{"id":206,"date":"2023-06-21T13:22:44","date_gmt":"2023-06-21T13:22:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/graphs-of-polynomial-functions\/"},"modified":"2024-01-08T19:11:40","modified_gmt":"2024-01-08T19:11:40","slug":"graphs-of-polynomial-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/graphs-of-polynomial-functions\/","title":{"raw":"R4.2   Graphs of Polynomial Functions","rendered":"R4.2   Graphs of Polynomial Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify graphs of polynomial functions<\/li>\r\n \t<li>Identify general characteristics of a polynomial function from its graph<\/li>\r\n<\/ul>\r\n<\/div>\r\nPlotting polynomial functions using tables of values can be misleading because of some of the inherent characteristics of polynomials. Additionally, the algebra of finding points like x-intercepts for higher degree polynomials can get very messy and oftentimes be impossible to find\u00a0by hand. We have therefore developed some techniques for describing the general behavior of polynomial graphs.\r\n\r\nPolynomial functions of degree\u00a0[latex]2[\/latex] or more have graphs that do not have sharp corners. These types of graphs are called smooth curves. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous. The figure below\u00a0shows\u00a0a graph that represents a <strong>polynomial function<\/strong> and a graph that represents a function that is not a polynomial.\r\n\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201418\/CNX_Precalc_Figure_03_04_0012.jpg\" alt=\"Graph of f(x)=x^3-0.01x.\" width=\"900\" height=\"409\" \/>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWhich of the graphs below\u00a0represents a polynomial function?\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201420\/CNX_Precalc_Figure_03_04_0022.jpg\" alt=\"Two graphs in which one has a polynomial function and the other has a function closely resembling a polynomial but is not.\" width=\"731\" height=\"766\" \/>\r\n\r\n[reveal-answer q=\"207827\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"207827\"]\r\n<p id=\"fs-id1165134129608\">The graphs of <em>f<\/em>\u00a0and <em>h<\/em>\u00a0are graphs of polynomial functions. They are smooth and <strong>continuous<\/strong>.<\/p>\r\n<p id=\"fs-id1165134188794\">The graphs of <em>g<\/em>\u00a0and <em>k\u00a0<\/em>are graphs of functions that are not polynomials. The graph of function <em>g<\/em>\u00a0has a sharp corner. The graph of function <em>k<\/em>\u00a0is not continuous.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div id=\"Example_03_04_01\" class=\"example\"><\/div>\r\n<div id=\"fs-id1165134164967\" class=\"note precalculus qa textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<p id=\"fs-id1165135496631\"><strong>Do all polynomial functions have \"all real numbers\" as their domain?<\/strong><\/p>\r\n<p id=\"fs-id1165134342693\"><em>Yes. Any real number is a valid input for a polynomial function.<\/em><\/p>\r\n\r\n<\/div>\r\n<h2>Identifying\u00a0the Shape of the Graph of a Polynomial Function<\/h2>\r\n<p id=\"fs-id1165137601421\">Knowing the degree of a polynomial function is useful in helping us predict what its graph will look like.\u00a0Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as <em>x<\/em>\u00a0gets very large or very small, so its behavior will dominate the graph. For any polynomial, the\u00a0graph\u00a0of the polynomial will match the end behavior of the term of highest degree.<\/p>\r\nAs an example, we compare the outputs of a degree\u00a0[latex]2[\/latex] polynomial and a degree\u00a0[latex]5[\/latex] polynomial in the following table.\r\n<table>\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">x<\/td>\r\n<td style=\"height: 15px;\">[latex]f(x)=2x^2-2x+4[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]g(x)=x^5+2x^3-12x+3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]4[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]8[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">[latex]10[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]184[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]98117[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">[latex]100[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]19804[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]9998001197[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"height: 15px;\">[latex]1000[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]1998004[\/latex]<\/td>\r\n<td style=\"height: 15px;\">[latex]9999980000000000[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nAs the inputs for both functions get larger, the degree [latex]5[\/latex] polynomial outputs get much larger than the degree\u00a0[latex]2[\/latex] polynomial outputs. This is why we use the leading term to get a rough idea of the behavior of polynomial graphs.\r\n\r\nThere are two other important features of polynomials that influence the shape of its graph. The first is whether the degree is even or odd, and the second is whether the leading term is negative.\r\n<h3>Even Degree Polynomials<\/h3>\r\n<p id=\"fs-id1165135436540\">In the figure below, we show\u00a0the 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 all have even degrees. Notice that these graphs have similar shapes, very much like that of a\u00a0quadratic function. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.<\/p>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201324\/CNX_Precalc_Figure_03_03_0022.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\" \/>\r\n<h3>Odd Degree Polynomials<\/h3>\r\n<p id=\"fs-id1165137533222\">The next figure\u00a0shows 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 all have odd degrees. <img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201325\/CNX_Precalc_Figure_03_03_0032.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=\"312\" height=\"366\" \/><\/p>\r\n<p id=\"fs-id1165137533222\">Notice that one arm of the graph points down and the other points up. This is because\u00a0when your input is negative, you will get a negative output if the degree is odd.\u00a0The following table of values shows this.<\/p>\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>x<\/td>\r\n<td>[latex]f(x)=x^4[\/latex]<\/td>\r\n<td>[latex]h(x)=x^5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]-1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]16[\/latex]<\/td>\r\n<td>[latex]-32[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]81[\/latex]<\/td>\r\n<td>[latex]-243[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow you try it.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIdentify whether each graph represents a polynomial function that has a degree that is even or odd.\r\n\r\na)\r\n\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201335\/CNX_Precalc_Figure_03_03_0112.jpg\" alt=\"Graph of f(x)=5x^4+2x^3-x-4.\" width=\"231\" height=\"245\" \/>\r\n\r\nb)\r\n\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201338\/CNX_Precalc_Figure_03_03_0132.jpg\" alt=\"Graph of f(x)=3x^5-4x^4+2x^2+1.\" width=\"227\" height=\"241\" \/>\r\n[reveal-answer q=\"657906\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"657906\"]\r\n\r\na) Both arms of this polynomial point upward, similar to a quadratic polynomial, therefore the degree must be even. \u00a0If you apply negative inputs to an even degree polynomial, you will get positive outputs back.\r\n\r\nb) As the inputs of this polynomial become more negative the outputs also become negative. The only way this is possible is with an odd degree polynomial. Therefore, this polynomial must have an odd degree.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>\u00a0The Sign of the Leading Term<\/h3>\r\nWhat would happen if we change the sign of the leading term of an even degree polynomial? \u00a0For example, let us say that the leading term of a polynomial is [latex]-3x^4[\/latex]. \u00a0We will use a table of values to compare the outputs for a polynomial with leading term\u00a0[latex]-3x^4[\/latex] and\u00a0[latex]3x^4[\/latex].\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>x<\/td>\r\n<td>[latex]-3x^4[\/latex]<\/td>\r\n<td>[latex]3x^4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]-48[\/latex]<\/td>\r\n<td>[latex]48[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: left;\">[latex]-1[\/latex]<\/td>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]-48[\/latex]<\/td>\r\n<td>[latex]48[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe grid below shows a plot with these points. The red points indicate a negative leading coefficient, and the blue points indicate a positive leading coefficient:\r\n\r\n<img class=\"wp-image-2649 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15212141\/Screen-Shot-2016-07-15-at-2.21.36-PM-140x300.png\" alt=\"Screen Shot 2016-07-15 at 2.21.36 PM\" width=\"266\" height=\"570\" \/>\r\n\r\nThe negative sign creates a reflection of [latex]3x^4[\/latex] across the x-axis. \u00a0The arms of a polynomial with a leading term of\u00a0[latex]-3x^4[\/latex] will point down, whereas the arms of a polynomial with leading term\u00a0[latex]3x^4[\/latex] will point up.\r\n\r\nNow you try it.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIdentify whether the leading term is positive or negative and whether the degree is even or odd for the following graphs of polynomial functions.\r\n\r\na)\r\n\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201336\/CNX_Precalc_Figure_03_03_0122.jpg\" alt=\"Graph of f(x)=-2x^6-x^5+3x^4+x^3.\" width=\"214\" height=\"227\" \/>\r\n\r\nb)\r\n\r\n<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201339\/CNX_Precalc_Figure_03_03_0142.jpg\" alt=\"Graph of f(x)=-6x^3+7x^2+3x+1.\" width=\"217\" height=\"230\" \/>\r\n[reveal-answer q=\"317874\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"317874\"]\r\n\r\na) Both arms of this polynomial point in the same direction so it must have an even degree. \u00a0The leading term of the polynomial must be negative since the arms are pointing downward.\r\n\r\nb) The arms of this polynomial point in different directions, so the degree must be odd. As the inputs get really big and positive, the outputs get really big and negative, so the leading coefficient must be negative.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify graphs of polynomial functions<\/li>\n<li>Identify general characteristics of a polynomial function from its graph<\/li>\n<\/ul>\n<\/div>\n<p>Plotting polynomial functions using tables of values can be misleading because of some of the inherent characteristics of polynomials. Additionally, the algebra of finding points like x-intercepts for higher degree polynomials can get very messy and oftentimes be impossible to find\u00a0by hand. We have therefore developed some techniques for describing the general behavior of polynomial graphs.<\/p>\n<p>Polynomial functions of degree\u00a0[latex]2[\/latex] or more have graphs that do not have sharp corners. These types of graphs are called smooth curves. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous. The figure below\u00a0shows\u00a0a graph that represents a <strong>polynomial function<\/strong> and a graph that represents a function that is not a polynomial.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201418\/CNX_Precalc_Figure_03_04_0012.jpg\" alt=\"Graph of f(x)=x^3-0.01x.\" width=\"900\" height=\"409\" \/><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Which of the graphs below\u00a0represents a polynomial function?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201420\/CNX_Precalc_Figure_03_04_0022.jpg\" alt=\"Two graphs in which one has a polynomial function and the other has a function closely resembling a polynomial but is not.\" width=\"731\" height=\"766\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q207827\">Show Solution<\/span><\/p>\n<div id=\"q207827\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165134129608\">The graphs of <em>f<\/em>\u00a0and <em>h<\/em>\u00a0are graphs of polynomial functions. They are smooth and <strong>continuous<\/strong>.<\/p>\n<p id=\"fs-id1165134188794\">The graphs of <em>g<\/em>\u00a0and <em>k\u00a0<\/em>are graphs of functions that are not polynomials. The graph of function <em>g<\/em>\u00a0has a sharp corner. The graph of function <em>k<\/em>\u00a0is not continuous.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div id=\"Example_03_04_01\" class=\"example\"><\/div>\n<div id=\"fs-id1165134164967\" class=\"note precalculus qa textbox\">\n<h3>Q &amp; A<\/h3>\n<p id=\"fs-id1165135496631\"><strong>Do all polynomial functions have &#8220;all real numbers&#8221; as their domain?<\/strong><\/p>\n<p id=\"fs-id1165134342693\"><em>Yes. Any real number is a valid input for a polynomial function.<\/em><\/p>\n<\/div>\n<h2>Identifying\u00a0the Shape of the Graph of a Polynomial Function<\/h2>\n<p id=\"fs-id1165137601421\">Knowing the degree of a polynomial function is useful in helping us predict what its graph will look like.\u00a0Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as <em>x<\/em>\u00a0gets very large or very small, so its behavior will dominate the graph. For any polynomial, the\u00a0graph\u00a0of the polynomial will match the end behavior of the term of highest degree.<\/p>\n<p>As an example, we compare the outputs of a degree\u00a0[latex]2[\/latex] polynomial and a degree\u00a0[latex]5[\/latex] polynomial in the following table.<\/p>\n<table>\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">x<\/td>\n<td style=\"height: 15px;\">[latex]f(x)=2x^2-2x+4[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]g(x)=x^5+2x^3-12x+3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">[latex]1[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]4[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]8[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">[latex]10[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]184[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]98117[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">[latex]100[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]19804[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]9998001197[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"height: 15px;\">[latex]1000[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]1998004[\/latex]<\/td>\n<td style=\"height: 15px;\">[latex]9999980000000000[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>As the inputs for both functions get larger, the degree [latex]5[\/latex] polynomial outputs get much larger than the degree\u00a0[latex]2[\/latex] polynomial outputs. This is why we use the leading term to get a rough idea of the behavior of polynomial graphs.<\/p>\n<p>There are two other important features of polynomials that influence the shape of its graph. The first is whether the degree is even or odd, and the second is whether the leading term is negative.<\/p>\n<h3>Even Degree Polynomials<\/h3>\n<p id=\"fs-id1165135436540\">In the figure below, we show\u00a0the 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 all have even degrees. Notice that these graphs have similar shapes, very much like that of a\u00a0quadratic function. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201324\/CNX_Precalc_Figure_03_03_0022.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\" \/><\/p>\n<h3>Odd Degree Polynomials<\/h3>\n<p id=\"fs-id1165137533222\">The next figure\u00a0shows 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 all have odd degrees. <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201325\/CNX_Precalc_Figure_03_03_0032.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=\"312\" height=\"366\" \/><\/p>\n<p id=\"fs-id1165137533222\">Notice that one arm of the graph points down and the other points up. This is because\u00a0when your input is negative, you will get a negative output if the degree is odd.\u00a0The following table of values shows this.<\/p>\n<table>\n<tbody>\n<tr>\n<td>x<\/td>\n<td>[latex]f(x)=x^4[\/latex]<\/td>\n<td>[latex]h(x)=x^5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]16[\/latex]<\/td>\n<td>[latex]-32[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]81[\/latex]<\/td>\n<td>[latex]-243[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now you try it.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Identify whether each graph represents a polynomial function that has a degree that is even or odd.<\/p>\n<p>a)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201335\/CNX_Precalc_Figure_03_03_0112.jpg\" alt=\"Graph of f(x)=5x^4+2x^3-x-4.\" width=\"231\" height=\"245\" \/><\/p>\n<p>b)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201338\/CNX_Precalc_Figure_03_03_0132.jpg\" alt=\"Graph of f(x)=3x^5-4x^4+2x^2+1.\" width=\"227\" height=\"241\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q657906\">Show Solution<\/span><\/p>\n<div id=\"q657906\" class=\"hidden-answer\" style=\"display: none\">\n<p>a) Both arms of this polynomial point upward, similar to a quadratic polynomial, therefore the degree must be even. \u00a0If you apply negative inputs to an even degree polynomial, you will get positive outputs back.<\/p>\n<p>b) As the inputs of this polynomial become more negative the outputs also become negative. The only way this is possible is with an odd degree polynomial. Therefore, this polynomial must have an odd degree.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h3>\u00a0The Sign of the Leading Term<\/h3>\n<p>What would happen if we change the sign of the leading term of an even degree polynomial? \u00a0For example, let us say that the leading term of a polynomial is [latex]-3x^4[\/latex]. \u00a0We will use a table of values to compare the outputs for a polynomial with leading term\u00a0[latex]-3x^4[\/latex] and\u00a0[latex]3x^4[\/latex].<\/p>\n<table>\n<tbody>\n<tr>\n<td>x<\/td>\n<td>[latex]-3x^4[\/latex]<\/td>\n<td>[latex]3x^4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-48[\/latex]<\/td>\n<td>[latex]48[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: left;\">[latex]-1[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]-48[\/latex]<\/td>\n<td>[latex]48[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The grid below shows a plot with these points. The red points indicate a negative leading coefficient, and the blue points indicate a positive leading coefficient:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2649 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/07\/15212141\/Screen-Shot-2016-07-15-at-2.21.36-PM-140x300.png\" alt=\"Screen Shot 2016-07-15 at 2.21.36 PM\" width=\"266\" height=\"570\" \/><\/p>\n<p>The negative sign creates a reflection of [latex]3x^4[\/latex] across the x-axis. \u00a0The arms of a polynomial with a leading term of\u00a0[latex]-3x^4[\/latex] will point down, whereas the arms of a polynomial with leading term\u00a0[latex]3x^4[\/latex] will point up.<\/p>\n<p>Now you try it.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Identify whether the leading term is positive or negative and whether the degree is even or odd for the following graphs of polynomial functions.<\/p>\n<p>a)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201336\/CNX_Precalc_Figure_03_03_0122.jpg\" alt=\"Graph of f(x)=-2x^6-x^5+3x^4+x^3.\" width=\"214\" height=\"227\" \/><\/p>\n<p>b)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/11\/25201339\/CNX_Precalc_Figure_03_03_0142.jpg\" alt=\"Graph of f(x)=-6x^3+7x^2+3x+1.\" width=\"217\" height=\"230\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q317874\">Show Solution<\/span><\/p>\n<div id=\"q317874\" class=\"hidden-answer\" style=\"display: none\">\n<p>a) Both arms of this polynomial point in the same direction so it must have an even degree. \u00a0The leading term of the polynomial must be negative since the arms are pointing downward.<\/p>\n<p>b) The arms of this polynomial point in different directions, so the degree must be odd. As the inputs get really big and positive, the outputs get really big and negative, so the leading coefficient must be negative.<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-206\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay, et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":395986,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay, et al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at: http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-206","chapter","type-chapter","status-publish","hentry"],"part":202,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/206","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/users\/395986"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/206\/revisions"}],"predecessor-version":[{"id":1513,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/206\/revisions\/1513"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/202"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/206\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/media?parent=206"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=206"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/contributor?post=206"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/license?post=206"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}