{"id":622,"date":"2022-01-19T17:47:36","date_gmt":"2022-01-19T17:47:36","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=622"},"modified":"2026-01-17T00:48:42","modified_gmt":"2026-01-17T00:48:42","slug":"1-4-the-algebraic-forms-of-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/1-4-the-algebraic-forms-of-functions\/","title":{"raw":"1.4.1: Algebraic Forms of Functions","rendered":"1.4.1: Algebraic Forms of Functions"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Objectives<\/h1>\r\nIdentify types of functions based on their algebraic forms\r\n<ul>\r\n \t<li style=\"margin-top: 0.5em;\">The algebraic form of polynomial functions<\/li>\r\n \t<li>The algebraic form of exponential functions<\/li>\r\n \t<li>The algebraic form of logarithmic functions<\/li>\r\n \t<li>The algebraic form of rational functions<\/li>\r\n<\/ul>\r\nUse graphing software to graph functions.\r\n\r\n<\/div>\r\n<h2>Types of Functions<\/h2>\r\nThere are many types of functions that can be written in algebraic notation. <span style=\"font-size: 1em;\">Functions can be broadly classified according to<\/span><span style=\"font-size: 1em;\">\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">the position of the independent variable in its algebraic form. The following four types of functions are defined by the position of the independent variable. Throughout this course we will use the Desmos Graphing Calculator to visualize functions. This is a free online calculator that is found at\u00a0<a href=\"https:\/\/www.desmos.com\/calculator\">desmos calculator<\/a>.\u00a0 You can create a free account that will enable you to save and print your graphs.<\/span>\r\n<h3>Polynomial Functions<\/h3>\r\nA <em><strong>polynomial function<\/strong><\/em> is a function where the variable is in the \"base\" position and the exponent on the variable is\u00a0a whole number. Recall that the set of whole numbers start at 0 and include all of the positive integers:\u00a0 [latex]\\mathbb{W} = \\{0, 1, 2, 3, 4, ...\\}[\/latex]. Examples of polynomial functions are [latex]f(x)=x^2,\\;g(x)=3x^6-7x^5+2x^4+3,\\;h(x)=5x-7,\\;T(x)=8[\/latex]. The polynomial function [latex]T(x)=8[\/latex] is an example of a <em><strong>constant function<\/strong><\/em> where the input value is irrelevant as the output will always be 8. The graph of a constant function is always a horizontal line (see figure 1).\u00a0 [latex]h(x)=5x-7[\/latex] is an example of a linear function where the exponent on the variable is 1.\u00a0The graph of a <em><strong>linear function<\/strong><\/em> is always a line (see figure 1).\r\n\r\n[caption id=\"attachment_1736\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1736 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-54-300x300.png\" alt=\"A constant function (horizontal line), and a non-constant linear function increasing function.\" width=\"300\" height=\"300\" \/> Figure 1. Constant function and linear function[\/caption]\r\n\r\nWhen the highest exponent on a variable is at least 2, the graph is a curve.\u00a0Figure 2 illustrates the graphs of the two polynomial functions [latex]\\color{blue}{f(x)=x^2}[\/latex] and [latex]\\color{green}{f(x)=x^3}[\/latex]. Notice that these functions are a completely different shape. However, they both pass through the points (0, 0) and (1, 1). All polynomial functions are continuous with a domain of [latex](-\\infty, +\\infty)[\/latex], which means that the independent variable [latex]x[\/latex] can take on any real number value.\r\n\r\n[caption id=\"attachment_1737\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1737 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-55-300x300.png\" alt=\"Two polynomial functions. One with a high power of 2 that goes up on both ends, and one with a high power of 3 that goes up on the right and down on the left.\" width=\"300\" height=\"300\" \/> Figure 2. Polynomial functions[\/caption]\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>POLYNOMIAL FUNCTIONS<\/h3>\r\nA polynomial function is any function of the form [latex]f(x)=a_nx^n+a_{n-1}x^{n-1} +...+a_1x+a_0[\/latex] where [latex]n[\/latex] in a whole number and [latex]a_i[\/latex] are real number constants.\r\n\r\nAll polynomial functions of the form [latex]f(x)=ax^n[\/latex] with [latex]n\u22651[\/latex] pass through the point [latex](0, 0)[\/latex] and [latex](1, a)[\/latex].\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nUse the <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos Graphing Calculator<\/a> to determine the shape of the functions:\r\n\r\n[latex]f(x)=x^4[\/latex]\r\n\r\n[latex]g(x)=x^3-4x^2+5x-2[\/latex]\r\n\r\n[latex]h(x)=x^6-2x^3+x[\/latex]\r\n<h4><strong>Solution<\/strong><\/h4>\r\nGo to the <a href=\"https:\/\/www.desmos.com\/calculator\/ko1xkeqwhk\">Desmos Graphing Calculator<\/a> and type each function into a box on the left side of the screen. Use the ^ (carat) key on your keyboard for exponents.\r\n\r\n\u200b\r\n<table class=\"aligncenter\" style=\"border-collapse: collapse; width: 100.355%; height: 268px;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 33.3333%; vertical-align: bottom; text-align: center;\">\r\n<div class=\"mceTemp\">4th degree only<\/div><\/th>\r\n<th style=\"width: 33.3333%; vertical-align: bottom; text-align: center;\">\r\n<div class=\"mceTemp\">3rd degree with extra terms<\/div><\/th>\r\n<th style=\"width: 33.3333%; vertical-align: bottom; text-align: center;\">\r\n<div class=\"mceTemp\">4th degree with extra terms<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%; vertical-align: bottom;\">[caption id=\"attachment_1005\" align=\"aligncenter\" width=\"165\"]<img class=\"wp-image-1005\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09200602\/fxx%5E4-258x300.png\" alt=\"Graph of the function f(x)=x^4. Abroad U with steep sides.\" width=\"165\" height=\"192\" \/> [latex]f(x)=x^4[\/latex][\/caption]<\/td>\r\n<td style=\"width: 33.3333%; vertical-align: bottom;\">[caption id=\"attachment_1004\" align=\"aligncenter\" width=\"147\"]<img class=\"wp-image-1004\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09200557\/cubic-function-233x300.png\" alt=\"Graph of [latex]g(x)=x^3-4x^2+5x-2[\/latex]. Up on right, down on left, a local max and a local min.\" width=\"147\" height=\"189\" \/> [latex]g(x)=x^3-4x^2+5x-2[\/latex][\/caption]<\/td>\r\n<td style=\"width: 33.3333%; vertical-align: bottom;\">[caption id=\"attachment_4231\" align=\"aligncenter\" width=\"174\"]<img class=\"wp-image-4231\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/26220928\/desmos-graph-2022-08-26T160836.875-300x300.png\" alt=\"h(x)=x^6-2x^3+x, Up on left, up on right, with two local mins and one local max.\" width=\"174\" height=\"174\" \/> [latex]h(x)=x^6-2x^3+x[\/latex][\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nUse the <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos Graphing Calculator<\/a> to determine the shape of the functions:\r\n\r\n[latex]f(x)=x^5[\/latex]\r\n\r\n[latex]g(x)=3x-4[\/latex]\r\n\r\n[latex]h(x)=x^6-3x^3+2x-5[\/latex]\r\n\r\n[reveal-answer q=\"hjm438\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm438\"]\r\n\r\n\u200b\u200b\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">5th degree function<\/div><\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">1st degree function<\/div><\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">4th degree function<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1742\" align=\"aligncenter\" width=\"270\"]<img class=\"wp-image-1742\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/22202424\/desmos-graph-56-300x300.png\" alt=\"x^5\" width=\"270\" height=\"270\" \/> [latex]f(x)=x^5[\/latex][\/caption]<\/td>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1744\" align=\"aligncenter\" width=\"270\"]<img class=\"wp-image-1744\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/22202624\/desmos-graph-57-300x300.png\" alt=\"3x-4\" width=\"270\" height=\"270\" \/> [latex]g(x)=3x-4[\/latex][\/caption]<\/td>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1746\" align=\"aligncenter\" width=\"270\"]<img class=\"wp-image-1746\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/22204349\/desmos-graph-58-300x300.png\" alt=\"6th degree polynomial\" width=\"270\" height=\"270\" \/> [latex]h(x)=x^6-3x^3+2x-5[\/latex][\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<h3>Exponential Functions<\/h3>\r\nAn <em><strong>exponential function<\/strong><\/em> is a function where the independent variable sits at the exponent position. The functions [latex]f(x)=4^x,\\;g(x)=(3.2)^x, \\;h(x)=\\left (\\frac{1}{5}\\right )^x[\/latex] are all examples of exponential functions: a positive real number raised to a power of [latex]x[\/latex]. Figure 3 illustrates the graphs of four exponential functions: [latex]f(x)=2^x, \\;g(x)=3^x, \\;r(x)=5^x, s(x)=\\;10^x[\/latex]. Notice that each of these graphs have the same basic shape. They slowly come up from the [latex]x[\/latex]-axis then quickly move towards [latex]+\\infty[\/latex].\u00a0 The graphs never quite make it to the line [latex]y=0[\/latex] (the [latex]x[\/latex]-axis) and this line is referred to as a horizontal asymptote; a horizontal line that the graph never quite reaches as it moves towards either positive or negative (in this case) infinity. In addition, they all pass through the point (0, 1).\r\n\r\n[caption id=\"attachment_844\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-844 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-ExpFun1-300x300.png\" alt=\"Four exponential graphs described above.\" width=\"300\" height=\"300\" \/> Figure 3. Exponential functions.[\/caption]\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>exponential FUNCTIONS<\/h3>\r\nAn exponential function is any function of the form [latex]f(x)=a^x[\/latex] where [latex]a[\/latex] is a positive real number and [latex]a\\ne1[\/latex].\r\n\r\nAll exponential functions of the form [latex]f(x)=a^x[\/latex] pass through the point (0, 1) and the line [latex]y=0[\/latex] is a horizontal asymptote.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nUse the <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos Graphing Calculator<\/a> to determine the shape of the functions:\r\n\r\n[latex]f(x)=4^x[\/latex]\r\n\r\n[latex]g(x)=\\left (\\frac{1}{4}\\right )^x[\/latex]\r\n\r\n[latex]h(x)=(3.5)^x[\/latex]\r\n<h4><strong>Solution<\/strong><\/h4>\r\nGo to the <a href=\"https:\/\/www.desmos.com\/calculator\/bjxla254vm\">Desmos Graphing Calculator<\/a> and type each function into a box on the left side of the screen.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">Exponential<\/div><\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">Exponential reflected in y<\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">Exponential<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1010\" align=\"aligncenter\" width=\"249\"]<img class=\"wp-image-1010\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09203859\/fx4%5Ex-300x241.png\" alt=\"Graph of f(x)=4^x. Approaches an asymptote of the negative x axis, passes through (0,1), and increases quickly on the right.\" width=\"249\" height=\"200\" \/> [latex]f(x)=4^x[\/latex][\/caption]<\/td>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1011\" align=\"aligncenter\" width=\"236\"]<img class=\"wp-image-1011\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09204129\/gx14%5Ex-300x253.png\" alt=\"Graph of g(x)=(1\/4)^x. Increases quickly on the left, passes through (0,1), and approaches the positive x axis as an asymptote.\" width=\"236\" height=\"199\" \/> [latex]g(x)=\\left (\\frac{1}{4}\\right )^x[\/latex][\/caption]<\/td>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1012\" align=\"aligncenter\" width=\"250\"]<img class=\"wp-image-1012\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09204335\/hx35%5Ex-300x240.png\" alt=\"Graph of h(x)=3.5^x. Approaches an asymptote of the negative x axis, passes through (0,1), and increases quickly on the right.\" width=\"250\" height=\"200\" \/> [latex]h(x)=(3.5)^x[\/latex][\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nDid you notice that the line [latex]y=0[\/latex] is a horizontal asymptote for all of the graphs; that they all pass through (0, 1); and that they all pass through [latex](1, a)[\/latex] where [latex]a[\/latex] is the base? [latex]f(x)=4^x[\/latex] passes through (1, 4); [latex]g(x)=\\left (\\frac{1}{4}\\right )^x[\/latex] passes through [latex](1, \\frac{1}{4})[\/latex]; and [latex]h(x)=(3.5)^x[\/latex] passes through (1, 3.5).\r\n\r\nWhat makes the graphs of[latex]f(x)=4^x[\/latex] and [latex]h(x)=(3.5)^x[\/latex] different from [latex]g(x)=\\left (\\frac{1}{4}\\right )^x[\/latex]? The first two are continuously increasing while the last is continuously decreasing. Why? When the base of the exponential expression is &gt; 1 the graph will increase. When the base of the graph lies between 0 and 1, the graph will decrease.\r\n\r\nWhat happens if we try a base of 0, 1 or a negative number?\u00a0 Try it in Desmos and see!\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nUse the <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos Graphing Calculator<\/a> to determine the shape of the functions:\r\n\r\n[latex]f(x)=7^x[\/latex]\r\n\r\n[latex]g(x)=\\left (\\frac{2}{5}\\right )^x[\/latex]\r\n\r\n[latex]h(x)=(1.5)^x[\/latex]\r\n\r\n[reveal-answer q=\"hjm495\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm495\"]\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">Higher base<\/div><\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">Fraction base<\/div><\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">Smaller base<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1016\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1016 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/7^x-300x244.png\" alt=\"Graph of f(x)=7^x. Fast approach to the negative x axis as an asymptote, passes through (0,1), very quickly climbs to infinity on the right.\" width=\"300\" height=\"244\" \/> [latex]f(x)=7^x[\/latex][\/caption]<\/td>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1017\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1017 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/0.4^x-300x252.png\" alt=\"Graph of 2\/5 to the power x. decreases quickly on the left, passes through (0,1), quickly approaches the positive x axis on the right as an asymptote.\" width=\"300\" height=\"252\" \/> [latex]g(x)=\\left (\\frac{2}{5}\\right )^x[\/latex][\/caption]<\/td>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1018\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1018 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1.5^x-300x225.png\" alt=\"Graph of 1.5 to the power x. Gradually approaches the negative x axis as an asymptote, passes through (0,1), then increases to infinity.\" width=\"300\" height=\"225\" \/> [latex]h(x)=(1.5)^x[\/latex][\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Logarithmic Functions<\/h3>\r\nThe inverse of an exponential function is referred to as a <em><strong>logarithmic function<\/strong><\/em>. If we were to find the inverse of each of the functions in figure 2 by reflecting them across the line [latex]y=x[\/latex] we would get their inverse functions. Recall that to find an inverse function, all we do is switch the variables. So if we let [latex]y=f(x)[\/latex] in the function [latex]f(x)=2^x[\/latex], then [latex]y=2^x[\/latex] and the inverse function is simply [latex]x=2^y[\/latex]. Typically, we write an inverse function in function notation, which would require solving for [latex]y[\/latex]. We will learn how to do this later in the course, but for now it will suffice to know that the notation for this inverse function requires the use of a <em><strong>logarithm<\/strong><\/em>. No one wants to write logarithm more than once, so we truncate it to log. To indicate the base of the exponent in the function where [latex]x=2^y[\/latex], we write [latex]f(x)=log_2x[\/latex]. This is read, \"[latex]f[\/latex] of [latex]x[\/latex] equals log base 2 of [latex]x[\/latex]. The base is always written as a subscript attached to log. A logarithm always has a base. Had the function been\u00a0[latex]x=3^y[\/latex] we would write [latex]g(x)=log_3x[\/latex] and if the function was [latex]x=10^y[\/latex] we would write [latex]h(x)=log_{10}x[\/latex].\r\n\r\nFigure 4 shows graphs of three logarithmic functions: [latex]f(x)=log_2 x, g(x)=log_3 x, s(x)=log_{10} x[\/latex]. Notice that each of the graphs follows the same basic shape; coming up fast from [latex]y=-\\infty[\/latex] close to [latex]x=0[\/latex] then slowly heading continuously upwards towards [latex]+\\infty[\/latex]. Notice also that each graph passes through the point (1, 0) and none of these graphs cross the [latex]y[\/latex]-axis. In this case, the line [latex]x=0[\/latex] (the [latex]y[\/latex]-axis) is an example of a\u00a0<em><strong>vertical<\/strong><strong> asymptote<\/strong><\/em>; a vertical line that the graph never crosses.\r\n\r\n[caption id=\"attachment_852\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-852 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-LogFun-300x300.png\" alt=\"Three logarithmic graphs demonstrating a vertical asymptote of the negative y axis, all passing through (1,0), and slowly increasing on the right.\" width=\"300\" height=\"300\" \/> Figure 4. Logarithmic functions.[\/caption]\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>logarithmic FUNCTIONS<\/h3>\r\nA logarithmic function is any function of the form [latex]f(x)=log_a(x)[\/latex] where [latex]a[\/latex] is a positive real number and [latex]a\\ne1[\/latex].\r\n\r\nAll logarithmic functions of the form [latex]f(x)=log_a(x)[\/latex] pass through the points (1, 0) and ([latex]a[\/latex], 1). In addition, the line [latex]x=0[\/latex] is a vertical asymptote.\r\n\r\nLogarithmic functions are inverse functions of exponential functions: If [latex]g(x)=a^x[\/latex], then [latex]g^{-1}(x)=log_a (x)[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nUse the <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos Graphing Calculator<\/a> to determine the shape of the functions:\r\n\r\n[latex]f(x)=log_{10}x[\/latex]\r\n\r\n[latex]g(x)=log_{\\frac{1}{4}}x[\/latex]\r\n\r\n[latex]h(x)=log_8 x[\/latex]\r\n<h4><strong>Solution<\/strong><\/h4>\r\nTo type a subscript for the base of the log, use the underscore key (shift -). After you enter the base you must click past the base to insert [latex]x[\/latex]. So [latex]f(x)=log_{10}x[\/latex] will be typed in as f(x)=log_10 \"click past 10 to get back to standard script\" x\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">Common log<\/div><\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">inverted log<\/div><\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">log base 8<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1019\" align=\"aligncenter\" width=\"225\"]<img class=\"wp-image-1019\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09212702\/logx-300x179.png\" alt=\"Graph of log base 10 of x\" width=\"225\" height=\"134\" \/> [latex]f(x)=log_{10}x[\/latex][\/caption]<\/td>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1020\" align=\"aligncenter\" width=\"225\"]<img class=\"wp-image-1020\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09212957\/log14x-300x182.png\" alt=\"Graph of log base 1\/4 of x\" width=\"225\" height=\"137\" \/> [latex]g(x)=log_{\\frac{1}{4}}x[\/latex][\/caption]<\/td>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1021\" align=\"aligncenter\" width=\"225\"]<img class=\"wp-image-1021\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09213338\/log_8x-300x167.png\" alt=\"Graph of log base 2 of x\" width=\"225\" height=\"125\" \/> [latex]h(x)=log_8 x[\/latex][\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nDid you notice that the line [latex]x=0[\/latex] is a vertical asymptote for all of the graphs; that they all pass through (1, 0); and that they all pass through [latex](a, 1)[\/latex] where [latex]a[\/latex] is the base?\u00a0[latex]f(x)=log_{10}x[\/latex] passes through (10, 1);\u00a0[latex]g(x)=log_{\\frac{1}{4}} x[\/latex] passes through [latex]\\left ( \\frac{1}{4}, 1\\right )[\/latex]; and\u00a0[latex]h(x)=log_8 x[\/latex] passes through (8, 1).\r\n\r\nWhat makes the graphs of\u00a0 [latex]f(x)=log_{10}x[\/latex] and [latex]h(x)=log_8 x[\/latex] different from\u00a0[latex]g(x)=log_{\\frac{1}{4}}x[\/latex]? The first two are continuously increasing while the last is continuously decreasing. Why? When the base of the logarithm is &gt; 1 the graph will increase. When the base of the graph lies between 0 and 1, the graph will decrease.\r\n\r\nWhat happens if we try a base of 0, 1 or a negative number?\u00a0 Try it in Desmos and see!\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nUse the <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos Graphing Calculator<\/a> to determine the shape of the functions:\r\n\r\n[latex]f(x)=log_{12}x[\/latex]\r\n\r\n[latex]g(x)=log_{\\frac{1}{8}}x[\/latex]\r\n\r\n[latex]h(x)=log_2 x[\/latex]\r\n\r\n[reveal-answer q=\"hjm592\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm592\"]\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 33.3333%; text-align: center;\">log base 12<\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">log base 1\/8<\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">Log base 2<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1022\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1022 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09214127\/log-base-12-x-300x144.png\" alt=\"Graph of f(x) = log base 12 of x\" width=\"300\" height=\"144\" \/> [latex]f(x)=log_{12}x[\/latex][\/caption]<\/td>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1023\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1023 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09214341\/log-18-x-300x182.png\" alt=\"Graph of log base 1\/8 of x\" width=\"300\" height=\"182\" \/> [latex]g(x)=log_{\\frac{1}{8}}x[\/latex][\/caption]<\/td>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1024\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-1024\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09214535\/log_2x-300x241.png\" alt=\"Graph of log base 2 of x\" width=\"300\" height=\"241\" \/> [latex]h(x)=log_2 x[\/latex][\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>Rational Functions<\/h3>\r\nA rational function is a function where the independent variable is in the denominator of a fraction. Technically it is one polynomial function divided by another polynomial function. It can be as basic as [latex]f(x)=\\frac{1}{x}[\/latex] or more complicated like [latex]g(x)=\\frac{5x^5-3x^4+2x^3-x^2+6}{x^6-2x+8}[\/latex]. Figure 5 shows the graphs of two basic rational functions. Later in this course we will go into more detail on these basic functions. The graphs of more complicated functions are shown in figures 6 and 7. Notice that the graphs of both functions in figure 5 have horizontal and vertical asymptotes. The graph of the function in figure 7 has only a horizontal asymptote, and the graph of the function in figure 6 has a vertical and a slant asymptote. If you take College Algebra, you will learn more about these asymptotes.\r\n<table style=\"border-collapse: collapse; width: 100%; height: 402px;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">Basic Rational Functions<\/div><\/th>\r\n<th style=\"width: 50%; text-align: center;\">\r\n<div class=\"mceTemp\">Special Rational Function<\/div><\/th>\r\n<\/tr>\r\n<tr style=\"height: 402px;\">\r\n<td style=\"width: 50%; height: 402px;\">\r\n\r\n[caption id=\"attachment_853\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-853 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-RationalFun-300x300.png\" alt=\"Rational functions with both vertical and horizontal asymptotes.\" width=\"300\" height=\"300\" \/> Figure 5. Rational functions.[\/caption]<\/td>\r\n<td style=\"width: 50%; height: 402px;\">\r\n\r\n[caption id=\"attachment_1026\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-1026 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Rational-function-with-asymptotes-300x286.png\" alt=\"Graph of a rational function with vertical and slant asymptotes\" width=\"300\" height=\"286\" \/> Figure 6. A rational function with vertical and slant asymptotes.[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 100%; text-align: center;\">\r\n<div class=\"mceTemp\">More complicated Rational Function<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 100%;\">\r\n\r\n[caption id=\"attachment_1025\" align=\"aligncenter\" width=\"800\"]<img class=\"wp-image-1025\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09220728\/rational-function-1024x341.png\" alt=\"Graph of a complicated rational function\" width=\"800\" height=\"266\" \/> Figure 7. Rational function with a horizontal asymptote.[\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox shaded\">\r\n<h3>RATIONAL\u00a0 FUNCTIONS<\/h3>\r\nA rational function is any function of the form [latex]f(x)=\\frac{p(x)}{q(x)}[\/latex] where [latex]p(x)[\/latex] and [latex]q(x)[\/latex] are polynomial functions.\r\n\r\nRational functions can have horizontal, vertical, and slant asymptotes.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\nUse the <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos Graphing Calculator<\/a> to determine the shape of the functions:\r\n\r\n[latex]f(x)=\\frac{1}{3x}[\/latex]\r\n\r\n[latex]g(x)=\\frac{6}{x}[\/latex]\r\n\r\n[latex]h(x)=\\frac{x^2}{x-3}[\/latex]\r\n<h4><strong>Solution<\/strong><\/h4>\r\nTo type a rational function into desmos, type \"frac\" and a fraction will appear. Type the numerator of the function, then click on the denominator and type the denominator of the function.\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">Compressed Rational function<\/div><\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">Stretched Rational Function<\/div><\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">Special Rational Function<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1027\" align=\"aligncenter\" width=\"250\"]<img class=\"wp-image-1027\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09222143\/13x-298x300.png\" alt=\"Graph of 1 over 3x\" width=\"250\" height=\"251\" \/> [latex]f(x)=\\frac{1}{3x}[\/latex][\/caption]<\/td>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1028\" align=\"aligncenter\" width=\"250\"]<img class=\"wp-image-1028\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09222336\/6x-300x300.png\" alt=\"Graph of 6 over x\" width=\"250\" height=\"249\" \/> [latex]g(x)=\\frac{6}{x}[\/latex][\/caption]<\/td>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1029\" align=\"aligncenter\" width=\"250\"]<img class=\"wp-image-1029\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09222506\/x%5E2x-3-300x300.png\" alt=\"Graph of x^2 over (x-3)\" width=\"250\" height=\"250\" \/> [latex]h(x)=\\frac{x^2}{x-3}[\/latex][\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nDid you notice that the graphs of [latex]f(x)=\\frac{1}{3x}[\/latex] and [latex]g(x)=\\frac{6}{x}[\/latex] have a vertical asymptote and a horizontal asymptote? On the other hand, the graph of\u00a0[latex]h(x)=\\frac{x^2}{x-3}[\/latex] has a vertical asymptote and a slant asymptote.\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nUse the <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos Graphing Calculator<\/a> to determine the shape of the functions:\r\n\r\n[latex]f(x)=\\frac{4}{x}[\/latex]\r\n\r\n[latex]g(x)=\\frac{3}{8x}[\/latex]\r\n\r\n[latex]h(x)=\\frac{x+4}{x^{2}-5x+6}[\/latex]\r\n\r\n[reveal-answer q=\"hjm956\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm956\"]\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">Stretched Rational Function<\/div><\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">Compressed Rational Function<\/div><\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">\r\n<div class=\"mceTemp\">Special Rational Function<\/div><\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1030\" align=\"aligncenter\" width=\"250\"]<img class=\"wp-image-1030\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09223124\/4x-298x300.png\" alt=\"Graph of 4\/x\" width=\"250\" height=\"252\" \/> [latex]f(x)=\\frac{4}{x}[\/latex][\/caption]<\/td>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1031\" align=\"aligncenter\" width=\"250\"]<img class=\"wp-image-1031\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09223301\/38x-300x300.png\" alt=\"Graph of 3 over 8x\" width=\"250\" height=\"251\" \/> [latex]g(x)=\\frac{3}{8x}[\/latex][\/caption]<\/td>\r\n<td style=\"width: 33.3333%;\">[caption id=\"attachment_1032\" align=\"aligncenter\" width=\"250\"]<img class=\"wp-image-1032\" style=\"font-size: 16px; orphans: 1;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09223507\/rational-fn-300x221.png\" alt=\"Graph of The function in the caption.\" width=\"250\" height=\"184\" \/> [latex]h(x)=\\frac{x+4}{x^{2}-5x+6}[\/latex][\/caption]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nFunctions take on many shapes. A graphing calculator like Desmos helps us visualize the graphs of functions letting us see asymptotes, intercepts, turning points and other graphical features. In this section, we highlighted the types of graphs and functions we are going to look at in more detail in this course.\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 5<\/h3>\r\nIdentify the type of function from its graph.\r\n<table style=\"border-collapse: collapse; width: 100%; height: 300px;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 33.3333%; text-align: center;\">No asymptote<\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">Vertical asymptote<\/th>\r\n<th style=\"width: 33.3333%; text-align: center;\">Horizontal asymptote<\/th>\r\n<\/tr>\r\n<tr style=\"height: 300px;\">\r\n<td style=\"width: 33.3333%; height: 300px;\"><img class=\"aligncenter wp-image-1036 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Poly-fun-153x300.png\" alt=\"Continuous function with no asymptote.\" width=\"153\" height=\"300\" \/><\/td>\r\n<td style=\"width: 33.3333%; height: 300px;\"><img class=\"aligncenter wp-image-1037\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09225707\/log-fun-300x294.png\" alt=\"Continuous function with a Vertical Asymptote.\" width=\"250\" height=\"245\" \/><\/td>\r\n<td style=\"width: 33.3333%; height: 300px;\"><img class=\"aligncenter wp-image-1038\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09225827\/Expo-fun-300x251.png\" alt=\"Continuous function with a horizontal asymptote.\" width=\"250\" height=\"210\" \/><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"hjm078\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm078\"]\r\n<ol>\r\n \t<li>Polynomial function<\/li>\r\n \t<li>Logarithmic function<\/li>\r\n \t<li>Exponential function<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"textbox learning-objectives\">\n<h1>Learning Objectives<\/h1>\n<p>Identify types of functions based on their algebraic forms<\/p>\n<ul>\n<li style=\"margin-top: 0.5em;\">The algebraic form of polynomial functions<\/li>\n<li>The algebraic form of exponential functions<\/li>\n<li>The algebraic form of logarithmic functions<\/li>\n<li>The algebraic form of rational functions<\/li>\n<\/ul>\n<p>Use graphing software to graph functions.<\/p>\n<\/div>\n<h2>Types of Functions<\/h2>\n<p>There are many types of functions that can be written in algebraic notation. <span style=\"font-size: 1em;\">Functions can be broadly classified according to<\/span><span style=\"font-size: 1em;\">\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">the position of the independent variable in its algebraic form. The following four types of functions are defined by the position of the independent variable. Throughout this course we will use the Desmos Graphing Calculator to visualize functions. This is a free online calculator that is found at\u00a0<a href=\"https:\/\/www.desmos.com\/calculator\">desmos calculator<\/a>.\u00a0 You can create a free account that will enable you to save and print your graphs.<\/span><\/p>\n<h3>Polynomial Functions<\/h3>\n<p>A <em><strong>polynomial function<\/strong><\/em> is a function where the variable is in the &#8220;base&#8221; position and the exponent on the variable is\u00a0a whole number. Recall that the set of whole numbers start at 0 and include all of the positive integers:\u00a0 [latex]\\mathbb{W} = \\{0, 1, 2, 3, 4, ...\\}[\/latex]. Examples of polynomial functions are [latex]f(x)=x^2,\\;g(x)=3x^6-7x^5+2x^4+3,\\;h(x)=5x-7,\\;T(x)=8[\/latex]. The polynomial function [latex]T(x)=8[\/latex] is an example of a <em><strong>constant function<\/strong><\/em> where the input value is irrelevant as the output will always be 8. The graph of a constant function is always a horizontal line (see figure 1).\u00a0 [latex]h(x)=5x-7[\/latex] is an example of a linear function where the exponent on the variable is 1.\u00a0The graph of a <em><strong>linear function<\/strong><\/em> is always a line (see figure 1).<\/p>\n<div id=\"attachment_1736\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1736\" class=\"wp-image-1736 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-54-300x300.png\" alt=\"A constant function (horizontal line), and a non-constant linear function increasing function.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-54-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-54-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-54-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-54-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-54-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-54-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-54.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1736\" class=\"wp-caption-text\">Figure 1. Constant function and linear function<\/p>\n<\/div>\n<p>When the highest exponent on a variable is at least 2, the graph is a curve.\u00a0Figure 2 illustrates the graphs of the two polynomial functions [latex]\\color{blue}{f(x)=x^2}[\/latex] and [latex]\\color{green}{f(x)=x^3}[\/latex]. Notice that these functions are a completely different shape. However, they both pass through the points (0, 0) and (1, 1). All polynomial functions are continuous with a domain of [latex](-\\infty, +\\infty)[\/latex], which means that the independent variable [latex]x[\/latex] can take on any real number value.<\/p>\n<div id=\"attachment_1737\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1737\" class=\"wp-image-1737 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-55-300x300.png\" alt=\"Two polynomial functions. One with a high power of 2 that goes up on both ends, and one with a high power of 3 that goes up on the right and down on the left.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-55-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-55-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-55-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-55-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-55-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-55-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/desmos-graph-55.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1737\" class=\"wp-caption-text\">Figure 2. Polynomial functions<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>POLYNOMIAL FUNCTIONS<\/h3>\n<p>A polynomial function is any function of the form [latex]f(x)=a_nx^n+a_{n-1}x^{n-1} +...+a_1x+a_0[\/latex] where [latex]n[\/latex] in a whole number and [latex]a_i[\/latex] are real number constants.<\/p>\n<p>All polynomial functions of the form [latex]f(x)=ax^n[\/latex] with [latex]n\u22651[\/latex] pass through the point [latex](0, 0)[\/latex] and [latex](1, a)[\/latex].<\/p>\n<\/div>\n<div>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Use the <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos Graphing Calculator<\/a> to determine the shape of the functions:<\/p>\n<p>[latex]f(x)=x^4[\/latex]<\/p>\n<p>[latex]g(x)=x^3-4x^2+5x-2[\/latex]<\/p>\n<p>[latex]h(x)=x^6-2x^3+x[\/latex]<\/p>\n<h4><strong>Solution<\/strong><\/h4>\n<p>Go to the <a href=\"https:\/\/www.desmos.com\/calculator\/ko1xkeqwhk\">Desmos Graphing Calculator<\/a> and type each function into a box on the left side of the screen. Use the ^ (carat) key on your keyboard for exponents.<\/p>\n<p>\u200b<\/p>\n<table class=\"aligncenter\" style=\"border-collapse: collapse; width: 100.355%; height: 268px;\">\n<tbody>\n<tr>\n<th style=\"width: 33.3333%; vertical-align: bottom; text-align: center;\">\n<div class=\"mceTemp\">4th degree only<\/div>\n<\/th>\n<th style=\"width: 33.3333%; vertical-align: bottom; text-align: center;\">\n<div class=\"mceTemp\">3rd degree with extra terms<\/div>\n<\/th>\n<th style=\"width: 33.3333%; vertical-align: bottom; text-align: center;\">\n<div class=\"mceTemp\">4th degree with extra terms<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%; vertical-align: bottom;\">\n<div id=\"attachment_1005\" style=\"width: 175px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1005\" class=\"wp-image-1005\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09200602\/fxx%5E4-258x300.png\" alt=\"Graph of the function f(x)=x^4. Abroad U with steep sides.\" width=\"165\" height=\"192\" \/><\/p>\n<p id=\"caption-attachment-1005\" class=\"wp-caption-text\">[latex]f(x)=x^4[\/latex]<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 33.3333%; vertical-align: bottom;\">\n<div id=\"attachment_1004\" style=\"width: 157px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1004\" class=\"wp-image-1004\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09200557\/cubic-function-233x300.png\" alt=\"Graph of [latex]g(x)=x^3-4x^2+5x-2[\/latex]. Up on right, down on left, a local max and a local min.\" width=\"147\" height=\"189\" \/><\/p>\n<p id=\"caption-attachment-1004\" class=\"wp-caption-text\">[latex]g(x)=x^3-4x^2+5x-2[\/latex]<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 33.3333%; vertical-align: bottom;\">\n<div id=\"attachment_4231\" style=\"width: 184px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-4231\" class=\"wp-image-4231\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/26220928\/desmos-graph-2022-08-26T160836.875-300x300.png\" alt=\"h(x)=x^6-2x^3+x, Up on left, up on right, with two local mins and one local max.\" width=\"174\" height=\"174\" \/><\/p>\n<p id=\"caption-attachment-4231\" class=\"wp-caption-text\">[latex]h(x)=x^6-2x^3+x[\/latex]<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>Use the <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos Graphing Calculator<\/a> to determine the shape of the functions:<\/p>\n<p>[latex]f(x)=x^5[\/latex]<\/p>\n<p>[latex]g(x)=3x-4[\/latex]<\/p>\n<p>[latex]h(x)=x^6-3x^3+2x-5[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm438\">Show Answer<\/span><\/p>\n<div id=\"qhjm438\" class=\"hidden-answer\" style=\"display: none\">\n<p>\u200b\u200b<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">5th degree function<\/div>\n<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">1st degree function<\/div>\n<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">4th degree function<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1742\" style=\"width: 280px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1742\" class=\"wp-image-1742\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/22202424\/desmos-graph-56-300x300.png\" alt=\"x^5\" width=\"270\" height=\"270\" \/><\/p>\n<p id=\"caption-attachment-1742\" class=\"wp-caption-text\">[latex]f(x)=x^5[\/latex]<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1744\" style=\"width: 280px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1744\" class=\"wp-image-1744\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/22202624\/desmos-graph-57-300x300.png\" alt=\"3x-4\" width=\"270\" height=\"270\" \/><\/p>\n<p id=\"caption-attachment-1744\" class=\"wp-caption-text\">[latex]g(x)=3x-4[\/latex]<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1746\" style=\"width: 280px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1746\" class=\"wp-image-1746\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/22204349\/desmos-graph-58-300x300.png\" alt=\"6th degree polynomial\" width=\"270\" height=\"270\" \/><\/p>\n<p id=\"caption-attachment-1746\" class=\"wp-caption-text\">[latex]h(x)=x^6-3x^3+2x-5[\/latex]<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h3>Exponential Functions<\/h3>\n<p>An <em><strong>exponential function<\/strong><\/em> is a function where the independent variable sits at the exponent position. The functions [latex]f(x)=4^x,\\;g(x)=(3.2)^x, \\;h(x)=\\left (\\frac{1}{5}\\right )^x[\/latex] are all examples of exponential functions: a positive real number raised to a power of [latex]x[\/latex]. Figure 3 illustrates the graphs of four exponential functions: [latex]f(x)=2^x, \\;g(x)=3^x, \\;r(x)=5^x, s(x)=\\;10^x[\/latex]. Notice that each of these graphs have the same basic shape. They slowly come up from the [latex]x[\/latex]-axis then quickly move towards [latex]+\\infty[\/latex].\u00a0 The graphs never quite make it to the line [latex]y=0[\/latex] (the [latex]x[\/latex]-axis) and this line is referred to as a horizontal asymptote; a horizontal line that the graph never quite reaches as it moves towards either positive or negative (in this case) infinity. In addition, they all pass through the point (0, 1).<\/p>\n<div id=\"attachment_844\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-844\" class=\"wp-image-844 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-ExpFun1-300x300.png\" alt=\"Four exponential graphs described above.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-ExpFun1-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-ExpFun1-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-ExpFun1-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-ExpFun1-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-ExpFun1-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-ExpFun1-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-ExpFun1.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-844\" class=\"wp-caption-text\">Figure 3. Exponential functions.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>exponential FUNCTIONS<\/h3>\n<p>An exponential function is any function of the form [latex]f(x)=a^x[\/latex] where [latex]a[\/latex] is a positive real number and [latex]a\\ne1[\/latex].<\/p>\n<p>All exponential functions of the form [latex]f(x)=a^x[\/latex] pass through the point (0, 1) and the line [latex]y=0[\/latex] is a horizontal asymptote.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>Use the <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos Graphing Calculator<\/a> to determine the shape of the functions:<\/p>\n<p>[latex]f(x)=4^x[\/latex]<\/p>\n<p>[latex]g(x)=\\left (\\frac{1}{4}\\right )^x[\/latex]<\/p>\n<p>[latex]h(x)=(3.5)^x[\/latex]<\/p>\n<h4><strong>Solution<\/strong><\/h4>\n<p>Go to the <a href=\"https:\/\/www.desmos.com\/calculator\/bjxla254vm\">Desmos Graphing Calculator<\/a> and type each function into a box on the left side of the screen.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">Exponential<\/div>\n<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">Exponential reflected in y<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">Exponential<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1010\" style=\"width: 259px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1010\" class=\"wp-image-1010\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09203859\/fx4%5Ex-300x241.png\" alt=\"Graph of f(x)=4^x. Approaches an asymptote of the negative x axis, passes through (0,1), and increases quickly on the right.\" width=\"249\" height=\"200\" \/><\/p>\n<p id=\"caption-attachment-1010\" class=\"wp-caption-text\">[latex]f(x)=4^x[\/latex]<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1011\" style=\"width: 246px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1011\" class=\"wp-image-1011\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09204129\/gx14%5Ex-300x253.png\" alt=\"Graph of g(x)=(1\/4)^x. Increases quickly on the left, passes through (0,1), and approaches the positive x axis as an asymptote.\" width=\"236\" height=\"199\" \/><\/p>\n<p id=\"caption-attachment-1011\" class=\"wp-caption-text\">[latex]g(x)=\\left (\\frac{1}{4}\\right )^x[\/latex]<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1012\" style=\"width: 260px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1012\" class=\"wp-image-1012\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09204335\/hx35%5Ex-300x240.png\" alt=\"Graph of h(x)=3.5^x. Approaches an asymptote of the negative x axis, passes through (0,1), and increases quickly on the right.\" width=\"250\" height=\"200\" \/><\/p>\n<p id=\"caption-attachment-1012\" class=\"wp-caption-text\">[latex]h(x)=(3.5)^x[\/latex]<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Did you notice that the line [latex]y=0[\/latex] is a horizontal asymptote for all of the graphs; that they all pass through (0, 1); and that they all pass through [latex](1, a)[\/latex] where [latex]a[\/latex] is the base? [latex]f(x)=4^x[\/latex] passes through (1, 4); [latex]g(x)=\\left (\\frac{1}{4}\\right )^x[\/latex] passes through [latex](1, \\frac{1}{4})[\/latex]; and [latex]h(x)=(3.5)^x[\/latex] passes through (1, 3.5).<\/p>\n<p>What makes the graphs of[latex]f(x)=4^x[\/latex] and [latex]h(x)=(3.5)^x[\/latex] different from [latex]g(x)=\\left (\\frac{1}{4}\\right )^x[\/latex]? The first two are continuously increasing while the last is continuously decreasing. Why? When the base of the exponential expression is &gt; 1 the graph will increase. When the base of the graph lies between 0 and 1, the graph will decrease.<\/p>\n<p>What happens if we try a base of 0, 1 or a negative number?\u00a0 Try it in Desmos and see!<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Use the <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos Graphing Calculator<\/a> to determine the shape of the functions:<\/p>\n<p>[latex]f(x)=7^x[\/latex]<\/p>\n<p>[latex]g(x)=\\left (\\frac{2}{5}\\right )^x[\/latex]<\/p>\n<p>[latex]h(x)=(1.5)^x[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm495\">Show Answer<\/span><\/p>\n<div id=\"qhjm495\" class=\"hidden-answer\" style=\"display: none\">\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">Higher base<\/div>\n<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">Fraction base<\/div>\n<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">Smaller base<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1016\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1016\" class=\"wp-image-1016 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/7^x-300x244.png\" alt=\"Graph of f(x)=7^x. Fast approach to the negative x axis as an asymptote, passes through (0,1), very quickly climbs to infinity on the right.\" width=\"300\" height=\"244\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/7^x-300x244.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/7^x-768x625.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/7^x-1024x833.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/7^x-65x53.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/7^x-225x183.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/7^x-350x285.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/7^x.png 1502w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1016\" class=\"wp-caption-text\">[latex]f(x)=7^x[\/latex]<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1017\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1017\" class=\"wp-image-1017 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/0.4^x-300x252.png\" alt=\"Graph of 2\/5 to the power x. decreases quickly on the left, passes through (0,1), quickly approaches the positive x axis on the right as an asymptote.\" width=\"300\" height=\"252\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/0.4^x-300x252.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/0.4^x-768x645.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/0.4^x-1024x861.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/0.4^x-65x55.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/0.4^x-225x189.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/0.4^x-350x294.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/0.4^x.png 1542w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1017\" class=\"wp-caption-text\">[latex]g(x)=\\left (\\frac{2}{5}\\right )^x[\/latex]<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1018\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1018\" class=\"wp-image-1018 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1.5^x-300x225.png\" alt=\"Graph of 1.5 to the power x. Gradually approaches the negative x axis as an asymptote, passes through (0,1), then increases to infinity.\" width=\"300\" height=\"225\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1.5^x-300x225.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1.5^x-768x576.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1.5^x-1024x767.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1.5^x-65x49.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1.5^x-225x169.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1.5^x-350x262.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1.5^x.png 1668w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1018\" class=\"wp-caption-text\">[latex]h(x)=(1.5)^x[\/latex]<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<h3>Logarithmic Functions<\/h3>\n<p>The inverse of an exponential function is referred to as a <em><strong>logarithmic function<\/strong><\/em>. If we were to find the inverse of each of the functions in figure 2 by reflecting them across the line [latex]y=x[\/latex] we would get their inverse functions. Recall that to find an inverse function, all we do is switch the variables. So if we let [latex]y=f(x)[\/latex] in the function [latex]f(x)=2^x[\/latex], then [latex]y=2^x[\/latex] and the inverse function is simply [latex]x=2^y[\/latex]. Typically, we write an inverse function in function notation, which would require solving for [latex]y[\/latex]. We will learn how to do this later in the course, but for now it will suffice to know that the notation for this inverse function requires the use of a <em><strong>logarithm<\/strong><\/em>. No one wants to write logarithm more than once, so we truncate it to log. To indicate the base of the exponent in the function where [latex]x=2^y[\/latex], we write [latex]f(x)=log_2x[\/latex]. This is read, &#8220;[latex]f[\/latex] of [latex]x[\/latex] equals log base 2 of [latex]x[\/latex]. The base is always written as a subscript attached to log. A logarithm always has a base. Had the function been\u00a0[latex]x=3^y[\/latex] we would write [latex]g(x)=log_3x[\/latex] and if the function was [latex]x=10^y[\/latex] we would write [latex]h(x)=log_{10}x[\/latex].<\/p>\n<p>Figure 4 shows graphs of three logarithmic functions: [latex]f(x)=log_2 x, g(x)=log_3 x, s(x)=log_{10} x[\/latex]. Notice that each of the graphs follows the same basic shape; coming up fast from [latex]y=-\\infty[\/latex] close to [latex]x=0[\/latex] then slowly heading continuously upwards towards [latex]+\\infty[\/latex]. Notice also that each graph passes through the point (1, 0) and none of these graphs cross the [latex]y[\/latex]-axis. In this case, the line [latex]x=0[\/latex] (the [latex]y[\/latex]-axis) is an example of a\u00a0<em><strong>vertical<\/strong><strong> asymptote<\/strong><\/em>; a vertical line that the graph never crosses.<\/p>\n<div id=\"attachment_852\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-852\" class=\"wp-image-852 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-LogFun-300x300.png\" alt=\"Three logarithmic graphs demonstrating a vertical asymptote of the negative y axis, all passing through (1,0), and slowly increasing on the right.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-LogFun-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-LogFun-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-LogFun-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-LogFun-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-LogFun-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-LogFun-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-LogFun.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-852\" class=\"wp-caption-text\">Figure 4. Logarithmic functions.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>logarithmic FUNCTIONS<\/h3>\n<p>A logarithmic function is any function of the form [latex]f(x)=log_a(x)[\/latex] where [latex]a[\/latex] is a positive real number and [latex]a\\ne1[\/latex].<\/p>\n<p>All logarithmic functions of the form [latex]f(x)=log_a(x)[\/latex] pass through the points (1, 0) and ([latex]a[\/latex], 1). In addition, the line [latex]x=0[\/latex] is a vertical asymptote.<\/p>\n<p>Logarithmic functions are inverse functions of exponential functions: If [latex]g(x)=a^x[\/latex], then [latex]g^{-1}(x)=log_a (x)[\/latex].<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>Use the <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos Graphing Calculator<\/a> to determine the shape of the functions:<\/p>\n<p>[latex]f(x)=log_{10}x[\/latex]<\/p>\n<p>[latex]g(x)=log_{\\frac{1}{4}}x[\/latex]<\/p>\n<p>[latex]h(x)=log_8 x[\/latex]<\/p>\n<h4><strong>Solution<\/strong><\/h4>\n<p>To type a subscript for the base of the log, use the underscore key (shift -). After you enter the base you must click past the base to insert [latex]x[\/latex]. So [latex]f(x)=log_{10}x[\/latex] will be typed in as f(x)=log_10 &#8220;click past 10 to get back to standard script&#8221; x<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">Common log<\/div>\n<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">inverted log<\/div>\n<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">log base 8<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1019\" style=\"width: 235px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1019\" class=\"wp-image-1019\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09212702\/logx-300x179.png\" alt=\"Graph of log base 10 of x\" width=\"225\" height=\"134\" \/><\/p>\n<p id=\"caption-attachment-1019\" class=\"wp-caption-text\">[latex]f(x)=log_{10}x[\/latex]<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1020\" style=\"width: 235px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1020\" class=\"wp-image-1020\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09212957\/log14x-300x182.png\" alt=\"Graph of log base 1\/4 of x\" width=\"225\" height=\"137\" \/><\/p>\n<p id=\"caption-attachment-1020\" class=\"wp-caption-text\">[latex]g(x)=log_{\\frac{1}{4}}x[\/latex]<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1021\" style=\"width: 235px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1021\" class=\"wp-image-1021\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09213338\/log_8x-300x167.png\" alt=\"Graph of log base 2 of x\" width=\"225\" height=\"125\" \/><\/p>\n<p id=\"caption-attachment-1021\" class=\"wp-caption-text\">[latex]h(x)=log_8 x[\/latex]<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Did you notice that the line [latex]x=0[\/latex] is a vertical asymptote for all of the graphs; that they all pass through (1, 0); and that they all pass through [latex](a, 1)[\/latex] where [latex]a[\/latex] is the base?\u00a0[latex]f(x)=log_{10}x[\/latex] passes through (10, 1);\u00a0[latex]g(x)=log_{\\frac{1}{4}} x[\/latex] passes through [latex]\\left ( \\frac{1}{4}, 1\\right )[\/latex]; and\u00a0[latex]h(x)=log_8 x[\/latex] passes through (8, 1).<\/p>\n<p>What makes the graphs of\u00a0 [latex]f(x)=log_{10}x[\/latex] and [latex]h(x)=log_8 x[\/latex] different from\u00a0[latex]g(x)=log_{\\frac{1}{4}}x[\/latex]? The first two are continuously increasing while the last is continuously decreasing. Why? When the base of the logarithm is &gt; 1 the graph will increase. When the base of the graph lies between 0 and 1, the graph will decrease.<\/p>\n<p>What happens if we try a base of 0, 1 or a negative number?\u00a0 Try it in Desmos and see!<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Use the <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos Graphing Calculator<\/a> to determine the shape of the functions:<\/p>\n<p>[latex]f(x)=log_{12}x[\/latex]<\/p>\n<p>[latex]g(x)=log_{\\frac{1}{8}}x[\/latex]<\/p>\n<p>[latex]h(x)=log_2 x[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm592\">Show Answer<\/span><\/p>\n<div id=\"qhjm592\" class=\"hidden-answer\" style=\"display: none\">\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 33.3333%; text-align: center;\">log base 12<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">log base 1\/8<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">Log base 2<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1022\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1022\" class=\"wp-image-1022 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09214127\/log-base-12-x-300x144.png\" alt=\"Graph of f(x) = log base 12 of x\" width=\"300\" height=\"144\" \/><\/p>\n<p id=\"caption-attachment-1022\" class=\"wp-caption-text\">[latex]f(x)=log_{12}x[\/latex]<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1023\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1023\" class=\"wp-image-1023 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09214341\/log-18-x-300x182.png\" alt=\"Graph of log base 1\/8 of x\" width=\"300\" height=\"182\" \/><\/p>\n<p id=\"caption-attachment-1023\" class=\"wp-caption-text\">[latex]g(x)=log_{\\frac{1}{8}}x[\/latex]<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1024\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1024\" class=\"size-medium wp-image-1024\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09214535\/log_2x-300x241.png\" alt=\"Graph of log base 2 of x\" width=\"300\" height=\"241\" \/><\/p>\n<p id=\"caption-attachment-1024\" class=\"wp-caption-text\">[latex]h(x)=log_2 x[\/latex]<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<h3>Rational Functions<\/h3>\n<p>A rational function is a function where the independent variable is in the denominator of a fraction. Technically it is one polynomial function divided by another polynomial function. It can be as basic as [latex]f(x)=\\frac{1}{x}[\/latex] or more complicated like [latex]g(x)=\\frac{5x^5-3x^4+2x^3-x^2+6}{x^6-2x+8}[\/latex]. Figure 5 shows the graphs of two basic rational functions. Later in this course we will go into more detail on these basic functions. The graphs of more complicated functions are shown in figures 6 and 7. Notice that the graphs of both functions in figure 5 have horizontal and vertical asymptotes. The graph of the function in figure 7 has only a horizontal asymptote, and the graph of the function in figure 6 has a vertical and a slant asymptote. If you take College Algebra, you will learn more about these asymptotes.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 402px;\">\n<tbody>\n<tr>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">Basic Rational Functions<\/div>\n<\/th>\n<th style=\"width: 50%; text-align: center;\">\n<div class=\"mceTemp\">Special Rational Function<\/div>\n<\/th>\n<\/tr>\n<tr style=\"height: 402px;\">\n<td style=\"width: 50%; height: 402px;\">\n<div id=\"attachment_853\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-853\" class=\"wp-image-853 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-RationalFun-300x300.png\" alt=\"Rational functions with both vertical and horizontal asymptotes.\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-RationalFun-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-RationalFun-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-RationalFun-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-RationalFun-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-RationalFun-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-RationalFun-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/1-4-RationalFun.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-853\" class=\"wp-caption-text\">Figure 5. Rational functions.<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 50%; height: 402px;\">\n<div id=\"attachment_1026\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1026\" class=\"wp-image-1026 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Rational-function-with-asymptotes-300x286.png\" alt=\"Graph of a rational function with vertical and slant asymptotes\" width=\"300\" height=\"286\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Rational-function-with-asymptotes-300x286.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Rational-function-with-asymptotes-768x732.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Rational-function-with-asymptotes-1024x977.png 1024w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Rational-function-with-asymptotes-65x62.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Rational-function-with-asymptotes-225x215.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Rational-function-with-asymptotes-350x334.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Rational-function-with-asymptotes.png 1510w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-1026\" class=\"wp-caption-text\">Figure 6. A rational function with vertical and slant asymptotes.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 100%; text-align: center;\">\n<div class=\"mceTemp\">More complicated Rational Function<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 100%;\">\n<div id=\"attachment_1025\" style=\"width: 810px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1025\" class=\"wp-image-1025\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09220728\/rational-function-1024x341.png\" alt=\"Graph of a complicated rational function\" width=\"800\" height=\"266\" \/><\/p>\n<p id=\"caption-attachment-1025\" class=\"wp-caption-text\">Figure 7. Rational function with a horizontal asymptote.<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox shaded\">\n<h3>RATIONAL\u00a0 FUNCTIONS<\/h3>\n<p>A rational function is any function of the form [latex]f(x)=\\frac{p(x)}{q(x)}[\/latex] where [latex]p(x)[\/latex] and [latex]q(x)[\/latex] are polynomial functions.<\/p>\n<p>Rational functions can have horizontal, vertical, and slant asymptotes.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p>Use the <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos Graphing Calculator<\/a> to determine the shape of the functions:<\/p>\n<p>[latex]f(x)=\\frac{1}{3x}[\/latex]<\/p>\n<p>[latex]g(x)=\\frac{6}{x}[\/latex]<\/p>\n<p>[latex]h(x)=\\frac{x^2}{x-3}[\/latex]<\/p>\n<h4><strong>Solution<\/strong><\/h4>\n<p>To type a rational function into desmos, type &#8220;frac&#8221; and a fraction will appear. Type the numerator of the function, then click on the denominator and type the denominator of the function.<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">Compressed Rational function<\/div>\n<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">Stretched Rational Function<\/div>\n<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">Special Rational Function<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1027\" style=\"width: 260px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1027\" class=\"wp-image-1027\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09222143\/13x-298x300.png\" alt=\"Graph of 1 over 3x\" width=\"250\" height=\"251\" \/><\/p>\n<p id=\"caption-attachment-1027\" class=\"wp-caption-text\">[latex]f(x)=\\frac{1}{3x}[\/latex]<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1028\" style=\"width: 260px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1028\" class=\"wp-image-1028\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09222336\/6x-300x300.png\" alt=\"Graph of 6 over x\" width=\"250\" height=\"249\" \/><\/p>\n<p id=\"caption-attachment-1028\" class=\"wp-caption-text\">[latex]g(x)=\\frac{6}{x}[\/latex]<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1029\" style=\"width: 260px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1029\" class=\"wp-image-1029\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09222506\/x%5E2x-3-300x300.png\" alt=\"Graph of x^2 over (x-3)\" width=\"250\" height=\"250\" \/><\/p>\n<p id=\"caption-attachment-1029\" class=\"wp-caption-text\">[latex]h(x)=\\frac{x^2}{x-3}[\/latex]<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Did you notice that the graphs of [latex]f(x)=\\frac{1}{3x}[\/latex] and [latex]g(x)=\\frac{6}{x}[\/latex] have a vertical asymptote and a horizontal asymptote? On the other hand, the graph of\u00a0[latex]h(x)=\\frac{x^2}{x-3}[\/latex] has a vertical asymptote and a slant asymptote.<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>Use the <a href=\"https:\/\/www.desmos.com\/calculator\">Desmos Graphing Calculator<\/a> to determine the shape of the functions:<\/p>\n<p>[latex]f(x)=\\frac{4}{x}[\/latex]<\/p>\n<p>[latex]g(x)=\\frac{3}{8x}[\/latex]<\/p>\n<p>[latex]h(x)=\\frac{x+4}{x^{2}-5x+6}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm956\">Show Answer<\/span><\/p>\n<div id=\"qhjm956\" class=\"hidden-answer\" style=\"display: none\">\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">Stretched Rational Function<\/div>\n<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">Compressed Rational Function<\/div>\n<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">\n<div class=\"mceTemp\">Special Rational Function<\/div>\n<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1030\" style=\"width: 260px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1030\" class=\"wp-image-1030\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09223124\/4x-298x300.png\" alt=\"Graph of 4\/x\" width=\"250\" height=\"252\" \/><\/p>\n<p id=\"caption-attachment-1030\" class=\"wp-caption-text\">[latex]f(x)=\\frac{4}{x}[\/latex]<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1031\" style=\"width: 260px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1031\" class=\"wp-image-1031\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09223301\/38x-300x300.png\" alt=\"Graph of 3 over 8x\" width=\"250\" height=\"251\" \/><\/p>\n<p id=\"caption-attachment-1031\" class=\"wp-caption-text\">[latex]g(x)=\\frac{3}{8x}[\/latex]<\/p>\n<\/div>\n<\/td>\n<td style=\"width: 33.3333%;\">\n<div id=\"attachment_1032\" style=\"width: 260px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1032\" class=\"wp-image-1032\" style=\"font-size: 16px; orphans: 1;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09223507\/rational-fn-300x221.png\" alt=\"Graph of The function in the caption.\" width=\"250\" height=\"184\" \/><\/p>\n<p id=\"caption-attachment-1032\" class=\"wp-caption-text\">[latex]h(x)=\\frac{x+4}{x^{2}-5x+6}[\/latex]<\/p>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Functions take on many shapes. A graphing calculator like Desmos helps us visualize the graphs of functions letting us see asymptotes, intercepts, turning points and other graphical features. In this section, we highlighted the types of graphs and functions we are going to look at in more detail in this course.<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It 5<\/h3>\n<p>Identify the type of function from its graph.<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 300px;\">\n<tbody>\n<tr>\n<th style=\"width: 33.3333%; text-align: center;\">No asymptote<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">Vertical asymptote<\/th>\n<th style=\"width: 33.3333%; text-align: center;\">Horizontal asymptote<\/th>\n<\/tr>\n<tr style=\"height: 300px;\">\n<td style=\"width: 33.3333%; height: 300px;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1036 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Poly-fun-153x300.png\" alt=\"Continuous function with no asymptote.\" width=\"153\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Poly-fun-153x300.png 153w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Poly-fun-524x1024.png 524w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Poly-fun-65x127.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Poly-fun-225x440.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Poly-fun-350x684.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/01\/Poly-fun.png 706w\" sizes=\"auto, (max-width: 153px) 100vw, 153px\" \/><\/td>\n<td style=\"width: 33.3333%; height: 300px;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1037\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09225707\/log-fun-300x294.png\" alt=\"Continuous function with a Vertical Asymptote.\" width=\"250\" height=\"245\" \/><\/td>\n<td style=\"width: 33.3333%; height: 300px;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1038\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/01\/09225827\/Expo-fun-300x251.png\" alt=\"Continuous function with a horizontal asymptote.\" width=\"250\" height=\"210\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm078\">Show Answer<\/span><\/p>\n<div id=\"qhjm078\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Polynomial function<\/li>\n<li>Logarithmic function<\/li>\n<li>Exponential function<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\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-622\">\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>Algebraic Forms of Functions. <strong>Authored by<\/strong>: Hazel McKenna and Leo Chang. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>All graphs created using desmos graphing calculator. <strong>Authored by<\/strong>: Hazel McKenna and Leo Chang. <strong>Provided by<\/strong>: Utah Valley University. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\">https:\/\/www.desmos.com\/calculator<\/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":422608,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Algebraic Forms of Functions\",\"author\":\"Hazel McKenna and Leo Chang\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All graphs created using desmos graphing calculator\",\"author\":\"Hazel McKenna and Leo Chang\",\"organization\":\"Utah Valley University\",\"url\":\"https:\/\/www.desmos.com\/calculator\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-622","chapter","type-chapter","status-publish","hentry"],"part":613,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/622","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/users\/422608"}],"version-history":[{"count":49,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/622\/revisions"}],"predecessor-version":[{"id":4818,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/622\/revisions\/4818"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/613"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/622\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=622"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=622"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=622"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=622"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}