)=x^4 in orange, and f(x)=x^6 in blue.\" width=\"487\" height=\"253\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2. <\/strong>Even-power functions<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137911555\">To describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbol[latex]\\,\\infty \\,[\/latex]for positive infinity and[latex]\\,\\mathrm{-\\infty }\\,[\/latex]for negative infinity. When we say that \u201c[latex]x\\,[\/latex]approaches infinity,\u201d which can be symbolically written as[latex]\\,x\\to \\infty ,\\,[\/latex]we are describing a behavior; we are saying that[latex]\\,x\\,[\/latex]is increasing without bound.<\/p>\n<p id=\"fs-id1165137658268\">With the positive even-power function, as the input increases or decreases without bound, the output values become very large, positive numbers. Equivalently, we could describe this behavior by saying that as[latex]\\,x\\,[\/latex]approaches positive or negative infinity, the[latex]\\,f\\left(x\\right)\\,[\/latex]values increase without bound. In symbolic form, we could write<\/p>\n<div id=\"eip-742\" class=\"unnumbered aligncenter\">[latex]\\text{as }x\\to \u00b1\\infty , f\\left(x\\right)\\to \\infty[\/latex]<\/div>\n<p id=\"fs-id1165137533222\"><a class=\"autogenerated-content\" href=\"#Figure_03_03_003\">(Figure)<\/a> shows the graphs of[latex]\\,f\\left(x\\right)={x}^{3},\\,g\\left(x\\right)={x}^{5},[\/latex]and[latex]\\,h\\left(x\\right)={x}^{7},[\/latex]which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function in the toolkit. Again, as the power increases, the graphs flatten near the origin and become steeper away from the origin.<\/p>\n<div id=\"Figure_03_03_003\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135711\/CNX_Precalc_Figure_03_03_003.jpg\" alt=\"Graph of three functions, f(x)=x^3 in green, g(x)=x^5 in orange, and h(x)=x^7 in blue.\" width=\"487\" height=\"366\" \/><figcaption class=\"wp-caption-text\"><strong> Figure 3.<\/strong> Odd-power functions<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137730237\">These examples illustrate that functions of the form[latex]\\,f\\left(x\\right)={x}^{n}\\,[\/latex]reveal symmetry of one kind or another. First, in <a class=\"autogenerated-content\" href=\"#Figure_03_03_002\">(Figure)<\/a> we see that even functions of the form[latex]\\,f\\left(x\\right)={x}^{n}\\text{, }n\\,[\/latex]even, are symmetric about the[latex]\\,y\\text{-}[\/latex]axis. In <a class=\"autogenerated-content\" href=\"#Figure_03_03_003\">(Figure)<\/a> we see that odd functions of the form[latex]\\,f\\left(x\\right)={x}^{n}\\text{, }n\\,[\/latex] odd, are symmetric about the origin.<\/p>\n<p id=\"fs-id1165137812578\">For these odd power functions, as[latex]\\,x\\,[\/latex] approaches negative infinity,[latex]\\,f\\left(x\\right)\\,[\/latex] decreases without bound. As[latex]\\,x\\,[\/latex] approaches positive infinity,[latex]\\,f\\left(x\\right)\\,[\/latex] increases without bound. In symbolic form we write<\/p>\n<div id=\"eip-77\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{as} x\\to -\\infty , f\\left(x\\right)\\to -\\infty \\\\ \\text{as} x\\to \\infty , f\\left(x\\right)\\to \\infty \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137425284\">The behavior of the graph of a function as the input values get very small ([latex]\\,x\\to -\\infty \\,[\/latex]) and get very large ([latex]\\,x\\to \\infty \\,[\/latex]) is referred to as the end behavior of the function. We can use words or symbols to describe end behavior.<\/p>\n<p id=\"fs-id1165137433212\"><a class=\"autogenerated-content\" href=\"#Figure_03_03_004abcd\">(Figure)<\/a> shows the end behavior of power functions in the form[latex]\\,f\\left(x\\right)=k{x}^{n}\\,[\/latex]where[latex]\\,n\\,[\/latex]is a non-negative integer depending on the power and the constant.<\/p>\n<div id=\"Figure_03_03_004abcd\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135713\/CNX_Precalc_Figure_03_03_004abcd.jpg\" alt=\"Graph of an even-powered function with a positive constant. As x goes to negative infinity, the function goes to positive infinity; as x goes to positive infinity, the function goes to positive infinity. Graph of an odd-powered function with a positive constant. As x goes to negative infinity, the function goes to positive infinity; as x goes to positive infinity, the function goes to negative infinity. Graph of an even-powered function with a negative constant. As x goes to negative infinity, the function goes to negative infinity; as x goes to positive infinity, the function goes to negative infinity. Graph of an odd-powered function with a negative constant. As x goes to negative infinity, the function goes to negative infinity; as x goes to positive infinity, the function goes to negative infinity.\" width=\"731\" height=\"734\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id1165135161436\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137415258\"><strong>Given a power function[latex]\\,f\\left(x\\right)=k{x}^{n}\\,[\/latex]where<\/strong>[latex]\\,n\\,[\/latex]<strong>is a non-negative integer, identify the end behavior.<\/strong><\/p>\n<ol id=\"fs-id1165137409522\" type=\"1\">\n<li>Determine whether the power is even or odd.<\/li>\n<li>Determine whether the constant is positive or negative.<\/li>\n<li>Use <a class=\"autogenerated-content\" href=\"#Figure_03_03_004abcd\">(Figure)<\/a> to identify the end behavior.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_03_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137923491\">\n<div id=\"fs-id1165137599768\">\n<h3>Identifying the End Behavior of a Power Function<\/h3>\n<p id=\"fs-id1165137644554\">Describe the end behavior of the graph of[latex]\\,f\\left(x\\right)={x}^{8}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135169237\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137502379\">The coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). As[latex]\\,x\\,[\/latex]approaches infinity, the output (value of[latex]\\,f\\left(x\\right)\\,[\/latex]) increases without bound. We write as[latex]\\,x\\to \\infty ,f\\left(x\\right)\\to \\infty .\\,[\/latex]As[latex]\\,x\\,[\/latex]approaches negative infinity, the output increases without bound. In symbolic form, as[latex]x\\to -\\infty , f\\left(x\\right)\\to \\infty .[\/latex] We can graphically represent the function as shown in <a class=\"autogenerated-content\" href=\"#Figure_03_03_008\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_03_03_008\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135720\/CNX_Precalc_Figure_03_03_008.jpg\" alt=\"Graph of f(x)=x^8.\" width=\"487\" height=\"330\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 5.<\/strong><\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137535914\">\n<div id=\"fs-id1165137811997\">\n<h3>Identifying the End Behavior of a Power Function.<\/h3>\n<p id=\"fs-id1165137453217\">Describe the end behavior of the graph of[latex]\\,f\\left(x\\right)=-{x}^{9}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137722696\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137409513\">The exponent of the power function is 9 (an odd number). Because the coefficient is[latex]\\,\u20131\\,[\/latex](negative), the graph is the reflection about the[latex]\\,x\\text{-}[\/latex]axis of the graph of[latex]\\,f\\left(x\\right)={x}^{9}.\\,[\/latex]<a class=\"autogenerated-content\" href=\"#Figure_03_03_009\">(Figure)<\/a> shows that as[latex]\\,x\\,[\/latex]approaches infinity, the output decreases without bound. As[latex]\\,x\\,[\/latex]approaches negative infinity, the output increases without bound. In symbolic form, we would write<\/p>\n<div id=\"eip-id1165134384400\" class=\"unnumbered\">[latex]\\begin{array}{l}\\text{as} x\\to -\\infty , f\\left(x\\right)\\to \\infty \\\\ \\text{as} x\\to \\infty , f\\left(x\\right)\\to -\\infty \\end{array}[\/latex]<\/div>\n<div id=\"Figure_03_03_009\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135741\/CNX_Precalc_Figure_03_03_009.jpg\" alt=\"Graph of f(x)=-x^9.\" width=\"487\" height=\"667\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 6.<\/strong><\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135259565\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137548471\">We can check our work by using the table feature on a graphing utility.<\/p>\n<table id=\"Table_03_03_03\" summary=\"..\">\n<thead>\n<tr>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]f\\left(x\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>\u201310<\/td>\n<td>1,000,000,000<\/td>\n<\/tr>\n<tr>\n<td>\u20135<\/td>\n<td>1,953,125<\/td>\n<\/tr>\n<tr>\n<td>0<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>\u20131,953,125<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>\u20131,000,000,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137644426\">We can see from <a class=\"autogenerated-content\" href=\"#Table_03_03_03\">(Figure)<\/a> that, when we substitute very small values for[latex]\\,x,\\,[\/latex]the output is very large, and when we substitute very large values for[latex]\\,x,\\,[\/latex]the output is very small (meaning that it is a very large negative value).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137626838\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165137734867\">\n<p id=\"fs-id1165137734868\">Describe in words and symbols the end behavior of[latex]\\,f\\left(x\\right)=-5{x}^{4}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137647550\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137647551\">As[latex]\\,x\\,[\/latex]approaches positive or negative infinity,[latex]\\,f\\left(x\\right)\\,[\/latex]decreases without bound: as[latex]\\,x\\to \u00b1\\infty , f\\left(x\\right)\\to -\\infty \\,[\/latex]because of the negative coefficient.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134069294\" class=\"bc-section section\">\n<h3>Identifying Polynomial Functions<\/h3>\n<p id=\"fs-id1165135689465\">An oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius[latex]\\,r\\,[\/latex]<br \/>\nof the spill depends on the number of weeks[latex]\\,w\\,[\/latex]<br \/>\nthat have passed. This relationship is linear.<\/p>\n<div class=\"unnumbered\">[latex]r\\left(w\\right)=24+8w[\/latex]<\/div>\n<p id=\"fs-id1165133432974\">We can combine this with the formula for the area[latex]\\,A\\,[\/latex]<br \/>\nof a circle.<\/p>\n<div id=\"eip-731\" class=\"unnumbered aligncenter\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/div>\n<p id=\"fs-id1165137704887\">Composing these functions gives a formula for the area in terms of weeks.<\/p>\n<div id=\"eip-645\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill A\\left(w\\right)& =& A\\left(r\\left(w\\right)\\right)\\hfill \\\\ & =& A\\left(24+8w\\right)\\hfill \\\\ & =& \\pi {\\left(24+8w\\right)}^{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137835475\">Multiplying gives the formula.<\/p>\n<div id=\"eip-290\" class=\"unnumbered aligncenter\">[latex]A\\left(w\\right)=576\\pi +384\\pi w+64\\pi {w}^{2}[\/latex]<\/div>\n<p id=\"fs-id1165135205726\">This formula is an example of a <strong>polynomial function<\/strong>. A polynomial function consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.<\/p>\n<div id=\"fs-id1165137715427\" class=\"textbox key-takeaways\">\n<h3>Polynomial Functions<\/h3>\n<p id=\"fs-id1165137823247\">Let[latex]\\,n\\,[\/latex]<br \/>\nbe a non-negative integer. A polynomial function is a function that can be written in the form<\/p>\n<div id=\"fs-id1165131937978\">[latex]f\\left(x\\right)={a}_{n}{x}^{n}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/div>\n<p id=\"eip-id1165137832690\">This is called the general form of a polynomial function. Each[latex]\\,{a}_{i}\\,[\/latex]<br \/>\nis a coefficient and can be any real number, but<br \/>\n[latex]\\,{a}_{n}\\,[\/latex]cannot = 0. Each expression[latex]\\,{a}_{i}{x}^{i}\\,[\/latex]<br \/>\nis a term of a polynomial function.<\/p>\n<\/div>\n<div id=\"Example_03_03_04\" class=\"textbox examples\">\n<div id=\"fs-id1165137817691\">\n<div id=\"fs-id1165137817693\">\n<h3>Identifying Polynomial Functions<\/h3>\n<p id=\"fs-id1165135262000\">Which of the following are polynomial functions?<\/p>\n<div><\/div>\n<div id=\"eip-id1165134474011\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill f\\left(x\\right)& =& 2{x}^{3}\\cdot 3x+4\\hfill \\\\ \\hfill g\\left(x\\right)& =& -x\\left({x}^{2}-4\\right)\\hfill \\\\ \\hfill h\\left(x\\right)& =& 5\\sqrt{x+2}\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165134221783\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134094645\">The first two functions are examples of polynomial functions because they can be written in the form[latex]\\,f\\left(x\\right)={a}_{n}{x}^{n}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0},\\,[\/latex]<br \/>\nwhere the powers are non-negative integers and the coefficients are real numbers.<\/p>\n<ul id=\"fs-id1165137864157\">\n<li>[latex]f\\left(x\\right)\\,[\/latex]<br \/>\ncan be written as[latex]\\,f\\left(x\\right)=6{x}^{4}+4.[\/latex]<\/li>\n<li>[latex]g\\left(x\\right)\\,[\/latex]<br \/>\ncan be written as[latex]\\,g\\left(x\\right)=-{x}^{3}+4x.[\/latex]<\/li>\n<li>[latex]h\\left(x\\right)\\,[\/latex]<br \/>\ncannot be written in this form and is therefore not a polynomial function.<\/li>\n<\/ul>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135508329\" class=\"bc-section section\">\n<h3>Identifying the Degree and Leading Coefficient of a Polynomial Function<\/h3>\n<p id=\"fs-id1165137831216\">Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order of power, or in general form. The degree of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form. The leading term is the term containing the highest power of the variable, or the term with the highest degree. The leading coefficient is the coefficient of the leading term.<\/p>\n<div id=\"fs-id1165135193124\" class=\"textbox key-takeaways\">\n<h3>Terminology of Polynomial Functions<\/h3>\n<p id=\"fs-id1165137921667\">We often rearrange polynomials so that the powers are descending.<\/p>\n<p><span id=\"fs-id1165137406148\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135744\/CNX_Precalc_Figure_03_03_010n.jpg\" alt=\"Diagram to show what the components of the leading term in a function are. The leading coefficient is a_n and the degree of the variable is the exponent in x^n. Both the leading coefficient and highest degree variable make up the leading term. So the function looks like f(x)=a_nx^n +\u2026+a_2x^2+a_1x+a_0.\" \/><\/span><\/p>\n<p id=\"fs-id1165137482568\">When a polynomial is written in this way, we say that it is in general form.<\/p>\n<\/div>\n<div id=\"fs-id1165134031372\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137803898\"><strong>Given a polynomial function, identify the degree and leading coefficient.<\/strong><\/p>\n<ol id=\"fs-id1165135587816\" type=\"1\">\n<li>Find the highest power of[latex]\\,x\\,[\/latex]<br \/>\nto determine the degree function.<\/li>\n<li>Identify the term containing the highest power of[latex]\\,x\\,[\/latex]<br \/>\nto find the leading term.<\/li>\n<li>Identify the coefficient of the leading term.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_03_05\" class=\"textbox examples\">\n<div id=\"fs-id1165137401820\">\n<div id=\"fs-id1165137862379\">\n<h3>Identifying the Degree and Leading Coefficient of a Polynomial Function<\/h3>\n<p id=\"fs-id1165137435372\">Identify the degree, leading term, and leading coefficient of the following polynomial functions.<\/p>\n<div id=\"eip-id1165134242117\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill f\\left(x\\right)& =& 3+2{x}^{2}-4{x}^{3}\\hfill \\\\ \\hfill g\\left(t\\right)& =& 5{t}^{5}-2{t}^{3}+7t\\hfill \\\\ h\\left(p\\right)\\hfill & =& 6p-{p}^{3}-2\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135527012\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137722510\">For the function[latex]\\,f\\left(x\\right),\\,[\/latex]the highest power of[latex]\\,x\\,[\/latex]is 3, so the degree is 3. The leading term is the term containing that degree,[latex]\\,-4{x}^{3}.\\,[\/latex]The leading coefficient is the coefficient of that term,[latex]\\,-4.[\/latex]<\/p>\n<p id=\"fs-id1165135457771\">For the function[latex]\\,g\\left(t\\right),\\,[\/latex]the highest power of[latex]\\,t\\,[\/latex]is[latex]\\,5,\\,[\/latex]so the degree is[latex]\\,5.\\,[\/latex]The leading term is the term containing that degree,[latex]\\,5{t}^{5}.\\,[\/latex]The leading coefficient is the coefficient of that term,[latex]\\,5.[\/latex]<\/p>\n<p id=\"fs-id1165135503949\">For the function[latex]\\,h\\left(p\\right),\\,[\/latex]the highest power of[latex]\\,p\\,[\/latex]is[latex]\\,3,\\,[\/latex]so the degree is[latex]\\,3.\\,[\/latex]The leading term is the term containing that degree,[latex]\\,-{p}^{3}.\\,[\/latex]The leading coefficient is the coefficient of that term,[latex]\\,-1.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137534935\" class=\"textbox tryit\">\n<div id=\"ti_03_03_03\">\n<div id=\"fs-id1165137424483\">\n<p id=\"fs-id1165137424484\">Identify the degree, leading term, and leading coefficient of the polynomial[latex]\\,f\\left(x\\right)=4{x}^{2}-{x}^{6}+2x-6.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135701674\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135701675\">The degree is 6. The leading term is[latex]\\,-{x}^{6}.\\,[\/latex]The leading coefficient is[latex]\\,-1.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137702213\" class=\"bc-section section\">\n<h4>Identifying End Behavior of Polynomial Functions<\/h4>\n<p id=\"fs-id1165137601421\">Knowing the degree of a polynomial function is useful in helping us predict its end behavior. To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as[latex]\\,x\\,[\/latex] gets very large or very small, so its behavior will dominate the graph. For any polynomial, the end behavior of the polynomial will match the end behavior of the power function consisting of the leading term. See <a class=\"autogenerated-content\" href=\"#Table_03_03_04\">(Figure)<\/a>.<\/p>\n<table id=\"Table_03_03_04\" summary=\"..\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr>\n<th>Polynomial Function<\/th>\n<th>Leading Term<\/th>\n<th>Graph of Polynomial Function<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]f\\left(x\\right)=5{x}^{4}+2{x}^{3}-x-4[\/latex]<\/td>\n<td>[latex]5{x}^{4}[\/latex]<\/td>\n<td><span id=\"fs-id1165137768814\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135746\/CNX_Precalc_Figure_03_03_011.jpg\" alt=\"Graph of f(x)=5x^4+2x^3-x-4.\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>[latex]f\\left(x\\right)=-2{x}^{6}-{x}^{5}+3{x}^{4}+{x}^{3}[\/latex]<\/td>\n<td>[latex]-2{x}^{6}[\/latex]<\/td>\n<td><span id=\"fs-id1165137714206\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135752\/CNX_Precalc_Figure_03_03_012.jpg\" alt=\"Graph of f(x)=-2x^6-x^5+3x^4+x^3.\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>[latex]f\\left(x\\right)=3{x}^{5}-4{x}^{4}+2{x}^{2}+1[\/latex]<\/td>\n<td>[latex]3{x}^{5}[\/latex]<\/td>\n<td><span id=\"fs-id1165137540879\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135755\/CNX_Precalc_Figure_03_03_013.jpg\" alt=\"Graph of f(x)=3x^5-4x^4+2x^2+1.\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td>[latex]f\\left(x\\right)=-6{x}^{3}+7{x}^{2}+3x+1[\/latex]<\/td>\n<td>[latex]-6{x}^{3}[\/latex]<\/td>\n<td><span id=\"fs-id1165137600670\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135801\/CNX_Precalc_Figure_03_03_014.jpg\" alt=\"Graph of f(x)=-6x^3+7x^2+3x+1.\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"Example_03_03_06\" class=\"textbox examples\">\n<div id=\"fs-id1165137452413\">\n<div id=\"fs-id1165137452415\">\n<h3>Identifying End Behavior and Degree of a Polynomial Function<\/h3>\n<p id=\"fs-id1165137831279\">Describe the end behavior and determine a possible degree of the polynomial function in <a class=\"autogenerated-content\" href=\"#Figure_03_03_015\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_03_03_015\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135812\/CNX_Precalc_Figure_03_03_015.jpg\" alt=\"Graph of an odd-degree polynomial.\" width=\"487\" height=\"443\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 7.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135251309\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135251312\">As the input values[latex]\\,x\\,[\/latex]<br \/>\nget very large, the output values[latex]\\,f\\left(x\\right)\\,[\/latex]increase without bound. As the input values[latex]\\,x\\,[\/latex]<br \/>\nget very small, the output values[latex]\\,f\\left(x\\right)\\,[\/latex]decrease without bound. We can describe the end behavior symbolically by writing<\/p>\n<div id=\"eip-id1165137778911\" class=\"unnumbered\">[latex]\\begin{array}{l}\\text{as} x\\to -\\infty , f\\left(x\\right)\\to -\\infty \\\\ \\text{as} x\\to \\infty , f\\left(x\\right)\\to \\infty \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137454991\">In words, we could say that as[latex]\\,x\\,[\/latex]values approach infinity, the function values approach infinity, and as[latex]\\,x\\,[\/latex]values approach negative infinity, the function values approach negative infinity.<\/p>\n<p id=\"fs-id1165134113949\">We can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polynomial creating this graph must be odd and the leading coefficient must be positive.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137470875\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_03_04\">\n<div id=\"fs-id1165137732301\">\n<p id=\"fs-id1165135460938\">Describe the end behavior, and determine a possible degree of the polynomial function in <a class=\"autogenerated-content\" href=\"#Figure_03_03_016\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_03_03_016\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135827\/CNX_Precalc_Figure_03_03_016n.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"487\" height=\"440\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 8.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134047710\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134047711\">As[latex]\\,x\\to \\infty , f\\left(x\\right)\\to -\\infty ; as x\\to -\\infty , f\\left(x\\right)\\to -\\infty .\\,[\/latex]It has the shape of an even degree power function with a negative coefficient.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_07\" class=\"textbox examples\">\n<div id=\"fs-id1165137470361\">\n<div id=\"fs-id1165137470363\">\n<h3>Identifying End Behavior and Degree of a Polynomial Function<\/h3>\n<p id=\"fs-id1165132011287\">Given the function[latex]\\,f\\left(x\\right)=-3{x}^{2}\\left(x-1\\right)\\left(x+4\\right),\\,[\/latex]express the function as a polynomial in general form, and determine the leading term, degree, and end behavior of the function.<\/p>\n<\/div>\n<div id=\"fs-id1165137401107\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137401109\">Obtain the general form by expanding the given expression for[latex]\\,f\\left(x\\right).[\/latex]<\/p>\n<div id=\"eip-id1165132051075\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill f\\left(x\\right)& =& -3{x}^{2}\\left(x-1\\right)\\left(x+4\\right)\\hfill \\\\ & =& -3{x}^{2}\\left({x}^{2}+3x-4\\right)\\hfill \\\\ & =& -3{x}^{4}-9{x}^{3}+12{x}^{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137634030\">The general form is[latex]\\,f\\left(x\\right)=-3{x}^{4}-9{x}^{3}+12{x}^{2}.\\,[\/latex]<br \/>\nThe leading term is[latex]\\,-3{x}^{4};\\,[\/latex]<br \/>\ntherefore, the degree of the polynomial is 4. The degree is even (4) and the leading coefficient is negative (\u20133), so the end behavior is<\/p>\n<div id=\"eip-id1165133007607\" class=\"unnumbered\">[latex]\\begin{array}{l}\\text{as} x\\to -\\infty , f\\left(x\\right)\\to -\\infty \\\\ \\text{as} x\\to \\infty , f\\left(x\\right)\\to -\\infty \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_03_05\">\n<div id=\"fs-id1165137722131\">\n<p id=\"fs-id1165137416652\">Given the function[latex]\\,f\\left(x\\right)=0.2\\left(x-2\\right)\\left(x+1\\right)\\left(x-5\\right),\\,[\/latex]express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.<\/p>\n<\/div>\n<div id=\"fs-id1165135409431\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137749856\">The leading term is[latex]\\,0.2{x}^{3},\\,[\/latex]so it is a degree 3 polynomial. As[latex]\\,x\\,[\/latex]approaches positive infinity,[latex]\\,f\\left(x\\right)\\,[\/latex]increases without bound; as[latex]\\,x\\,[\/latex]approaches negative infinity,[latex]\\,f\\left(x\\right)\\,[\/latex]decreases without bound.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137735781\" class=\"bc-section section\">\n<h4>Identifying Local Behavior of Polynomial Functions<\/h4>\n<p id=\"fs-id1165134054039\">In addition to the end behavior of polynomial functions, we are also interested in what happens in the \u201cmiddle\u201d of the function. In particular, we are interested in locations where graph behavior changes. A <strong>turning point <\/strong>is a point at which the function values change from increasing to decreasing or decreasing to increasing.<\/p>\n<p id=\"fs-id1165137417044\">We are also interested in the intercepts. As with all functions, the <em>y-<\/em>intercept is the point at which the graph intersects the vertical axis. The point corresponds to the coordinate pair in which the input value is zero. Because a polynomial is a function, only one output value corresponds to each input value so there can be only one <em>y-<\/em>intercept[latex]\\,\\left(0,{a}_{0}\\right).\\,[\/latex]The <em>x-<\/em>intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one <em>x-<\/em>intercept. See <a class=\"autogenerated-content\" href=\"#Figure_03_03_017\">(Figure)<\/a><strong>.<\/strong><\/p>\n<div id=\"Figure_03_03_017\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135837\/CNX_Precalc_Figure_03_03_017.jpg\" alt=\"\" width=\"731\" height=\"629\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 9.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Intercepts and Turning Points of Polynomial Functions<\/h3>\n<p id=\"fs-id1165137638552\">A turning point of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. The <em>y-<\/em>intercept is the point at which the function has an input value of zero. The <em>x<\/em>-intercepts are the points at which the output value is zero.<\/p>\n<\/div>\n<div id=\"fs-id1165137766902\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137645233\"><strong>Given a polynomial function, determine the intercepts.<\/strong><\/p>\n<ol id=\"fs-id1165137571388\" type=\"1\">\n<li>Determine the <em>y-<\/em>intercept by setting [latex]\\,x=0\\,[\/latex] and finding the corresponding output value.<\/li>\n<li>Determine the <em>x<\/em>-intercepts by solving for the input values that yield an output value of zero.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_03_03_08\" class=\"textbox examples\">\n<div id=\"fs-id1165137435581\">\n<div id=\"fs-id1165137803210\">\n<h3>Determining the Intercepts of a Polynomial Function<\/h3>\n<p id=\"fs-id1165137441767\">Given the polynomial function[latex]\\,f\\left(x\\right)=\\left(x-2\\right)\\left(x+1\\right)\\left(x-4\\right),\\,[\/latex]written in factored form for your convenience, determine the <em>y<\/em>&#8211; and <em>x<\/em>-intercepts.<\/p>\n<\/div>\n<div id=\"fs-id1165135251466\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135251468\">The <em>y-<\/em>intercept occurs when the input is zero so substitute 0 for[latex]\\,x.[\/latex]<\/p>\n<div id=\"eip-id1165133032876\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill f\\left(0\\right)& =& {\\left(0\\right)}^{4}-4{\\left(0\\right)}^{2}-45\\hfill \\\\ & =& -45\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135689436\">The <em>y-<\/em>intercept is (0, 8).<\/p>\n<p id=\"fs-id1165137863224\">The <em>x<\/em>-intercepts occur when the output is zero.<\/p>\n<div id=\"eip-id1165134380311\" class=\"unnumbered\">[latex]0=\\left(x-2\\right)\\left(x+1\\right)\\left(x-4\\right)[\/latex]<\/div>\n<div id=\"fs-id2036938\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccccccccccc}\\hfill x-2& =& 0\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{or}\\phantom{\\rule{2em}{0ex}}& \\hfill x+1& =& 0\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{or}\\phantom{\\rule{2em}{0ex}}& \\hfill x-4& =& 0\\hfill \\\\ \\hfill x& =& 2\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{or}\\phantom{\\rule{2em}{0ex}}& \\hfill x& =& -1\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{or}\\phantom{\\rule{2em}{0ex}}& \\hfill x& =& 4\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135316178\">The <em>x<\/em>-intercepts are[latex]\\,\\left(2,0\\right),\\left(\u20131,0\\right),\\,[\/latex]and[latex]\\,\\left(4,0\\right).[\/latex]<\/p>\n<p id=\"fs-id1165134380385\">We can see these intercepts on the graph of the function shown in <a class=\"autogenerated-content\" href=\"#Figure_03_03_018\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_03_03_018\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135844\/CNX_Precalc_Figure_03_03_018.jpg\" alt=\"Graph of f(x)=(x-2)(x+1)(x-4), which labels all the intercepts.\" width=\"487\" height=\"630\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 10.<\/strong><\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_09\" class=\"textbox examples\">\n<div id=\"fs-id1165137834894\">\n<div id=\"fs-id1165137834896\">\n<h3>Determining the Intercepts of a Polynomial Function with Factoring<\/h3>\n<p>Given the polynomial function[latex]\\,f\\left(x\\right)={x}^{4}-4{x}^{2}-45,\\,[\/latex]determine the <em>y<\/em>&#8211; and <em>x<\/em>-intercepts.<\/p>\n<\/div>\n<div id=\"fs-id1165137634473\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137634475\">The <em>y-<\/em>intercept occurs when the input is zero.<\/p>\n<div id=\"eip-id1165132943488\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill f\\left(0\\right)& =& {\\left(0\\right)}^{4}-4{\\left(0\\right)}^{2}-45\\hfill \\\\ & =& -45\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135653967\">The <em>y-<\/em>intercept is[latex]\\,\\left(0,-45\\right).[\/latex]<\/p>\n<p id=\"fs-id1165135152099\">The <em>x<\/em>-intercepts occur when the output is zero. To determine when the output is zero, we will need to factor the polynomial.<\/p>\n<p>[latex]\\begin{array}{ccc}\\hfill f\\left(x\\right)& =& {x}^{4}-4{x}^{2}-45\\hfill \\\\ & =& \\left({x}^{2}-9\\right)\\left({x}^{2}+5\\right)\\hfill \\\\ & =& \\left(x-3\\right)\\left(x+3\\right)\\left({x}^{2}+5\\right)\\hfill \\end{array}[\/latex]<br \/>\n[latex]\\phantom{\\rule{2em}{0ex}}0=\\left(x-3\\right)\\left(x+3\\right)\\left({x}^{2}+5\\right)[\/latex]<\/p>\n<p>[latex]\\begin{array}{ccccccccc}\\hfill x-3& =& 0\\hfill & \\text{or}& \\hfill x+3& =& 0\\hfill &\\text{or}& {x}^{2}+5=0\\\\ \\hfill x& =& 3\\hfill & \\text{or} & \\hfill x& =& -3\\hfill & \\text{or}& (\\text{no real solution)}\\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165135436471\">The <em>x<\/em>-intercepts are[latex]\\,\\left(3,0\\right)\\,[\/latex]and[latex]\\,\\left(\u20133,0\\right).[\/latex]<\/p>\n<p>We can see these intercepts on the graph of the function shown in <a class=\"autogenerated-content\" href=\"#Figure_03_03_019\">(Figure)<\/a>. We can see that the function is even because[latex]\\,f\\left(x\\right)=f\\left(-x\\right).[\/latex]<\/p>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135849\/CNX_Precalc_Figure_03_03_019.jpg\" alt=\"Graph of f(x)=x^4-4x^2-45, which labels all the intercepts at (-3, 0), (3, 0), and (0, -45).\" width=\"487\" height=\"426\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 11.<\/strong><\/figcaption><\/figure>\n<\/details>\n<p><span id=\"fs-id1165137803348\"><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137749604\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_03_06\">\n<div id=\"fs-id1165137405243\">\n<p id=\"fs-id1165137405244\">Given the polynomial function[latex]\\,f\\left(x\\right)=2{x}^{3}-6{x}^{2}-20x,\\,[\/latex]determine the <em>y<\/em>&#8211; and <em>x<\/em>-intercepts.<\/p>\n<\/div>\n<div id=\"fs-id1165137762370\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137762371\"><em>y<\/em>-intercept[latex]\\,\\left(0,0\\right);\\,[\/latex]<em>x<\/em>-intercepts[latex]\\,\\left(0,0\\right),\\left(\u20132,0\\right),\\,[\/latex]and[latex]\\left(5,0\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134080932\" class=\"bc-section section\">\n<h4>Comparing Smooth and Continuous Graphs<\/h4>\n<p id=\"fs-id1165137692509\">The degree of a polynomial function helps us to determine the number of <em>x<\/em>-intercepts and the number of turning points. A polynomial function of[latex]\\,n\\text{th}\\,[\/latex]degree is the product of[latex]\\,n\\,[\/latex]factors, so it will have at most[latex]\\,n\\,[\/latex]roots or zeros, or <em>x<\/em>-intercepts. The graph of the polynomial function of degree[latex]\\,n\\,[\/latex]must have at most[latex]\\,n\u20131\\,[\/latex]turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.<\/p>\n<p id=\"fs-id1165137657937\">A continuous function has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A smooth curve is a graph that has no sharp corners. The turning points of a smooth graph must always occur at rounded curves. The graphs of polynomial functions are both continuous and smooth.<\/p>\n<div id=\"fs-id1165137847104\" class=\"textbox key-takeaways\">\n<h3>Intercepts and Turning Points of Polynomials<\/h3>\n<p id=\"fs-id1165137405499\">A polynomial of degree[latex]\\,n\\,[\/latex]will have, at most,[latex]\\,n\\,[\/latex]<em>x<\/em>-intercepts and[latex]\\,n-1\\,[\/latex]turning points.<\/p>\n<\/div>\n<div id=\"Example_03_03_10\" class=\"textbox examples\">\n<div id=\"fs-id1165135237034\">\n<div id=\"fs-id1165135237036\">\n<h3>Determining the Number of Intercepts and Turning Points of a Polynomial<\/h3>\n<p id=\"fs-id1165134152759\">Without graphing the function, determine the local behavior of the function by finding the maximum number of <em>x<\/em>-intercepts and turning points for[latex]\\,f\\left(x\\right)=-3{x}^{10}+4{x}^{7}-{x}^{4}+2{x}^{3}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135414339\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135414341\">The polynomial has a degree of[latex]\\,10,\\,[\/latex]so there are at most 10 <em>x<\/em>-intercepts and at most 9 turning points.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137628834\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_03_07\">\n<div id=\"fs-id1165135188273\">\n<p id=\"fs-id1165135188274\">Without graphing the function, determine the maximum number of <em>x<\/em>-intercepts and turning points for[latex]\\,f\\left(x\\right)=108-13{x}^{9}-8{x}^{4}+14{x}^{12}+2{x}^{3}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137660801\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137660802\">There are at most 12[latex]\\,x\\text{-}[\/latex]intercepts and at most 11 turning points.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_11\" class=\"textbox examples\">\n<div id=\"fs-id1165137435064\">\n<div id=\"fs-id1165137435066\">\n<h3>Drawing Conclusions about a Polynomial Function from the Graph<\/h3>\n<p id=\"fs-id1165137843783\">What can we conclude about the polynomial represented by the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_03_03_020\">(Figure)<\/a> based on its intercepts and turning points?<\/p>\n<div id=\"Figure_03_03_020\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135851\/CNX_Precalc_Figure_03_03_020.jpg\" alt=\"Graph of an even-degree polynomial.\" width=\"487\" height=\"367\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 12.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137737264\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165131926327\">The end behavior of the graph tells us this is the graph of an even-degree polynomial. See <a class=\"autogenerated-content\" href=\"#Figure_03_03_021\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_03_03_021\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135900\/CNX_Precalc_Figure_03_03_021.jpg\" alt=\"Graph of an even-degree polynomial that denotes the turning points and intercepts.\" width=\"487\" height=\"368\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 13.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165135670389\">The graph has 2 <em>x<\/em>-intercepts, suggesting a degree of 2 or greater, and 3 turning points, suggesting a degree of 4 or greater. Based on this, it would be reasonable to conclude that the degree is even and at least 4.<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137871106\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_03_08\">\n<div id=\"fs-id1165137834183\">\n<p id=\"fs-id1165137454180\">What can we conclude about the polynomial represented by the graph shown in <a class=\"autogenerated-content\" href=\"#Figure_03_03_022\">(Figure)<\/a> based on its intercepts and turning points?<\/p>\n<div id=\"Figure_03_03_022\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135910\/CNX_Precalc_Figure_03_03_022.jpg\" alt=\"Graph of an odd-degree polynomial.\" width=\"487\" height=\"442\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 14.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137666790\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137666791\">The end behavior indicates an odd-degree polynomial function; there are 3[latex]\\,x\\text{-}[\/latex]intercepts and 2 turning points, so the degree is odd and at least 3. Because of the end behavior, we know that the lead coefficient must be negative.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_03_03_12\" class=\"textbox examples\">\n<div id=\"fs-id1165135184013\">\n<div id=\"fs-id1165137725458\">\n<h3>Drawing Conclusions about a Polynomial Function from the Factors<\/h3>\n<p id=\"fs-id1165135435639\">Given the function[latex]\\,f\\left(x\\right)=-4x\\left(x+3\\right)\\left(x-4\\right),\\,[\/latex]<br \/>\ndetermine the local behavior.<\/p>\n<\/div>\n<div id=\"fs-id1165135457721\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135457723\">The <em>y<\/em>-intercept is found by evaluating[latex]\\,f\\left(0\\right).[\/latex]<\/p>\n<div id=\"eip-id1165134587897\" class=\"unnumbered\">[latex]\\begin{array}{ccc}\\hfill f\\left(0\\right)& =& -4\\left(0\\right)\\left(0+3\\right)\\left(0-4\\\\ & =& 0\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135245749\">The <em>y<\/em>-intercept is[latex]\\,\\left(0,0\\right).[\/latex]<\/p>\n<p id=\"fs-id1165135203755\">The <em>x<\/em>-intercepts are found by determining the zeros of the function.<\/p>\n<div id=\"eip-id1165135401630\" class=\"unnumbered\">[latex]0=-4x\\left(x+3\\right)\\left(x-4\\right)[\/latex]<\/div>\n<div id=\"fs-id1178321\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccccccccccc}\\hfill x& =& 0\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{or}\\phantom{\\rule{2em}{0ex}}& \\hfill x+3& =& 0\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{or}\\phantom{\\rule{2em}{0ex}}& \\hfill x-4& =& 0\\hfill \\\\ x& =& 0& \\phantom{\\rule{2em}{0ex}}\\text{or}\\phantom{\\rule{2em}{0ex}}& x& =& -3& \\phantom{\\rule{2em}{0ex}}\\text{or}\\phantom{\\rule{2em}{0ex}}& x& =& 4\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135431016\">The <em>x<\/em>-intercepts are[latex]\\,\\left(0,0\\right),\\left(\u20133,0\\right),\\,[\/latex]and[latex]\\,\\left(4,0\\right).[\/latex]<\/p>\n<p id=\"fs-id1165137472984\">The degree is 3 so the graph has at most 2 turning points.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137661075\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_03_03_09\">\n<div id=\"fs-id1165137575430\">\n<p id=\"fs-id1165137575431\">Given the function[latex]\\,f\\left(x\\right)=0.2\\left(x-2\\right)\\left(x+1\\right)\\left(x-5\\right),\\,[\/latex]determine the local behavior.<\/p>\n<\/div>\n<div id=\"fs-id1165137833005\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137833006\">The[latex]\\,x\\text{-}[\/latex]intercepts are[latex]\\,\\left(2,0\\right),\\left(-1,0\\right),[\/latex]and[latex]\\,\\left(5,0\\right),\\,[\/latex]the <em>y-<\/em>intercept is[latex]\\,\\left(0,\\text{2}\\right),\\,[\/latex]and the graph has at most 2 turning points.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137653058\" class=\"precalculus media\">\n<p id=\"fs-id1165135456729\">Access these online resources for additional instruction and practice with power and polinomial functions.<\/p>\n<ul id=\"fs-id1165137410802\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/keyinfopoly\">Find Key Information about a Given Polynomial Function<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/endbehavior\">End Behavior of a Polynomial Function<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/turningpoints\">Turning Points and[latex]\\,x\\text{-}[\/latex]intercepts of Polynomial Functions<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/leastposdegree\">Least Possible Degree of a Polynomial Function<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id1165134063974\" summary=\"..\">\n<tbody>\n<tr>\n<td>general form of a polynomial function<\/td>\n<td>[latex]f\\left(x\\right)={a}_{n}{x}^{n}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137731646\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165135438864\">\n<li>A power function is a variable base raised to a number power. See <a class=\"autogenerated-content\" href=\"#Example_03_03_01\">(Figure)<\/a>.<\/li>\n<li>The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.<\/li>\n<li>The end behavior depends on whether the power is even or odd. See <a class=\"autogenerated-content\" href=\"#Example_03_03_02\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_03_03_03\">(Figure)<\/a>.<\/li>\n<li>A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power. See <a class=\"autogenerated-content\" href=\"#Example_03_03_04\">(Figure)<\/a>.<\/li>\n<li>The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient. See <a class=\"autogenerated-content\" href=\"#Example_03_03_05\">(Figure)<\/a>.<\/li>\n<li>The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function. See <a class=\"autogenerated-content\" href=\"#Example_03_03_06\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_03_03_07\">(Figure)<\/a>.<\/li>\n<li>A polynomial of degree[latex]\\,n\\,[\/latex]<br \/>\nwill have at most[latex]\\,n\\,[\/latex]<br \/>\n<em>x-<\/em>intercepts and at most[latex]\\,n-1\\,[\/latex]<br \/>\nturning points. See <a class=\"autogenerated-content\" href=\"#Example_03_03_08\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_03_03_09\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_03_03_10\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_03_03_11\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_03_03_12\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137553381\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165137761271\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165137651029\">\n<div id=\"fs-id1165137651030\">\n<p id=\"fs-id1165137651031\">Explain the difference between the coefficient of a power function and its degree.<\/p>\n<\/div>\n<div id=\"fs-id1165137423709\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137423710\">The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137784642\">\n<div id=\"fs-id1165137784643\">\n<p>If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137794232\">\n<div id=\"fs-id1165137794233\">\n<p id=\"fs-id1165137794234\">In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.<\/p>\n<\/div>\n<div id=\"fs-id1165135205853\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135205854\">As[latex]\\,x\\,[\/latex]<br \/>\ndecreases without bound, so does[latex]\\,f\\left(x\\right).\\,[\/latex]<br \/>\nAs[latex]\\,x\\,[\/latex]<br \/>\nincreases without bound, so does[latex]\\,f\\left(x\\right).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137409144\">\n<div id=\"fs-id1165137409146\">\n<p id=\"fs-id1165137628457\">What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137628462\">\n<div id=\"fs-id1165137628463\">\n<p>What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As[latex]\\,x\\to -\\infty ,\\,f\\left(x\\right)\\to -\\infty \\,[\/latex]<br \/>\nand as[latex]\\,x\\to \\infty ,\\,f\\left(x\\right)\\to -\\infty .\\,[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137679007\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137679008\">The polynomial function is of even degree and leading coefficient is negative.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135444041\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165137807114\">For the following exercises, identify the function as a power function, a polynomial function, or neither.<\/p>\n<div id=\"fs-id1165137807118\">\n<div id=\"fs-id1165137807119\">\n<p id=\"fs-id1165137431407\">[latex]f\\left(x\\right)={x}^{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137585053\">\n<div id=\"fs-id1165137585054\">\n<p id=\"fs-id1165137585055\">[latex]f\\left(x\\right)={\\left({x}^{2}\\right)}^{3}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134043753\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137771535\">Power function<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137771538\">\n<div id=\"fs-id1165137771539\">\n<p id=\"fs-id1165137771540\">[latex]f\\left(x\\right)=x-{x}^{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135339598\">\n<div id=\"fs-id1165135339599\">\n<p id=\"fs-id1165134475621\">[latex]f\\left(x\\right)=\\frac{{x}^{2}}{{x}^{2}-1}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137723151\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137723152\">Neither<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137723156\">\n<div id=\"fs-id1165137723157\">\n<p id=\"fs-id1165134043631\">[latex]f\\left(x\\right)=2x\\left(x+2\\right){\\left(x-1\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134149794\">\n<div id=\"fs-id1165134149795\">\n<p id=\"fs-id1165134149796\">[latex]f\\left(x\\right)={3}^{x+1}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137619578\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137619580\">Neither<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137619583\">For the following exercises, find the degree and leading coefficient for the given polynomial.<\/p>\n<div id=\"fs-id1165137464314\">\n<div id=\"fs-id1165137464315\">\n<p id=\"fs-id1165137464316\">[latex]-3x{}^{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134380913\">\n<div id=\"fs-id1165134380914\">\n<p id=\"fs-id1165134380915\">[latex]7-2{x}^{2}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137424474\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137424475\">Degree = 2, Coefficient = \u20132<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137425870\">\n<p id=\"fs-id1165137425871\">[latex]-2{x}^{2}-3{x}^{5}+x-6[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137742433\">\n<div id=\"fs-id1165137742434\">\n<p id=\"fs-id1165137742435\">[latex]x\\left(4-{x}^{2}\\right)\\left(2x+1\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137842483\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165132943532\">Degree =4, Coefficient = \u20132<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132943536\">\n<div id=\"fs-id1165132943537\">\n<p id=\"fs-id1165132943538\">[latex]{x}^{2}{\\left(2x-3\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137455116\">For the following exercises, determine the end behavior of the functions.<\/p>\n<div id=\"fs-id1165137455119\">\n<div id=\"fs-id1165133078101\">\n<p id=\"fs-id1165133078102\">[latex]f\\left(x\\right)={x}^{4}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137445856\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137445857\">[latex]\\text{As}\\,x\\to \\infty ,\\,\\,f\\left(x\\right)\\to \\infty ,\\,\\text{as}\\,x\\to -\\infty ,\\,f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137423948\">\n<div id=\"fs-id1165137423949\">\n<p id=\"fs-id1165137423950\">[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137678592\">\n<div id=\"fs-id1165137678593\">\n<p id=\"fs-id1165137678594\">[latex]f\\left(x\\right)=-{x}^{4}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137643430\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137643431\">[latex]\\text{As}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to -\\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to -\\infty[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135557775\">\n<div id=\"fs-id1165135457738\">\n<p id=\"fs-id1165135457739\">[latex]f\\left(x\\right)=-{x}^{9}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137526614\">\n<div id=\"fs-id1165137526615\">\n<p id=\"fs-id1165137526616\">[latex]f\\left(x\\right)=-2{x}^{4}-3{x}^{2}+x-1[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137656888\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137656889\">[latex]\\text{As}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to -\\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to -\\infty[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135696746\">\n<div id=\"fs-id1165135696747\">\n<p id=\"fs-id1165135696748\">[latex]f\\left(x\\right)=3{x}^{2}+x-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137627936\">\n<div id=\"fs-id1165137627937\">\n<p id=\"fs-id1165137806389\">[latex]f\\left(x\\right)={x}^{2}\\left(2{x}^{3}-x+1\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137794123\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135333271\">[latex]\\text{As}\\,x\\to \\infty ,\\,\\,f\\left(x\\right)\\to \\infty ,\\,\\text{as}\\,x\\to -\\infty ,\\,f\\left(x\\right)\\to -\\infty[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137761673\">\n<div id=\"fs-id1165137761674\">\n<p id=\"fs-id1165137761675\">[latex]f\\left(x\\right)={\\left(2-x\\right)}^{7}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135650636\">For the following exercises, find the intercepts of the functions.<\/p>\n<div id=\"fs-id1165135650639\">\n<div id=\"fs-id1165135650640\">\n<p id=\"fs-id1165135436503\">[latex]f\\left(t\\right)=2\\left(t-1\\right)\\left(t+2\\right)\\left(t-3\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135697918\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135697919\"><em>y<\/em>-intercept is[latex]\\,\\left(0,12\\right),\\,[\/latex]<em>t<\/em>-intercepts are[latex]\\,\\left(1,0\\right);\\left(\u20132,0\\right);\\text{and }\\left(3,0\\right).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137634317\">\n<div id=\"fs-id1165137634318\">\n<p id=\"fs-id1165137634319\">[latex]g\\left(n\\right)=-2\\left(3n-1\\right)\\left(2n+1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134113935\">\n<div id=\"fs-id1165137645540\">\n<p id=\"fs-id1165137645541\">[latex]f\\left(x\\right)={x}^{4}-16[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137728286\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137728287\"><em>y<\/em>-intercept is[latex]\\,\\left(0,-16\\right).\\,[\/latex]<em>x<\/em>-intercepts are[latex]\\,\\left(2,0\\right)\\,[\/latex]and[latex]\\,\\left(-2,0\\right).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137783610\">\n<div id=\"fs-id1165137783611\">\n<p id=\"fs-id1165137783612\">[latex]f\\left(x\\right)={x}^{3}+27[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137823331\">\n<div id=\"fs-id1165137734444\">\n<p id=\"fs-id1165137734445\">[latex]f\\left(x\\right)=x\\left({x}^{2}-2x-8\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137896961\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137896962\"><em>y<\/em>-intercept is[latex]\\,\\left(0,0\\right).\\,[\/latex]<em>x<\/em>-intercepts are[latex]\\,\\left(0,0\\right),\\left(4,0\\right),\\,[\/latex]and[latex]\\,\\left(-2, 0\\right).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134223322\">\n<div id=\"fs-id1165134223324\">\n<p id=\"fs-id1165137456270\">[latex]f\\left(x\\right)=\\left(x+3\\right)\\left(4{x}^{2}-1\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135416569\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165134486753\">For the following exercises, determine the least possible degree of the polynomial function shown.<\/p>\n<div id=\"fs-id1165134486757\">\n<div id=\"fs-id1165134486758\"><span id=\"fs-id1165137942457\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135918\/CNX_Precalc_Figure_03_03_201.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/><\/span><\/div>\n<div id=\"fs-id1165135388469\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135388470\">3<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137823453\">\n<div id=\"fs-id1165137823454\"><span id=\"fs-id1165135530394\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135921\/CNX_Precalc_Figure_03_03_202.jpg\" alt=\"Graph of an even-degree polynomial.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165135436619\">\n<div id=\"fs-id1165135436620\"><span id=\"fs-id1165135543346\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135936\/CNX_Precalc_Figure_03_03_203.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/><\/span><\/div>\n<div id=\"fs-id1165135181683\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135181684\">5<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137781545\">\n<div id=\"fs-id1165137781546\"><span id=\"fs-id1165137433119\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135938\/CNX_Precalc_Figure_03_03_204.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165135457734\">\n<div id=\"fs-id1165137833940\"><span id=\"fs-id1165137833945\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135940\/CNX_Precalc_Figure_03_03_205.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/><\/span><\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135255943\">3<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137410264\">\n<div id=\"fs-id1165137410266\"><span id=\"fs-id1165134323766\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135944\/CNX_Precalc_Figure_03_03_206.jpg\" alt=\"Graph of an even-degree polynomial.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165134129751\">\n<div id=\"fs-id1165134129752\"><span id=\"fs-id1165137405932\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135955\/CNX_Precalc_Figure_03_03_207.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/><\/span><\/div>\n<div id=\"fs-id1165135191388\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135191389\">5<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135191392\">\n<div id=\"fs-id1165135191393\"><span id=\"fs-id1165137653349\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19135957\/CNX_Precalc_Figure_03_03_208.jpg\" alt=\"Graph of an even-degree polynomial.\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1165134262467\">For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.<\/p>\n<div id=\"fs-id1165135332833\">\n<div id=\"fs-id1165135332834\"><span id=\"fs-id1165135678606\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140002\/CNX_Precalc_Figure_03_03_209.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/><\/span><\/div>\n<div id=\"fs-id1165137444866\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137444867\">Yes. Number of turning points is 2. Least possible degree is 3.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137444870\">\n<div id=\"fs-id1165137444871\"><span id=\"fs-id1165135264631\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140011\/CNX_Precalc_Figure_03_03_210.jpg\" alt=\"Graph of an equation.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165137704867\">\n<div id=\"fs-id1165137704868\"><span id=\"fs-id1165134089461\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140017\/CNX_Precalc_Figure_03_03_211.jpg\" alt=\"Graph of an even-degree polynomial.\" \/><\/span><\/div>\n<div id=\"fs-id1165137570516\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137570517\">Yes. Number of turning points is 1. Least possible degree is 2.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135191925\">\n<div id=\"fs-id1165135191926\"><span id=\"fs-id1165135591035\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140023\/CNX_Precalc_Figure_03_03_212.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165134313927\">\n<div id=\"fs-id1165137476631\"><span id=\"fs-id1165135319603\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140025\/CNX_Precalc_Figure_03_03_213.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/><\/span><\/div>\n<div id=\"fs-id1165134042451\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134042452\">Yes. Number of turning points is 0. Least possible degree is 1.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137436429\">\n<div id=\"fs-id1165137436430\"><span id=\"fs-id1165137772222\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140031\/CNX_Precalc_Figure_03_03_214.jpg\" alt=\"Graph of an equation.\" \/><\/span><\/div>\n<div id=\"fs-id1165133310452\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133310453\">No.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133310456\">\n<div><span id=\"fs-id1165137642864\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140035\/CNX_Precalc_Figure_03_03_215.jpg\" alt=\"Graph of an odd-degree polynomial.\" \/><\/span><\/div>\n<div id=\"fs-id1165137569703\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137569704\">Yes. Number of turning points is 0. Least possible degree is 1.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134130094\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<p id=\"fs-id1165137828131\">For the following exercises, make a table to confirm the end behavior of the function.<\/p>\n<div id=\"fs-id1165137828134\">\n<div id=\"fs-id1165137416865\">\n<p id=\"fs-id1165137416866\">[latex]f\\left(x\\right)=-{x}^{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137734861\">\n<div id=\"fs-id1165137734862\">\n<p id=\"fs-id1165137734863\">[latex]f\\left(x\\right)={x}^{4}-5{x}^{2}[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<table id=\"fs-id1165137654655\" class=\"unnumbered\" summary=\"..\">\n<caption>&nbsp;<\/caption>\n<thead>\n<tr>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]f\\left(x\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>10<\/td>\n<td>9,500<\/td>\n<\/tr>\n<tr>\n<td>100<\/td>\n<td>99,950,000<\/td>\n<\/tr>\n<tr>\n<td>\u201310<\/td>\n<td>9,500<\/td>\n<\/tr>\n<tr>\n<td>\u2013100<\/td>\n<td>99,950,000<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137654654\">[latex]\\text{as}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to \\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133045350\">\n<div id=\"fs-id1165133045351\">\n<p id=\"fs-id1165133045352\">[latex]f\\left(x\\right)={x}^{2}{\\left(1-x\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135456684\">\n<div id=\"fs-id1165135456685\">\n<p id=\"fs-id1165135456686\">[latex]f\\left(x\\right)=\\left(x-1\\right)\\left(x-2\\right)\\left(3-x\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134122928\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<table id=\"fs-id1165134122930\" class=\"unnumbered\" summary=\"..\">\n<caption>&nbsp;<\/caption>\n<thead>\n<tr>\n<th>[latex]x[\/latex]<\/th>\n<th>[latex]f\\left(x\\right)[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>10<\/td>\n<td>\u2013504<\/td>\n<\/tr>\n<tr>\n<td>100<\/td>\n<td>\u2013941,094<\/td>\n<\/tr>\n<tr>\n<td>\u201310<\/td>\n<td>1,716<\/td>\n<\/tr>\n<tr>\n<td>\u2013100<\/td>\n<td>1,061,106<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165134122929\">[latex]\\text{as}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to \\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to -\\infty[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134122886\">\n<div id=\"fs-id1165134122887\">\n<p id=\"fs-id1165134122888\">[latex]f\\left(x\\right)=\\frac{{x}^{5}}{10}-{x}^{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137755588\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1165133238486\">For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.<\/p>\n<div id=\"fs-id1165135317475\">\n<div>\n<p id=\"fs-id1165135317477\">[latex]f\\left(x\\right)={x}^{3}\\left(x-2\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137629480\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135168133\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140037\/CNX_Precalc_Figure_03_03_216.jpg\" alt=\"Graph of f(x)=x^3(x-2).\" \/><\/span><\/p>\n<p id=\"fs-id1165135168128\">The[latex]\\,y\\text{-}[\/latex]intercept is[latex]\\,\\left(0, 0\\right).\\,[\/latex]<br \/>\nThe[latex]\\,x\\text{-}[\/latex]intercepts are[latex]\\,\\left(0, 0\\right),\\text{ }\\left(2, 0\\right).\\,[\/latex]<br \/>\n[latex]\\text{As}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to \\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<p id=\"fs-id1165135168128\"><\/details>\n<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135388458\">\n<div id=\"fs-id1165135388459\">\n<p id=\"fs-id1165135388460\">[latex]f\\left(x\\right)=x\\left(x-3\\right)\\left(x+3\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137920714\">\n<div id=\"fs-id1165137920716\">\n<p id=\"fs-id1165137920717\">[latex]f\\left(x\\right)=x\\left(14-2x\\right)\\left(10-2x\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135436598\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137438048\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140052\/CNX_Precalc_Figure_03_03_218.jpg\" alt=\"Graph of f(x)=x(14-2x)(10-2x).\" \/><\/span><\/p>\n<p id=\"fs-id1165135436600\">The[latex]\\,y\\text{-}[\/latex]intercept is[latex]\\,\\left(0,0\\right)[\/latex]<br \/>\n. The[latex]\\,x\\text{-}[\/latex]intercepts are[latex]\\,\\left(0, 0\\right),\\text{ }\\left(5, 0\\right),\\text{ }\\left(7, 0\\right).\\,[\/latex]<br \/>\n[latex]\\text{As}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to -\\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<p id=\"fs-id1165135436600\"><\/details>\n<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133448049\">\n<div>\n<p id=\"fs-id1165133448052\">[latex]f\\left(x\\right)=x\\left(14-2x\\right){\\left(10-2x\\right)}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134417902\">\n<div id=\"fs-id1165134417903\">\n<p id=\"fs-id1165134417904\">[latex]f\\left(x\\right)={x}^{3}-16x[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137676496\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137676502\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140059\/CNX_Precalc_Figure_03_03_220.jpg\" alt=\"\" \/><\/span><\/p>\n<p id=\"fs-id1165137676497\">The[latex]\\,y\\text{-}[\/latex]intercept is[latex]\\,\\left(0, 0\\right).\\,[\/latex] The[latex]\\,x\\text{-}[\/latex]intercept is[latex]\\,\\left(-4, 0\\right),\\text{ }\\left(0, 0\\right),\\text{ }\\left(4, 0\\right).\\,[\/latex]<br \/>\n[latex]As\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to -\\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<p id=\"fs-id1165137676497\"><\/details>\n<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137837875\">\n<div id=\"fs-id1165137837876\">\n<p id=\"fs-id1165137837878\">[latex]f\\left(x\\right)={x}^{3}-27[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135160696\">\n<div id=\"fs-id1165135160697\">\n<p id=\"fs-id1165131891794\">[latex]f\\left(x\\right)={x}^{4}-81[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165131891798\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135555475\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140101\/CNX_Precalc_Figure_03_03_222.jpg\" alt=\"Graph of f(x)=x^3-27.\" \/><\/span><\/p>\n<p id=\"fs-id1165131891799\">The[latex]\\,y\\text{-}[\/latex]intercept is[latex]\\,\\left(0, -81\\right).\\,[\/latex]<br \/>\nThe[latex]\\,x\\text{-}[\/latex]intercept are[latex]\\,\\left(3, 0\\right),\\text{ }\\left(-3, 0\\right).\\,[\/latex]<br \/>\n[latex]\\text{As}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to \\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<p id=\"fs-id1165131891799\"><\/details>\n<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135484565\">\n<div id=\"fs-id1165135484566\">\n<p id=\"fs-id1165135484567\">[latex]f\\left(x\\right)=-{x}^{3}+{x}^{2}+2x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133092744\">\n<div id=\"fs-id1165133092745\">\n<p id=\"fs-id1165133092746\">[latex]f\\left(x\\right)={x}^{3}-2{x}^{2}-15x[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<div id=\"fs-id1165133092745\">\n<p id=\"fs-id1165133092746\"><span id=\"fs-id1165137401080\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140107\/CNX_Precalc_Figure_03_03_224.jpg\" alt=\"Graph of f(x)=-x^3+x^2+2x.\" \/>The[latex]\\,y\\text{-}[\/latex]intercept is[latex]\\,\\left(0, 0\\right).\\,[\/latex] The[latex]\\,x\\text{-}[\/latex]intercepts are[latex]\\,\\left(-3, 0\\right),\\text{ }\\left(0, 0\\right),\\text{ }\\left(5, 0\\right).\\,[\/latex]<br \/>\n[latex]\\text{As}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to -\\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty[\/latex]<\/details>\n<p><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134378662\">\n<div id=\"fs-id1165134378663\">\n<p id=\"fs-id1165134378664\">[latex]f\\left(x\\right)={x}^{3}-0.01x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137811752\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id1165133111641\">For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or \u20131. There may be more than one correct answer.<\/p>\n<div id=\"fs-id1165133448009\">\n<div id=\"fs-id1165133448010\">\n<p id=\"fs-id1165133448012\">The[latex]\\,y-[\/latex]intercept is[latex]\\,\\left(0,-4\\right).\\,[\/latex]The[latex]\\,x-[\/latex]intercepts are[latex]\\,\\left(-2,0\\right),\\,\\left(2,0\\right).\\,[\/latex]Degree is 2.<\/p>\n<p id=\"eip-id1165134540124\">End behavior:[latex]\\,\\text{as}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to \\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty .[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135160156\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135160157\">[latex]f\\left(x\\right)={x}^{2}-4[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137894256\">\n<div id=\"fs-id1165137894257\">\n<p id=\"fs-id1165137894258\">The[latex]\\,y-[\/latex]intercept is[latex]\\,\\left(0,9\\right).\\,[\/latex]The[latex]\\,x\\text{-}[\/latex]intercepts are[latex]\\,\\left(-3,0\\right),\\,\\left(3,0\\right).\\,[\/latex]Degree is 2.<\/p>\n<p id=\"eip-id1165134566570\">End behavior:[latex]\\,\\text{as}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to -\\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to -\\infty .[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137897907\">\n<div id=\"fs-id1165137897908\">\n<p id=\"fs-id1165137897909\">The[latex]\\,y-[\/latex]intercept is[latex]\\,\\left(0,0\\right).\\,[\/latex]The[latex]\\,x-[\/latex]intercepts are[latex]\\,\\left(0,0\\right),\\,\\left(2,0\\right).\\,[\/latex]Degree is 3.<\/p>\n<p id=\"eip-id1165137749966\">End behavior:[latex]\\,\\text{as}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to -\\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty .[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134389974\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134389976\">[latex]f\\left(x\\right)={x}^{3}-4{x}^{2}+4x[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137921541\">\n<div id=\"fs-id1165137921542\">\n<p id=\"fs-id1165137921544\">The[latex]\\,y-[\/latex]intercept is[latex]\\,\\left(0,1\\right).\\,[\/latex]The[latex]\\,x-[\/latex]intercept is[latex]\\,\\left(1,0\\right).\\,[\/latex]Degree is 3.<\/p>\n<p id=\"eip-id1165137895228\">End behavior:[latex]\\,\\text{as}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to \\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to -\\infty .[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135536216\">\n<div id=\"fs-id1165135536217\">\n<p id=\"fs-id1165135536218\">The[latex]\\,y-[\/latex]intercept is[latex]\\,\\left(0,1\\right).\\,[\/latex]There is no[latex]\\,x-[\/latex]intercept. Degree is 4.<\/p>\n<p id=\"eip-id1165135434911\">End behavior:[latex]\\,\\text{as}\\,x\\to -\\infty ,\\,\\,f\\left(x\\right)\\to \\infty ,\\,\\text{as}\\,x\\to \\infty ,\\,f\\left(x\\right)\\to \\infty .[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137922537\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135386380\">[latex]f\\left(x\\right)={x}^{4}+1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134085635\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<p id=\"fs-id1165134085641\">For the following exercises, use the written statements to construct a polynomial function that represents the required information.<\/p>\n<div id=\"fs-id1165135581116\">\n<div id=\"fs-id1165135581117\">\n<p id=\"fs-id1165135581118\">An oil slick is expanding as a circle. The radius of the circle is increasing at the rate of 20 meters per day. Express the area of the circle as a function of[latex]\\,d,\\,[\/latex]the number of days elapsed.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137892519\">\n<div id=\"fs-id1165137892520\">\n<p id=\"fs-id1165137892521\">A cube has an edge of 3 feet. The edge is increasing at the rate of 2 feet per minute. Express the volume of the cube as a function of[latex]\\,m,\\,[\/latex]the number of minutes elapsed.<\/p>\n<\/div>\n<div id=\"fs-id1165134032299\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137704612\">[latex]V\\left(m\\right)=8{m}^{3}+36{m}^{2}+54m+27[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137848042\">\n<div id=\"fs-id1165137848043\">\n<p id=\"fs-id1165137848044\">A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased by[latex]\\,x\\,[\/latex]inches and the width increased by twice that amount, express the area of the rectangle as a function of[latex]\\,x.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137891257\">\n<div id=\"fs-id1165137891258\">\n<p id=\"fs-id1165137891259\">An open box is to be constructed by cutting out square corners of [latex]\\,x-[\/latex]inch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function of[latex]\\,x.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135571779\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135571780\">[latex]V\\left(x\\right)=4{x}^{3}-32{x}^{2}+64x[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135190825\">\n<div id=\"fs-id1165135190826\">\n<p id=\"fs-id1165135190827\">A rectangle is twice as long as it is wide. Squares of side 2 feet are cut out from each corner. Then the sides are folded up to make an open box. Express the volume of the box as a function of the width ([latex]x[\/latex]).<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165137668266\">\n<dt>coefficient<\/dt>\n<dd id=\"fs-id1165135194915\">a nonzero real number multiplied by a variable raised to an exponent<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135194918\">\n<dt>continuous function<\/dt>\n<dd id=\"fs-id1165135194921\">a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137832108\">\n<dt>degree<\/dt>\n<dd id=\"fs-id1165137832112\">the highest power of the variable that occurs in a polynomial<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137832115\">\n<dt>end behavior<\/dt>\n<dd>the behavior of the graph of a function as the input decreases without bound and increases without bound<\/dd>\n<\/dl>\n<dl id=\"fs-id1165131990658\">\n<dt>leading coefficient<\/dt>\n<dd id=\"fs-id1165131990661\">the coefficient of the leading term<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943522\">\n<dt>leading term<\/dt>\n<dd id=\"fs-id1165132943525\">the term containing the highest power of the variable<\/dd>\n<\/dl>\n<dl id=\"fs-id1165132943528\">\n<dt>polynomial function<\/dt>\n<dd id=\"fs-id1165134297639\">a function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134297646\">\n<dt>power function<\/dt>\n<dd id=\"fs-id1165135486042\">a function that can be represented in the form[latex]\\,f\\left(x\\right)=k{x}^{p}\\,[\/latex]where[latex]\\,k\\,[\/latex]is a constant, the base is a variable, and the exponent,[latex]\\,p[\/latex], is a constant<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137833929\">\n<dt>smooth curve<\/dt>\n<dd>a graph with no sharp corners<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\">\n<dt>term of a polynomial function<\/dt>\n<dd id=\"fs-id1165137644990\">any[latex]\\,{a}_{i}{x}^{i}\\,[\/latex]of a polynomial function in the form[latex]\\,f\\left(x\\right)={a}_{n}{x}^{n}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\">\n<dt>turning point<\/dt>\n<dd id=\"fs-id1165133085665\">the location at which the graph of a function changes direction<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":291,"menu_order":3,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-81","chapter","type-chapter","status-publish","hentry"],"part":76,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/81","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/users\/291"}],"version-history":[{"count":1,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/81\/revisions"}],"predecessor-version":[{"id":82,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/81\/revisions\/82"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/parts\/76"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/81\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=81"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=81"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=81"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=81"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}