{"id":100,"date":"2019-08-20T17:02:28","date_gmt":"2019-08-20T21:02:28","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/graphs-of-exponential-functions\/"},"modified":"2022-06-01T10:39:28","modified_gmt":"2022-06-01T14:39:28","slug":"graphs-of-exponential-functions","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/graphs-of-exponential-functions\/","title":{"raw":"Graphs of Exponential Functions","rendered":"Graphs of Exponential Functions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n \t<li>Graph exponential functions.<\/li>\n \t<li>Graph exponential functions using transformations.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137442020\">As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool. It gives us another layer of insight for predicting future events.<\/p>\n\n<div id=\"fs-id1165135407520\" class=\"bc-section section\">\n<h3>Graphing Exponential Functions<\/h3>\n<p id=\"fs-id1165137592823\">Before we begin graphing, it is helpful to review the behavior of exponential growth. Recall the table of values for a function of the form[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]whose base is greater than one. We\u2019ll use the function[latex]\\,f\\left(x\\right)={2}^{x}.\\,[\/latex]Observe how the output values in <a class=\"autogenerated-content\" href=\"#Table_04_02_01\">(Figure)<\/a> change as the input increases by[latex]\\,1.[\/latex]<\/p>\n\n<table id=\"Table_04_02_01\" summary=\"Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=2^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, 1\/8), (-2, 1\/4), (-1, 1\/2), (0, 1), (1, 2), (2, 4), and (3, 8).\"><colgroup> <col> <col> <col> <col> <col> <col> <col> <col><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)={2}^{x}[\/latex]<\/strong><\/td>\n<td>[latex]\\frac{1}{8}[\/latex]<\/td>\n<td>[latex]\\frac{1}{4}[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137432031\">Each output value is the product of the previous output and the base,[latex]\\,2.\\,[\/latex]We call the base[latex]\\,2\\,[\/latex]the <em>constant ratio<\/em>. In fact, for any exponential function with the form[latex]\\,f\\left(x\\right)=a{b}^{x},[\/latex][latex]\\,b\\,[\/latex]is the constant ratio of the function. This means that as the input increases by 1, the output value will be the product of the base and the previous output, regardless of the value of[latex]\\,a.[\/latex]<\/p>\n<p id=\"fs-id1165137585799\">Notice from the table that<\/p>\n\n<ul id=\"fs-id1165137658509\">\n \t<li>the output values are positive for all values of [latex]x;[\/latex]<\/li>\n \t<li>as[latex]\\,x\\,[\/latex]increases, the output values increase without bound; and<\/li>\n \t<li>as[latex]\\,x\\,[\/latex]decreases, the output values grow smaller, approaching zero.<\/li>\n<\/ul>\n<p id=\"fs-id1165137647215\"><a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_001\">(Figure)<\/a> shows the exponential growth function [latex]\\,f\\left(x\\right)={2}^{x}.[\/latex]<\/p>\n\n<div id=\"CNX_Precalc_Figure_04_02_001\" class=\"small aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140149\/CNX_Precalc_Figure_04_02_001.jpg\" alt=\"Graph of the exponential function, 2^(x), with labeled points at (-3, 1\/8), (-2, \u00bc), (-1, \u00bd), (0, 1), (1, 2), (2, 4), and (3, 8). The graph notes that the x-axis is an asymptote.\" width=\"487\" height=\"520\"> <strong>Figure 1. <\/strong>Notice that the graph gets close to the x-axis, but never touches it.[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137459614\">The domain of[latex]\\,f\\left(x\\right)={2}^{x}\\,[\/latex]is all real numbers, the range is[latex]\\,\\left(0,\\infty \\right),[\/latex] and the horizontal asymptote is[latex]\\,y=0.[\/latex]<\/p>\n<p id=\"fs-id1165137838249\">To get a sense of the behavior of <span class=\"no-emphasis\">exponential decay<\/span>, we can create a table of values for a function of the form[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]whose base is between zero and one. We\u2019ll use the function[latex]\\,g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}.\\,[\/latex]Observe how the output values in <a class=\"autogenerated-content\" href=\"#Table_04_02_02\">(Figure)<\/a> change as the input increases by[latex]\\,1.[\/latex]<\/p>\n\n<table summary=\"Two rows and eight columns. The first row is labeled, \u201cf(x)=2^x\u201d, with the following values: (-3, 1\/8), (-2, 1\/4), (-1, 1\/2), (0, 1), (1, 2), (2, 4), and (3, 8). The second row is labeled, \u201cg(x)=log_2(x)\u201d, with the following values: (1\/8, -3), (1\/4, -2), (1\/2, -1), (1, 0), (2, 1), (4, 2), and (8, 3).\">\n<tbody>\n<tr>\n<td style=\"width: 90px\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td style=\"width: 59px\">[latex]-3[\/latex]<\/td>\n<td style=\"width: 59px\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 59px\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 57px\">[latex]0[\/latex]<\/td>\n<td style=\"width: 68px\">[latex]1[\/latex]<\/td>\n<td style=\"width: 67px\">[latex]2[\/latex]<\/td>\n<td style=\"width: 67px\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 90px\">[latex]g(x)=\\left(\\frac{1}{2}\\right)^{x}[\/latex]<\/td>\n<td style=\"width: 59px\">[latex]8[\/latex]<\/td>\n<td style=\"width: 59px\">[latex]4[\/latex]<\/td>\n<td style=\"width: 59px\">[latex]2[\/latex]<\/td>\n<td style=\"width: 57px\">[latex]1[\/latex]<\/td>\n<td style=\"width: 68px\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 67px\">[latex]\\frac{1}{4}[\/latex]<\/td>\n<td style=\"width: 67px\">[latex]\\frac{1}{8}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165135347846\">Again, because the input is increasing by 1, each output value is the product of the previous output and the base, or constant ratio[latex]\\,\\frac{1}{2}.[\/latex]<\/p>\n<p id=\"fs-id1165137452063\">Notice from the table that<\/p>\n\n<ul id=\"fs-id1165135499992\">\n \t<li>the output values are positive for all values of[latex]\\,x;[\/latex]<\/li>\n \t<li>as[latex]\\,x\\,[\/latex]increases, the output values grow smaller, approaching zero; and<\/li>\n \t<li>as[latex]\\,x\\,[\/latex]decreases, the output values grow without bound.<\/li>\n<\/ul>\n<p id=\"fs-id1165137405421\"><a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_002\">(Figure)<\/a> shows the exponential decay function,[latex]\\,g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}.[\/latex]<\/p>\n\n<div id=\"CNX_Precalc_Figure_04_02_002\" class=\"small aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140156\/CNX_Precalc_Figure_04_02_002.jpg\" alt=\"Graph of decreasing exponential function, (1\/2)^x, with labeled points at (-3, 8), (-2, 4), (-1, 2), (0, 1), (1, 1\/2), (2, 1\/4), and (3, 1\/8). The graph notes that the x-axis is an asymptote.\" width=\"487\" height=\"520\"> <strong>Figure 2.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137723586\">The domain of[latex]\\,g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}\\,[\/latex]is all real numbers, the range is[latex]\\,\\left(0,\\infty \\right),[\/latex]and the horizontal asymptote is[latex]\\,y=0.[\/latex]<\/p>\n\n<div id=\"fs-id1165135571835\" class=\"textbox key-takeaways\">\n<h3>Characteristics of the Graph of the Parent Function <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\n<p id=\"fs-id1165137848929\">An exponential function with the form[latex]\\,f\\left(x\\right)={b}^{x},[\/latex][latex]\\,b&gt;0,[\/latex][latex]\\,b\\ne 1,[\/latex]has these characteristics:<\/p>\n\n<ul id=\"fs-id1165135186684\">\n \t<li><span class=\"no-emphasis\">one-to-one<\/span> function<\/li>\n \t<li>horizontal asymptote:[latex]\\,y=0[\/latex]<\/li>\n \t<li>domain:[latex]\\,\\left(\u2013\\infty , \\infty \\right)[\/latex]<\/li>\n \t<li>range:[latex]\\,\\left(0,\\infty \\right)[\/latex]<\/li>\n \t<li><em>x-<\/em>intercept: none<\/li>\n \t<li><em>y-<\/em>intercept:[latex]\\,\\left(0,1\\right)\\,[\/latex]<\/li>\n \t<li>increasing if[latex]\\,b&gt;1[\/latex]<\/li>\n \t<li>decreasing if[latex]\\,b&lt;1[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137471878\"><a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_003\">(Figure)<\/a> compares the graphs of <span class=\"no-emphasis\">exponential growth<\/span> and decay functions.<\/p>\n\n<div id=\"CNX_Precalc_Figure_04_02_003\" class=\"medium aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140158\/CNX_Precalc_Figure_04_02_003new.jpg\" alt=\"\" width=\"731\" height=\"407\"> <strong>Figure 3.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134195243\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135194093\"><strong>Given an exponential function of the form[latex]\\,f\\left(x\\right)={b}^{x},[\/latex]graph the function.<\/strong><\/p>\n\n<ol id=\"fs-id1165137435782\" type=\"1\">\n \t<li>Create a table of points.<\/li>\n \t<li>Plot at least[latex]\\,3\\,[\/latex]point from the table, including the <em>y<\/em>-intercept[latex]\\,\\left(0,1\\right).[\/latex]<\/li>\n \t<li>Draw a smooth curve through the points.<\/li>\n \t<li>State the domain,[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex]the range,[latex]\\,\\left(0,\\infty \\right),[\/latex]and the horizontal asymptote, [latex]\\,y=0.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165135208984\">\n<div id=\"fs-id1165137453336\">\n<h3>Sketching the Graph of an Exponential Function of the Form <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\n<p id=\"fs-id1165137767671\">Sketch a graph of[latex]\\,f\\left(x\\right)={0.25}^{x}.\\,[\/latex]State the domain, range, and asymptote.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135696740\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135696740\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135696740\"]\n<p id=\"fs-id1165137734539\">Before graphing, identify the behavior and create a table of points for the graph.<\/p>\n\n<ul id=\"fs-id1165137566570\">\n \t<li>Since[latex]\\,b=0.25\\,[\/latex]is between zero and one, we know the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote[latex]\\,y=0.[\/latex]<\/li>\n \t<li>Create a table of points as in <a class=\"autogenerated-content\" href=\"#Table_04_02_03\">(Figure)<\/a>.\n<table summary=\"Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=(0.25)^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, 64), (-2, 16), (-1, 4), (0, 1), (1, 0.25), (2, 0.0625), and (3, Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=(0.25)^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, 64), (-2, 16), (-1, 4), (0, 1), (1, 0.25), (2, 0.0625), and (3, 0.015625).\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)={0.25}^{x}[\/latex]<\/strong><\/td>\n<td>[latex]64[\/latex]<\/td>\n<td>[latex]16[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]0.25[\/latex]<\/td>\n<td>[latex]0.0625[\/latex]<\/td>\n<td>[latex]0.015625[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n \t<li>Plot the <em>y<\/em>-intercept,[latex]\\,\\left(0,1\\right),[\/latex]along with two other points. We can use[latex]\\,\\left(-1,4\\right)\\,[\/latex]and[latex]\\,\\left(1,0.25\\right).[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137482830\">Draw a smooth curve connecting the points as in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_004\">(Figure)<\/a>.<\/p>\n\n<div id=\"CNX_Precalc_Figure_04_02_004\" class=\"small aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140213\/CNX_Precalc_Figure_04_02_004.jpg\" alt=\"Graph of the decaying exponential function f(x) = 0.25^x with labeled points at (-1, 4), (0, 1), and (1, 0.25).\" width=\"487\" height=\"332\"> <strong>Figure 4.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137548870\">The domain is[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]the range is[latex]\\,\\left(0,\\infty \\right);\\,[\/latex]the horizontal asymptote is[latex]\\,y=0.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135499977\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165137761245\">\n<p id=\"fs-id1165137548853\">Sketch the graph of[latex]\\,f\\left(x\\right)={4}^{x}.\\,[\/latex]State the domain, range, and asymptote.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137731723\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137731723\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137731723\"]\n<p id=\"fs-id1165137500954\">The domain is[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]the range is[latex]\\,\\left(0,\\infty \\right);\\,[\/latex]the horizontal asymptote is[latex]\\,y=0.[\/latex]<\/p>\n<span id=\"fs-id1165137437648\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140215\/CNX_Precalc_Figure_04_02_005.jpg\" alt=\"Graph of the increasing exponential function f(x) = 4^x with labeled points at (-1, 0.25), (0, 1), and (1, 4).\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137694074\" class=\"bc-section section\">\n<h3>Graphing Transformations of Exponential Functions<\/h3>\n<p id=\"fs-id1165137575238\">Transformations of exponential graphs behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations\u2014shifts, reflections, stretches, and compressions\u2014to the parent function[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]without loss of shape. For instance, just as the quadratic function maintains its parabolic shape when shifted, reflected, stretched, or compressed, the exponential function also maintains its general shape regardless of the transformations applied.<\/p>\n\n<div id=\"fs-id1165134312214\" class=\"bc-section section\">\n<h4>Graphing a Vertical Shift<\/h4>\n<p id=\"fs-id1165137911544\">The first transformation occurs when we add a constant[latex]\\,d\\,[\/latex]to the parent function[latex]\\,f\\left(x\\right)={b}^{x},[\/latex] giving us a <span class=\"no-emphasis\">vertical shift<\/span>[latex]\\,d\\,[\/latex]units in the same direction as the sign. For example, if we begin by graphing a parent function,[latex]\\,f\\left(x\\right)={2}^{x},[\/latex] we can then graph two vertical shifts alongside it, using[latex]\\,d=3:\\,[\/latex]the upward shift,[latex]\\,g\\left(x\\right)={2}^{x}+3\\,[\/latex]and the downward shift,[latex]\\,h\\left(x\\right)={2}^{x}-3.\\,[\/latex]Both vertical shifts are shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_006\">(Figure)<\/a>.<\/p>\n\n<div id=\"CNX_Precalc_Figure_04_02_006\" class=\"small aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140221\/CNX_Precalc_Figure_04_02_006.jpg\" alt=\"Graph of three functions, g(x) = 2^x+3 in blue with an asymptote at y=3, f(x) = 2^x in orange with an asymptote at y=0, and h(x)=2^x-3 with an asymptote at y=-3. Note that each functions\u2019 transformations are described in the text.\" width=\"487\" height=\"628\"> <strong>Figure 5.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137464499\">Observe the results of shifting[latex]\\,f\\left(x\\right)={2}^{x}\\,[\/latex]vertically:<\/p>\n\n<ul id=\"fs-id1165135203774\">\n \t<li>The domain,[latex]\\,\\left(-\\infty ,\\infty \\right)\\,[\/latex]remains unchanged.<\/li>\n \t<li>When the function is shifted up[latex]\\,3\\,[\/latex]units to[latex]\\,g\\left(x\\right)={2}^{x}+3:[\/latex]\n<ul id=\"fs-id1165137601587\">\n \t<li>The <em>y-<\/em>intercept shifts up[latex]\\,3\\,[\/latex]units to[latex]\\,\\left(0,4\\right).[\/latex]<\/li>\n \t<li>The asymptote shifts up[latex]\\,3\\,[\/latex]units to[latex]\\,y=3.[\/latex]<\/li>\n \t<li>The range becomes[latex]\\,\\left(3,\\infty \\right).[\/latex]<\/li>\n<\/ul>\n<\/li>\n \t<li>When the function is shifted down[latex]\\,3\\,[\/latex]units to[latex]\\,h\\left(x\\right)={2}^{x}-3:[\/latex]\n<ul id=\"fs-id1165137784817\">\n \t<li>The <em>y-<\/em>intercept shifts down[latex]\\,3\\,[\/latex]units to[latex]\\,\\left(0,-2\\right).[\/latex]<\/li>\n \t<li>The asymptote also shifts down[latex]\\,3\\,[\/latex]units to[latex]\\,y=-3.[\/latex]<\/li>\n \t<li>The range becomes[latex]\\,\\left(-3,\\infty \\right).[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137566517\" class=\"bc-section section\">\n<h4>Graphing a Horizontal Shift<\/h4>\n<p id=\"fs-id1165137748336\">The next transformation occurs when we add a constant[latex]\\,c\\,[\/latex]to the input of the parent function[latex]\\,f\\left(x\\right)={b}^{x},[\/latex] giving us a <span class=\"no-emphasis\">horizontal shift<\/span>[latex]\\,c\\,[\/latex]units in the <em>opposite<\/em> direction of the sign. For example, if we begin by graphing the parent function[latex]\\,f\\left(x\\right)={2}^{x},[\/latex] we can then graph two horizontal shifts alongside it, using[latex]\\,c=3:\\,[\/latex]the shift left,[latex]\\,g\\left(x\\right)={2}^{x+3},[\/latex] and the shift right,[latex]\\,h\\left(x\\right)={2}^{x-3}.\\,[\/latex]Both horizontal shifts are shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_007\">(Figure)<\/a>.<\/p>\n\n<div id=\"CNX_Precalc_Figure_04_02_007\" class=\"medium aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140233\/CNX_Precalc_Figure_04_02_007.jpg\" alt=\"Graph of three functions, g(x) = 2^(x+3) in blue, f(x) = 2^x in orange, and h(x)=2^(x-3). Each functions\u2019 asymptotes are at y=0Note that each functions\u2019 transformations are described in the text.\" width=\"731\" height=\"478\"> <strong>Figure 6.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137411256\">Observe the results of shifting[latex]\\,f\\left(x\\right)={2}^{x}\\,[\/latex]horizontally:<\/p>\n\n<ul id=\"fs-id1165135187815\">\n \t<li>The domain,[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex]remains unchanged.<\/li>\n \t<li>The asymptote,[latex]\\,y=0,[\/latex]remains unchanged.<\/li>\n \t<li>The <em>y-<\/em>intercept shifts such that:\n<ul id=\"fs-id1165137482879\">\n \t<li>When the function is shifted left[latex]\\,3\\,[\/latex]units to[latex]\\,g\\left(x\\right)={2}^{x+3},[\/latex]the <em>y<\/em>-intercept becomes[latex]\\,\\left(0,8\\right).\\,[\/latex]This is because[latex]\\,{2}^{x+3}=\\left(8\\right){2}^{x},[\/latex]so the initial value of the function is[latex]\\,8.[\/latex]<\/li>\n \t<li>When the function is shifted right[latex]\\,3\\,[\/latex]units to[latex]\\,h\\left(x\\right)={2}^{x-3},[\/latex]the <em>y<\/em>-intercept becomes[latex]\\,\\left(0,\\frac{1}{8}\\right).\\,[\/latex]Again, see that[latex]\\,{2}^{x-3}=\\left(\\frac{1}{8}\\right){2}^{x},[\/latex]so the initial value of the function is[latex]\\,\\frac{1}{8}.[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<div id=\"fs-id1165134042183\" class=\"textbox key-takeaways\">\n<h3>Shifts of the Parent Function <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\n<p id=\"fs-id1165134037589\">For any constants[latex]\\,c\\,[\/latex]and[latex]\\,d,[\/latex]the function[latex]\\,f\\left(x\\right)={b}^{x+c}+d\\,[\/latex]shifts the parent function[latex]\\,f\\left(x\\right)={b}^{x}[\/latex]<\/p>\n\n<ul id=\"fs-id1165137638569\">\n \t<li>vertically[latex]\\,d\\,[\/latex]units, in the <em>same<\/em> direction of the sign of[latex]\\,d.[\/latex]<\/li>\n \t<li>horizontally[latex]\\,c\\,[\/latex]units, in the <em>opposite<\/em> direction of the sign of[latex]\\,c.[\/latex]<\/li>\n \t<li>The <em>y<\/em>-intercept becomes[latex]\\,\\left(0,{b}^{c}+d\\right).[\/latex]<\/li>\n \t<li>The horizontal asymptote becomes[latex]\\,y=d.[\/latex]<\/li>\n \t<li>The range becomes[latex]\\,\\left(d,\\infty \\right).[\/latex]<\/li>\n \t<li>The domain,[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex]remains unchanged.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135500732\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135500706\"><strong>Given an exponential function with the form[latex]\\,f\\left(x\\right)={b}^{x+c}+d,[\/latex]graph the translation.<\/strong><\/p>\n\n<ol id=\"fs-id1165137767676\" type=\"1\">\n \t<li>Draw the horizontal asymptote[latex]\\,y=d.[\/latex]<\/li>\n \t<li>Identify the shift as[latex]\\,\\left(-c,d\\right).\\,[\/latex]Shift the graph of[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]left[latex]\\,c\\,[\/latex]units if[latex]\\,c\\,[\/latex]is positive, and right[latex]\\,c\\,[\/latex]units if[latex]c\\,[\/latex]is negative.<\/li>\n \t<li>Shift the graph of[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]up[latex]\\,d\\,[\/latex]units if[latex]\\,d\\,[\/latex]is positive, and down[latex]\\,d\\,[\/latex]units if[latex]\\,d\\,[\/latex]is negative.<\/li>\n \t<li>State the domain,[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex]the range,[latex]\\,\\left(d,\\infty \\right),[\/latex]and the horizontal asymptote[latex]\\,y=d.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137834201\">\n<div id=\"fs-id1165137416701\">\n<h3>Graphing a Shift of an Exponential Function<\/h3>\n<p id=\"fs-id1165137563667\">Graph[latex]\\,f\\left(x\\right)={2}^{x+1}-3.\\,[\/latex]State the domain, range, and asymptote.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135175234\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135175234\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135175234\"]\n<p id=\"fs-id1165137923482\">We have an exponential equation of the form[latex]\\,f\\left(x\\right)={b}^{x+c}+d,[\/latex] with[latex]\\,b=2,[\/latex][latex]\\,c=1,[\/latex] and[latex]\\,d=-3.[\/latex]<\/p>\n<p id=\"fs-id1165137469681\">Draw the horizontal asymptote[latex]\\,y=d[\/latex], so draw[latex]\\,y=-3.[\/latex]<\/p>\n<p id=\"fs-id1165137661814\">Identify the shift as[latex]\\,\\left(-c,d\\right),[\/latex] so the shift is[latex]\\,\\left(-1,-3\\right).[\/latex]<\/p>\n<p id=\"fs-id1165137693953\">Shift the graph of[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]left 1 units and down 3 units.<\/p>\n\n<div id=\"CNX_Precalc_Figure_04_02_008\" class=\"small aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140245\/CNX_Precalc_Figure_04_02_008.jpg\" alt=\"Graph of the function, f(x) = 2^(x+1)-3, with an asymptote at y=-3. Labeled points in the graph are (-1, -2), (0, -1), and (1, 1).\" width=\"487\" height=\"519\"> <strong>Figure 7.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165134199602\">The domain is[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]the range is[latex]\\,\\left(-3,\\infty \\right);\\,[\/latex]the horizontal asymptote is[latex]\\,y=-3.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135241073\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165137805939\">\n<p id=\"fs-id1165137805941\">Graph[latex]\\,f\\left(x\\right)={2}^{x-1}+3.\\,[\/latex]State domain, range, and asymptote.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137731918\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137731918\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137731918\"]\n<p id=\"fs-id1165135513714\">The domain is[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]the range is[latex]\\,\\left(3,\\infty \\right);\\,[\/latex]the horizontal asymptote is[latex]\\,y=3.[\/latex]<\/p>\n<span id=\"fs-id1165137628194\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140252\/CNX_Precalc_Figure_04_02_009.jpg\" alt=\"Graph of the function, f(x) = 2^(x-1)+3, with an asymptote at y=3. Labeled points in the graph are (-1, 3.25), (0, 3.5), and (1, 4).\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137639988\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137756810\"><strong>Given an equation of the form[latex]\\,f\\left(x\\right)={b}^{x+c}+d\\,[\/latex]for[latex]\\,x,[\/latex] use a graphing calculator to approximate the solution.<\/strong><\/p>\n\n<ul id=\"fs-id1165137842461\">\n \t<li>Press <strong>[Y=]<\/strong>. Enter the given exponential equation in the line headed \u201c<strong>Y<sub>1<\/sub>=<\/strong>\u201d.<\/li>\n \t<li>Enter the given value for[latex]\\,f\\left(x\\right)\\,[\/latex]in the line headed \u201c<strong>Y<sub>2<\/sub>=<\/strong>\u201d.<\/li>\n \t<li>Press <strong>[WINDOW]<\/strong>. Adjust the <em>y<\/em>-axis so that it includes the value entered for \u201c<strong>Y<sub>2<\/sub>=<\/strong>\u201d.<\/li>\n \t<li>Press <strong>[GRAPH]<\/strong> to observe the graph of the exponential function along with the line for the specified value of[latex]\\,f\\left(x\\right).[\/latex]<\/li>\n \t<li>To find the value of[latex]\\,x,[\/latex]we compute the point of intersection. Press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select \u201cintersect\u201d and press <strong>[ENTER]<\/strong> three times. The point of intersection gives the value of <em>x <\/em>for the indicated value of the function.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137618985\">\n<div id=\"fs-id1165137618987\">\n<h3>Approximating the Solution of an Exponential Equation<\/h3>\n<p id=\"fs-id1165135449598\">Solve[latex]\\,42=1.2{\\left(5\\right)}^{x}+2.8\\,[\/latex]graphically. Round to the nearest thousandth.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137653309\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137653309\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137653309\"]\n<p id=\"fs-id1165137737383\">Press <strong>[Y=]<\/strong> and enter[latex]\\,1.2{\\left(5\\right)}^{x}+2.8\\,[\/latex]next to <strong>Y<sub>1<\/sub><\/strong>=. Then enter 42 next to <strong>Y2=<\/strong>. For a window, use the values \u20133 to 3 for[latex]\\,x\\,[\/latex]and \u20135 to 55 for[latex]\\,y.\\,[\/latex]Press <strong>[GRAPH]<\/strong>. The graphs should intersect somewhere near[latex]\\,x=2.[\/latex]<\/p>\n<p id=\"fs-id1165137460953\">For a better approximation, press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select <strong>[5: intersect]<\/strong> and press <strong>[ENTER]<\/strong> three times. The <em>x<\/em>-coordinate of the point of intersection is displayed as 2.1661943. (Your answer may be different if you use a different window or use a different value for <strong>Guess?<\/strong>) To the nearest thousandth,[latex]\\,x\\approx 2.166.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135545893\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165137838712\">\n<p id=\"fs-id1165137838714\">Solve[latex]\\,4=7.85{\\left(1.15\\right)}^{x}-2.27\\,[\/latex]graphically. Round to the nearest thousandth.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137854192\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137854192\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137854192\"]\n<p id=\"fs-id1165137854194\">[latex]x\\approx -1.608[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137431154\" class=\"bc-section section\">\n<h4>Graphing a Stretch or Compression<\/h4>\n<p id=\"fs-id1165137863514\">While horizontal and vertical shifts involve adding constants to the input or to the function itself, a <span class=\"no-emphasis\">stretch<\/span> or <span class=\"no-emphasis\">compression<\/span> occurs when we multiply the parent function[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]by a constant[latex]\\,|a|&gt;0.\\,[\/latex]For example, if we begin by graphing the parent function[latex]\\,f\\left(x\\right)={2}^{x},[\/latex]we can then graph the stretch, using[latex]\\,a=3,[\/latex]to get[latex]\\,g\\left(x\\right)=3{\\left(2\\right)}^{x}\\,[\/latex]as shown on the left in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_010\">(Figure)<\/a>, and the compression, using[latex]\\,a=\\frac{1}{3},[\/latex]to get[latex]\\,h\\left(x\\right)=\\frac{1}{3}{\\left(2\\right)}^{x}\\,[\/latex]as shown on the right in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_010\">(Figure)<\/a>.<\/p>\n\n<div id=\"CNX_Precalc_Figure_04_02_010\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140255\/CNX_Precalc_Figure_04_02_010.jpg\" alt=\"Two graphs where graph a is an example of vertical stretch and graph b is an example of vertical compression.\" width=\"975\" height=\"445\"> <strong>Figure 8.<\/strong> (a)[latex]\\,g\\left(x\\right)=3{\\left(2\\right)}^{x}\\,[\/latex]stretches the graph of[latex]\\,f\\left(x\\right)={2}^{x}\\,[\/latex]vertically by a factor of[latex]\\,3.\\,[\/latex](b)[latex]\\,h\\left(x\\right)=\\frac{1}{3}{\\left(2\\right)}^{x}\\,[\/latex]compresses the graph of[latex]\\,f\\left(x\\right)={2}^{x}\\,[\/latex]vertically by a factor of[latex]\\,\\frac{1}{3}.[\/latex][\/caption]\n<div id=\"fs-id1165137627908\" class=\"textbox key-takeaways\">\n<h3>Stretches and Compressions of the Parent Function <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\n<p id=\"fs-id1165137696285\">For any factor[latex]\\,a&gt;0,[\/latex]the function[latex]\\,f\\left(x\\right)=a{\\left(b\\right)}^{x}[\/latex]<\/p>\n\n<ul id=\"fs-id1165137476370\">\n \t<li>is stretched vertically by a factor of[latex]\\,a\\,[\/latex]if[latex]\\,|a|&gt;1.[\/latex]<\/li>\n \t<li>is compressed vertically by a factor of[latex]\\,a\\,[\/latex]if[latex]\\,|a|&lt;1.[\/latex]<\/li>\n \t<li>has a <em>y<\/em>-intercept of[latex]\\,\\left(0,a\\right).[\/latex]<\/li>\n \t<li>has a horizontal asymptote at[latex]\\,y=0,[\/latex] a range of[latex]\\,\\left(0,\\infty \\right),[\/latex] and a domain of[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex]which are unchanged from the parent function.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165135528997\">\n<div id=\"fs-id1165135656098\">\n<h3 id=\"fs-id1165135656100\">Graphing the Stretch of an Exponential Function<\/h3>\n<p id=\"fs-id1165135656104\">Sketch a graph of[latex]\\,f\\left(x\\right)=4{\\left(\\frac{1}{2}\\right)}^{x}.\\,[\/latex]State the domain, range, and asymptote.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137657436\" class=\"solution textbox shaded\">\n\n[reveal-answer q=\"274741\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"274741\"]\n\nBefore graphing, identify the behavior and key points on the graph.\n<ul id=\"fs-id1165137657441\">\n \t<li>Since[latex]\\,b=\\frac{1}{2}\\,[\/latex]is between zero and one, the left tail of the graph will increase without bound as[latex]\\,x\\,[\/latex]decreases, and the right tail will approach the <em>x<\/em>-axis as[latex]\\,x\\,[\/latex]increases.<\/li>\n \t<li>Since[latex]\\,a=4,[\/latex]the graph of[latex]\\,f\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}\\,[\/latex]will be stretched by a factor of[latex]\\,4.[\/latex]<\/li>\n \t<li>Create a table of points as shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2414&amp;action=edit#Table_04_02_04\">(Figure)<\/a>.\n<table summary=\"Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=4(0.25)^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, 32), (-2, 16), (-1, 8), (0, 4), (1, 2), (2, 1), and (3, 0.5).\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]f\\left(x\\right)=4\\left(\\frac{1}{2}\\right)^{x}[\/latex]<\/td>\n<td>[latex]32[\/latex]<\/td>\n<td>[latex]16[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]0.5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n \t<li>Plot the <em>y-<\/em>intercept,[latex]\\,\\left(0,4\\right),[\/latex]along with two other points. We can use[latex]\\,\\left(-1,8\\right)\\,[\/latex]and[latex]\\,\\left(1,2\\right).[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165135319502\">Draw a smooth curve connecting the points, as shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2414&amp;action=edit#CNX_Precalc_Figure_04_02_011\">(Figure)<\/a>.<\/p>\n\n<div id=\"CNX_Precalc_Figure_04_02_011\" class=\"small aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140302\/CNX_Precalc_Figure_04_02_011.jpg\" alt=\"Graph of the function, f(x) = 4(1\/2)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 8), (0, 4), and (1, 2).\" width=\"487\" height=\"482\"> <strong>Figure 9.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137442037\">The domain is[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]the range is[latex]\\,\\left(0,\\infty \\right);\\,[\/latex]the horizontal asymptote is[latex]\\,y=0.[\/latex]<\/p>\n<p id=\"fs-id1165137442037\">[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135541809\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_02_04\">\n<div>\n<p id=\"fs-id1165137452032\">Sketch the graph of[latex]\\,f\\left(x\\right)=\\frac{1}{2}{\\left(4\\right)}^{x}.\\,[\/latex]State the domain, range, and asymptote.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137694067\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137694067\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137694067\"]\n<p id=\"fs-id1165137653325\">The domain is[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]the range is[latex]\\,\\left(0,\\infty \\right);\\,[\/latex]the horizontal asymptote is[latex]\\,y=0.\\,[\/latex]<span id=\"fs-id1165135417835\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140309\/CNX_Precalc_Figure_04_02_012.jpg\" alt=\"Graph of the function, f(x) = (1\/2)(4)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 0.125), (0, 0.5), and (1, 2).\"><\/span>[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135433028\" class=\"bc-section section\">\n<h4>Graphing Reflections<\/h4>\n<p id=\"fs-id1165137452750\">In addition to shifting, compressing, and stretching a graph, we can also reflect it about the <em>x<\/em>-axis or the <em>y<\/em>-axis. When we multiply the parent function[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]by[latex]\\,-1,[\/latex]we get a reflection about the <em>x<\/em>-axis. When we multiply the input by[latex]\\,-1,[\/latex]we get a <span class=\"no-emphasis\">reflection<\/span> about the <em>y<\/em>-axis. For example, if we begin by graphing the parent function[latex]\\,f\\left(x\\right)={2}^{x},[\/latex] we can then graph the two reflections alongside it. The reflection about the <em>x<\/em>-axis,[latex]\\,g\\left(x\\right)={-2}^{x},[\/latex]is shown on the left side of <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_013\">(Figure)<\/a>, and the reflection about the <em>y<\/em>-axis[latex]\\,h\\left(x\\right)={2}^{-x},[\/latex] is shown on the right side of <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_013\">(Figure)<\/a>.<\/p>\n\n<div id=\"CNX_Precalc_Figure_04_02_013\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140318\/CNX_Precalc_Figure_04_02_013.jpg\" alt=\"Two graphs where graph a is an example of a reflection about the x-axis and graph b is an example of a reflection about the y-axis.\" width=\"975\" height=\"628\"> <strong>Figure 10. <\/strong>(a)[latex]\\,g\\left(x\\right)=-{2}^{x}\\,[\/latex]reflects the graph of[latex]\\,f\\left(x\\right)={2}^{x}\\,[\/latex]about the x-axis. (b)[latex]\\,g\\left(x\\right)={2}^{-x}\\,[\/latex]reflects the graph of[latex]\\,f\\left(x\\right)={2}^{x}\\,[\/latex]about the y-axis.[\/caption]\n<div id=\"fs-id1165135477501\" class=\"textbox key-takeaways\">\n<h3>Reflections of the Parent Function <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\n<p id=\"fs-id1165137455888\">The function[latex]\\,f\\left(x\\right)=-{b}^{x}[\/latex]<\/p>\n\n<ul>\n \t<li>reflects the parent function[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]about the <em>x<\/em>-axis.<\/li>\n \t<li>has a <em>y<\/em>-intercept of[latex]\\,\\left(0,-1\\right).[\/latex]<\/li>\n \t<li>has a range of[latex]\\,\\left(-\\infty ,0\\right)[\/latex]<\/li>\n \t<li>has a horizontal asymptote at[latex]\\,y=0\\,[\/latex]and domain of[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex]which are unchanged from the parent function.<\/li>\n<\/ul>\n<p id=\"fs-id1165137742185\">The function[latex]\\,f\\left(x\\right)={b}^{-x}[\/latex]<\/p>\n\n<ul id=\"fs-id1165137551240\">\n \t<li>reflects the parent function[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]about the <em>y<\/em>-axis.<\/li>\n \t<li>has a <em>y<\/em>-intercept of[latex]\\,\\left(0,1\\right),[\/latex] a horizontal asymptote at[latex]\\,y=0,[\/latex] a range of[latex]\\,\\left(0,\\infty \\right),[\/latex] and a domain of[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex] which are unchanged from the parent function.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Example_04_02_05\" class=\"textbox examples\">\n<div id=\"fs-id1165137406134\">\n<div id=\"fs-id1165137406136\">\n<h3>Writing and Graphing the Reflection of an Exponential Function<\/h3>\n<p id=\"fs-id1165137896193\">Find and graph the equation for a function,[latex]\\,g\\left(x\\right),[\/latex]that reflects[latex]\\,f\\left(x\\right)={\\left(\\frac{1}{4}\\right)}^{x}\\,[\/latex]about the <em>x<\/em>-axis. State its domain, range, and asymptote.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137937537\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137937537\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137937537\"]\n<p id=\"fs-id1165137937539\">Since we want to reflect the parent function[latex]\\,f\\left(x\\right)={\\left(\\frac{1}{4}\\right)}^{x}\\,[\/latex]about the <em>x-<\/em>axis, we multiply[latex]\\,f\\left(x\\right)\\,[\/latex]by[latex]\\,-1\\,[\/latex]to get,[latex]\\,g\\left(x\\right)=-{\\left(\\frac{1}{4}\\right)}^{x}.\\,[\/latex]Next we create a table of points as in <a class=\"autogenerated-content\" href=\"#Table_04_02_005\">(Figure)<\/a>.<\/p>\n\n<table id=\"Table_04_02_005\" summary=\"Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=-(1\/4)^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, -64), (-2, -16), (-1, -4), (0, -1), (1, -0.25), (2, -0.0625), and (3, -0.0156).\"><colgroup> <col> <col> <col> <col> <col> <col> <col> <col><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]g\\left(x\\right)=-\\left(\\frac{1}{4}\\right)^{x}[\/latex]<\/td>\n<td>[latex]-64[\/latex]<\/td>\n<td>[latex]-16[\/latex]<\/td>\n<td>[latex]-4[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]-0.25[\/latex]<\/td>\n<td>[latex]-0.0625[\/latex]<\/td>\n<td>[latex]-0.0156[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"eip-id1167546794019\">Plot the <em>y-<\/em>intercept,[latex]\\,\\left(0,-1\\right),[\/latex]along with two other points. We can use[latex]\\,\\left(-1,-4\\right)\\,[\/latex]and[latex]\\,\\left(1,-0.25\\right).[\/latex]<\/p>\n<p id=\"fs-id1165135369275\">Draw a smooth curve connecting the points:<\/p>\n\n<div id=\"CNX_Precalc_Figure_04_02_014\" class=\"small aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140330\/CNX_Precalc_Figure_04_02_014.jpg\" alt=\"Graph of the function, g(x) = -(0.25)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, -4), (0, -1), and (1, -0.25).\" width=\"487\" height=\"407\"> <strong>Figure 11.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165137828154\">The domain is[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]the range is[latex]\\,\\left(-\\infty ,0\\right);\\,[\/latex]the horizontal asymptote is[latex]\\,y=0.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135205992\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_02_05\">\n<div id=\"fs-id1165135254653\">\n<p id=\"fs-id1165135254655\">Find and graph the equation for a function,[latex]\\,g\\left(x\\right),[\/latex] that reflects[latex]\\,f\\left(x\\right)={1.25}^{x}\\,[\/latex]about the <em>y<\/em>-axis. State its domain, range, and asymptote.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135368458\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135368458\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135368458\"]\n<p id=\"fs-id1165135368461\">The domain is[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]the range is[latex]\\,\\left(0,\\infty \\right);\\,[\/latex]the horizontal asymptote is[latex]\\,y=0.[\/latex]<\/p>\n<span id=\"fs-id1165137828034\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140340\/CNX_Precalc_Figure_04_02_015.jpg\" alt=\"Graph of the function, g(x) = -(1.25)^(-x), with an asymptote at y=0. Labeled points in the graph are (-1, 1.25), (0, 1), and (1, 0.8).\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135501015\" class=\"bc-section section\">\n<h4>Summarizing Translations of the Exponential Function<\/h4>\n<p id=\"fs-id1165135501021\">Now that we have worked with each type of translation for the exponential function, we can summarize them in <a class=\"autogenerated-content\" href=\"#Table_04_02_006\">(Figure)<\/a> to arrive at the general equation for translating exponential functions.<\/p>\n1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept remains (1, 0), the key point changes to (b^(-1), 1), the domain remains (0, infinity), and the range remains (-infinity, infinity). The second column shows the left shift of the equation g(x)=log_b(x) when b&gt;1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept changes to (-1, 0), the key point changes to (-b, 1), the domain changes to (-infinity, 0), and the range remains (-infinity, infinity).\"&gt;\n<table id=\"Table_04_02_006\">\n<thead>\n<tr>\n<th colspan=\"2\">Translations of the Parent Function [latex]\\,f\\left(x\\right)={b}^{x}[\/latex]<\/th>\n<\/tr>\n<tr>\n<th>Translation<\/th>\n<th>Form<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Shift\n<ul id=\"fs-id1165137640731\">\n \t<li>Horizontally[latex]\\,c\\,[\/latex]units to the left<\/li>\n \t<li>Vertically[latex]\\,d\\,[\/latex]units up<\/li>\n<\/ul>\n<\/td>\n<td>[latex]f\\left(x\\right)={b}^{x+c}+d[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Stretch and Compress\n<ul id=\"fs-id1165134074993\">\n \t<li>Stretch if[latex]\\,|a|&gt;1[\/latex]<\/li>\n \t<li>Compression if[latex]\\,0&lt;|a|&lt;1[\/latex]<\/li>\n<\/ul>\n<\/td>\n<td>[latex]f\\left(x\\right)=a{b}^{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reflect about the <em>x<\/em>-axis<\/td>\n<td>[latex]f\\left(x\\right)=-{b}^{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reflect about the <em>y<\/em>-axis<\/td>\n<td>[latex]f\\left(x\\right)={b}^{-x}={\\left(\\frac{1}{b}\\right)}^{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>General equation for all translations<\/td>\n<td>[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165137635134\" class=\"textbox key-takeaways\">\n<h3>Translations of Exponential Functions<\/h3>\n<p id=\"fs-id1165137806521\">A translation of an exponential function has the form<\/p>\n\n<div id=\"fs-id1165137806525\" class=\"unnumered aligncenter\">[latex] f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/div>\n<p id=\"fs-id1165137805520\">Where the parent function,[latex]\\,y={b}^{x},[\/latex][latex]\\,b&gt;1,[\/latex]is<\/p>\n\n<ul id=\"fs-id1165137678290\">\n \t<li>shifted horizontally[latex]\\,c\\,[\/latex]units to the left.<\/li>\n \t<li>stretched vertically by a factor of[latex]\\,|a|\\,[\/latex]if[latex]\\,|a|&gt;0.[\/latex]<\/li>\n \t<li>compressed vertically by a factor of[latex]\\,|a|\\,[\/latex]if[latex]\\,0&lt;|a|&lt;1.[\/latex]<\/li>\n \t<li>shifted vertically[latex]\\,d\\,[\/latex]units.<\/li>\n \t<li>reflected about the <em>x-<\/em>axis when[latex]\\,a&lt;0.[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137937613\">Note the order of the shifts, transformations, and reflections follow the order of operations.<\/p>\n\n<\/div>\n<div id=\"Example_04_02_06\" class=\"textbox examples\">\n<div id=\"fs-id1165137937623\">\n<div id=\"fs-id1165135250578\">\n<h3 id=\"fs-id1165135250580\">Writing a Function from a Description<\/h3>\n<p id=\"fs-id1165135250584\">Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.<\/p>\n\n<ul id=\"fs-id1165137724821\">\n \t<li>[latex]f\\left(x\\right)={e}^{x}\\,[\/latex]is vertically stretched by a factor of[latex]\\,2\\,[\/latex], reflected across the <em>y<\/em>-axis, and then shifted up[latex]\\,4\\,[\/latex]units.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135532412\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135532412\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135532412\"]\n<p id=\"fs-id1165135532414\">We want to find an equation of the general form[latex] \\,f\\left(x\\right)=a{b}^{x+c}+d.\\,[\/latex]We use the description provided to find[latex]\\,a,[\/latex] [latex]b,[\/latex] [latex]c,[\/latex] and [latex]\\,d.[\/latex]<\/p>\n\n<ul id=\"fs-id1165137807102\">\n \t<li>We are given the parent function[latex]\\,f\\left(x\\right)={e}^{x},[\/latex] so[latex]\\,b=e.[\/latex]<\/li>\n \t<li>The function is stretched by a factor of[latex]\\,2[\/latex], so[latex]\\,a=2.[\/latex]<\/li>\n \t<li>The function is reflected about the <em>y<\/em>-axis. We replace[latex]\\,x\\,[\/latex]with[latex]\\,-x\\,[\/latex]to get:[latex]\\,{e}^{-x}.[\/latex]<\/li>\n \t<li>The graph is shifted vertically 4 units, so[latex]\\,d=4.[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137634849\">Substituting in the general form we get,<\/p>\n\n<div id=\"eip-id1165137832492\" class=\"unnumbered\">[latex]\\begin{array}{ll} f\\left(x\\right)\\hfill &amp; =a{b}^{x+c}+d\\hfill \\\\ \\hfill &amp; =2{e}^{-x+0}+4\\hfill \\\\ \\hfill &amp; =2{e}^{-x}+4\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137665666\">The domain is[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]the range is[latex]\\,\\left(4,\\infty \\right);\\,[\/latex]the horizontal asymptote is[latex]\\,y=4.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137553895\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_02_06\">\n<div id=\"fs-id1165137724079\">\n<p id=\"fs-id1165137724081\">Write the equation for function described below. Give the horizontal asymptote, the domain, and the range.<\/p>\n\n<ul id=\"fs-id1165137539693\">\n \t<li>[latex]f\\left(x\\right)={e}^{x}\\,[\/latex]is compressed vertically by a factor of[latex]\\,\\frac{1}{3},[\/latex] reflected across the <em>x<\/em>-axis and then shifted down [latex]\\,2[\/latex] units.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137724110\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137724110\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137724110\"]\n<p id=\"fs-id1165137724112\">[latex]f\\left(x\\right)=-\\frac{1}{3}{e}^{x}-2;\\,[\/latex]the domain is[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]the range is[latex]\\,\\left(-\\infty ,2\\right);\\,[\/latex]the horizontal asymptote is[latex]\\,y=2.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135560786\" class=\"precalculus media\">\n<p id=\"fs-id1165137785000\">Access this online resource for additional instruction and practice with graphing exponential functions.<\/p>\n\n<ul id=\"fs-id1165137785004\">\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphexpfunc\">Graph Exponential Functions<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137661989\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"fs-id2055298\" summary=\"...\">\n<tbody>\n<tr>\n<td>General Form for the Translation of the Parent Function[latex]\\text{ }f\\left(x\\right)={b}^{x}[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137447701\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137447708\">\n \t<li>The graph of the function[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]has a <em>y-<\/em>intercept at[latex]\\,\\left(0, 1\\right),[\/latex]domain[latex]\\,\\left(-\\infty , \\infty \\right),[\/latex]range[latex]\\,\\left(0, \\infty \\right),[\/latex] and horizontal asymptote[latex]\\,y=0.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_04_02_01\">(Figure)<\/a>.<\/li>\n \t<li>If[latex]\\,b&gt;1,[\/latex]the function is increasing. The left tail of the graph will approach the asymptote[latex]\\,y=0,[\/latex] and the right tail will increase without bound.<\/li>\n \t<li>If[latex]\\,0&lt;b&lt;1,[\/latex] the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote[latex]\\,y=0.[\/latex]<\/li>\n \t<li>The equation[latex]\\,f\\left(x\\right)={b}^{x}+d\\,[\/latex]represents a vertical shift of the parent function[latex]\\,f\\left(x\\right)={b}^{x}.[\/latex]<\/li>\n \t<li>The equation[latex]\\,f\\left(x\\right)={b}^{x+c}\\,[\/latex]represents a horizontal shift of the parent function[latex]\\,f\\left(x\\right)={b}^{x}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_04_02_02\">(Figure)<\/a>.<\/li>\n \t<li>Approximate solutions of the equation[latex]\\,f\\left(x\\right)={b}^{x+c}+d\\,[\/latex]can be found using a graphing calculator. See <a class=\"autogenerated-content\" href=\"#Example_04_02_03\">(Figure)<\/a>.<\/li>\n \t<li>The equation[latex]\\,f\\left(x\\right)=a{b}^{x},[\/latex] where[latex]\\,a&gt;0,[\/latex] represents a vertical stretch if[latex]\\,|a|&gt;1\\,[\/latex]or compression if[latex]\\,0&lt;|a|&lt;1\\,[\/latex]of the parent function[latex]\\,f\\left(x\\right)={b}^{x}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_04_02_04\">(Figure)<\/a>.<\/li>\n \t<li>When the parent function[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]is multiplied by[latex]\\,-1,[\/latex]the result,[latex]\\,f\\left(x\\right)=-{b}^{x},[\/latex] is a reflection about the <em>x<\/em>-axis. When the input is multiplied by[latex]\\,-1,[\/latex]the result,[latex]\\,f\\left(x\\right)={b}^{-x},[\/latex] is a reflection about the <em>y<\/em>-axis. See <a class=\"autogenerated-content\" href=\"#Example_04_02_05\">(Figure)<\/a>.<\/li>\n \t<li>All translations of the exponential function can be summarized by the general equation[latex]\\,f\\left(x\\right)=a{b}^{x+c}+d.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Table_04_02_03\">(Figure)<\/a>.<\/li>\n \t<li>Using the general equation[latex]\\,f\\left(x\\right)=a{b}^{x+c}+d,[\/latex] we can write the equation of a function given its description. See <a class=\"autogenerated-content\" href=\"#Example_04_02_06\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137634271\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165137634275\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165135386454\">\n<div id=\"fs-id1165135386456\">\n<p id=\"fs-id1165135386458\">What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph?<\/p>\n\n<\/div>\n<div id=\"fs-id1165135386464\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135386464\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135386464\"]\n<p id=\"fs-id1165135386466\">An asymptote is a line that the graph of a function approaches, as[latex]\\,x\\,[\/latex]either increases or decreases without bound. The horizontal asymptote of an exponential function tells us the limit of the function\u2019s values as the independent variable gets either extremely large or extremely small.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137724992\">\n<div id=\"fs-id1165137724994\">\n<p id=\"fs-id1165137769966\">What is the advantage of knowing how to recognize transformations of the graph of a parent function algebraically?<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137769974\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<div id=\"fs-id1165135696183\">\n<div id=\"fs-id1165135696185\">\n<p id=\"fs-id1165135696187\">The graph of[latex]\\,f\\left(x\\right)={3}^{x}\\,[\/latex]is reflected about the <em>y<\/em>-axis and stretched vertically by a factor of[latex]\\,4.\\,[\/latex]What is the equation of the new function,[latex]\\,g\\left(x\\right)?\\,[\/latex]State its <em>y<\/em>-intercept, domain, and range.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135369259\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135369259\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135369259\"]\n<p id=\"fs-id1165135369261\">[latex]g\\left(x\\right)=4{\\left(3\\right)}^{-x};\\,[\/latex]<em>y<\/em>-intercept:[latex]\\,\\left(0,4\\right);\\,[\/latex]Domain: all real numbers; Range: all real numbers greater than[latex]\\,0.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137471014\">\n<div id=\"fs-id1165137471016\">\n<p id=\"fs-id1165137824073\">The graph of[latex]\\,f\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{-x}\\,[\/latex]is reflected about the <em>y<\/em>-axis and compressed vertically by a factor of[latex]\\,\\frac{1}{5}.\\,[\/latex]What is the equation of the new function,[latex]\\,g\\left(x\\right)?\\,[\/latex]State its <em>y<\/em>-intercept, domain, and range.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135459835\">\n<div id=\"fs-id1165135169299\">\n<p id=\"fs-id1165135169301\">The graph of[latex]\\,f\\left(x\\right)={10}^{x}\\,[\/latex]is reflected about the <em>x<\/em>-axis and shifted upward[latex]\\,7\\,[\/latex]units. What is the equation of the new function,[latex]\\,g\\left(x\\right)?\\,[\/latex]State its <em>y<\/em>-intercept, domain, and range.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137851416\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137851416\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137851416\"]\n<p id=\"fs-id1165137851418\">[latex]g\\left(x\\right)=-{10}^{x}+7;\\,[\/latex]<em>y<\/em>-intercept:[latex]\\,\\left(0,6\\right);\\,[\/latex]Domain: all real numbers; Range: all real numbers less than[latex]\\,7.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135459852\">\n<div id=\"fs-id1165135459854\">\n<p id=\"fs-id1165137675394\">The graph of[latex]\\,f\\left(x\\right)={\\left(1.68\\right)}^{x}\\,[\/latex]is shifted right[latex]\\,3\\,[\/latex]units, stretched vertically by a factor of[latex]\\,2,[\/latex]reflected about the <em>x<\/em>-axis, and then shifted downward[latex]\\,3\\,[\/latex]units. What is the equation of the new function,[latex]\\,g\\left(x\\right)?\\,[\/latex]State its <em>y<\/em>-intercept (to the nearest thousandth), domain, and range.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137874974\">\n<div id=\"fs-id1165137874976\">\n<p id=\"fs-id1165137874978\">The graph of[latex]\\,f\\left(x\\right)=2{\\left(\\frac{1}{4}\\right)}^{x-20}[\/latex] is shifted left[latex]\\,2\\,[\/latex]units, stretched vertically by a factor of[latex]\\,4,[\/latex]reflected about the <em>x<\/em>-axis, and then shifted downward[latex]\\,4\\,[\/latex]units. What is the equation of the new function,[latex]\\,g\\left(x\\right)?\\,[\/latex]State its <em>y<\/em>-intercept, domain, and range.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137724981\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137724981\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137724981\"]\n<p id=\"fs-id1165137863050\">[latex]g\\left(x\\right)=2{\\left(\\frac{1}{4}\\right)}^{x};\\,[\/latex]<em> y<\/em>-intercept:[latex]\\,\\left(0,\\text{ 2}\\right);\\,[\/latex]Domain: all real numbers; Range: all real numbers greater than[latex]\\,0.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137597358\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\nFor the following exercises, graph the function and its reflection about the <em>y<\/em>-axis on the same axes, and give the <em>y<\/em>-intercept.\n<div id=\"fs-id1165137731815\">\n<div id=\"fs-id1165137731818\">\n<p id=\"fs-id1165137731820\">[latex]f\\left(x\\right)=3{\\left(\\frac{1}{2}\\right)}^{x}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137758922\">\n<div id=\"fs-id1165137758924\">\n<p id=\"fs-id1165137758926\">[latex]g\\left(x\\right)=-2{\\left(0.25\\right)}^{x}[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"159818\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"159818\"]<span id=\"fs-id1165137693544\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140348\/CNX_PreCalc_Figure_04_02_202.jpg\" alt=\"Graph of two functions, g(-x)=-2(0.25)^(-x) in blue and g(x)=-2(0.25)^x in orange.\"><\/span>\n<p id=\"fs-id1165135581056\"><em>y<\/em>-intercept:[latex]\\,\\left(0,-2\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137556927\">\n<div id=\"fs-id1165135383139\">\n<p id=\"fs-id1165135383141\">[latex]h\\left(x\\right)=6{\\left(1.75\\right)}^{-x}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137432021\">For the following exercises, graph each set of functions on the same axes.<\/p>\n\n<div id=\"fs-id1165137579050\">\n<div id=\"fs-id1165137579053\">\n<p id=\"fs-id1165137579055\">[latex]f\\left(x\\right)=3{\\left(\\frac{1}{4}\\right)}^{x},[\/latex][latex]g\\left(x\\right)=3{\\left(2\\right)}^{x},[\/latex]and[latex]\\,h\\left(x\\right)=3{\\left(4\\right)}^{x}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137639767\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137639767\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137639767\"]<span id=\"fs-id1165135500738\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140350\/CNX_PreCalc_Figure_04_02_204.jpg\" alt=\"Graph of three functions, g(x)=3(2)^(x) in blue, h(x)=3(4)^(x) in green, and f(x)=3(1\/4)^(x) in orange.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137610823\">\n<div id=\"fs-id1165137610825\">\n<p id=\"fs-id1165137610827\">[latex]f\\left(x\\right)=\\frac{1}{4}{\\left(3\\right)}^{x},[\/latex][latex]g\\left(x\\right)=2{\\left(3\\right)}^{x},[\/latex]and[latex]\\,h\\left(x\\right)=4{\\left(3\\right)}^{x}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137731675\">For the following exercises, match each function with one of the graphs in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_206\">(Figure)<\/a>.<\/p>\n\n<div id=\"CNX_Precalc_Figure_04_02_206\" class=\"small aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"425\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140357\/CNX_PreCalc_Figure_04_02_206.jpg\" alt=\"Graph of six exponential functions.\" width=\"425\" height=\"487\"> <strong>Figure 12.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"fs-id1165137758080\">\n<div id=\"fs-id1165137758082\">\n<p id=\"fs-id1165137758084\">[latex]f\\left(x\\right)=2{\\left(0.69\\right)}^{x}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137722409\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137722409\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137722409\"]\n<p id=\"fs-id1165137722412\">B<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135543453\">\n<div>[latex]f\\left(x\\right)=2{\\left(1.28\\right)}^{x}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137447034\">\n<div id=\"fs-id1165137762800\">\n<p id=\"fs-id1165137762802\">[latex]f\\left(x\\right)=2{\\left(0.81\\right)}^{x}[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165137767451\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137767451\"]\n<p id=\"fs-id1165137767451\">A<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137767457\">\n<div id=\"fs-id1165135193784\">\n<p id=\"fs-id1165135193786\">[latex]f\\left(x\\right)=4{\\left(1.28\\right)}^{x}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137692364\">\n<div id=\"fs-id1165137692366\">\n<p id=\"fs-id1165137692368\">[latex]f\\left(x\\right)=2{\\left(1.59\\right)}^{x}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135541572\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135541572\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135541572\"]\n<p id=\"fs-id1165137705256\">E<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137705261\">\n<div id=\"fs-id1165137705263\">\n<p id=\"fs-id1165137705266\">[latex]f\\left(x\\right)=4{\\left(0.69\\right)}^{x}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135161045\">For the following exercises, use the graphs shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_207\">(Figure)<\/a>. All have the form[latex]\\,f\\left(x\\right)=a{b}^{x}.[\/latex]<\/p>\n\n<div id=\"CNX_Precalc_Figure_04_02_207\" class=\"small aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140407\/CNX_PreCalc_Figure_04_02_207.jpg\" alt=\"Graph of six exponential functions.\" width=\"487\" height=\"470\"> <strong>Figure 13.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"fs-id1165137817442\">\n<div id=\"fs-id1165137817444\">\n<p id=\"fs-id1165137817446\">Which graph has the largest value for[latex]\\,b?[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134040575\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134040575\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134040575\"]\n<p id=\"fs-id1165134040577\">D<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134040583\">\n<div id=\"fs-id1165134040585\">\n<p id=\"fs-id1165137645213\">Which graph has the smallest value for[latex]\\,b?[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137531608\">\n<div id=\"fs-id1165137531610\">\n<p id=\"fs-id1165137531612\">Which graph has the largest value for[latex]\\,a?[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137836509\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137836509\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137836509\"]\n<p id=\"fs-id1165137836511\">C<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1165137666444\">Which graph has the smallest value for[latex]\\,a?[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137936780\">For the following exercises, graph the function and its reflection about the <em>x<\/em>-axis on the same axes.<\/p>\n\n<div id=\"fs-id1165137936789\">\n<div id=\"fs-id1165137936791\">\n<p id=\"fs-id1165137936793\">[latex]f\\left(x\\right)=\\frac{1}{2}{\\left(4\\right)}^{x}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135581221\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135581221\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135581221\"]<span id=\"fs-id1165137736410\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140421\/CNX_PreCalc_Figure_04_02_208.jpg\" alt=\"Graph of two functions, f(x)=(1\/2)(4)^(x) in blue and -f(x)=(-1\/2)(4)^x in orange.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137760861\">\n<div id=\"fs-id1165137760864\">\n<p id=\"fs-id1165137760866\">[latex]f\\left(x\\right)=3{\\left(0.75\\right)}^{x}-1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137727219\">\n<div id=\"fs-id1165137727221\">\n<p id=\"fs-id1165137727223\">[latex]f\\left(x\\right)=-4{\\left(2\\right)}^{x}+2[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135187279\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135187279\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135187279\"]<span id=\"fs-id1165137806564\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140436\/CNX_PreCalc_Figure_04_02_210.jpg\" alt=\"Graph of two functions, -f(x)=(4)(2)^(x)-2 in blue and f(x)=(-4)(2)^x+1 in orange.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<p id=\"fs-id1165137736387\">For the following exercises, graph the transformation of[latex]\\,f\\left(x\\right)={2}^{x}.\\,[\/latex]Give the horizontal asymptote, the domain, and the range.<\/p>\n\n<div id=\"fs-id1165135388489\">\n<div id=\"fs-id1165135388491\">\n<p id=\"fs-id1165135388493\">[latex]f\\left(x\\right)={2}^{-x}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135700147\">\n<div id=\"fs-id1165135700149\">\n<p id=\"fs-id1165135700151\">[latex]h\\left(x\\right)={2}^{x}+3[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137704844\" class=\"solution textbox shaded\">\n\n[reveal-answer q=\"979418\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"979418\"]\n\n<span id=\"fs-id1165135188613\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140445\/CNX_PreCalc_Figure_04_02_212.jpg\" alt=\"Graph of h(x)=2^(x)+3.\"><\/span>\n<p id=\"fs-id1165135481142\">Horizontal asymptote:[latex]\\,h\\left(x\\right)=3;[\/latex] Domain: all real numbers; Range: all real numbers strictly greater than[latex]\\,3.[\/latex]<\/p>\n<p id=\"fs-id1165135481142\">[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137849554\">\n<div>\n<p id=\"fs-id1165137849559\">[latex]f\\left(x\\right)={2}^{x-2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135435788\">For the following exercises, describe the end behavior of the graphs of the functions.<\/p>\n\n<div id=\"fs-id1165135435791\">\n<div id=\"fs-id1165135241045\">\n<p id=\"fs-id1165135241047\">[latex]f\\left(x\\right)=-5{\\left(4\\right)}^{x}-1[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135160376\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135160376\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135160376\"]As [latex]x\\to \\infty [\/latex],\n[latex]f\\left(x\\right)\\to -\\infty [\/latex];[\/hidden-answer]<\/div>\n<div id=\"fs-id1165137628660\">\n<div id=\"fs-id1165137628662\">\n<p id=\"fs-id1165137628664\">[latex]f\\left(x\\right)=3{\\left(\\frac{1}{2}\\right)}^{x}-2[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137514785\">\n<div id=\"fs-id1165137514787\">\n<p id=\"fs-id1165137514789\">[latex]f\\left(x\\right)=3{\\left(4\\right)}^{-x}+2[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135543075\" class=\"solution textbox shaded\">\n\n[reveal-answer q=\"fs-id1165135543075\"]Show Solution[\/reveal-answer]\n\n[hidden-answer a=\"fs-id1165135543075\"]\n\nAs [latex]x\\to \\infty [\/latex],\n[latex]f\\left(x\\right)\\to 2[\/latex];[\/hidden-answer]\n\n<\/div>\n<p id=\"fs-id1165135417905\">For the following exercises, start with the graph of[latex]\\,f\\left(x\\right)={4}^{x}.\\,[\/latex]Then write a function that results from the given transformation.<\/p>\n\n<div id=\"fs-id1165135529096\">\n<div id=\"fs-id1165135529098\">\n<p id=\"fs-id1165135529100\">Shift [latex]f\\left(x\\right)[\/latex] 4 units upward<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137652886\">\n<div id=\"fs-id1165137652888\">\n<p id=\"fs-id1165137652890\">Shift[latex]\\,f\\left(x\\right)\\,[\/latex]3 units downward<\/p>\n\n<\/div>\n<div id=\"fs-id1165137731311\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137731311\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137731311\"]\n<p id=\"fs-id1165137731314\">[latex]f\\left(x\\right)={4}^{x}-3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137762972\">\n<div id=\"fs-id1165137762974\">\n<p id=\"fs-id1165137762976\">Shift[latex]\\,f\\left(x\\right)\\,[\/latex]2 units left<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135437143\">\n<div id=\"fs-id1165135437145\">\n<p id=\"fs-id1165135437147\">Shift[latex]\\,f\\left(x\\right)\\,[\/latex]5 units right<\/p>\n\n<\/div>\n<div id=\"fs-id1165137572737\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137572737\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137572737\"]\n<p id=\"fs-id1165137827124\">[latex]f\\left(x\\right)={4}^{x-5}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137526797\">\n<div id=\"fs-id1165137526799\">\n<p id=\"fs-id1165137526801\">Reflect[latex]\\,f\\left(x\\right)\\,[\/latex]about the <em>x<\/em>-axis<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135432987\">\n<div id=\"fs-id1165135432989\">\n<p id=\"fs-id1165135432991\">Reflect[latex]\\,f\\left(x\\right)\\,[\/latex]about the <em>y<\/em>-axis<\/p>\n\n<\/div>\n<div id=\"fs-id1165135209553\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135209553\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135209553\"]\n[latex]f\\left(x\\right)={4}^{-x}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<p id=\"fs-id1165137936583\">For the following exercises, each graph is a transformation of[latex]\\,y={2}^{x}.\\,[\/latex]Write an equation describing the transformation.<\/p>\n\n<div id=\"fs-id1165137838408\">\n<div id=\"fs-id1165137838410\">\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140449\/CNX_PreCalc_Figure_04_02_214.jpg\" alt=\"Graph of f(x)=2^(x) with the following translations: vertical stretch of 4, a reflection about the x-axis, and a shift up by 1.\">\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135191679\">\n<div id=\"fs-id1165135191681\">\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140452\/CNX_PreCalc_Figure_04_02_215.jpg\" alt=\"Graph of f(x)=2^(x) with the following translations: a reflection about the x-axis, and a shift up by 3.\">\n\n<\/div>\n<div id=\"fs-id1165135536375\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135536375\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135536375\"]\n<p id=\"fs-id1165137444723\">[latex]y=-{2}^{x}+3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137692628\">\n<div id=\"fs-id1165137692630\">\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140454\/CNX_PreCalc_Figure_04_02_216.jpg\" alt=\"Graph of f(x)=2^(x) with the following translations: vertical stretch of 2, a reflection about the x-axis and y-axis, and a shift up by 3.\">\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137408027\">For the following exercises, find an exponential equation for the graph.<\/p>\n\n<div id=\"fs-id1165137550967\">\n<div id=\"fs-id1165137550969\">\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140511\/CNX_PreCalc_Figure_04_02_217.jpg\" alt=\"Graph of f(x)=3^(x) with the following translations: vertical stretch of 2, a reflection about the x-axis, and a shift up by 7.\">\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165135199467\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135199467\"]\n<p id=\"fs-id1165135199467\">[latex]y=-2{\\left(3\\right)}^{x}+7[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134341508\">\n<div id=\"fs-id1165134341510\">\n\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140514\/CNX_PreCalc_Figure_04_02_218.jpg\" alt=\"Graph of f(x)=(1\/2)^(x) with the following translations: vertical stretch of 2, and a shift down by 4.\">\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135560731\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<p id=\"fs-id1165135560673\">For the following exercises, evaluate the exponential functions for the indicated value of[latex]\\,x.[\/latex]<\/p>\n\n<div id=\"fs-id1165135332695\">\n<div id=\"fs-id1165135332697\">\n<p id=\"fs-id1165135332699\">[latex]g\\left(x\\right)=\\frac{1}{3}{\\left(7\\right)}^{x-2}\\,[\/latex]for[latex]\\,g\\left(6\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135175180\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135175180\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135175180\"]\n<p id=\"fs-id1165135175182\">[latex]g\\left(6\\right)=800+\\frac{1}{3}\\approx 800.3333[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134313940\">\n<div id=\"fs-id1165134313943\">\n<p id=\"fs-id1165135409814\">[latex]f\\left(x\\right)=4{\\left(2\\right)}^{x-1}-2\\,[\/latex]for[latex]\\,f\\left(5\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135519341\">\n<div id=\"fs-id1165135519343\">\n<p id=\"fs-id1165135519345\">[latex]h\\left(x\\right)=-\\frac{1}{2}{\\left(\\frac{1}{2}\\right)}^{x}+6\\,[\/latex]for[latex]\\,h\\left(-7\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134044680\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134044680\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134044680\"]\n<p id=\"fs-id1165134044682\">[latex]h\\left(-7\\right)=-58[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135321925\" class=\"bc-section section\">\n<h4>Technology<\/h4>\nFor the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth.\n<div id=\"fs-id1165135190288\">\n<div id=\"fs-id1165135190290\">\n<p id=\"fs-id1165135190292\">[latex]-50=-{\\left(\\frac{1}{2}\\right)}^{-x}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137476627\">\n<div id=\"fs-id1165137476629\">[latex]116=\\frac{1}{4}{\\left(\\frac{1}{8}\\right)}^{x}[\/latex]<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165135309920\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135309920\"]\n<p id=\"fs-id1165135309920\">[latex]x\\approx -2.953[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137838260\">\n<div id=\"fs-id1165137838262\">\n<p id=\"fs-id1165137838264\">[latex]12=2{\\left(3\\right)}^{x}+1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137838096\">\n<div id=\"fs-id1165137838098\">\n<p id=\"fs-id1165137838100\">[latex]5=3{\\left(\\frac{1}{2}\\right)}^{x-1}-2[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137605832\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137605832\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137605832\"]\n<p id=\"fs-id1165137605835\">[latex]x\\approx -0.222[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137737011\">\n<div id=\"fs-id1165137737013\">\n<p id=\"fs-id1165137737015\">[latex]-30=-4{\\left(2\\right)}^{x+2}+2[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137697128\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1165135187157\">\n<div id=\"fs-id1165135187160\">\n<p id=\"fs-id1165135187162\">Explore and discuss the graphs of[latex]\\,F\\left(x\\right)={\\left(b\\right)}^{x}\\,[\/latex]and[latex]\\,G\\left(x\\right)={\\left(\\frac{1}{b}\\right)}^{x}.\\,[\/latex]Then make a conjecture about the relationship between the graphs of the functions[latex]\\,{b}^{x}\\,[\/latex]and[latex]\\,{\\left(\\frac{1}{b}\\right)}^{x}\\,[\/latex]for any real number[latex]\\,b&gt;0.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137635181\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137635181\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137635181\"]\n<p id=\"fs-id1165137635183\">The graph of[latex]\\,G\\left(x\\right)={\\left(\\frac{1}{b}\\right)}^{x}\\,[\/latex]is the refelction about the <em>y<\/em>-axis of the graph of[latex]\\,F\\left(x\\right)={b}^{x};\\,[\/latex]For any real number[latex]\\,b&gt;0\\,[\/latex]and function[latex]\\,f\\left(x\\right)={b}^{x},[\/latex]the graph of[latex]\\,{\\left(\\frac{1}{b}\\right)}^{x}\\,[\/latex]is the the reflection about the <em>y<\/em>-axis,[latex]\\,F\\left(-x\\right).[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135456802\">\n<div id=\"fs-id1165135456804\">\n<p id=\"fs-id1165135456806\">Prove the conjecture made in the previous exercise.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135456811\">\n<div id=\"fs-id1165135456813\">\n<p id=\"fs-id1165135456816\">Explore and discuss the graphs of[latex]\\,f\\left(x\\right)={4}^{x},[\/latex][latex]\\,g\\left(x\\right)={4}^{x-2},[\/latex]and[latex]\\,h\\left(x\\right)=\\left(\\frac{1}{16}\\right){4}^{x}.\\,[\/latex]Then make a conjecture about the relationship between the graphs of the functions[latex]\\,{b}^{x}\\,[\/latex]and[latex]\\,\\left(\\frac{1}{{b}^{n}}\\right){b}^{x}\\,[\/latex]for any real number <em>n <\/em>and real number[latex]\\,b&gt;0.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135693738\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135693738\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135693738\"]\n<p id=\"fs-id1165135693740\">The graphs of[latex]\\,g\\left(x\\right)\\,[\/latex]and[latex]\\,h\\left(x\\right)\\,[\/latex]are the same and are a horizontal shift to the right of the graph of[latex]\\,f\\left(x\\right);\\,[\/latex]For any real number <em>n<\/em>, real number[latex]\\,b&gt;0,[\/latex] and function[latex]\\,f\\left(x\\right)={b}^{x},[\/latex] the graph of[latex]\\,\\left(\\frac{1}{{b}^{n}}\\right){b}^{x}\\,[\/latex]is the horizontal shift[latex]\\,f\\left(x-n\\right).[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137641366\">\n<div id=\"fs-id1165137641369\">\n<p id=\"fs-id1165137641371\">Prove the conjecture made in the previous exercise.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Graph exponential functions.<\/li>\n<li>Graph exponential functions using transformations.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137442020\">As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool. It gives us another layer of insight for predicting future events.<\/p>\n<div id=\"fs-id1165135407520\" class=\"bc-section section\">\n<h3>Graphing Exponential Functions<\/h3>\n<p id=\"fs-id1165137592823\">Before we begin graphing, it is helpful to review the behavior of exponential growth. Recall the table of values for a function of the form[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]whose base is greater than one. We\u2019ll use the function[latex]\\,f\\left(x\\right)={2}^{x}.\\,[\/latex]Observe how the output values in <a class=\"autogenerated-content\" href=\"#Table_04_02_01\">(Figure)<\/a> change as the input increases by[latex]\\,1.[\/latex]<\/p>\n<table id=\"Table_04_02_01\" summary=\"Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=2^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, 1\/8), (-2, 1\/4), (-1, 1\/2), (0, 1), (1, 2), (2, 4), and (3, 8).\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)={2}^{x}[\/latex]<\/strong><\/td>\n<td>[latex]\\frac{1}{8}[\/latex]<\/td>\n<td>[latex]\\frac{1}{4}[\/latex]<\/td>\n<td>[latex]\\frac{1}{2}[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165137432031\">Each output value is the product of the previous output and the base,[latex]\\,2.\\,[\/latex]We call the base[latex]\\,2\\,[\/latex]the <em>constant ratio<\/em>. In fact, for any exponential function with the form[latex]\\,f\\left(x\\right)=a{b}^{x},[\/latex][latex]\\,b\\,[\/latex]is the constant ratio of the function. This means that as the input increases by 1, the output value will be the product of the base and the previous output, regardless of the value of[latex]\\,a.[\/latex]<\/p>\n<p id=\"fs-id1165137585799\">Notice from the table that<\/p>\n<ul id=\"fs-id1165137658509\">\n<li>the output values are positive for all values of [latex]x;[\/latex]<\/li>\n<li>as[latex]\\,x\\,[\/latex]increases, the output values increase without bound; and<\/li>\n<li>as[latex]\\,x\\,[\/latex]decreases, the output values grow smaller, approaching zero.<\/li>\n<\/ul>\n<p id=\"fs-id1165137647215\"><a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_001\">(Figure)<\/a> shows the exponential growth function [latex]\\,f\\left(x\\right)={2}^{x}.[\/latex]<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_001\" class=\"small aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140149\/CNX_Precalc_Figure_04_02_001.jpg\" alt=\"Graph of the exponential function, 2^(x), with labeled points at (-3, 1\/8), (-2, \u00bc), (-1, \u00bd), (0, 1), (1, 2), (2, 4), and (3, 8). The graph notes that the x-axis is an asymptote.\" width=\"487\" height=\"520\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1. <\/strong>Notice that the graph gets close to the x-axis, but never touches it.<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137459614\">The domain of[latex]\\,f\\left(x\\right)={2}^{x}\\,[\/latex]is all real numbers, the range is[latex]\\,\\left(0,\\infty \\right),[\/latex] and the horizontal asymptote is[latex]\\,y=0.[\/latex]<\/p>\n<p id=\"fs-id1165137838249\">To get a sense of the behavior of <span class=\"no-emphasis\">exponential decay<\/span>, we can create a table of values for a function of the form[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]whose base is between zero and one. We\u2019ll use the function[latex]\\,g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}.\\,[\/latex]Observe how the output values in <a class=\"autogenerated-content\" href=\"#Table_04_02_02\">(Figure)<\/a> change as the input increases by[latex]\\,1.[\/latex]<\/p>\n<table summary=\"Two rows and eight columns. The first row is labeled, \u201cf(x)=2^x\u201d, with the following values: (-3, 1\/8), (-2, 1\/4), (-1, 1\/2), (0, 1), (1, 2), (2, 4), and (3, 8). The second row is labeled, \u201cg(x)=log_2(x)\u201d, with the following values: (1\/8, -3), (1\/4, -2), (1\/2, -1), (1, 0), (2, 1), (4, 2), and (8, 3).\">\n<tbody>\n<tr>\n<td style=\"width: 90px\"><strong>[latex]x[\/latex]<\/strong><\/td>\n<td style=\"width: 59px\">[latex]-3[\/latex]<\/td>\n<td style=\"width: 59px\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 59px\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 57px\">[latex]0[\/latex]<\/td>\n<td style=\"width: 68px\">[latex]1[\/latex]<\/td>\n<td style=\"width: 67px\">[latex]2[\/latex]<\/td>\n<td style=\"width: 67px\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 90px\">[latex]g(x)=\\left(\\frac{1}{2}\\right)^{x}[\/latex]<\/td>\n<td style=\"width: 59px\">[latex]8[\/latex]<\/td>\n<td style=\"width: 59px\">[latex]4[\/latex]<\/td>\n<td style=\"width: 59px\">[latex]2[\/latex]<\/td>\n<td style=\"width: 57px\">[latex]1[\/latex]<\/td>\n<td style=\"width: 68px\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td style=\"width: 67px\">[latex]\\frac{1}{4}[\/latex]<\/td>\n<td style=\"width: 67px\">[latex]\\frac{1}{8}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1165135347846\">Again, because the input is increasing by 1, each output value is the product of the previous output and the base, or constant ratio[latex]\\,\\frac{1}{2}.[\/latex]<\/p>\n<p id=\"fs-id1165137452063\">Notice from the table that<\/p>\n<ul id=\"fs-id1165135499992\">\n<li>the output values are positive for all values of[latex]\\,x;[\/latex]<\/li>\n<li>as[latex]\\,x\\,[\/latex]increases, the output values grow smaller, approaching zero; and<\/li>\n<li>as[latex]\\,x\\,[\/latex]decreases, the output values grow without bound.<\/li>\n<\/ul>\n<p id=\"fs-id1165137405421\"><a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_002\">(Figure)<\/a> shows the exponential decay function,[latex]\\,g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}.[\/latex]<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_002\" class=\"small aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140156\/CNX_Precalc_Figure_04_02_002.jpg\" alt=\"Graph of decreasing exponential function, (1\/2)^x, with labeled points at (-3, 8), (-2, 4), (-1, 2), (0, 1), (1, 1\/2), (2, 1\/4), and (3, 1\/8). The graph notes that the x-axis is an asymptote.\" width=\"487\" height=\"520\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137723586\">The domain of[latex]\\,g\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}\\,[\/latex]is all real numbers, the range is[latex]\\,\\left(0,\\infty \\right),[\/latex]and the horizontal asymptote is[latex]\\,y=0.[\/latex]<\/p>\n<div id=\"fs-id1165135571835\" class=\"textbox key-takeaways\">\n<h3>Characteristics of the Graph of the Parent Function <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\n<p id=\"fs-id1165137848929\">An exponential function with the form[latex]\\,f\\left(x\\right)={b}^{x},[\/latex][latex]\\,b>0,[\/latex][latex]\\,b\\ne 1,[\/latex]has these characteristics:<\/p>\n<ul id=\"fs-id1165135186684\">\n<li><span class=\"no-emphasis\">one-to-one<\/span> function<\/li>\n<li>horizontal asymptote:[latex]\\,y=0[\/latex]<\/li>\n<li>domain:[latex]\\,\\left(\u2013\\infty , \\infty \\right)[\/latex]<\/li>\n<li>range:[latex]\\,\\left(0,\\infty \\right)[\/latex]<\/li>\n<li><em>x-<\/em>intercept: none<\/li>\n<li><em>y-<\/em>intercept:[latex]\\,\\left(0,1\\right)\\,[\/latex]<\/li>\n<li>increasing if[latex]\\,b>1[\/latex]<\/li>\n<li>decreasing if[latex]\\,b<1[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137471878\"><a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_003\">(Figure)<\/a> compares the graphs of <span class=\"no-emphasis\">exponential growth<\/span> and decay functions.<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_003\" class=\"medium aligncenter\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140158\/CNX_Precalc_Figure_04_02_003new.jpg\" alt=\"\" width=\"731\" height=\"407\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134195243\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135194093\"><strong>Given an exponential function of the form[latex]\\,f\\left(x\\right)={b}^{x},[\/latex]graph the function.<\/strong><\/p>\n<ol id=\"fs-id1165137435782\" type=\"1\">\n<li>Create a table of points.<\/li>\n<li>Plot at least[latex]\\,3\\,[\/latex]point from the table, including the <em>y<\/em>-intercept[latex]\\,\\left(0,1\\right).[\/latex]<\/li>\n<li>Draw a smooth curve through the points.<\/li>\n<li>State the domain,[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex]the range,[latex]\\,\\left(0,\\infty \\right),[\/latex]and the horizontal asymptote, [latex]\\,y=0.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165135208984\">\n<div id=\"fs-id1165137453336\">\n<h3>Sketching the Graph of an Exponential Function of the Form <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\n<p id=\"fs-id1165137767671\">Sketch a graph of[latex]\\,f\\left(x\\right)={0.25}^{x}.\\,[\/latex]State the domain, range, and asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165135696740\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137734539\">Before graphing, identify the behavior and create a table of points for the graph.<\/p>\n<ul id=\"fs-id1165137566570\">\n<li>Since[latex]\\,b=0.25\\,[\/latex]is between zero and one, we know the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote[latex]\\,y=0.[\/latex]<\/li>\n<li>Create a table of points as in <a class=\"autogenerated-content\" href=\"#Table_04_02_03\">(Figure)<\/a>.<br \/>\n<table summary=\"Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=(0.25)^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, 64), (-2, 16), (-1, 4), (0, 1), (1, 0.25), (2, 0.0625), and (3, Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=(0.25)^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, 64), (-2, 16), (-1, 4), (0, 1), (1, 0.25), (2, 0.0625), and (3, 0.015625).\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong>[latex]f\\left(x\\right)={0.25}^{x}[\/latex]<\/strong><\/td>\n<td>[latex]64[\/latex]<\/td>\n<td>[latex]16[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]0.25[\/latex]<\/td>\n<td>[latex]0.0625[\/latex]<\/td>\n<td>[latex]0.015625[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Plot the <em>y<\/em>-intercept,[latex]\\,\\left(0,1\\right),[\/latex]along with two other points. We can use[latex]\\,\\left(-1,4\\right)\\,[\/latex]and[latex]\\,\\left(1,0.25\\right).[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137482830\">Draw a smooth curve connecting the points as in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_004\">(Figure)<\/a>.<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_004\" class=\"small aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140213\/CNX_Precalc_Figure_04_02_004.jpg\" alt=\"Graph of the decaying exponential function f(x) = 0.25^x with labeled points at (-1, 4), (0, 1), and (1, 0.25).\" width=\"487\" height=\"332\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137548870\">The domain is[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]the range is[latex]\\,\\left(0,\\infty \\right);\\,[\/latex]the horizontal asymptote is[latex]\\,y=0.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135499977\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165137761245\">\n<p id=\"fs-id1165137548853\">Sketch the graph of[latex]\\,f\\left(x\\right)={4}^{x}.\\,[\/latex]State the domain, range, and asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137731723\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137500954\">The domain is[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]the range is[latex]\\,\\left(0,\\infty \\right);\\,[\/latex]the horizontal asymptote is[latex]\\,y=0.[\/latex]<\/p>\n<p><span id=\"fs-id1165137437648\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140215\/CNX_Precalc_Figure_04_02_005.jpg\" alt=\"Graph of the increasing exponential function f(x) = 4^x with labeled points at (-1, 0.25), (0, 1), and (1, 4).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137694074\" class=\"bc-section section\">\n<h3>Graphing Transformations of Exponential Functions<\/h3>\n<p id=\"fs-id1165137575238\">Transformations of exponential graphs behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations\u2014shifts, reflections, stretches, and compressions\u2014to the parent function[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]without loss of shape. For instance, just as the quadratic function maintains its parabolic shape when shifted, reflected, stretched, or compressed, the exponential function also maintains its general shape regardless of the transformations applied.<\/p>\n<div id=\"fs-id1165134312214\" class=\"bc-section section\">\n<h4>Graphing a Vertical Shift<\/h4>\n<p id=\"fs-id1165137911544\">The first transformation occurs when we add a constant[latex]\\,d\\,[\/latex]to the parent function[latex]\\,f\\left(x\\right)={b}^{x},[\/latex] giving us a <span class=\"no-emphasis\">vertical shift<\/span>[latex]\\,d\\,[\/latex]units in the same direction as the sign. For example, if we begin by graphing a parent function,[latex]\\,f\\left(x\\right)={2}^{x},[\/latex] we can then graph two vertical shifts alongside it, using[latex]\\,d=3:\\,[\/latex]the upward shift,[latex]\\,g\\left(x\\right)={2}^{x}+3\\,[\/latex]and the downward shift,[latex]\\,h\\left(x\\right)={2}^{x}-3.\\,[\/latex]Both vertical shifts are shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_006\">(Figure)<\/a>.<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_006\" class=\"small aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140221\/CNX_Precalc_Figure_04_02_006.jpg\" alt=\"Graph of three functions, g(x) = 2^x+3 in blue with an asymptote at y=3, f(x) = 2^x in orange with an asymptote at y=0, and h(x)=2^x-3 with an asymptote at y=-3. Note that each functions\u2019 transformations are described in the text.\" width=\"487\" height=\"628\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 5.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137464499\">Observe the results of shifting[latex]\\,f\\left(x\\right)={2}^{x}\\,[\/latex]vertically:<\/p>\n<ul id=\"fs-id1165135203774\">\n<li>The domain,[latex]\\,\\left(-\\infty ,\\infty \\right)\\,[\/latex]remains unchanged.<\/li>\n<li>When the function is shifted up[latex]\\,3\\,[\/latex]units to[latex]\\,g\\left(x\\right)={2}^{x}+3:[\/latex]\n<ul id=\"fs-id1165137601587\">\n<li>The <em>y-<\/em>intercept shifts up[latex]\\,3\\,[\/latex]units to[latex]\\,\\left(0,4\\right).[\/latex]<\/li>\n<li>The asymptote shifts up[latex]\\,3\\,[\/latex]units to[latex]\\,y=3.[\/latex]<\/li>\n<li>The range becomes[latex]\\,\\left(3,\\infty \\right).[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>When the function is shifted down[latex]\\,3\\,[\/latex]units to[latex]\\,h\\left(x\\right)={2}^{x}-3:[\/latex]\n<ul id=\"fs-id1165137784817\">\n<li>The <em>y-<\/em>intercept shifts down[latex]\\,3\\,[\/latex]units to[latex]\\,\\left(0,-2\\right).[\/latex]<\/li>\n<li>The asymptote also shifts down[latex]\\,3\\,[\/latex]units to[latex]\\,y=-3.[\/latex]<\/li>\n<li>The range becomes[latex]\\,\\left(-3,\\infty \\right).[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137566517\" class=\"bc-section section\">\n<h4>Graphing a Horizontal Shift<\/h4>\n<p id=\"fs-id1165137748336\">The next transformation occurs when we add a constant[latex]\\,c\\,[\/latex]to the input of the parent function[latex]\\,f\\left(x\\right)={b}^{x},[\/latex] giving us a <span class=\"no-emphasis\">horizontal shift<\/span>[latex]\\,c\\,[\/latex]units in the <em>opposite<\/em> direction of the sign. For example, if we begin by graphing the parent function[latex]\\,f\\left(x\\right)={2}^{x},[\/latex] we can then graph two horizontal shifts alongside it, using[latex]\\,c=3:\\,[\/latex]the shift left,[latex]\\,g\\left(x\\right)={2}^{x+3},[\/latex] and the shift right,[latex]\\,h\\left(x\\right)={2}^{x-3}.\\,[\/latex]Both horizontal shifts are shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_007\">(Figure)<\/a>.<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_007\" class=\"medium aligncenter\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140233\/CNX_Precalc_Figure_04_02_007.jpg\" alt=\"Graph of three functions, g(x) = 2^(x+3) in blue, f(x) = 2^x in orange, and h(x)=2^(x-3). Each functions\u2019 asymptotes are at y=0Note that each functions\u2019 transformations are described in the text.\" width=\"731\" height=\"478\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 6.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137411256\">Observe the results of shifting[latex]\\,f\\left(x\\right)={2}^{x}\\,[\/latex]horizontally:<\/p>\n<ul id=\"fs-id1165135187815\">\n<li>The domain,[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex]remains unchanged.<\/li>\n<li>The asymptote,[latex]\\,y=0,[\/latex]remains unchanged.<\/li>\n<li>The <em>y-<\/em>intercept shifts such that:\n<ul id=\"fs-id1165137482879\">\n<li>When the function is shifted left[latex]\\,3\\,[\/latex]units to[latex]\\,g\\left(x\\right)={2}^{x+3},[\/latex]the <em>y<\/em>-intercept becomes[latex]\\,\\left(0,8\\right).\\,[\/latex]This is because[latex]\\,{2}^{x+3}=\\left(8\\right){2}^{x},[\/latex]so the initial value of the function is[latex]\\,8.[\/latex]<\/li>\n<li>When the function is shifted right[latex]\\,3\\,[\/latex]units to[latex]\\,h\\left(x\\right)={2}^{x-3},[\/latex]the <em>y<\/em>-intercept becomes[latex]\\,\\left(0,\\frac{1}{8}\\right).\\,[\/latex]Again, see that[latex]\\,{2}^{x-3}=\\left(\\frac{1}{8}\\right){2}^{x},[\/latex]so the initial value of the function is[latex]\\,\\frac{1}{8}.[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<div id=\"fs-id1165134042183\" class=\"textbox key-takeaways\">\n<h3>Shifts of the Parent Function <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\n<p id=\"fs-id1165134037589\">For any constants[latex]\\,c\\,[\/latex]and[latex]\\,d,[\/latex]the function[latex]\\,f\\left(x\\right)={b}^{x+c}+d\\,[\/latex]shifts the parent function[latex]\\,f\\left(x\\right)={b}^{x}[\/latex]<\/p>\n<ul id=\"fs-id1165137638569\">\n<li>vertically[latex]\\,d\\,[\/latex]units, in the <em>same<\/em> direction of the sign of[latex]\\,d.[\/latex]<\/li>\n<li>horizontally[latex]\\,c\\,[\/latex]units, in the <em>opposite<\/em> direction of the sign of[latex]\\,c.[\/latex]<\/li>\n<li>The <em>y<\/em>-intercept becomes[latex]\\,\\left(0,{b}^{c}+d\\right).[\/latex]<\/li>\n<li>The horizontal asymptote becomes[latex]\\,y=d.[\/latex]<\/li>\n<li>The range becomes[latex]\\,\\left(d,\\infty \\right).[\/latex]<\/li>\n<li>The domain,[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex]remains unchanged.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135500732\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135500706\"><strong>Given an exponential function with the form[latex]\\,f\\left(x\\right)={b}^{x+c}+d,[\/latex]graph the translation.<\/strong><\/p>\n<ol id=\"fs-id1165137767676\" type=\"1\">\n<li>Draw the horizontal asymptote[latex]\\,y=d.[\/latex]<\/li>\n<li>Identify the shift as[latex]\\,\\left(-c,d\\right).\\,[\/latex]Shift the graph of[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]left[latex]\\,c\\,[\/latex]units if[latex]\\,c\\,[\/latex]is positive, and right[latex]\\,c\\,[\/latex]units if[latex]c\\,[\/latex]is negative.<\/li>\n<li>Shift the graph of[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]up[latex]\\,d\\,[\/latex]units if[latex]\\,d\\,[\/latex]is positive, and down[latex]\\,d\\,[\/latex]units if[latex]\\,d\\,[\/latex]is negative.<\/li>\n<li>State the domain,[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex]the range,[latex]\\,\\left(d,\\infty \\right),[\/latex]and the horizontal asymptote[latex]\\,y=d.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137834201\">\n<div id=\"fs-id1165137416701\">\n<h3>Graphing a Shift of an Exponential Function<\/h3>\n<p id=\"fs-id1165137563667\">Graph[latex]\\,f\\left(x\\right)={2}^{x+1}-3.\\,[\/latex]State the domain, range, and asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165135175234\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137923482\">We have an exponential equation of the form[latex]\\,f\\left(x\\right)={b}^{x+c}+d,[\/latex] with[latex]\\,b=2,[\/latex][latex]\\,c=1,[\/latex] and[latex]\\,d=-3.[\/latex]<\/p>\n<p id=\"fs-id1165137469681\">Draw the horizontal asymptote[latex]\\,y=d[\/latex], so draw[latex]\\,y=-3.[\/latex]<\/p>\n<p id=\"fs-id1165137661814\">Identify the shift as[latex]\\,\\left(-c,d\\right),[\/latex] so the shift is[latex]\\,\\left(-1,-3\\right).[\/latex]<\/p>\n<p id=\"fs-id1165137693953\">Shift the graph of[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]left 1 units and down 3 units.<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_008\" class=\"small aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140245\/CNX_Precalc_Figure_04_02_008.jpg\" alt=\"Graph of the function, f(x) = 2^(x+1)-3, with an asymptote at y=-3. Labeled points in the graph are (-1, -2), (0, -1), and (1, 1).\" width=\"487\" height=\"519\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 7.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165134199602\">The domain is[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]the range is[latex]\\,\\left(-3,\\infty \\right);\\,[\/latex]the horizontal asymptote is[latex]\\,y=-3.[\/latex]<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135241073\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165137805939\">\n<p id=\"fs-id1165137805941\">Graph[latex]\\,f\\left(x\\right)={2}^{x-1}+3.\\,[\/latex]State domain, range, and asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137731918\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135513714\">The domain is[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]the range is[latex]\\,\\left(3,\\infty \\right);\\,[\/latex]the horizontal asymptote is[latex]\\,y=3.[\/latex]<\/p>\n<p><span id=\"fs-id1165137628194\"><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140252\/CNX_Precalc_Figure_04_02_009.jpg\" alt=\"Graph of the function, f(x) = 2^(x-1)+3, with an asymptote at y=3. Labeled points in the graph are (-1, 3.25), (0, 3.5), and (1, 4).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137639988\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165137756810\"><strong>Given an equation of the form[latex]\\,f\\left(x\\right)={b}^{x+c}+d\\,[\/latex]for[latex]\\,x,[\/latex] use a graphing calculator to approximate the solution.<\/strong><\/p>\n<ul id=\"fs-id1165137842461\">\n<li>Press <strong>[Y=]<\/strong>. Enter the given exponential equation in the line headed \u201c<strong>Y<sub>1<\/sub>=<\/strong>\u201d.<\/li>\n<li>Enter the given value for[latex]\\,f\\left(x\\right)\\,[\/latex]in the line headed \u201c<strong>Y<sub>2<\/sub>=<\/strong>\u201d.<\/li>\n<li>Press <strong>[WINDOW]<\/strong>. Adjust the <em>y<\/em>-axis so that it includes the value entered for \u201c<strong>Y<sub>2<\/sub>=<\/strong>\u201d.<\/li>\n<li>Press <strong>[GRAPH]<\/strong> to observe the graph of the exponential function along with the line for the specified value of[latex]\\,f\\left(x\\right).[\/latex]<\/li>\n<li>To find the value of[latex]\\,x,[\/latex]we compute the point of intersection. Press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select \u201cintersect\u201d and press <strong>[ENTER]<\/strong> three times. The point of intersection gives the value of <em>x <\/em>for the indicated value of the function.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165137618985\">\n<div id=\"fs-id1165137618987\">\n<h3>Approximating the Solution of an Exponential Equation<\/h3>\n<p id=\"fs-id1165135449598\">Solve[latex]\\,42=1.2{\\left(5\\right)}^{x}+2.8\\,[\/latex]graphically. Round to the nearest thousandth.<\/p>\n<\/div>\n<div id=\"fs-id1165137653309\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137737383\">Press <strong>[Y=]<\/strong> and enter[latex]\\,1.2{\\left(5\\right)}^{x}+2.8\\,[\/latex]next to <strong>Y<sub>1<\/sub><\/strong>=. Then enter 42 next to <strong>Y2=<\/strong>. For a window, use the values \u20133 to 3 for[latex]\\,x\\,[\/latex]and \u20135 to 55 for[latex]\\,y.\\,[\/latex]Press <strong>[GRAPH]<\/strong>. The graphs should intersect somewhere near[latex]\\,x=2.[\/latex]<\/p>\n<p id=\"fs-id1165137460953\">For a better approximation, press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select <strong>[5: intersect]<\/strong> and press <strong>[ENTER]<\/strong> three times. The <em>x<\/em>-coordinate of the point of intersection is displayed as 2.1661943. (Your answer may be different if you use a different window or use a different value for <strong>Guess?<\/strong>) To the nearest thousandth,[latex]\\,x\\approx 2.166.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135545893\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1165137838712\">\n<p id=\"fs-id1165137838714\">Solve[latex]\\,4=7.85{\\left(1.15\\right)}^{x}-2.27\\,[\/latex]graphically. Round to the nearest thousandth.<\/p>\n<\/div>\n<div id=\"fs-id1165137854192\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137854194\">[latex]x\\approx -1.608[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137431154\" class=\"bc-section section\">\n<h4>Graphing a Stretch or Compression<\/h4>\n<p id=\"fs-id1165137863514\">While horizontal and vertical shifts involve adding constants to the input or to the function itself, a <span class=\"no-emphasis\">stretch<\/span> or <span class=\"no-emphasis\">compression<\/span> occurs when we multiply the parent function[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]by a constant[latex]\\,|a|>0.\\,[\/latex]For example, if we begin by graphing the parent function[latex]\\,f\\left(x\\right)={2}^{x},[\/latex]we can then graph the stretch, using[latex]\\,a=3,[\/latex]to get[latex]\\,g\\left(x\\right)=3{\\left(2\\right)}^{x}\\,[\/latex]as shown on the left in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_010\">(Figure)<\/a>, and the compression, using[latex]\\,a=\\frac{1}{3},[\/latex]to get[latex]\\,h\\left(x\\right)=\\frac{1}{3}{\\left(2\\right)}^{x}\\,[\/latex]as shown on the right in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_010\">(Figure)<\/a>.<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_010\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140255\/CNX_Precalc_Figure_04_02_010.jpg\" alt=\"Two graphs where graph a is an example of vertical stretch and graph b is an example of vertical compression.\" width=\"975\" height=\"445\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 8.<\/strong> (a)[latex]\\,g\\left(x\\right)=3{\\left(2\\right)}^{x}\\,[\/latex]stretches the graph of[latex]\\,f\\left(x\\right)={2}^{x}\\,[\/latex]vertically by a factor of[latex]\\,3.\\,[\/latex](b)[latex]\\,h\\left(x\\right)=\\frac{1}{3}{\\left(2\\right)}^{x}\\,[\/latex]compresses the graph of[latex]\\,f\\left(x\\right)={2}^{x}\\,[\/latex]vertically by a factor of[latex]\\,\\frac{1}{3}.[\/latex]<\/figcaption><\/figure>\n<div id=\"fs-id1165137627908\" class=\"textbox key-takeaways\">\n<h3>Stretches and Compressions of the Parent Function <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\n<p id=\"fs-id1165137696285\">For any factor[latex]\\,a>0,[\/latex]the function[latex]\\,f\\left(x\\right)=a{\\left(b\\right)}^{x}[\/latex]<\/p>\n<ul id=\"fs-id1165137476370\">\n<li>is stretched vertically by a factor of[latex]\\,a\\,[\/latex]if[latex]\\,|a|>1.[\/latex]<\/li>\n<li>is compressed vertically by a factor of[latex]\\,a\\,[\/latex]if[latex]\\,|a|<1.[\/latex]<\/li>\n<li>has a <em>y<\/em>-intercept of[latex]\\,\\left(0,a\\right).[\/latex]<\/li>\n<li>has a horizontal asymptote at[latex]\\,y=0,[\/latex] a range of[latex]\\,\\left(0,\\infty \\right),[\/latex] and a domain of[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex]which are unchanged from the parent function.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1165135528997\">\n<div id=\"fs-id1165135656098\">\n<h3 id=\"fs-id1165135656100\">Graphing the Stretch of an Exponential Function<\/h3>\n<p id=\"fs-id1165135656104\">Sketch a graph of[latex]\\,f\\left(x\\right)=4{\\left(\\frac{1}{2}\\right)}^{x}.\\,[\/latex]State the domain, range, and asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137657436\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>Before graphing, identify the behavior and key points on the graph.<\/p>\n<ul id=\"fs-id1165137657441\">\n<li>Since[latex]\\,b=\\frac{1}{2}\\,[\/latex]is between zero and one, the left tail of the graph will increase without bound as[latex]\\,x\\,[\/latex]decreases, and the right tail will approach the <em>x<\/em>-axis as[latex]\\,x\\,[\/latex]increases.<\/li>\n<li>Since[latex]\\,a=4,[\/latex]the graph of[latex]\\,f\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{x}\\,[\/latex]will be stretched by a factor of[latex]\\,4.[\/latex]<\/li>\n<li>Create a table of points as shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2414&amp;action=edit#Table_04_02_04\">(Figure)<\/a>.<br \/>\n<table summary=\"Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=4(0.25)^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, 32), (-2, 16), (-1, 8), (0, 4), (1, 2), (2, 1), and (3, 0.5).\">\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]f\\left(x\\right)=4\\left(\\frac{1}{2}\\right)^{x}[\/latex]<\/td>\n<td>[latex]32[\/latex]<\/td>\n<td>[latex]16[\/latex]<\/td>\n<td>[latex]8[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]0.5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li>Plot the <em>y-<\/em>intercept,[latex]\\,\\left(0,4\\right),[\/latex]along with two other points. We can use[latex]\\,\\left(-1,8\\right)\\,[\/latex]and[latex]\\,\\left(1,2\\right).[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165135319502\">Draw a smooth curve connecting the points, as shown in <a class=\"autogenerated-content\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/wp-admin\/post.php?post=2414&amp;action=edit#CNX_Precalc_Figure_04_02_011\">(Figure)<\/a>.<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_011\" class=\"small aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140302\/CNX_Precalc_Figure_04_02_011.jpg\" alt=\"Graph of the function, f(x) = 4(1\/2)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 8), (0, 4), and (1, 2).\" width=\"487\" height=\"482\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 9.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137442037\">The domain is[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]the range is[latex]\\,\\left(0,\\infty \\right);\\,[\/latex]the horizontal asymptote is[latex]\\,y=0.[\/latex]<\/p>\n<p id=\"fs-id1165137442037\"><\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135541809\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_02_04\">\n<div>\n<p id=\"fs-id1165137452032\">Sketch the graph of[latex]\\,f\\left(x\\right)=\\frac{1}{2}{\\left(4\\right)}^{x}.\\,[\/latex]State the domain, range, and asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137694067\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137653325\">The domain is[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]the range is[latex]\\,\\left(0,\\infty \\right);\\,[\/latex]the horizontal asymptote is[latex]\\,y=0.\\,[\/latex]<span id=\"fs-id1165135417835\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140309\/CNX_Precalc_Figure_04_02_012.jpg\" alt=\"Graph of the function, f(x) = (1\/2)(4)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 0.125), (0, 0.5), and (1, 2).\" \/><\/span><\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135433028\" class=\"bc-section section\">\n<h4>Graphing Reflections<\/h4>\n<p id=\"fs-id1165137452750\">In addition to shifting, compressing, and stretching a graph, we can also reflect it about the <em>x<\/em>-axis or the <em>y<\/em>-axis. When we multiply the parent function[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]by[latex]\\,-1,[\/latex]we get a reflection about the <em>x<\/em>-axis. When we multiply the input by[latex]\\,-1,[\/latex]we get a <span class=\"no-emphasis\">reflection<\/span> about the <em>y<\/em>-axis. For example, if we begin by graphing the parent function[latex]\\,f\\left(x\\right)={2}^{x},[\/latex] we can then graph the two reflections alongside it. The reflection about the <em>x<\/em>-axis,[latex]\\,g\\left(x\\right)={-2}^{x},[\/latex]is shown on the left side of <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_013\">(Figure)<\/a>, and the reflection about the <em>y<\/em>-axis[latex]\\,h\\left(x\\right)={2}^{-x},[\/latex] is shown on the right side of <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_013\">(Figure)<\/a>.<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_013\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140318\/CNX_Precalc_Figure_04_02_013.jpg\" alt=\"Two graphs where graph a is an example of a reflection about the x-axis and graph b is an example of a reflection about the y-axis.\" width=\"975\" height=\"628\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 10. <\/strong>(a)[latex]\\,g\\left(x\\right)=-{2}^{x}\\,[\/latex]reflects the graph of[latex]\\,f\\left(x\\right)={2}^{x}\\,[\/latex]about the x-axis. (b)[latex]\\,g\\left(x\\right)={2}^{-x}\\,[\/latex]reflects the graph of[latex]\\,f\\left(x\\right)={2}^{x}\\,[\/latex]about the y-axis.<\/figcaption><\/figure>\n<div id=\"fs-id1165135477501\" class=\"textbox key-takeaways\">\n<h3>Reflections of the Parent Function <em>f<\/em>(<em>x<\/em>) = <em>b<\/em><sup><em>x<\/em><\/sup><\/h3>\n<p id=\"fs-id1165137455888\">The function[latex]\\,f\\left(x\\right)=-{b}^{x}[\/latex]<\/p>\n<ul>\n<li>reflects the parent function[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]about the <em>x<\/em>-axis.<\/li>\n<li>has a <em>y<\/em>-intercept of[latex]\\,\\left(0,-1\\right).[\/latex]<\/li>\n<li>has a range of[latex]\\,\\left(-\\infty ,0\\right)[\/latex]<\/li>\n<li>has a horizontal asymptote at[latex]\\,y=0\\,[\/latex]and domain of[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex]which are unchanged from the parent function.<\/li>\n<\/ul>\n<p id=\"fs-id1165137742185\">The function[latex]\\,f\\left(x\\right)={b}^{-x}[\/latex]<\/p>\n<ul id=\"fs-id1165137551240\">\n<li>reflects the parent function[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]about the <em>y<\/em>-axis.<\/li>\n<li>has a <em>y<\/em>-intercept of[latex]\\,\\left(0,1\\right),[\/latex] a horizontal asymptote at[latex]\\,y=0,[\/latex] a range of[latex]\\,\\left(0,\\infty \\right),[\/latex] and a domain of[latex]\\,\\left(-\\infty ,\\infty \\right),[\/latex] which are unchanged from the parent function.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Example_04_02_05\" class=\"textbox examples\">\n<div id=\"fs-id1165137406134\">\n<div id=\"fs-id1165137406136\">\n<h3>Writing and Graphing the Reflection of an Exponential Function<\/h3>\n<p id=\"fs-id1165137896193\">Find and graph the equation for a function,[latex]\\,g\\left(x\\right),[\/latex]that reflects[latex]\\,f\\left(x\\right)={\\left(\\frac{1}{4}\\right)}^{x}\\,[\/latex]about the <em>x<\/em>-axis. State its domain, range, and asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165137937537\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137937539\">Since we want to reflect the parent function[latex]\\,f\\left(x\\right)={\\left(\\frac{1}{4}\\right)}^{x}\\,[\/latex]about the <em>x-<\/em>axis, we multiply[latex]\\,f\\left(x\\right)\\,[\/latex]by[latex]\\,-1\\,[\/latex]to get,[latex]\\,g\\left(x\\right)=-{\\left(\\frac{1}{4}\\right)}^{x}.\\,[\/latex]Next we create a table of points as in <a class=\"autogenerated-content\" href=\"#Table_04_02_005\">(Figure)<\/a>.<\/p>\n<table id=\"Table_04_02_005\" summary=\"Two rows and eight columns. The first row is labeled, \u201cx\u201d, and the second row is labeled, \u201cf(x)=-(1\/4)^x\u201d. Reading the columns as ordered pairs, we have the following values: (-3, -64), (-2, -16), (-1, -4), (0, -1), (1, -0.25), (2, -0.0625), and (3, -0.0156).\">\n<colgroup>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]g\\left(x\\right)=-\\left(\\frac{1}{4}\\right)^{x}[\/latex]<\/td>\n<td>[latex]-64[\/latex]<\/td>\n<td>[latex]-16[\/latex]<\/td>\n<td>[latex]-4[\/latex]<\/td>\n<td>[latex]-1[\/latex]<\/td>\n<td>[latex]-0.25[\/latex]<\/td>\n<td>[latex]-0.0625[\/latex]<\/td>\n<td>[latex]-0.0156[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"eip-id1167546794019\">Plot the <em>y-<\/em>intercept,[latex]\\,\\left(0,-1\\right),[\/latex]along with two other points. We can use[latex]\\,\\left(-1,-4\\right)\\,[\/latex]and[latex]\\,\\left(1,-0.25\\right).[\/latex]<\/p>\n<p id=\"fs-id1165135369275\">Draw a smooth curve connecting the points:<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_014\" class=\"small aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140330\/CNX_Precalc_Figure_04_02_014.jpg\" alt=\"Graph of the function, g(x) = -(0.25)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, -4), (0, -1), and (1, -0.25).\" width=\"487\" height=\"407\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 11.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137828154\">The domain is[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]the range is[latex]\\,\\left(-\\infty ,0\\right);\\,[\/latex]the horizontal asymptote is[latex]\\,y=0.[\/latex]<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135205992\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_02_05\">\n<div id=\"fs-id1165135254653\">\n<p id=\"fs-id1165135254655\">Find and graph the equation for a function,[latex]\\,g\\left(x\\right),[\/latex] that reflects[latex]\\,f\\left(x\\right)={1.25}^{x}\\,[\/latex]about the <em>y<\/em>-axis. State its domain, range, and asymptote.<\/p>\n<\/div>\n<div id=\"fs-id1165135368458\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135368461\">The domain is[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]the range is[latex]\\,\\left(0,\\infty \\right);\\,[\/latex]the horizontal asymptote is[latex]\\,y=0.[\/latex]<\/p>\n<p><span id=\"fs-id1165137828034\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140340\/CNX_Precalc_Figure_04_02_015.jpg\" alt=\"Graph of the function, g(x) = -(1.25)^(-x), with an asymptote at y=0. Labeled points in the graph are (-1, 1.25), (0, 1), and (1, 0.8).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135501015\" class=\"bc-section section\">\n<h4>Summarizing Translations of the Exponential Function<\/h4>\n<p id=\"fs-id1165135501021\">Now that we have worked with each type of translation for the exponential function, we can summarize them in <a class=\"autogenerated-content\" href=\"#Table_04_02_006\">(Figure)<\/a> to arrive at the general equation for translating exponential functions.<\/p>\n<p>1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept remains (1, 0), the key point changes to (b^(-1), 1), the domain remains (0, infinity), and the range remains (-infinity, infinity). The second column shows the left shift of the equation g(x)=log_b(x) when b&gt;1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept changes to (-1, 0), the key point changes to (-b, 1), the domain changes to (-infinity, 0), and the range remains (-infinity, infinity).&#8221;&gt;<\/p>\n<table id=\"Table_04_02_006\">\n<thead>\n<tr>\n<th colspan=\"2\">Translations of the Parent Function [latex]\\,f\\left(x\\right)={b}^{x}[\/latex]<\/th>\n<\/tr>\n<tr>\n<th>Translation<\/th>\n<th>Form<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Shift<\/p>\n<ul id=\"fs-id1165137640731\">\n<li>Horizontally[latex]\\,c\\,[\/latex]units to the left<\/li>\n<li>Vertically[latex]\\,d\\,[\/latex]units up<\/li>\n<\/ul>\n<\/td>\n<td>[latex]f\\left(x\\right)={b}^{x+c}+d[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Stretch and Compress<\/p>\n<ul id=\"fs-id1165134074993\">\n<li>Stretch if[latex]\\,|a|>1[\/latex]<\/li>\n<li>Compression if[latex]\\,0<|a|<1[\/latex]<\/li>\n<\/ul>\n<\/td>\n<td>[latex]f\\left(x\\right)=a{b}^{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reflect about the <em>x<\/em>-axis<\/td>\n<td>[latex]f\\left(x\\right)=-{b}^{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reflect about the <em>y<\/em>-axis<\/td>\n<td>[latex]f\\left(x\\right)={b}^{-x}={\\left(\\frac{1}{b}\\right)}^{x}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>General equation for all translations<\/td>\n<td>[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-id1165137635134\" class=\"textbox key-takeaways\">\n<h3>Translations of Exponential Functions<\/h3>\n<p id=\"fs-id1165137806521\">A translation of an exponential function has the form<\/p>\n<div id=\"fs-id1165137806525\" class=\"unnumered aligncenter\">[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/div>\n<p id=\"fs-id1165137805520\">Where the parent function,[latex]\\,y={b}^{x},[\/latex][latex]\\,b>1,[\/latex]is<\/p>\n<ul id=\"fs-id1165137678290\">\n<li>shifted horizontally[latex]\\,c\\,[\/latex]units to the left.<\/li>\n<li>stretched vertically by a factor of[latex]\\,|a|\\,[\/latex]if[latex]\\,|a|>0.[\/latex]<\/li>\n<li>compressed vertically by a factor of[latex]\\,|a|\\,[\/latex]if[latex]\\,0<|a|<1.[\/latex]<\/li>\n<li>shifted vertically[latex]\\,d\\,[\/latex]units.<\/li>\n<li>reflected about the <em>x-<\/em>axis when[latex]\\,a<0.[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137937613\">Note the order of the shifts, transformations, and reflections follow the order of operations.<\/p>\n<\/div>\n<div id=\"Example_04_02_06\" class=\"textbox examples\">\n<div id=\"fs-id1165137937623\">\n<div id=\"fs-id1165135250578\">\n<h3 id=\"fs-id1165135250580\">Writing a Function from a Description<\/h3>\n<p id=\"fs-id1165135250584\">Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.<\/p>\n<ul id=\"fs-id1165137724821\">\n<li>[latex]f\\left(x\\right)={e}^{x}\\,[\/latex]is vertically stretched by a factor of[latex]\\,2\\,[\/latex], reflected across the <em>y<\/em>-axis, and then shifted up[latex]\\,4\\,[\/latex]units.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165135532412\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135532414\">We want to find an equation of the general form[latex]\\,f\\left(x\\right)=a{b}^{x+c}+d.\\,[\/latex]We use the description provided to find[latex]\\,a,[\/latex] [latex]b,[\/latex] [latex]c,[\/latex] and [latex]\\,d.[\/latex]<\/p>\n<ul id=\"fs-id1165137807102\">\n<li>We are given the parent function[latex]\\,f\\left(x\\right)={e}^{x},[\/latex] so[latex]\\,b=e.[\/latex]<\/li>\n<li>The function is stretched by a factor of[latex]\\,2[\/latex], so[latex]\\,a=2.[\/latex]<\/li>\n<li>The function is reflected about the <em>y<\/em>-axis. We replace[latex]\\,x\\,[\/latex]with[latex]\\,-x\\,[\/latex]to get:[latex]\\,{e}^{-x}.[\/latex]<\/li>\n<li>The graph is shifted vertically 4 units, so[latex]\\,d=4.[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137634849\">Substituting in the general form we get,<\/p>\n<div id=\"eip-id1165137832492\" class=\"unnumbered\">[latex]\\begin{array}{ll} f\\left(x\\right)\\hfill & =a{b}^{x+c}+d\\hfill \\\\ \\hfill & =2{e}^{-x+0}+4\\hfill \\\\ \\hfill & =2{e}^{-x}+4\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137665666\">The domain is[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]the range is[latex]\\,\\left(4,\\infty \\right);\\,[\/latex]the horizontal asymptote is[latex]\\,y=4.[\/latex]<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137553895\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_04_02_06\">\n<div id=\"fs-id1165137724079\">\n<p id=\"fs-id1165137724081\">Write the equation for function described below. Give the horizontal asymptote, the domain, and the range.<\/p>\n<ul id=\"fs-id1165137539693\">\n<li>[latex]f\\left(x\\right)={e}^{x}\\,[\/latex]is compressed vertically by a factor of[latex]\\,\\frac{1}{3},[\/latex] reflected across the <em>x<\/em>-axis and then shifted down [latex]\\,2[\/latex] units.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137724110\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137724112\">[latex]f\\left(x\\right)=-\\frac{1}{3}{e}^{x}-2;\\,[\/latex]the domain is[latex]\\,\\left(-\\infty ,\\infty \\right);\\,[\/latex]the range is[latex]\\,\\left(-\\infty ,2\\right);\\,[\/latex]the horizontal asymptote is[latex]\\,y=2.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135560786\" class=\"precalculus media\">\n<p id=\"fs-id1165137785000\">Access this online resource for additional instruction and practice with graphing exponential functions.<\/p>\n<ul id=\"fs-id1165137785004\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/graphexpfunc\">Graph Exponential Functions<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137661989\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"fs-id2055298\" summary=\"...\">\n<tbody>\n<tr>\n<td>General Form for the Translation of the Parent Function[latex]\\text{ }f\\left(x\\right)={b}^{x}[\/latex]<\/td>\n<td>[latex]f\\left(x\\right)=a{b}^{x+c}+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137447701\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137447708\">\n<li>The graph of the function[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]has a <em>y-<\/em>intercept at[latex]\\,\\left(0, 1\\right),[\/latex]domain[latex]\\,\\left(-\\infty , \\infty \\right),[\/latex]range[latex]\\,\\left(0, \\infty \\right),[\/latex] and horizontal asymptote[latex]\\,y=0.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_04_02_01\">(Figure)<\/a>.<\/li>\n<li>If[latex]\\,b>1,[\/latex]the function is increasing. The left tail of the graph will approach the asymptote[latex]\\,y=0,[\/latex] and the right tail will increase without bound.<\/li>\n<li>If[latex]\\,0<b<1,[\/latex] the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote[latex]\\,y=0.[\/latex]<\/li>\n<li>The equation[latex]\\,f\\left(x\\right)={b}^{x}+d\\,[\/latex]represents a vertical shift of the parent function[latex]\\,f\\left(x\\right)={b}^{x}.[\/latex]<\/li>\n<li>The equation[latex]\\,f\\left(x\\right)={b}^{x+c}\\,[\/latex]represents a horizontal shift of the parent function[latex]\\,f\\left(x\\right)={b}^{x}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_04_02_02\">(Figure)<\/a>.<\/li>\n<li>Approximate solutions of the equation[latex]\\,f\\left(x\\right)={b}^{x+c}+d\\,[\/latex]can be found using a graphing calculator. See <a class=\"autogenerated-content\" href=\"#Example_04_02_03\">(Figure)<\/a>.<\/li>\n<li>The equation[latex]\\,f\\left(x\\right)=a{b}^{x},[\/latex] where[latex]\\,a>0,[\/latex] represents a vertical stretch if[latex]\\,|a|>1\\,[\/latex]or compression if[latex]\\,0<|a|<1\\,[\/latex]of the parent function[latex]\\,f\\left(x\\right)={b}^{x}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_04_02_04\">(Figure)<\/a>.<\/li>\n<li>When the parent function[latex]\\,f\\left(x\\right)={b}^{x}\\,[\/latex]is multiplied by[latex]\\,-1,[\/latex]the result,[latex]\\,f\\left(x\\right)=-{b}^{x},[\/latex] is a reflection about the <em>x<\/em>-axis. When the input is multiplied by[latex]\\,-1,[\/latex]the result,[latex]\\,f\\left(x\\right)={b}^{-x},[\/latex] is a reflection about the <em>y<\/em>-axis. See <a class=\"autogenerated-content\" href=\"#Example_04_02_05\">(Figure)<\/a>.<\/li>\n<li>All translations of the exponential function can be summarized by the general equation[latex]\\,f\\left(x\\right)=a{b}^{x+c}+d.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Table_04_02_03\">(Figure)<\/a>.<\/li>\n<li>Using the general equation[latex]\\,f\\left(x\\right)=a{b}^{x+c}+d,[\/latex] we can write the equation of a function given its description. See <a class=\"autogenerated-content\" href=\"#Example_04_02_06\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137634271\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165137634275\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165135386454\">\n<div id=\"fs-id1165135386456\">\n<p id=\"fs-id1165135386458\">What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph?<\/p>\n<\/div>\n<div id=\"fs-id1165135386464\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135386466\">An asymptote is a line that the graph of a function approaches, as[latex]\\,x\\,[\/latex]either increases or decreases without bound. The horizontal asymptote of an exponential function tells us the limit of the function\u2019s values as the independent variable gets either extremely large or extremely small.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137724992\">\n<div id=\"fs-id1165137724994\">\n<p id=\"fs-id1165137769966\">What is the advantage of knowing how to recognize transformations of the graph of a parent function algebraically?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137769974\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<div id=\"fs-id1165135696183\">\n<div id=\"fs-id1165135696185\">\n<p id=\"fs-id1165135696187\">The graph of[latex]\\,f\\left(x\\right)={3}^{x}\\,[\/latex]is reflected about the <em>y<\/em>-axis and stretched vertically by a factor of[latex]\\,4.\\,[\/latex]What is the equation of the new function,[latex]\\,g\\left(x\\right)?\\,[\/latex]State its <em>y<\/em>-intercept, domain, and range.<\/p>\n<\/div>\n<div id=\"fs-id1165135369259\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135369261\">[latex]g\\left(x\\right)=4{\\left(3\\right)}^{-x};\\,[\/latex]<em>y<\/em>-intercept:[latex]\\,\\left(0,4\\right);\\,[\/latex]Domain: all real numbers; Range: all real numbers greater than[latex]\\,0.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137471014\">\n<div id=\"fs-id1165137471016\">\n<p id=\"fs-id1165137824073\">The graph of[latex]\\,f\\left(x\\right)={\\left(\\frac{1}{2}\\right)}^{-x}\\,[\/latex]is reflected about the <em>y<\/em>-axis and compressed vertically by a factor of[latex]\\,\\frac{1}{5}.\\,[\/latex]What is the equation of the new function,[latex]\\,g\\left(x\\right)?\\,[\/latex]State its <em>y<\/em>-intercept, domain, and range.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135459835\">\n<div id=\"fs-id1165135169299\">\n<p id=\"fs-id1165135169301\">The graph of[latex]\\,f\\left(x\\right)={10}^{x}\\,[\/latex]is reflected about the <em>x<\/em>-axis and shifted upward[latex]\\,7\\,[\/latex]units. What is the equation of the new function,[latex]\\,g\\left(x\\right)?\\,[\/latex]State its <em>y<\/em>-intercept, domain, and range.<\/p>\n<\/div>\n<div id=\"fs-id1165137851416\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137851418\">[latex]g\\left(x\\right)=-{10}^{x}+7;\\,[\/latex]<em>y<\/em>-intercept:[latex]\\,\\left(0,6\\right);\\,[\/latex]Domain: all real numbers; Range: all real numbers less than[latex]\\,7.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135459852\">\n<div id=\"fs-id1165135459854\">\n<p id=\"fs-id1165137675394\">The graph of[latex]\\,f\\left(x\\right)={\\left(1.68\\right)}^{x}\\,[\/latex]is shifted right[latex]\\,3\\,[\/latex]units, stretched vertically by a factor of[latex]\\,2,[\/latex]reflected about the <em>x<\/em>-axis, and then shifted downward[latex]\\,3\\,[\/latex]units. What is the equation of the new function,[latex]\\,g\\left(x\\right)?\\,[\/latex]State its <em>y<\/em>-intercept (to the nearest thousandth), domain, and range.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137874974\">\n<div id=\"fs-id1165137874976\">\n<p id=\"fs-id1165137874978\">The graph of[latex]\\,f\\left(x\\right)=2{\\left(\\frac{1}{4}\\right)}^{x-20}[\/latex] is shifted left[latex]\\,2\\,[\/latex]units, stretched vertically by a factor of[latex]\\,4,[\/latex]reflected about the <em>x<\/em>-axis, and then shifted downward[latex]\\,4\\,[\/latex]units. What is the equation of the new function,[latex]\\,g\\left(x\\right)?\\,[\/latex]State its <em>y<\/em>-intercept, domain, and range.<\/p>\n<\/div>\n<div id=\"fs-id1165137724981\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137863050\">[latex]g\\left(x\\right)=2{\\left(\\frac{1}{4}\\right)}^{x};\\,[\/latex]<em> y<\/em>-intercept:[latex]\\,\\left(0,\\text{ 2}\\right);\\,[\/latex]Domain: all real numbers; Range: all real numbers greater than[latex]\\,0.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137597358\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p>For the following exercises, graph the function and its reflection about the <em>y<\/em>-axis on the same axes, and give the <em>y<\/em>-intercept.<\/p>\n<div id=\"fs-id1165137731815\">\n<div id=\"fs-id1165137731818\">\n<p id=\"fs-id1165137731820\">[latex]f\\left(x\\right)=3{\\left(\\frac{1}{2}\\right)}^{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137758922\">\n<div id=\"fs-id1165137758924\">\n<p id=\"fs-id1165137758926\">[latex]g\\left(x\\right)=-2{\\left(0.25\\right)}^{x}[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137693544\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140348\/CNX_PreCalc_Figure_04_02_202.jpg\" alt=\"Graph of two functions, g(-x)=-2(0.25)^(-x) in blue and g(x)=-2(0.25)^x in orange.\" \/><\/span><\/p>\n<p id=\"fs-id1165135581056\"><em>y<\/em>-intercept:[latex]\\,\\left(0,-2\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137556927\">\n<div id=\"fs-id1165135383139\">\n<p id=\"fs-id1165135383141\">[latex]h\\left(x\\right)=6{\\left(1.75\\right)}^{-x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137432021\">For the following exercises, graph each set of functions on the same axes.<\/p>\n<div id=\"fs-id1165137579050\">\n<div id=\"fs-id1165137579053\">\n<p id=\"fs-id1165137579055\">[latex]f\\left(x\\right)=3{\\left(\\frac{1}{4}\\right)}^{x},[\/latex][latex]g\\left(x\\right)=3{\\left(2\\right)}^{x},[\/latex]and[latex]\\,h\\left(x\\right)=3{\\left(4\\right)}^{x}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137639767\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135500738\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140350\/CNX_PreCalc_Figure_04_02_204.jpg\" alt=\"Graph of three functions, g(x)=3(2)^(x) in blue, h(x)=3(4)^(x) in green, and f(x)=3(1\/4)^(x) in orange.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137610823\">\n<div id=\"fs-id1165137610825\">\n<p id=\"fs-id1165137610827\">[latex]f\\left(x\\right)=\\frac{1}{4}{\\left(3\\right)}^{x},[\/latex][latex]g\\left(x\\right)=2{\\left(3\\right)}^{x},[\/latex]and[latex]\\,h\\left(x\\right)=4{\\left(3\\right)}^{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137731675\">For the following exercises, match each function with one of the graphs in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_206\">(Figure)<\/a>.<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_206\" class=\"small aligncenter\">\n<figure style=\"width: 425px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140357\/CNX_PreCalc_Figure_04_02_206.jpg\" alt=\"Graph of six exponential functions.\" width=\"425\" height=\"487\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 12.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id1165137758080\">\n<div id=\"fs-id1165137758082\">\n<p id=\"fs-id1165137758084\">[latex]f\\left(x\\right)=2{\\left(0.69\\right)}^{x}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137722409\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137722412\">B<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135543453\">\n<div>[latex]f\\left(x\\right)=2{\\left(1.28\\right)}^{x}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137447034\">\n<div id=\"fs-id1165137762800\">\n<p id=\"fs-id1165137762802\">[latex]f\\left(x\\right)=2{\\left(0.81\\right)}^{x}[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137767451\">A<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137767457\">\n<div id=\"fs-id1165135193784\">\n<p id=\"fs-id1165135193786\">[latex]f\\left(x\\right)=4{\\left(1.28\\right)}^{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137692364\">\n<div id=\"fs-id1165137692366\">\n<p id=\"fs-id1165137692368\">[latex]f\\left(x\\right)=2{\\left(1.59\\right)}^{x}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135541572\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137705256\">E<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137705261\">\n<div id=\"fs-id1165137705263\">\n<p id=\"fs-id1165137705266\">[latex]f\\left(x\\right)=4{\\left(0.69\\right)}^{x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135161045\">For the following exercises, use the graphs shown in <a class=\"autogenerated-content\" href=\"#CNX_Precalc_Figure_04_02_207\">(Figure)<\/a>. All have the form[latex]\\,f\\left(x\\right)=a{b}^{x}.[\/latex]<\/p>\n<div id=\"CNX_Precalc_Figure_04_02_207\" class=\"small aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140407\/CNX_PreCalc_Figure_04_02_207.jpg\" alt=\"Graph of six exponential functions.\" width=\"487\" height=\"470\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 13.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id1165137817442\">\n<div id=\"fs-id1165137817444\">\n<p id=\"fs-id1165137817446\">Which graph has the largest value for[latex]\\,b?[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134040575\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134040577\">D<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134040583\">\n<div id=\"fs-id1165134040585\">\n<p id=\"fs-id1165137645213\">Which graph has the smallest value for[latex]\\,b?[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137531608\">\n<div id=\"fs-id1165137531610\">\n<p id=\"fs-id1165137531612\">Which graph has the largest value for[latex]\\,a?[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137836509\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137836511\">C<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1165137666444\">Which graph has the smallest value for[latex]\\,a?[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137936780\">For the following exercises, graph the function and its reflection about the <em>x<\/em>-axis on the same axes.<\/p>\n<div id=\"fs-id1165137936789\">\n<div id=\"fs-id1165137936791\">\n<p id=\"fs-id1165137936793\">[latex]f\\left(x\\right)=\\frac{1}{2}{\\left(4\\right)}^{x}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135581221\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137736410\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140421\/CNX_PreCalc_Figure_04_02_208.jpg\" alt=\"Graph of two functions, f(x)=(1\/2)(4)^(x) in blue and -f(x)=(-1\/2)(4)^x in orange.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137760861\">\n<div id=\"fs-id1165137760864\">\n<p id=\"fs-id1165137760866\">[latex]f\\left(x\\right)=3{\\left(0.75\\right)}^{x}-1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137727219\">\n<div id=\"fs-id1165137727221\">\n<p id=\"fs-id1165137727223\">[latex]f\\left(x\\right)=-4{\\left(2\\right)}^{x}+2[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135187279\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137806564\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140436\/CNX_PreCalc_Figure_04_02_210.jpg\" alt=\"Graph of two functions, -f(x)=(4)(2)^(x)-2 in blue and f(x)=(-4)(2)^x+1 in orange.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137736387\">For the following exercises, graph the transformation of[latex]\\,f\\left(x\\right)={2}^{x}.\\,[\/latex]Give the horizontal asymptote, the domain, and the range.<\/p>\n<div id=\"fs-id1165135388489\">\n<div id=\"fs-id1165135388491\">\n<p id=\"fs-id1165135388493\">[latex]f\\left(x\\right)={2}^{-x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135700147\">\n<div id=\"fs-id1165135700149\">\n<p id=\"fs-id1165135700151\">[latex]h\\left(x\\right)={2}^{x}+3[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137704844\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135188613\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140445\/CNX_PreCalc_Figure_04_02_212.jpg\" alt=\"Graph of h(x)=2^(x)+3.\" \/><\/span><\/p>\n<p id=\"fs-id1165135481142\">Horizontal asymptote:[latex]\\,h\\left(x\\right)=3;[\/latex] Domain: all real numbers; Range: all real numbers strictly greater than[latex]\\,3.[\/latex]<\/p>\n<p id=\"fs-id1165135481142\"><\/details>\n<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137849554\">\n<div>\n<p id=\"fs-id1165137849559\">[latex]f\\left(x\\right)={2}^{x-2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135435788\">For the following exercises, describe the end behavior of the graphs of the functions.<\/p>\n<div id=\"fs-id1165135435791\">\n<div id=\"fs-id1165135241045\">\n<p id=\"fs-id1165135241047\">[latex]f\\left(x\\right)=-5{\\left(4\\right)}^{x}-1[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135160376\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>As [latex]x\\to \\infty[\/latex],<br \/>\n[latex]f\\left(x\\right)\\to -\\infty[\/latex];<\/details>\n<\/div>\n<div id=\"fs-id1165137628660\">\n<div id=\"fs-id1165137628662\">\n<p id=\"fs-id1165137628664\">[latex]f\\left(x\\right)=3{\\left(\\frac{1}{2}\\right)}^{x}-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137514785\">\n<div id=\"fs-id1165137514787\">\n<p id=\"fs-id1165137514789\">[latex]f\\left(x\\right)=3{\\left(4\\right)}^{-x}+2[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135543075\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>As [latex]x\\to \\infty[\/latex],<br \/>\n[latex]f\\left(x\\right)\\to 2[\/latex];<\/details>\n<\/div>\n<p id=\"fs-id1165135417905\">For the following exercises, start with the graph of[latex]\\,f\\left(x\\right)={4}^{x}.\\,[\/latex]Then write a function that results from the given transformation.<\/p>\n<div id=\"fs-id1165135529096\">\n<div id=\"fs-id1165135529098\">\n<p id=\"fs-id1165135529100\">Shift [latex]f\\left(x\\right)[\/latex] 4 units upward<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137652886\">\n<div id=\"fs-id1165137652888\">\n<p id=\"fs-id1165137652890\">Shift[latex]\\,f\\left(x\\right)\\,[\/latex]3 units downward<\/p>\n<\/div>\n<div id=\"fs-id1165137731311\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137731314\">[latex]f\\left(x\\right)={4}^{x}-3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137762972\">\n<div id=\"fs-id1165137762974\">\n<p id=\"fs-id1165137762976\">Shift[latex]\\,f\\left(x\\right)\\,[\/latex]2 units left<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135437143\">\n<div id=\"fs-id1165135437145\">\n<p id=\"fs-id1165135437147\">Shift[latex]\\,f\\left(x\\right)\\,[\/latex]5 units right<\/p>\n<\/div>\n<div id=\"fs-id1165137572737\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137827124\">[latex]f\\left(x\\right)={4}^{x-5}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137526797\">\n<div id=\"fs-id1165137526799\">\n<p id=\"fs-id1165137526801\">Reflect[latex]\\,f\\left(x\\right)\\,[\/latex]about the <em>x<\/em>-axis<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135432987\">\n<div id=\"fs-id1165135432989\">\n<p id=\"fs-id1165135432991\">Reflect[latex]\\,f\\left(x\\right)\\,[\/latex]about the <em>y<\/em>-axis<\/p>\n<\/div>\n<div id=\"fs-id1165135209553\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]f\\left(x\\right)={4}^{-x}[\/latex]<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137936583\">For the following exercises, each graph is a transformation of[latex]\\,y={2}^{x}.\\,[\/latex]Write an equation describing the transformation.<\/p>\n<div id=\"fs-id1165137838408\">\n<div id=\"fs-id1165137838410\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140449\/CNX_PreCalc_Figure_04_02_214.jpg\" alt=\"Graph of f(x)=2^(x) with the following translations: vertical stretch of 4, a reflection about the x-axis, and a shift up by 1.\" \/><\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135191679\">\n<div id=\"fs-id1165135191681\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140452\/CNX_PreCalc_Figure_04_02_215.jpg\" alt=\"Graph of f(x)=2^(x) with the following translations: a reflection about the x-axis, and a shift up by 3.\" \/><\/p>\n<\/div>\n<div id=\"fs-id1165135536375\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137444723\">[latex]y=-{2}^{x}+3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137692628\">\n<div id=\"fs-id1165137692630\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140454\/CNX_PreCalc_Figure_04_02_216.jpg\" alt=\"Graph of f(x)=2^(x) with the following translations: vertical stretch of 2, a reflection about the x-axis and y-axis, and a shift up by 3.\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137408027\">For the following exercises, find an exponential equation for the graph.<\/p>\n<div id=\"fs-id1165137550967\">\n<div id=\"fs-id1165137550969\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140511\/CNX_PreCalc_Figure_04_02_217.jpg\" alt=\"Graph of f(x)=3^(x) with the following translations: vertical stretch of 2, a reflection about the x-axis, and a shift up by 7.\" \/><\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135199467\">[latex]y=-2{\\left(3\\right)}^{x}+7[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134341508\">\n<div id=\"fs-id1165134341510\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19140514\/CNX_PreCalc_Figure_04_02_218.jpg\" alt=\"Graph of f(x)=(1\/2)^(x) with the following translations: vertical stretch of 2, and a shift down by 4.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135560731\" class=\"bc-section section\">\n<h4>Numeric<\/h4>\n<p id=\"fs-id1165135560673\">For the following exercises, evaluate the exponential functions for the indicated value of[latex]\\,x.[\/latex]<\/p>\n<div id=\"fs-id1165135332695\">\n<div id=\"fs-id1165135332697\">\n<p id=\"fs-id1165135332699\">[latex]g\\left(x\\right)=\\frac{1}{3}{\\left(7\\right)}^{x-2}\\,[\/latex]for[latex]\\,g\\left(6\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135175180\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135175182\">[latex]g\\left(6\\right)=800+\\frac{1}{3}\\approx 800.3333[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134313940\">\n<div id=\"fs-id1165134313943\">\n<p id=\"fs-id1165135409814\">[latex]f\\left(x\\right)=4{\\left(2\\right)}^{x-1}-2\\,[\/latex]for[latex]\\,f\\left(5\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135519341\">\n<div id=\"fs-id1165135519343\">\n<p id=\"fs-id1165135519345\">[latex]h\\left(x\\right)=-\\frac{1}{2}{\\left(\\frac{1}{2}\\right)}^{x}+6\\,[\/latex]for[latex]\\,h\\left(-7\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134044680\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134044682\">[latex]h\\left(-7\\right)=-58[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135321925\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p>For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth.<\/p>\n<div id=\"fs-id1165135190288\">\n<div id=\"fs-id1165135190290\">\n<p id=\"fs-id1165135190292\">[latex]-50=-{\\left(\\frac{1}{2}\\right)}^{-x}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137476627\">\n<div id=\"fs-id1165137476629\">[latex]116=\\frac{1}{4}{\\left(\\frac{1}{8}\\right)}^{x}[\/latex]<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135309920\">[latex]x\\approx -2.953[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137838260\">\n<div id=\"fs-id1165137838262\">\n<p id=\"fs-id1165137838264\">[latex]12=2{\\left(3\\right)}^{x}+1[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137838096\">\n<div id=\"fs-id1165137838098\">\n<p id=\"fs-id1165137838100\">[latex]5=3{\\left(\\frac{1}{2}\\right)}^{x-1}-2[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137605832\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137605835\">[latex]x\\approx -0.222[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137737011\">\n<div id=\"fs-id1165137737013\">\n<p id=\"fs-id1165137737015\">[latex]-30=-4{\\left(2\\right)}^{x+2}+2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137697128\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1165135187157\">\n<div id=\"fs-id1165135187160\">\n<p id=\"fs-id1165135187162\">Explore and discuss the graphs of[latex]\\,F\\left(x\\right)={\\left(b\\right)}^{x}\\,[\/latex]and[latex]\\,G\\left(x\\right)={\\left(\\frac{1}{b}\\right)}^{x}.\\,[\/latex]Then make a conjecture about the relationship between the graphs of the functions[latex]\\,{b}^{x}\\,[\/latex]and[latex]\\,{\\left(\\frac{1}{b}\\right)}^{x}\\,[\/latex]for any real number[latex]\\,b>0.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137635181\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137635183\">The graph of[latex]\\,G\\left(x\\right)={\\left(\\frac{1}{b}\\right)}^{x}\\,[\/latex]is the refelction about the <em>y<\/em>-axis of the graph of[latex]\\,F\\left(x\\right)={b}^{x};\\,[\/latex]For any real number[latex]\\,b>0\\,[\/latex]and function[latex]\\,f\\left(x\\right)={b}^{x},[\/latex]the graph of[latex]\\,{\\left(\\frac{1}{b}\\right)}^{x}\\,[\/latex]is the the reflection about the <em>y<\/em>-axis,[latex]\\,F\\left(-x\\right).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135456802\">\n<div id=\"fs-id1165135456804\">\n<p id=\"fs-id1165135456806\">Prove the conjecture made in the previous exercise.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135456811\">\n<div id=\"fs-id1165135456813\">\n<p id=\"fs-id1165135456816\">Explore and discuss the graphs of[latex]\\,f\\left(x\\right)={4}^{x},[\/latex][latex]\\,g\\left(x\\right)={4}^{x-2},[\/latex]and[latex]\\,h\\left(x\\right)=\\left(\\frac{1}{16}\\right){4}^{x}.\\,[\/latex]Then make a conjecture about the relationship between the graphs of the functions[latex]\\,{b}^{x}\\,[\/latex]and[latex]\\,\\left(\\frac{1}{{b}^{n}}\\right){b}^{x}\\,[\/latex]for any real number <em>n <\/em>and real number[latex]\\,b>0.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135693738\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135693740\">The graphs of[latex]\\,g\\left(x\\right)\\,[\/latex]and[latex]\\,h\\left(x\\right)\\,[\/latex]are the same and are a horizontal shift to the right of the graph of[latex]\\,f\\left(x\\right);\\,[\/latex]For any real number <em>n<\/em>, real number[latex]\\,b>0,[\/latex] and function[latex]\\,f\\left(x\\right)={b}^{x},[\/latex] the graph of[latex]\\,\\left(\\frac{1}{{b}^{n}}\\right){b}^{x}\\,[\/latex]is the horizontal shift[latex]\\,f\\left(x-n\\right).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137641366\">\n<div id=\"fs-id1165137641369\">\n<p id=\"fs-id1165137641371\">Prove the conjecture made in the previous exercise.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":291,"menu_order":3,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-100","chapter","type-chapter","status-publish","hentry"],"part":95,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/100","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/users\/291"}],"version-history":[{"count":1,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/100\/revisions"}],"predecessor-version":[{"id":101,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/100\/revisions\/101"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/parts\/95"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/100\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=100"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=100"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=100"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=100"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}