{"id":209,"date":"2023-06-21T13:22:44","date_gmt":"2023-06-21T13:22:44","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/describe-the-end-behavior-of-power-functions\/"},"modified":"2023-10-05T15:50:25","modified_gmt":"2023-10-05T15:50:25","slug":"describe-the-end-behavior-of-power-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/describe-the-end-behavior-of-power-functions\/","title":{"raw":"\u25aa   End Behavior of Power Functions","rendered":"\u25aa   End Behavior of Power Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify a power function.<\/li>\r\n \t<li>Describe the end behavior of a power function given its equation or graph.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"mceTemp\"><\/div>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194447\/CNX_Precalc_Figure_03_03_0012.jpg\" alt=\"Three birds on a cliff with the sun rising in the background.\" width=\"488\" height=\"366\" \/> Three birds on a cliff with the sun rising in the background. Functions discussed in this module can be used to model populations of various animals, including birds. (credit: Jason Bay, Flickr)[\/caption]\r\n\r\nSuppose a certain species of bird thrives on a small island. Its population over the last few years is shown below.\r\n<table summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Year<\/strong><\/td>\r\n<td>2009<\/td>\r\n<td>2010<\/td>\r\n<td>2011<\/td>\r\n<td>2012<\/td>\r\n<td>2013<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong>Bird Population<\/strong><\/td>\r\n<td>800<\/td>\r\n<td>897<\/td>\r\n<td>992<\/td>\r\n<td>1,083<\/td>\r\n<td>1,169<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe 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 <i>t<\/i>\u00a0years 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.\r\n<h3>Identifying Power Functions<\/h3>\r\nIn order to better understand the bird problem, we need to understand a specific type of function. A <strong>power function <\/strong>is a function with a single term that is the product of a real number,\u00a0<strong>coefficient,<\/strong> and variable raised to a fixed real number power. Keep in mind a number that multiplies a variable raised to an exponent is known as a <strong>coefficient<\/strong>.\r\n\r\nAs an example, consider functions for area or volume. The function for the <strong>area of a circle<\/strong> with radius [latex]r[\/latex]<i> <\/i>is:\r\n<p style=\"text-align: center;\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/p>\r\nand the function for the <strong>volume of a sphere<\/strong> with radius <em>r<\/em>\u00a0is:\r\n<p style=\"text-align: center;\">[latex]V\\left(r\\right)=\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/p>\r\nBoth 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 <em>r<\/em>\u00a0raised to a power.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Power FunctionS<\/h3>\r\nA <strong>power function<\/strong> is a function that can be represented in the form\r\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{x}^{n}[\/latex]<\/p>\r\nwhere <i>a<\/i>\u00a0and <i>n<\/i>\u00a0are real numbers and <em>a<\/em><i>\u00a0<\/i>is known as the <strong>coefficient<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"textbox\">\r\n<h3>Q &amp; A<\/h3>\r\n<strong>Is [latex]f\\left(x\\right)={2}^{x}[\/latex] a power function?<\/strong>\r\n\r\n<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>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>tip for success<\/h3>\r\nUnlike a polynomial function, in which all the variable powers must be non-negative integers, a power function only requires the power on the exponent be a real number.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying Power Functions<\/h3>\r\nWhich of the following functions are power functions?\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)=1\\hfill &amp; \\text{Constant function}\\hfill \\\\ f\\left(x\\right)=x\\hfill &amp; \\text{Identity function}\\hfill \\\\ f\\left(x\\right)={x}^{2}\\hfill &amp; \\text{Quadratic}\\text{ }\\text{ function}\\hfill \\\\ f\\left(x\\right)={x}^{3}\\hfill &amp; \\text{Cubic function}\\hfill \\\\ f\\left(x\\right)=\\frac{1}{x} \\hfill &amp; \\text{Reciprocal function}\\hfill \\\\ f\\left(x\\right)=\\frac{1}{{x}^{2}}\\hfill &amp; \\text{Reciprocal squared function}\\hfill \\\\ f\\left(x\\right)=\\sqrt{x}\\hfill &amp; \\text{Square root function}\\hfill \\\\ f\\left(x\\right)=\\sqrt[3]{x}\\hfill &amp; \\text{Cube root function}\\hfill \\end{array}[\/latex]<\/p>\r\n[reveal-answer q=\"82786\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"82786\"]\r\n\r\nAll of the listed functions are power functions.\r\n\r\nThe 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.\r\n\r\nThe 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].\r\n\r\nThe <strong>reciprocal<\/strong> 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].\r\n\r\nThe square and <strong>cube root<\/strong> functions are power functions with fractional powers because they can be written as [latex]f\\left(x\\right)={x}^{1\/2}[\/latex] or [latex]f\\left(x\\right)={x}^{1\/3}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nWhich functions are power functions?\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)=2{x}^{2}\\cdot 4{x}^{3}\\hfill \\\\ g\\left(x\\right)=-{x}^{5}+5{x}^{3}-4x\\hfill \\\\ h\\left(x\\right)=\\frac{2{x}^{5}-1}{3{x}^{2}+4}\\hfill \\end{array}[\/latex]<\/p>\r\n[reveal-answer q=\"105254\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"105254\"]\r\n\r\n[latex]f\\left(x\\right)[\/latex]\u00a0is a power function because it can be written as [latex]f\\left(x\\right)=8{x}^{5}[\/latex].\u00a0The other functions are not power functions.[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Identifying End Behavior of Power Functions<\/h3>\r\nThe graph below shows the graphs of [latex]f\\left(x\\right)={x}^{2},g\\left(x\\right)={x}^{4}[\/latex], [latex]h\\left(x\\right)={x}^{6}[\/latex], [latex]k(x)=x^{8}[\/latex], and [latex]p(x)=x^{10}[\/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. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.\r\n\r\n<img class=\"wp-image-6761 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/11\/08231145\/Screen-Shot-2019-07-08-at-4.09.51-PM.png\" alt=\"\" width=\"145\" height=\"251\" \/>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<img class=\"wp-image-6762 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/11\/08231150\/Screen-Shot-2019-07-08-at-4.10.10-PM.png\" alt=\"\" width=\"465\" height=\"500\" \/>\r\n\r\nTo 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]-\\infty[\/latex] for negative infinity. When we say that \"<em>x<\/em> approaches infinity,\" which can be symbolically written as [latex]x\\to \\infty[\/latex], we are describing a behavior; we are saying that <em>x<\/em>\u00a0is increasing without bound.\r\n\r\nWith even-powered power functions, 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\r\n<p style=\"text-align: center;\">[latex]\\text{as }x\\to \\pm \\infty , f\\left(x\\right)\\to \\infty[\/latex]<\/p>\r\nThe graph below shows [latex]f\\left(x\\right)={x}^{3},g\\left(x\\right)={x}^{5},h\\left(x\\right)={x}^{7},k\\left(x\\right)={x}^{9},\\text{and }p\\left(x\\right)={x}^{11}[\/latex], which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function. As the power increases, the graphs flatten near the origin and become steeper away from the origin.\r\n\r\n<img class=\" wp-image-6764 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/11\/08231322\/Screen-Shot-2019-07-08-at-4.10.28-PM.png\" alt=\"\" width=\"146\" height=\"251\" \/> <img class=\" wp-image-6765 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/11\/08231327\/Screen-Shot-2019-07-08-at-4.11.02-PM.png\" alt=\"\" width=\"466\" height=\"500\" \/>\r\n\r\nThese examples illustrate that functions of the form [latex]f\\left(x\\right)={x}^{n}[\/latex] reveal symmetry of one kind or another. First, in the even-powered power functions, we see that even functions of the form [latex]f\\left(x\\right)={x}^{n}\\text{, }n\\text{ even,}[\/latex] are symmetric about the <em>y<\/em>-axis. In the odd-powered power functions, we see that odd functions of the form [latex]f\\left(x\\right)={x}^{n}\\text{, }n\\text{ odd,}[\/latex] are symmetric about the origin.\r\n\r\nFor these odd power functions, as <em>x<\/em>\u00a0approaches negative infinity, [latex]f\\left(x\\right)[\/latex]\u00a0decreases without bound. As <em>x<\/em>\u00a0approaches positive infinity, [latex]f\\left(x\\right)[\/latex]\u00a0increases without bound. In symbolic form we write\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\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]<\/p>\r\nThe 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 <strong>end behavior<\/strong> of the function. We can use words or symbols to describe end behavior.\r\n\r\nThe table\u00a0below shows the end behavior of power functions of the form [latex]f\\left(x\\right)=a{x}^{n}[\/latex] where [latex]n[\/latex] is a non-negative integer depending on the power and the constant.\r\n<table style=\"height: 479px;\">\r\n<thead>\r\n<tr style=\"height: 15px;\">\r\n<th style=\"height: 15px;\"><\/th>\r\n<th style=\"height: 15px; text-align: center;\">Even power<\/th>\r\n<th style=\"height: 15px; text-align: center;\">Odd power<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 230px;\">\r\n<td style=\"height: 230px;\"><strong>Positive constant<\/strong><strong><i>a<\/i> &gt; 0<\/strong><\/td>\r\n<td style=\"height: 230px;\"><img class=\"alignnone size-full wp-image-4485\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18231809\/Table1.png\" 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.\" width=\"356\" height=\"460\" \/><\/td>\r\n<td style=\"height: 230px;\"><img class=\"alignnone size-full wp-image-4487\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18231957\/Table2.png\" alt=\"Graph of an odd-powered function with a positive constant. As x goes to negative infinity, the function goes to negative infinity; as x goes to positive infinity, the function goes to positive infinity.\" width=\"359\" height=\"458\" \/><\/td>\r\n<\/tr>\r\n<tr style=\"height: 234px;\">\r\n<td style=\"height: 234px;\"><strong>Negative constant<\/strong><strong><i>a<\/i> &lt; 0<\/strong><\/td>\r\n<td style=\"height: 234px;\"><img class=\"alignnone size-full wp-image-4488\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18232026\/Table3.png\" alt=\"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.\" width=\"375\" height=\"460\" \/><\/td>\r\n<td style=\"height: 234px;\"><img class=\"alignnone size-full wp-image-4489\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18232106\/Table4.png\" alt=\"Graph of an odd-powered function with a negative 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.\" width=\"342\" height=\"464\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a power function [latex]f\\left(x\\right)=a{x}^{n}[\/latex] where [latex]n[\/latex]\u00a0is a non-negative integer, identify the end behavior.<\/h3>\r\n<ol id=\"fs-id1165137409522\">\r\n \t<li>Determine whether the power is even or odd.<\/li>\r\n \t<li>Determine whether the constant is positive or negative.<\/li>\r\n \t<li>Use the above graphs to identify the end behavior.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying the End Behavior of a Power Function<\/h3>\r\nDescribe the end behavior of the graph of [latex]f\\left(x\\right)={x}^{8}[\/latex].\r\n\r\n[reveal-answer q=\"556064\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"556064\"]\r\n\r\nThe coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). As <em>x\u00a0<\/em>(input)\u00a0approaches infinity, [latex]f\\left(x\\right)[\/latex] (output) increases without bound. We write as [latex]x\\to \\infty , f\\left(x\\right)\\to \\infty [\/latex]. As <em>x<\/em>\u00a0approaches 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.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194503\/CNX_Precalc_Figure_03_03_0082.jpg\" alt=\"Graph of f(x)=x^8.\" width=\"487\" height=\"330\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Identifying the End Behavior of a Power Function<\/h3>\r\nDescribe the end behavior of the graph of [latex]f\\left(x\\right)=-{x}^{9}[\/latex].\r\n\r\n[reveal-answer q=\"631242\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"631242\"]\r\n\r\nThe exponent of the power function is 9 (an odd number). Because the coefficient is \u20131 (negative), the graph is the reflection about the <em>x<\/em>-axis of the graph of [latex]f\\left(x\\right)={x}^{9}[\/latex]. The graph\u00a0shows that as <em>x<\/em>\u00a0approaches infinity, the output decreases without bound. As <em>x<\/em>\u00a0approaches negative infinity, the output increases without bound. In symbolic form, we would write as [latex]x\\to -\\infty , f\\left(x\\right)\\to \\infty[\/latex] and as [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty[\/latex].\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194505\/CNX_Precalc_Figure_03_03_0092.jpg\" alt=\"Graph of f(x)=-x^9.\" width=\"487\" height=\"667\" \/>\r\n<h4>Analysis of the Solution<\/h4>\r\nWe can check our work by using the table feature on an online graphing calculator.\r\n<ol>\r\n \t<li>Enter the function\u00a0[latex]f\\left(x\\right)=-{x}^{9}[\/latex] into an online graphing calculator<\/li>\r\n \t<li>Create a table with the following x values, and observe the sign of the outputs. [latex]-10,-5,0,5,10[\/latex]<\/li>\r\n \t<li>Now, enter the function\u00a0[latex]g\\left(x\\right)={x}^{9}[\/latex], and create a similar table. Compare the signs of the outputs for both functions.<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nDescribe in words and symbols the end behavior of [latex]f\\left(x\\right)=-5{x}^{4}[\/latex].\r\n\r\n[reveal-answer q=\"582534\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"582534\"]\r\n\r\nAs <em>x<\/em>\u00a0approaches positive or negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound: as [latex]x\\to \\pm \\infty , f\\left(x\\right)\\to -\\infty[\/latex] because of the negative coefficient.\r\n\r\n[\/hidden-answer]\r\n\r\n[ohm_question height=\"200\"]69337[\/ohm_question]\r\n\r\n[ohm_question height=\"200\"]15940[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify a power function.<\/li>\n<li>Describe the end behavior of a power function given its equation or graph.<\/li>\n<\/ul>\n<\/div>\n<div class=\"mceTemp\"><\/div>\n<div style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194447\/CNX_Precalc_Figure_03_03_0012.jpg\" alt=\"Three birds on a cliff with the sun rising in the background.\" width=\"488\" height=\"366\" \/><\/p>\n<p class=\"wp-caption-text\">Three birds on a cliff with the sun rising in the background. Functions discussed in this module can be used to model populations of various animals, including birds. (credit: Jason Bay, Flickr)<\/p>\n<\/div>\n<p>Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown below.<\/p>\n<table summary=\"..\">\n<tbody>\n<tr>\n<td><strong>Year<\/strong><\/td>\n<td>2009<\/td>\n<td>2010<\/td>\n<td>2011<\/td>\n<td>2012<\/td>\n<td>2013<\/td>\n<\/tr>\n<tr>\n<td><strong>Bird Population<\/strong><\/td>\n<td>800<\/td>\n<td>897<\/td>\n<td>992<\/td>\n<td>1,083<\/td>\n<td>1,169<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>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 <i>t<\/i>\u00a0years 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.<\/p>\n<h3>Identifying Power Functions<\/h3>\n<p>In order to better understand the bird problem, we need to understand a specific type of function. A <strong>power function <\/strong>is a function with a single term that is the product of a real number,\u00a0<strong>coefficient,<\/strong> and variable raised to a fixed real number power. Keep in mind a number that multiplies a variable raised to an exponent is known as a <strong>coefficient<\/strong>.<\/p>\n<p>As an example, consider functions for area or volume. The function for the <strong>area of a circle<\/strong> with radius [latex]r[\/latex]<i> <\/i>is:<\/p>\n<p style=\"text-align: center;\">[latex]A\\left(r\\right)=\\pi {r}^{2}[\/latex]<\/p>\n<p>and the function for the <strong>volume of a sphere<\/strong> with radius <em>r<\/em>\u00a0is:<\/p>\n<p style=\"text-align: center;\">[latex]V\\left(r\\right)=\\frac{4}{3}\\pi {r}^{3}[\/latex]<\/p>\n<p>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 <em>r<\/em>\u00a0raised to a power.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Power FunctionS<\/h3>\n<p>A <strong>power function<\/strong> is a function that can be represented in the form<\/p>\n<p style=\"text-align: center;\">[latex]f\\left(x\\right)=a{x}^{n}[\/latex]<\/p>\n<p>where <i>a<\/i>\u00a0and <i>n<\/i>\u00a0are real numbers and <em>a<\/em><i>\u00a0<\/i>is known as the <strong>coefficient<\/strong>.<\/p>\n<\/div>\n<div class=\"textbox\">\n<h3>Q &amp; A<\/h3>\n<p><strong>Is [latex]f\\left(x\\right)={2}^{x}[\/latex] a power function?<\/strong><\/p>\n<p><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 class=\"textbox examples\">\n<h3>tip for success<\/h3>\n<p>Unlike a polynomial function, in which all the variable powers must be non-negative integers, a power function only requires the power on the exponent be a real number.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying Power Functions<\/h3>\n<p>Which of the following functions are power functions?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)=1\\hfill & \\text{Constant function}\\hfill \\\\ f\\left(x\\right)=x\\hfill & \\text{Identity function}\\hfill \\\\ f\\left(x\\right)={x}^{2}\\hfill & \\text{Quadratic}\\text{ }\\text{ function}\\hfill \\\\ f\\left(x\\right)={x}^{3}\\hfill & \\text{Cubic function}\\hfill \\\\ f\\left(x\\right)=\\frac{1}{x} \\hfill & \\text{Reciprocal function}\\hfill \\\\ f\\left(x\\right)=\\frac{1}{{x}^{2}}\\hfill & \\text{Reciprocal squared function}\\hfill \\\\ f\\left(x\\right)=\\sqrt{x}\\hfill & \\text{Square root function}\\hfill \\\\ f\\left(x\\right)=\\sqrt[3]{x}\\hfill & \\text{Cube root function}\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q82786\">Show Solution<\/span><\/p>\n<div id=\"q82786\" class=\"hidden-answer\" style=\"display: none\">\n<p>All of the listed functions are power functions.<\/p>\n<p>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>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>The <strong>reciprocal<\/strong> 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>The square and <strong>cube root<\/strong> functions are power functions with fractional powers because they can be written as [latex]f\\left(x\\right)={x}^{1\/2}[\/latex] or [latex]f\\left(x\\right)={x}^{1\/3}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Which functions are power functions?<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}f\\left(x\\right)=2{x}^{2}\\cdot 4{x}^{3}\\hfill \\\\ g\\left(x\\right)=-{x}^{5}+5{x}^{3}-4x\\hfill \\\\ h\\left(x\\right)=\\frac{2{x}^{5}-1}{3{x}^{2}+4}\\hfill \\end{array}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q105254\">Show Solution<\/span><\/p>\n<div id=\"q105254\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f\\left(x\\right)[\/latex]\u00a0is a power function because it can be written as [latex]f\\left(x\\right)=8{x}^{5}[\/latex].\u00a0The other functions are not power functions.<\/p><\/div>\n<\/div>\n<\/div>\n<h3>Identifying End Behavior of Power Functions<\/h3>\n<p>The graph below shows the graphs of [latex]f\\left(x\\right)={x}^{2},g\\left(x\\right)={x}^{4}[\/latex], [latex]h\\left(x\\right)={x}^{6}[\/latex], [latex]k(x)=x^{8}[\/latex], and [latex]p(x)=x^{10}[\/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. 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=\"wp-image-6761 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/11\/08231145\/Screen-Shot-2019-07-08-at-4.09.51-PM.png\" alt=\"\" width=\"145\" height=\"251\" \/>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<img loading=\"lazy\" decoding=\"async\" class=\"wp-image-6762 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/11\/08231150\/Screen-Shot-2019-07-08-at-4.10.10-PM.png\" alt=\"\" width=\"465\" height=\"500\" \/><\/p>\n<p>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]-\\infty[\/latex] for negative infinity. When we say that &#8220;<em>x<\/em> approaches infinity,&#8221; which can be symbolically written as [latex]x\\to \\infty[\/latex], we are describing a behavior; we are saying that <em>x<\/em>\u00a0is increasing without bound.<\/p>\n<p>With even-powered power functions, 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<p style=\"text-align: center;\">[latex]\\text{as }x\\to \\pm \\infty , f\\left(x\\right)\\to \\infty[\/latex]<\/p>\n<p>The graph below shows [latex]f\\left(x\\right)={x}^{3},g\\left(x\\right)={x}^{5},h\\left(x\\right)={x}^{7},k\\left(x\\right)={x}^{9},\\text{and }p\\left(x\\right)={x}^{11}[\/latex], which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function. As the power increases, the graphs flatten near the origin and become steeper away from the origin.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-6764 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/11\/08231322\/Screen-Shot-2019-07-08-at-4.10.28-PM.png\" alt=\"\" width=\"146\" height=\"251\" \/> <img loading=\"lazy\" decoding=\"async\" class=\"wp-image-6765 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3794\/2016\/11\/08231327\/Screen-Shot-2019-07-08-at-4.11.02-PM.png\" alt=\"\" width=\"466\" height=\"500\" \/><\/p>\n<p>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 the even-powered power functions, we see that even functions of the form [latex]f\\left(x\\right)={x}^{n}\\text{, }n\\text{ even,}[\/latex] are symmetric about the <em>y<\/em>-axis. In the odd-powered power functions, we see that odd functions of the form [latex]f\\left(x\\right)={x}^{n}\\text{, }n\\text{ odd,}[\/latex] are symmetric about the origin.<\/p>\n<p>For these odd power functions, as <em>x<\/em>\u00a0approaches negative infinity, [latex]f\\left(x\\right)[\/latex]\u00a0decreases without bound. As <em>x<\/em>\u00a0approaches positive infinity, [latex]f\\left(x\\right)[\/latex]\u00a0increases without bound. In symbolic form we write<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}\\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]<\/p>\n<p>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 <strong>end behavior<\/strong> of the function. We can use words or symbols to describe end behavior.<\/p>\n<p>The table\u00a0below shows the end behavior of power functions of the form [latex]f\\left(x\\right)=a{x}^{n}[\/latex] where [latex]n[\/latex] is a non-negative integer depending on the power and the constant.<\/p>\n<table style=\"height: 479px;\">\n<thead>\n<tr style=\"height: 15px;\">\n<th style=\"height: 15px;\"><\/th>\n<th style=\"height: 15px; text-align: center;\">Even power<\/th>\n<th style=\"height: 15px; text-align: center;\">Odd power<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 230px;\">\n<td style=\"height: 230px;\"><strong>Positive constant<\/strong><strong><i>a<\/i> &gt; 0<\/strong><\/td>\n<td style=\"height: 230px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-4485\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18231809\/Table1.png\" 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.\" width=\"356\" height=\"460\" \/><\/td>\n<td style=\"height: 230px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-4487\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18231957\/Table2.png\" alt=\"Graph of an odd-powered function with a positive constant. As x goes to negative infinity, the function goes to negative infinity; as x goes to positive infinity, the function goes to positive infinity.\" width=\"359\" height=\"458\" \/><\/td>\n<\/tr>\n<tr style=\"height: 234px;\">\n<td style=\"height: 234px;\"><strong>Negative constant<\/strong><strong><i>a<\/i> &lt; 0<\/strong><\/td>\n<td style=\"height: 234px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-4488\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18232026\/Table3.png\" alt=\"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.\" width=\"375\" height=\"460\" \/><\/td>\n<td style=\"height: 234px;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-4489\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/1916\/2016\/11\/18232106\/Table4.png\" alt=\"Graph of an odd-powered function with a negative 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.\" width=\"342\" height=\"464\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox\">\n<h3>How To: Given a power function [latex]f\\left(x\\right)=a{x}^{n}[\/latex] where [latex]n[\/latex]\u00a0is a non-negative integer, identify the end behavior.<\/h3>\n<ol id=\"fs-id1165137409522\">\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 the above graphs to identify the end behavior.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying the End Behavior of a Power Function<\/h3>\n<p>Describe the end behavior of the graph of [latex]f\\left(x\\right)={x}^{8}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q556064\">Show Solution<\/span><\/p>\n<div id=\"q556064\" class=\"hidden-answer\" style=\"display: none\">\n<p>The coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). As <em>x\u00a0<\/em>(input)\u00a0approaches infinity, [latex]f\\left(x\\right)[\/latex] (output) increases without bound. We write as [latex]x\\to \\infty , f\\left(x\\right)\\to \\infty[\/latex]. As <em>x<\/em>\u00a0approaches 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.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194503\/CNX_Precalc_Figure_03_03_0082.jpg\" alt=\"Graph of f(x)=x^8.\" width=\"487\" height=\"330\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Identifying the End Behavior of a Power Function<\/h3>\n<p>Describe the end behavior of the graph of [latex]f\\left(x\\right)=-{x}^{9}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q631242\">Show Solution<\/span><\/p>\n<div id=\"q631242\" class=\"hidden-answer\" style=\"display: none\">\n<p>The exponent of the power function is 9 (an odd number). Because the coefficient is \u20131 (negative), the graph is the reflection about the <em>x<\/em>-axis of the graph of [latex]f\\left(x\\right)={x}^{9}[\/latex]. The graph\u00a0shows that as <em>x<\/em>\u00a0approaches infinity, the output decreases without bound. As <em>x<\/em>\u00a0approaches negative infinity, the output increases without bound. In symbolic form, we would write as [latex]x\\to -\\infty , f\\left(x\\right)\\to \\infty[\/latex] and as [latex]x\\to \\infty , f\\left(x\\right)\\to -\\infty[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/02194505\/CNX_Precalc_Figure_03_03_0092.jpg\" alt=\"Graph of f(x)=-x^9.\" width=\"487\" height=\"667\" \/><\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>We can check our work by using the table feature on an online graphing calculator.<\/p>\n<ol>\n<li>Enter the function\u00a0[latex]f\\left(x\\right)=-{x}^{9}[\/latex] into an online graphing calculator<\/li>\n<li>Create a table with the following x values, and observe the sign of the outputs. [latex]-10,-5,0,5,10[\/latex]<\/li>\n<li>Now, enter the function\u00a0[latex]g\\left(x\\right)={x}^{9}[\/latex], and create a similar table. Compare the signs of the outputs for both functions.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Describe in words and symbols the end behavior of [latex]f\\left(x\\right)=-5{x}^{4}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q582534\">Show Solution<\/span><\/p>\n<div id=\"q582534\" class=\"hidden-answer\" style=\"display: none\">\n<p>As <em>x<\/em>\u00a0approaches positive or negative infinity, [latex]f\\left(x\\right)[\/latex] decreases without bound: as [latex]x\\to \\pm \\infty , f\\left(x\\right)\\to -\\infty[\/latex] because of the negative coefficient.<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm69337\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=69337&theme=oea&iframe_resize_id=ohm69337&show_question_numbers\" width=\"100%\" height=\"200\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm15940\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=15940&theme=oea&iframe_resize_id=ohm15940&show_question_numbers\" width=\"100%\" height=\"200\"><\/iframe><\/p>\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-209\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Even Power Functions Interactive. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/rgdspbzldy\">https:\/\/www.desmos.com\/calculator\/rgdspbzldy<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><li>Odd Power Functions Interactive. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/4aeczmnp1w\">https:\/\/www.desmos.com\/calculator\/4aeczmnp1w<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 69337. <strong>Authored by<\/strong>: Shahbazian, Roy. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 15940. <strong>Authored by<\/strong>: Sousa, James. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Ex: End Behavior or Long Run Behavior of Functions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/Krjd_vU4Uvg\">https:\/\/youtu.be\/Krjd_vU4Uvg<\/a>. <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>\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":7,"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 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