{"id":2121,"date":"2022-05-16T20:39:55","date_gmt":"2022-05-16T20:39:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/?post_type=chapter&#038;p=2121"},"modified":"2026-01-22T18:33:44","modified_gmt":"2026-01-22T18:33:44","slug":"5-1-exponential-functions-and-their-graphs","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/chapter\/5-1-exponential-functions-and-their-graphs\/","title":{"raw":"5.1: Exponential Functions and their Graphs","rendered":"5.1: Exponential Functions and their Graphs"},"content":{"raw":"<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>\r\n<div class=\"textbox learning-objectives\">\r\n<h1>Learning Objectives<\/h1>\r\n<ul>\r\n \t<li>Determine the pattern of exponential growth<\/li>\r\n \t<li>Define and graph exponential functions<\/li>\r\n \t<li>Describe the asymptote of an exponential function<\/li>\r\n \t<li>Find the domain and range of an exponential function<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Exponential Growth<\/h2>\r\nIn Chapter 2, we looked at linear growth where there is a constant rate of change. Unlike linear growth that increases by <em>adding<\/em> a constant value to [latex]y[\/latex] for every unit increase in [latex]x[\/latex], <em><strong>exponential growth<\/strong><\/em> increases by <em>multiplying<\/em>\u00a0by a constant that is neither equal to 0 nor 1. Table 1 shows exponential growth as each [latex]y[\/latex]-value increases by a multiple of 2. Notice that the values in the [latex]x[\/latex] column correspond to the exponents in the fourth column. This is because the values in the\u00a0[latex]x[\/latex] column record the number of times we have multiplied by 2.\r\n<table style=\"border-collapse: collapse; width: 126.373%; height: 147px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 15.0718%; text-align: center; height: 12px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 9.70989%; text-align: center; height: 12px;\">[latex]y[\/latex]<\/th>\r\n<th style=\"width: 21.2648%; height: 12px; text-align: center;\">Number of times 2 gets multiplied<\/th>\r\n<th style=\"width: 12.87%; height: 12px; text-align: center;\">Exponential Notation<\/th>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\">1<\/td>\r\n<td style=\"width: 9.70989%; text-align: center; height: 12px;\">2<\/td>\r\n<td style=\"width: 21.2648%; text-align: left; height: 12px;\">[latex]= 2[\/latex]<\/td>\r\n<td style=\"width: 12.87%; text-align: left; height: 12px;\">[latex] =2^1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\">2<\/td>\r\n<td style=\"width: 9.70989%; text-align: center; height: 12px;\">4<\/td>\r\n<td style=\"width: 21.2648%; text-align: left; height: 12px;\">[latex]= 2 \\cdot 2[\/latex]<\/td>\r\n<td style=\"width: 12.87%; text-align: left; height: 12px;\">[latex] =2^2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 25px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 25px;\">3<\/td>\r\n<td style=\"width: 9.70989%; text-align: center; height: 25px;\">8<\/td>\r\n<td style=\"width: 21.2648%; text-align: left; height: 25px;\">[latex]= 2 \\cdot 2 \\cdot 2[\/latex]<\/td>\r\n<td style=\"width: 12.87%; text-align: left; height: 25px;\">[latex] =2^3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 25px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 25px;\">4<\/td>\r\n<td style=\"width: 9.70989%; text-align: center; height: 25px;\">16<\/td>\r\n<td style=\"width: 21.2648%; text-align: left; height: 25px;\">[latex]=2 \\cdot 2 \\cdot 2 \\cdot 2[\/latex]<\/td>\r\n<td style=\"width: 12.87%; text-align: left; height: 25px;\">[latex] =2^4[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 25px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 25px;\">5<\/td>\r\n<td style=\"width: 9.70989%; text-align: center; height: 25px;\">32<\/td>\r\n<td style=\"width: 21.2648%; text-align: left; height: 25px;\">[latex]=2 \\cdot 2 \\cdot 2 \\cdot 2 \\cdot 2[\/latex]<\/td>\r\n<td style=\"width: 12.87%; text-align: left; height: 25px;\">[latex] =2^5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"shaded\" style=\"width: 58.9165%; text-align: left;\" colspan=\"4\">Table 1. Exponential growth<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe pattern shown in the fourth column of table 1 indicates that exponential growth is repeated multiplication by a factor of 2 so the equation for the pattern is [latex]y= 2^x[\/latex].\r\n<h3>Zero and Negative Exponents<\/h3>\r\nTable 1 shows the pattern of exponential growth of [latex]2^x[\/latex] when [latex]x[\/latex] is a natural number. But what is the value of [latex]2^x[\/latex] when the exponent [latex]x[\/latex] is zero or a negative number? If we consider the pattern in Table 1 in\u00a0the reverse direction we can find the value of [latex]y=2^x[\/latex] when [latex]x=0, -1, -2,...[\/latex]. As the exponent decreases by 1,\u00a0the value of [latex]y=2^x[\/latex] is divided by 2.\r\n\r\nFor example, [latex]2^x=8[\/latex]\u00a0when the exponent [latex]x[\/latex]\u00a0is 3 (i.e., [latex]y=2^3=8[\/latex]). As [latex]x[\/latex] decreases by 1\u00a0to 2, the [latex]y[\/latex] value is 4 (See Table 2). In other words, 8 is divided by 2 to get 4.\u00a0Following this logic, the [latex]y[\/latex] value will be 4 \u00f7 2 = 2 when [latex]x[\/latex] decreases by 1 again to 1. Following this pattern, the [latex]y[\/latex] value will be 2 \u00f7 2 = 1\u00a0when [latex]x[\/latex] decreases by 1 again to zero. Therefore, [latex]2^0=1[\/latex]. In fact,\u00a0 [latex]a^0=1[\/latex]\u00a0for all values of [latex]a[\/latex], [latex]a\\ne 0[\/latex]. [latex]0^0[\/latex] is undefined.\r\n<div class=\"textbox shaded\">\r\n<h3>Exponent of Zero<\/h3>\r\n<p style=\"text-align: center;\">[latex]a^0=1[\/latex]\u00a0for all values of [latex]a[\/latex], [latex]a\\ne 0[\/latex]. [latex]0^0[\/latex] is undefined.<\/p>\r\n\r\n<\/div>\r\nContinuing this reverse pattern, the value of\u00a0[latex]y[\/latex] will be\u00a0[latex]1\\div 2=\\dfrac{1}{2}[\/latex]\u00a0when [latex]x=-1[\/latex]. Then, dividing by 2 again gives\u00a0[latex]y=\\dfrac{1}{2}\u00f72=\\dfrac{1}{2}\\cdot\\dfrac{1}{2}=\\dfrac{1}{4}[\/latex] when\u00a0[latex]x=-2[\/latex]. Recall that dividing by 2 is equivalent to multiplying by [latex]\\dfrac{1}{2}[\/latex]. This pattern shows that the value of\u00a0[latex]y[\/latex] is a fraction where\u00a0[latex]y=\\dfrac{1}{2^{|x|}}[\/latex] when the exponent\u00a0[latex]x[\/latex] is a negative number (See Table 2). For example,\u00a0[latex]y=2^{-3}=\\dfrac{1}{2^{|-3|}}=\\dfrac{1}{2^3}[\/latex]. In fact, the value of\u00a0[latex]y=a^x[\/latex] will alway be [latex]\\dfrac{1}{a^{|x|}}[\/latex] when the exponent [latex]x[\/latex] is a negative number and the base [latex]a \u2265 0[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Negative Exponents<\/h3>\r\n<p style=\"text-align: center;\">[latex]y=a^x=\\dfrac{1}{a^{|x|}}[\/latex] when the exponent [latex]x[\/latex] is a negative number and [latex]a \u2265 0[\/latex].<\/p>\r\n\r\n<\/div>\r\nTable 3 shows how the values of [latex]y=2^x[\/latex] for the exponents\u00a0[latex]x=0, -1, -2, -3...[\/latex] are obtained following the reverse pattern starting at [latex]x=0[\/latex].\r\n<table style=\"border-collapse: collapse; width: 126.373%; height: 189px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 15.0718%; text-align: center; height: 12px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 15.3369%; text-align: center; height: 12px;\">[latex]y[\/latex]<\/th>\r\n<th style=\"width: 15.6378%; height: 12px; text-align: left;\">Method for Obtaining\u00a0[latex]y[\/latex]<\/th>\r\n<th style=\"width: 12.87%; height: 12px; text-align: left;\">Equation<\/th>\r\n<\/tr>\r\n<tr style=\"height: 28px;\">\r\n<td style=\"width: 15.071770334928232%; text-align: center; height: 28px;\"><span style=\"color: #0000ff;\">-3<\/span><\/td>\r\n<td style=\"width: 15.336945869602815%; text-align: center; height: 28px;\"><span style=\"color: #0000ff;\">[latex]\\dfrac{1}{8}[\/latex]<\/span><\/td>\r\n<td style=\"width: 15.637757011498225%; height: 28px; text-align: left;\"><span style=\"color: #0000ff;\">[latex]=\\dfrac{1}{2} \\cdot \\dfrac{1}{2} \\cdot \\dfrac{1}{2}[\/latex]<\/span><\/td>\r\n<td style=\"width: 12.870007312340563%; height: 28px; text-align: left;\"><span style=\"color: #0000ff;\">[latex]2^{-3}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.071770334928232%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">-2<\/span><\/td>\r\n<td style=\"width: 15.336945869602815%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">[latex]\\dfrac{1}{4}[\/latex]<\/span><\/td>\r\n<td style=\"width: 15.637757011498225%; height: 12px; text-align: left;\"><span style=\"color: #0000ff;\">[latex]=\\dfrac{1}{2} \\cdot \\dfrac{1}{2}[\/latex]<\/span><\/td>\r\n<td style=\"width: 12.870007312340563%; height: 12px; text-align: left;\"><span style=\"color: #0000ff;\">[latex]2^{-2}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.071770334928232%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">-1<\/span><\/td>\r\n<td style=\"width: 15.336945869602815%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">[latex]\\dfrac{1}{2}[\/latex]<\/span><\/td>\r\n<td style=\"width: 15.6378%; height: 12px; text-align: left;\"><span style=\"color: #0000ff;\">[latex]=\\dfrac{1}{2}[\/latex]<\/span><\/td>\r\n<td style=\"width: 12.870007312340563%; height: 12px; text-align: left;\"><span style=\"color: #0000ff;\">[latex]2^{-1}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.071770334928232%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">0<\/span><\/td>\r\n<td style=\"width: 15.336945869602815%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">1<\/span><\/td>\r\n<td style=\"width: 15.6378%; height: 12px; text-align: left;\"><span style=\"color: #0000ff;\">[latex]= 2 \\cdot \\dfrac{1}{2}[\/latex] = 1<\/span><\/td>\r\n<td style=\"width: 12.870007312340563%; height: 12px; text-align: left;\"><span style=\"color: #0000ff;\">[latex] 2^0[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.071770334928232%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">1<\/span><\/td>\r\n<td style=\"width: 15.336945869602815%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">2<\/span><\/td>\r\n<td style=\"width: 15.637757011498225%; height: 12px; text-align: left;\"><span style=\"color: #ff0000;\">[latex]=2[\/latex]<\/span><\/td>\r\n<td style=\"width: 12.870007312340563%; height: 12px; text-align: left;\"><span style=\"color: #ff0000;\">[latex] 2^1[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.071770334928232%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">2<\/span><\/td>\r\n<td style=\"width: 15.336945869602815%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">4<\/span><\/td>\r\n<td style=\"width: 15.637757011498225%; height: 12px; text-align: left;\"><span style=\"color: #ff0000;\">[latex]= 2 \\cdot 2[\/latex]<\/span><\/td>\r\n<td style=\"width: 12.870007312340563%; height: 12px; text-align: left;\"><span style=\"color: #ff0000;\">[latex] 2^2[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 25px;\">\r\n<td style=\"width: 15.071770334928232%; text-align: center; height: 25px;\"><span style=\"color: #ff0000;\">3<\/span><\/td>\r\n<td style=\"width: 15.336945869602815%; text-align: center; height: 25px;\"><span style=\"color: #ff0000;\">8<\/span><\/td>\r\n<td style=\"width: 15.637757011498225%; height: 25px; text-align: left;\"><span style=\"color: #ff0000;\">[latex]=2 \\cdot 2 \\cdot 2[\/latex]<\/span><\/td>\r\n<td style=\"width: 12.870007312340563%; height: 25px; text-align: left;\"><span style=\"color: #ff0000;\">[latex] 2^3[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 25px;\">\r\n<td style=\"width: 15.071770334928232%; text-align: center; height: 25px;\"><span style=\"color: #ff0000;\">4<\/span><\/td>\r\n<td style=\"width: 15.336945869602815%; text-align: center; height: 25px;\"><span style=\"color: #ff0000;\">16<\/span><\/td>\r\n<td style=\"width: 15.637757011498225%; height: 25px; text-align: left;\"><span style=\"color: #ff0000;\">[latex]=2 \\cdot 2 \\cdot 2 \\cdot 2[\/latex]<\/span><\/td>\r\n<td style=\"width: 12.870007312340563%; height: 25px; text-align: left;\"><span style=\"color: #ff0000;\">[latex] 2^4[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 25px;\">\r\n<td style=\"width: 15.071770334928232%; text-align: center; height: 25px;\"><span style=\"color: #ff0000;\">5<\/span><\/td>\r\n<td style=\"width: 15.336945869602815%; text-align: center; height: 25px;\"><span style=\"color: #ff0000;\">32<\/span><\/td>\r\n<td style=\"width: 15.637757011498225%; height: 25px; text-align: left;\"><span style=\"color: #ff0000;\">[latex]=2 \\cdot 2 \\cdot 2 \\cdot 2 \\cdot 2[\/latex]<\/span><\/td>\r\n<td style=\"width: 12.870007312340563%; height: 25px; text-align: left;\"><span style=\"color: #ff0000;\">[latex] 2^5[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 15.071770334928232%; text-align: left; height: 14px;\" colspan=\"4\">Table 2. Exponential growth with base 2<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nZero and negative exponents will be discussed again in section 5.3.2 using the Product and Quotient Rule for Exponents.\r\n<div class=\"textbox examples\">\r\n<h3>Example 1<\/h3>\r\nComplete the table for the equation [latex]y=10^x[\/latex].\r\n<table style=\"border-collapse: collapse; width: 126.373%; height: 189px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 15.0718%; text-align: center; height: 12px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 15.3369%; text-align: center; height: 12px;\">[latex]y[\/latex]<\/th>\r\n<th style=\"width: 15.6378%; text-align: center; height: 12px;\">Method for Obtaining\u00a0[latex]y[\/latex]<\/th>\r\n<th style=\"width: 12.87%; text-align: center; height: 12px;\">Equation<\/th>\r\n<\/tr>\r\n<tr style=\"height: 28px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 28px;\"><span style=\"color: #0000ff;\">-3<\/span><\/td>\r\n<td style=\"width: 15.3369%; text-align: center; height: 28px;\"><\/td>\r\n<td style=\"width: 15.6378%; height: 28px; text-align: left;\"><\/td>\r\n<td style=\"width: 12.87%; height: 28px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=10^{-3}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">-2<\/span><\/td>\r\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\"><\/td>\r\n<td style=\"width: 15.6378%; height: 12px; text-align: left;\"><\/td>\r\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=10^{-2}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">-1<\/span><\/td>\r\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\"><\/td>\r\n<td style=\"width: 15.6378%; height: 12px; text-align: left;\"><\/td>\r\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=10^{-1}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">0<\/span><\/td>\r\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\"><\/td>\r\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\"><\/td>\r\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=10^0[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">1<\/span><\/td>\r\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">10<\/span><\/td>\r\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\"><span style=\"color: #ff0000;\">[latex]= 10[\/latex]<\/span><\/td>\r\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #ff0000;\">[latex]y=10^1[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Solution<\/h4>\r\nSince the base of the function is 10, we divide the [latex]y[\/latex]-value by 10 as [latex]x[\/latex] decreases by 1.\r\n<table style=\"border-collapse: collapse; width: 126.373%; height: 189px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 15.0718%; text-align: center; height: 12px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 15.3369%; text-align: center; height: 12px;\">[latex]y[\/latex]<\/th>\r\n<th style=\"width: 15.6378%; text-align: center; height: 12px;\">Method for Obtaining\u00a0[latex]y[\/latex]<\/th>\r\n<th style=\"width: 12.87%; text-align: center; height: 12px;\">Equation<\/th>\r\n<\/tr>\r\n<tr style=\"height: 28px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 28px;\"><span style=\"color: #0000ff;\">-3<\/span><\/td>\r\n<td style=\"width: 15.3369%; text-align: center; height: 28px;\">[latex]\\dfrac{1}{1000}[\/latex]<\/td>\r\n<td style=\"width: 15.6378%; height: 28px; text-align: center;\">[latex]\\dfrac{1}{100}\\div10=\\dfrac{1}{1000}[\/latex]<\/td>\r\n<td style=\"width: 12.87%; height: 28px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=10^{-3}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">-2<\/span><\/td>\r\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\">[latex]\\dfrac{1}{100}[\/latex]<\/td>\r\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\">[latex]\\dfrac{1}{10}\\div10=\\dfrac{1}{100}[\/latex]<\/td>\r\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=10^{-2}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">-1<\/span><\/td>\r\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\">[latex]\\dfrac{1}{10}[\/latex]<\/td>\r\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\">[latex]1\\div10=\\dfrac{1}{10}[\/latex]<\/td>\r\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=10^{-1}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">0<\/span><\/td>\r\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\">1<\/td>\r\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\">[latex]10\\div10=1[\/latex]<\/td>\r\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=10^0[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">1<\/span><\/td>\r\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">10<\/span><\/td>\r\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\"><span style=\"color: #ff0000;\">[latex]= 10[\/latex]<\/span><\/td>\r\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #ff0000;\">[latex]y=10^1[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 1<\/h3>\r\nComplete the table for the equation [latex]y=3^x[\/latex].\r\n<table style=\"border-collapse: collapse; width: 126.373%; height: 189px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 15.0718%; text-align: center; height: 12px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 15.3369%; text-align: center; height: 12px;\">[latex]y[\/latex]<\/th>\r\n<th style=\"width: 15.6378%; text-align: center; height: 12px;\">Method for Obtaining\u00a0[latex]y[\/latex]<\/th>\r\n<th style=\"width: 12.87%; text-align: center; height: 12px;\">Equation<\/th>\r\n<\/tr>\r\n<tr style=\"height: 28px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 28px;\"><span style=\"color: #0000ff;\">-3<\/span><\/td>\r\n<td style=\"width: 15.3369%; text-align: center; height: 28px;\"><\/td>\r\n<td style=\"width: 15.6378%; height: 28px; text-align: left;\"><\/td>\r\n<td style=\"width: 12.87%; height: 28px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=3^{-3}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">-2<\/span><\/td>\r\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\"><\/td>\r\n<td style=\"width: 15.6378%; height: 12px; text-align: left;\"><\/td>\r\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=3^{-2}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">-1<\/span><\/td>\r\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\"><\/td>\r\n<td style=\"width: 15.6378%; height: 12px; text-align: left;\"><\/td>\r\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=3^{-1}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">0<\/span><\/td>\r\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\"><\/td>\r\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\"><\/td>\r\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=3^0[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">1<\/span><\/td>\r\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">3<\/span><\/td>\r\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\"><span style=\"color: #ff0000;\">[latex]3[\/latex]<\/span><\/td>\r\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #ff0000;\">[latex]y=3^1[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Solution<\/h4>\r\nSince the base of the function is 3, we divide the [latex]y[\/latex]-value by 3 as [latex]x[\/latex] decreases by 1.\r\n<table style=\"border-collapse: collapse; width: 126.373%; height: 189px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 15.0718%; text-align: center; height: 12px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 15.3369%; text-align: center; height: 12px;\">[latex]y[\/latex]<\/th>\r\n<th style=\"width: 15.6378%; text-align: center; height: 12px;\">Method for Obtaining\u00a0[latex]y[\/latex]<\/th>\r\n<th style=\"width: 12.87%; text-align: center; height: 12px;\">Equation<\/th>\r\n<\/tr>\r\n<tr style=\"height: 28px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 28px;\"><span style=\"color: #0000ff;\">-3<\/span><\/td>\r\n<td style=\"width: 15.3369%; text-align: center; height: 28px;\">[latex]\\dfrac{1}{27}[\/latex]<\/td>\r\n<td style=\"width: 15.6378%; height: 28px; text-align: center;\">[latex]\\dfrac{1}{9}\\div3=\\dfrac{1}{27}[\/latex]<\/td>\r\n<td style=\"width: 12.87%; height: 28px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=3^{-3}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">-2<\/span><\/td>\r\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\">[latex]\\dfrac{1}{9}[\/latex]<\/td>\r\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\">[latex]\\dfrac{1}{3}\\div3=\\dfrac{1}{9}[\/latex]<\/td>\r\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=3^{-2}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">-1<\/span><\/td>\r\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\">[latex]\\dfrac{1}{3}[\/latex]<\/td>\r\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\">[latex]1\\div3=\\dfrac{1}{3}[\/latex]<\/td>\r\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=3^{-1}[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">0<\/span><\/td>\r\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\">1<\/td>\r\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\">[latex]3\\div3=1[\/latex]<\/td>\r\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=3^0[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">1<\/span><\/td>\r\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">3<\/span><\/td>\r\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\"><span style=\"color: #ff0000;\">[latex]=3[\/latex]<\/span><\/td>\r\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #ff0000;\">[latex]y=3^1[\/latex]<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 2<\/h3>\r\nEvaluate [latex]y=4^x[\/latex] when,\r\n<ul>\r\n \t<li>a) [latex]x=1[\/latex]<\/li>\r\n \t<li>b) [latex]x=3[\/latex]<\/li>\r\n \t<li>c) [latex]x=0[\/latex]<\/li>\r\n \t<li>d) [latex]x=-2[\/latex]<\/li>\r\n \t<li>e) [latex]x=-3[\/latex]<\/li>\r\n<\/ul>\r\n<h4>Solution<\/h4>\r\n<ul>\r\n \t<li>a) [latex]4^1=4[\/latex]<\/li>\r\n \t<li>b) [latex]4^3=4\\cdot4\\cdot4=64[\/latex]<\/li>\r\n \t<li>c) [latex]4^0=1[\/latex]<\/li>\r\n \t<li>d) [latex]4^{-2}=\\dfrac{1}{4^2}=\\dfrac{1}{16}[\/latex]<\/li>\r\n \t<li>e) [latex]4^{-3}=\\dfrac{1}{4^3}=\\dfrac{1}{64}[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 2<\/h3>\r\nEvaluate [latex]y=7^x[\/latex] when,\r\n<ul>\r\n \t<li>a) [latex]x=1[\/latex]<\/li>\r\n \t<li>b) [latex]x=2[\/latex]<\/li>\r\n \t<li>c) [latex]x=0[\/latex]<\/li>\r\n \t<li>d) [latex]x=-2[\/latex]<\/li>\r\n \t<li>e) [latex]x=-3[\/latex]<\/li>\r\n<\/ul>\r\n[reveal-answer q=\"hjm725\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm725\"]\r\n<ul>\r\n \t<li>a) [latex]7^1=7[\/latex]<\/li>\r\n \t<li>b) [latex]7^2=49[\/latex]<\/li>\r\n \t<li>c) [latex]7^0=1[\/latex]<\/li>\r\n \t<li>d) [latex]7^{-2}=\\dfrac{1}{49}[\/latex]<\/li>\r\n \t<li>e) [latex]7^{-3}=\\dfrac{1}{343}[\/latex]<\/li>\r\n<\/ul>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h3>The Difference between Exponential Growth and Power Growth<\/h3>\r\nIn exponential growth, the variable is the exponent (e.g., [latex]2^x[\/latex]). In power growth, the variable is the base (e.g., [latex]x^2[\/latex]). Exponential growth grows faster than power growth. For example, table 3 shows that the exponential growth [latex]2^x[\/latex] grows much faster than the power growth [latex]x^2[\/latex] while table 4 shows that the exponential growth [latex]3^x[\/latex] grows much faster than the power growth [latex]x^3[\/latex].\r\n<table style=\"border-collapse: collapse; width: 100%; height: 154px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 14px;\">\r\n<th style=\"width: 14.2857%; height: 14px;\"><span style=\"caret-color: #339966;\">[latex]x[\/latex]<\/span><\/th>\r\n<th style=\"width: 14.2857%; height: 14px;\"><span style=\"caret-color: #339966;\">[latex]x^2[\/latex]<\/span><\/th>\r\n<th style=\"width: 14.2857%; height: 14px;\"><span style=\"caret-color: #339966;\">[latex]2^x[\/latex]<\/span><\/th>\r\n<th style=\"width: 14.2857%; height: 14px;\">____<\/th>\r\n<th style=\"width: 14.2857%; height: 14px;\"><span style=\"caret-color: #339966;\">[latex]x[\/latex]<\/span><\/th>\r\n<th style=\"width: 14.2857%; height: 14px;\"><span style=\"caret-color: #339966;\">[latex]x^3[\/latex]<\/span><\/th>\r\n<th style=\"width: 14.2857%; height: 14px;\"><span style=\"caret-color: #339966;\">[latex]3^x[\/latex]<\/span><\/th>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 14.2857%; height: 14px;\">0<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">0<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">1<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\"><\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">0<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">0<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">1<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 14.2857%; height: 14px;\">1<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">1<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">2<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\"><\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">1<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">1<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">3<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 14.2857%; height: 14px;\">2<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">4<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">4<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\"><\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">2<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">8<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">9<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 14.2857%; height: 14px;\">3<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">9<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">8<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\"><\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">3<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">27<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">27<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 14.2857%; height: 14px;\">4<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">16<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">16<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\"><\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">4<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">64<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">81<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 14.2857%; height: 14px;\">5<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">25<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">32<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\"><\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">5<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">125<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">243<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 14.2857%; height: 14px;\">6<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">36<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">64<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\"><\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">6<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">216<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">729<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 14.2857%; height: 14px;\">7<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">49<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">128<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\"><\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">7<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">343<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">\u00a02187<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 14.2857%; height: 14px;\">8<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">64<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">256<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\"><\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">8<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">512<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\">6561<\/td>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<td style=\"width: 14.2857%; height: 14px;\" colspan=\"3\">Table 3. Exponential versus power growth<\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\"><\/td>\r\n<td style=\"width: 14.2857%; height: 14px;\" colspan=\"3\">Table 4.\u00a0Exponential versus power growth<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h3>Initial Value<\/h3>\r\nThe initial value of exponential growth occurs at [latex]x=0[\/latex].\u00a0So far we have considered only initial values of 1. Tables 1 and 2 show a pattern where [latex]y=1\\cdot 2^x[\/latex]. However, the initial value can be any real number.\r\n\r\nFor example, let's consider an initial value of 8 and an exponential growth rate of 2. We can create a table that illustrates this scenario starting with [latex]y=8[\/latex] when [latex]x=0[\/latex], then multiplying each [latex]y[\/latex]-value by 2:\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 33.3333%;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 33.3333%;\">[latex]y[\/latex]<\/th>\r\n<th style=\"width: 33.3333%;\">Equation<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">-2<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]4\\div 2=2[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]8\\cdot2^{-2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">-1<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]8\\div 2=4[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]8\\cdot2^{-1}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">0<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]8[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]y=8\\cdot2^0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">1<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]8\\cdot 2=16[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]8\\cdot2^1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">2<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]16\\cdot 2=32[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]8\\cdot2^2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 33.3333%;\">3<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]32\\cdot 2=64[\/latex]<\/td>\r\n<td style=\"width: 33.3333%;\">[latex]8\\cdot2^3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe equation that models an exponential growth rate of 2 with an initial value of 8 is [latex]y=8\\cdot2^x[\/latex].\r\n<div class=\"textbox shaded\">\r\n<h3>Exponential Growth<\/h3>\r\nThe equation that models an exponential growth rate of [latex]r[\/latex] with an initial value of [latex]a[\/latex] is [latex]y=a\\cdot r^x[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example 3<\/h3>\r\nComplete the table for an exponential growth of 4 and an initial value of 3.\u00a0 Then write an equation for the exponential growth pattern.\r\n<table style=\"border-collapse: collapse; width: 0%; height: 166px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 52.6328%; text-align: right; height: 12px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 143.5%; height: 12px; text-align: left;\">[latex]y[\/latex]<\/th>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 52.6328%; text-align: right; height: 12px;\">[latex]\u20132[\/latex]<\/td>\r\n<td style=\"width: 143.5%; height: 12px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 52.6328%; text-align: right; height: 12px;\">[latex]\u20131[\/latex]<\/td>\r\n<td style=\"width: 143.5%; height: 12px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 52.6328%; text-align: right; height: 12px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 143.5%; height: 12px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 52.6328%; text-align: right; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 143.5%; height: 12px;\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 52.6328%; text-align: right; height: 12px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 143.5%; height: 12px;\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 52.6328%; text-align: right;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 143.5%;\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Solution<\/h4>\r\nThe initial value of 3 occurs when [latex]x=0[\/latex] so we can add that to the table. Then we can multiply by 4 each time to determine the [latex]y[\/latex]-values for [latex]x=1,\\;2[\/latex]. To find the\u00a0[latex]y[\/latex]-values for [latex]x=-1,\\;-2[\/latex], we work backwards by dividing the initial value by 4.\r\n<table style=\"border-collapse: collapse; width: 0%; height: 72px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 52.2644%; text-align: right; height: 12px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 47.7356%; height: 12px; text-align: left;\">[latex]y[\/latex]<\/th>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 52.2644%; text-align: right; height: 12px;\">[latex]\u20132[\/latex]<\/td>\r\n<td style=\"width: 47.7356%; height: 12px;\">[latex]3\u00f74\u00f74=\\dfrac{3}{16}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 52.2644%; text-align: right; height: 12px;\">[latex]\u20131[\/latex]<\/td>\r\n<td style=\"width: 47.7356%; height: 12px;\">[latex]3\u00f74=\\dfrac{3}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 52.2644%; text-align: right; height: 12px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 47.7356%; height: 12px;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 52.2644%; text-align: right; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 47.7356%; height: 12px;\">[latex]3\\cdot 4=12[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 52.2644%; text-align: right; height: 12px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 47.7356%; height: 12px;\">[latex]3\\cdot 4\\cdot 4=48[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 52.2644%; text-align: right;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 47.7356%;\">[latex]3\\cdot 4\\cdot 4\\cdot 4=192[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe pattern is [latex]y=3\\cdot 4^x[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 3<\/h3>\r\nComplete the table for an exponential growth of 3 and an initial value of 5.\u00a0 Then write an equation for the exponential growth pattern.\r\n\r\n[reveal-answer q=\"hjm585\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm585\"]\r\n<table style=\"border-collapse: collapse; width: 0%; height: 72px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<th style=\"width: 46.5554%; text-align: right; height: 12px;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 53.4446%; text-align: right; height: 12px;\">[latex]y[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 46.5554%; text-align: right;\">[latex]\u20133[\/latex]<\/td>\r\n<td style=\"width: 53.4446%;\">[latex]\\dfrac{5}{27}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 46.5554%; text-align: right; height: 12px;\">[latex]\u20132[\/latex]<\/td>\r\n<td style=\"width: 53.4446%; height: 12px;\">[latex]\\dfrac{5}{9}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 46.5554%; text-align: right; height: 12px;\">[latex]\u20131[\/latex]<\/td>\r\n<td style=\"width: 53.4446%; height: 12px;\">[latex]\\dfrac{5}{3}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 46.5554%; text-align: right; height: 12px;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 53.4446%; height: 12px;\">[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 46.5554%; text-align: right; height: 12px;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 53.4446%; height: 12px;\">[latex]15[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"width: 46.5554%; text-align: right; height: 12px;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 53.4446%; height: 12px;\">[latex]45[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe pattern is [latex]y=5\\cdot 3^x[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 4<\/h3>\r\nWrite an equation for the following exponential growth patterns:\r\n<ol>\r\n \t<li>growth = 6; initial value = 4<\/li>\r\n \t<li>growth = 2; initial value = 9<\/li>\r\n \t<li>growth = [latex]\\dfrac{2}{3}[\/latex]; initial value = 5<\/li>\r\n \t<li>growth = 0.77; initial value = \u20135.4<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"hjm118\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm118\"]\r\n<ol>\r\n \t<li>[latex]y=4\\left(6^x\\right)[\/latex]<\/li>\r\n \t<li>[latex]y=9\\left(2^x\\right)[\/latex]<\/li>\r\n \t<li>[latex]y=5\\left(\\dfrac{2}{3}\\right)^x[\/latex]<\/li>\r\n \t<li>[latex]y=-5.4(0.77)^x[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2>Exponential Functions and Their Graphs<\/h2>\r\nExponential growth has an <strong><em>initial value<\/em><\/strong> and an <strong><em>exponential rate of change<\/em><\/strong>. <span style=\"color: #0000ff;\"><span style=\"color: #000000;\">The initial value occurs at [latex]x=0[\/latex].<\/span>\u00a0<\/span>In table 1, the initial value is 1 (when [latex]x=0[\/latex]), and the exponential rate of change is 2. This creates a pattern where [latex]y=1\\cdot 2^x[\/latex]. Consequently, the exponential growth in table 1 may be modeled or represented by the function [latex]f(x) = 2^x[\/latex].\r\n\r\nIf we graph the values [latex](x, y)[\/latex] from table 1, we can then connect the points to draw the graph the exponential function [latex]f(x)=2^x[\/latex] (figure 1).\r\n\r\n[caption id=\"attachment_2730\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-2730 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-16T164852.907-300x300.png\" alt=\"Curve increasing gradually from left to right until reaching (0,1), then steeply increasing\" width=\"300\" height=\"300\" \/> Figure 1. The graph of the function [latex]f(x)=2^x[\/latex].[\/caption]\r\n<div class=\"textbox examples\">\r\n<h3>Example 4<\/h3>\r\nCreate a table of values then graph the function [latex]f(x)=\\left(\\dfrac{1}{2}\\right)^x[\/latex].\r\n<h4>Solution<\/h4>\r\nWe can choose any [latex]x[\/latex]-values to create a table for [latex]y=f(x)[\/latex]:\r\n<table style=\"border-collapse: collapse; width: 50%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 15.8986%;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 26.4977%;\">[latex]y[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 15.8986%;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 26.4977%;\">[latex]\\left(\\dfrac{1}{2}\\right)^0=1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 15.8986%;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 26.4977%;\">[latex]\\left(\\dfrac{1}{2}\\right)^1=\\dfrac{1}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 15.8986%;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 26.4977%;\">[latex]\\left(\\dfrac{1}{2}\\right)^2=\\dfrac{1}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 15.8986%;\">[latex]-1[\/latex]<\/td>\r\n<td style=\"width: 26.4977%;\">[latex]\\left(\\dfrac{1}{2}\\right)^{-1}=2^1=2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 15.8986%;\">[latex]-2[\/latex]<\/td>\r\n<td style=\"width: 26.4977%;\">[latex]\\left(\\dfrac{1}{2}\\right)^{-2}=2^2=4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 15.8986%;\">[latex]-3[\/latex]<\/td>\r\n<td style=\"width: 26.4977%;\">\u00a0[latex]\\left(\\dfrac{1}{2}\\right)^{-3}=2^3=8[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe plot the [latex](x,y)[\/latex] points from the table:\r\n\r\n<img class=\"aligncenter size-medium wp-image-2614\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/10184730\/desmos-graph-2022-06-10T124619.629-300x300.png\" alt=\"Plotted points\" width=\"300\" height=\"300\" \/>\r\n\r\nThen join the points with a smooth curve:\r\n\r\n<img class=\"aligncenter wp-image-2615 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T124636.600-300x300.png\" alt=\"Exponential graph passing through points, decreasing by half as it moves left to right\" width=\"300\" height=\"300\" \/>\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It 5<\/h3>\r\nCreate a table of values then graph the function [latex]f(x)=4^x[\/latex].\r\n\r\n[reveal-answer q=\"hjm469\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm469\"]\r\n<table style=\"border-collapse: collapse; width: 24.2396%; height: 118px;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 48.1255%; text-align: right;\">[latex]x[\/latex]<\/th>\r\n<th style=\"width: 51.8745%;\">[latex]y[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 48.1255%; text-align: right;\">[latex]\u20133[\/latex]<\/td>\r\n<td style=\"width: 51.8745%;\">[latex]\\dfrac{1}{64}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 48.1255%; text-align: right;\">[latex]\u20132[\/latex]<\/td>\r\n<td style=\"width: 51.8745%;\">[latex]\\dfrac{1}{16}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 48.1255%; text-align: right;\">[latex]\u20131[\/latex]<\/td>\r\n<td style=\"width: 51.8745%;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 48.1255%; text-align: right;\">[latex]0[\/latex]<\/td>\r\n<td style=\"width: 51.8745%;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 48.1255%; text-align: right;\">[latex]1[\/latex]<\/td>\r\n<td style=\"width: 51.8745%;\">[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 48.1255%; text-align: right;\">[latex]2[\/latex]<\/td>\r\n<td style=\"width: 51.8745%;\">[latex]16[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 48.1255%; text-align: right;\">[latex]3[\/latex]<\/td>\r\n<td style=\"width: 51.8745%;\">[latex]64[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<img class=\"aligncenter size-medium wp-image-2538\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/06180547\/desmos-graph-2022-06-06T120534.071-300x300.png\" alt=\"Exponential graph\" width=\"300\" height=\"300\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>The Definition of an Exponential Function<\/h3>\r\nAn exponential function has the form\u00a0[latex]f(x) = r^x[\/latex], where [latex]r[\/latex] is a real number with [latex]r &gt;0[\/latex] and [latex]r \\neq 1[\/latex].\r\n\r\n<\/div>\r\nFigure 2 illustrates how the graph changes as the value of [latex]r[\/latex] changes. Move the red circle up or down to change the value of [latex]r[\/latex] and watch what happens to the function.\r\n<p style=\"text-align: center;\"><iframe style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/jgj3wtjxxx?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\r\n<p style=\"text-align: center;\">Figure 2. Interactive graph [latex]f(x)=r^x[\/latex]<\/p>\r\nManipulate the graph of [latex]f(x)=r^x[\/latex] in figure 2 to answer the following questions:\r\n\r\n1. What happens to the point (0, 1) as [latex]r[\/latex]changes?\r\n<p style=\"padding-left: 30px;\">The point (0, 1) never changes. The point (0, 1) is always on the graph of [latex]f(x)=r^x[\/latex].<\/p>\r\n<span style=\"font-size: 1em;\">2. What happens to the point on the graph at [latex]x=1[\/latex] as\u00a0[latex]r[\/latex]changes?<\/span>\r\n<p style=\"padding-left: 30px;\">At\u00a0[latex]x=1[\/latex], the point on the graph will always be\u00a0[latex](1, r)[\/latex], because [latex]f(1)=r^1=r[\/latex].<\/p>\r\n3. What happens to the graph when [latex]r=1[\/latex]?\r\n<p style=\"padding-left: 30px;\"><span style=\"font-size: 1em;\">If [latex]r=1[\/latex] we get a flat line; a linear equation [latex]y=1[\/latex]. This is why [latex]r[\/latex] is never allowed to equal 1.<\/span><\/p>\r\n4. What happens to the graph when [latex]r=0[\/latex]?\r\n<p style=\"padding-left: 30px;\">If [latex]r=0[\/latex] we get a flat line starting at [latex]x&gt;0[\/latex]. [latex]f(0)=0^0[\/latex] which is undefined.\u00a0 For any [latex]x&lt;0[\/latex], [latex]0^{\\text{negative number}}=\\dfrac{1}{0^{\\text{positive number}}}=\\dfrac{1}{0}[\/latex], which is undefined. This is why [latex]r[\/latex] is never allowed to equal 0.<\/p>\r\n5. What happens to the graph when [latex]r&lt;0[\/latex]?\r\n<p style=\"padding-left: 30px;\">When [latex]r&lt;0[\/latex], the graph disappears!! This is why [latex]r[\/latex] is a positive real number \u2260 0, 1.<\/p>\r\n6. What happens to the graph when [latex]r&gt;1[\/latex]?\r\n<p style=\"padding-left: 30px;\">The graph comes up from [latex]y=0[\/latex], passes through (0, 1) and (1, r), then quickly moves towards [latex]+\\infty[\/latex].<\/p>\r\n7. What happens to the graph when [latex]0&lt;r&lt;1[\/latex]?\r\n<p style=\"padding-left: 30px;\">When [latex]r[\/latex] is a proper fraction, the graph comes in from\u00a0[latex]+\\infty[\/latex], passes through (0, 1) and (1, r), then moves slowly towards\u00a0[latex]y=0[\/latex].<\/p>\r\n\r\n<h2>The Asymptote and Intercepts<\/h2>\r\nA significant feature on the graph of any exponential function\u00a0[latex]f(x) = r^x[\/latex] ([latex]r&gt;0,\\;r\\neq1[\/latex]) is that the graph never crosses the [latex]x[\/latex]-axis. It continually approaches the [latex]x[\/latex], getting closer and closer, but the graph never meets the\u00a0[latex]x[\/latex]-axis. Figure 2 illustrates that\u00a0<span style=\"font-size: 1em;\">when [latex]0&lt;r&lt;1,[\/latex] the graph gets closer and closer to the [latex]x[\/latex]-axis when [latex]x[\/latex] gets closer and closer to positive infinity (figure 3).\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">However,\u00a0<\/span><span style=\"font-size: 1em;\">when [latex]r&gt;1[\/latex], the graph gets closer and closer to the [latex]x[\/latex]-axis when [latex]x[\/latex] gets closer and closer to negative infinity (figure 4).<\/span>\r\n<table style=\"border-collapse: collapse; width: 100%;\" border=\"1\">\r\n<tbody>\r\n<tr>\r\n<th style=\"width: 50%;\" colspan=\"2\">Graphs with horizontal asymptotes<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\"><img class=\"aligncenter wp-image-2620 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T131850.316-300x300.png\" alt=\"r^x with 0, a decreasing exponential graph passing through (0,1)\" width=\"300\" height=\"300\" \/><\/td>\r\n<td style=\"width: 50%;\"><img class=\"aligncenter wp-image-2619 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T131830.3471-300x300.png\" alt=\"r^x with 0&lt;r&lt;1, an exponentially increasing curve passing through (0,1)\" width=\"300\" height=\"300\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"shaded\" style=\"width: 50%; text-align: center;\">Figure 3. [latex]f(x)=r^x[\/latex] with [latex]0&lt;r&lt;1[\/latex]<\/td>\r\n<td class=\"shaded\" style=\"width: 50%; text-align: center;\">Figure 4. [latex]f(x)=r^x[\/latex] with [latex]r&gt;1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWhy does the graph never meet the [latex]x[\/latex]-axis? Consider the following examples using the function [latex]f(x)=2^x[\/latex] that was graphed in figure 1:\r\n<p style=\"text-align: center;\">[latex]f(-10)=2^{-10}=\\dfrac{1}{2^{10}}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]f(-100)=2^{-100}=\\dfrac{1}{2^{100}}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]f(-100000)=2^{-100000}=\\dfrac{1}{2^{100000}}[\/latex]<\/p>\r\nAs the the value of [latex]x[\/latex] gets closer to negative infinity, the value of the function [latex]y[\/latex] is a fraction with a numerator of 1 and a denominator that is a very large positive number. The value of [latex]x[\/latex] gets more and more negative as it gets closer to negative infinity, so the value of the function will get smaller and smaller. It will get close to zero but will never be zero because [latex]\\dfrac{1}{\\text{very large positive number}}[\/latex] is always positive and therefore greater than zero.\r\n\r\nFigure 2 shows that for all values of [latex]r&gt;0[\/latex] and [latex]r\\neq1[\/latex], the graph gets close to but never crosses the [latex]x[\/latex]-axis, it is a <em><strong>horizontal asymptote<\/strong><\/em> of the function [latex]f(x)=r^x[\/latex]. Also, s<span style=\"font-size: 1rem; text-align: initial;\">ince the graph never meets the\u00a0[latex]x[\/latex]-axis, there is no\u00a0[latex]x[\/latex]-intercept for the function. The\u00a0[latex]y[\/latex]-intercept of the function\u00a0[latex]f(x)=r^x[\/latex] is always (0, 1).<\/span>\r\n\r\nWe use a dotted line to show that a graph has a horizontal asymptote (figure 5).\r\n\r\n[caption id=\"attachment_2732\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-2732 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-16T171836.699-300x300.png\" alt=\"Exponential function with asymptote at y=0\" width=\"300\" height=\"300\" \/> Figure 5. Exponential function with horizontal asymptote.[\/caption]\r\n<h2>Domain and Range<\/h2>\r\nFigure 5 shows the graph of [latex]f(x)=2^x[\/latex]. The domain of the function is the set of all possible [latex]x[\/latex]-values, so domain = [latex]\\{x\\;|\\;x\\in \\mathbb{R}\\}[\/latex]. Any [latex]x[\/latex]-value from [latex]-\\infty[\/latex] to\u00a0[latex]+\\infty[\/latex] has a corresponding function value. It's range, the set of all function values, lies above the line [latex]y=0[\/latex]. Consequently, the range =[latex]\\{f(x)\\;|\\;f(x)\\in\\mathbb{R}^+\\}[\/latex], where\u00a0[latex]\\mathbb{R}^+[\/latex] is the set of all positive real numbers.\r\n<div><\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>DOMAIN and RANGE<\/h3>\r\nThe domain of any exponential function [latex]f(x)=r^x[\/latex] is all real numbers, or [latex]\\{x | x \\in \\mathbb{R}\\}[\/latex], or [latex](-\\infty, \\infty)[\/latex]. The range of any exponential function[latex]f(x)=r^x[\/latex] is all real numbers that are above the horizontal asymptote. Range = [latex]\\{f(x)\\;|\\;f(x)\\in\\mathbb{R}^+\\}[\/latex], or [latex](0, \\infty)[\/latex].\r\n\r\n<\/div>\r\nThe exponential function\u00a0[latex]f(x)=r^x[\/latex] is the parent function of all exponential functions. In the next section, we will see what happens to the graph of the function when we transform the parent function.\r\n\r\n<script>\r\nwindow.embeddedChatbotConfig = {\r\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\r\ndomain: \"www.chatbase.co\"\r\n}\r\n<\/script>\r\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" chatbotId=\"ejVb5sgc1-w972OOCgl5x\" domain=\"www.chatbase.co\" defer>\r\n<\/script>\r\n\r\n<iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe>","rendered":"<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n<div class=\"textbox learning-objectives\">\n<h1>Learning Objectives<\/h1>\n<ul>\n<li>Determine the pattern of exponential growth<\/li>\n<li>Define and graph exponential functions<\/li>\n<li>Describe the asymptote of an exponential function<\/li>\n<li>Find the domain and range of an exponential function<\/li>\n<\/ul>\n<\/div>\n<h2>Exponential Growth<\/h2>\n<p>In Chapter 2, we looked at linear growth where there is a constant rate of change. Unlike linear growth that increases by <em>adding<\/em> a constant value to [latex]y[\/latex] for every unit increase in [latex]x[\/latex], <em><strong>exponential growth<\/strong><\/em> increases by <em>multiplying<\/em>\u00a0by a constant that is neither equal to 0 nor 1. Table 1 shows exponential growth as each [latex]y[\/latex]-value increases by a multiple of 2. Notice that the values in the [latex]x[\/latex] column correspond to the exponents in the fourth column. This is because the values in the\u00a0[latex]x[\/latex] column record the number of times we have multiplied by 2.<\/p>\n<table style=\"border-collapse: collapse; width: 126.373%; height: 147px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<th style=\"width: 15.0718%; text-align: center; height: 12px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 9.70989%; text-align: center; height: 12px;\">[latex]y[\/latex]<\/th>\n<th style=\"width: 21.2648%; height: 12px; text-align: center;\">Number of times 2 gets multiplied<\/th>\n<th style=\"width: 12.87%; height: 12px; text-align: center;\">Exponential Notation<\/th>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\">1<\/td>\n<td style=\"width: 9.70989%; text-align: center; height: 12px;\">2<\/td>\n<td style=\"width: 21.2648%; text-align: left; height: 12px;\">[latex]= 2[\/latex]<\/td>\n<td style=\"width: 12.87%; text-align: left; height: 12px;\">[latex]=2^1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\">2<\/td>\n<td style=\"width: 9.70989%; text-align: center; height: 12px;\">4<\/td>\n<td style=\"width: 21.2648%; text-align: left; height: 12px;\">[latex]= 2 \\cdot 2[\/latex]<\/td>\n<td style=\"width: 12.87%; text-align: left; height: 12px;\">[latex]=2^2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 25px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 25px;\">3<\/td>\n<td style=\"width: 9.70989%; text-align: center; height: 25px;\">8<\/td>\n<td style=\"width: 21.2648%; text-align: left; height: 25px;\">[latex]= 2 \\cdot 2 \\cdot 2[\/latex]<\/td>\n<td style=\"width: 12.87%; text-align: left; height: 25px;\">[latex]=2^3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 25px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 25px;\">4<\/td>\n<td style=\"width: 9.70989%; text-align: center; height: 25px;\">16<\/td>\n<td style=\"width: 21.2648%; text-align: left; height: 25px;\">[latex]=2 \\cdot 2 \\cdot 2 \\cdot 2[\/latex]<\/td>\n<td style=\"width: 12.87%; text-align: left; height: 25px;\">[latex]=2^4[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 25px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 25px;\">5<\/td>\n<td style=\"width: 9.70989%; text-align: center; height: 25px;\">32<\/td>\n<td style=\"width: 21.2648%; text-align: left; height: 25px;\">[latex]=2 \\cdot 2 \\cdot 2 \\cdot 2 \\cdot 2[\/latex]<\/td>\n<td style=\"width: 12.87%; text-align: left; height: 25px;\">[latex]=2^5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"shaded\" style=\"width: 58.9165%; text-align: left;\" colspan=\"4\">Table 1. Exponential growth<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The pattern shown in the fourth column of table 1 indicates that exponential growth is repeated multiplication by a factor of 2 so the equation for the pattern is [latex]y= 2^x[\/latex].<\/p>\n<h3>Zero and Negative Exponents<\/h3>\n<p>Table 1 shows the pattern of exponential growth of [latex]2^x[\/latex] when [latex]x[\/latex] is a natural number. But what is the value of [latex]2^x[\/latex] when the exponent [latex]x[\/latex] is zero or a negative number? If we consider the pattern in Table 1 in\u00a0the reverse direction we can find the value of [latex]y=2^x[\/latex] when [latex]x=0, -1, -2,...[\/latex]. As the exponent decreases by 1,\u00a0the value of [latex]y=2^x[\/latex] is divided by 2.<\/p>\n<p>For example, [latex]2^x=8[\/latex]\u00a0when the exponent [latex]x[\/latex]\u00a0is 3 (i.e., [latex]y=2^3=8[\/latex]). As [latex]x[\/latex] decreases by 1\u00a0to 2, the [latex]y[\/latex] value is 4 (See Table 2). In other words, 8 is divided by 2 to get 4.\u00a0Following this logic, the [latex]y[\/latex] value will be 4 \u00f7 2 = 2 when [latex]x[\/latex] decreases by 1 again to 1. Following this pattern, the [latex]y[\/latex] value will be 2 \u00f7 2 = 1\u00a0when [latex]x[\/latex] decreases by 1 again to zero. Therefore, [latex]2^0=1[\/latex]. In fact,\u00a0 [latex]a^0=1[\/latex]\u00a0for all values of [latex]a[\/latex], [latex]a\\ne 0[\/latex]. [latex]0^0[\/latex] is undefined.<\/p>\n<div class=\"textbox shaded\">\n<h3>Exponent of Zero<\/h3>\n<p style=\"text-align: center;\">[latex]a^0=1[\/latex]\u00a0for all values of [latex]a[\/latex], [latex]a\\ne 0[\/latex]. [latex]0^0[\/latex] is undefined.<\/p>\n<\/div>\n<p>Continuing this reverse pattern, the value of\u00a0[latex]y[\/latex] will be\u00a0[latex]1\\div 2=\\dfrac{1}{2}[\/latex]\u00a0when [latex]x=-1[\/latex]. Then, dividing by 2 again gives\u00a0[latex]y=\\dfrac{1}{2}\u00f72=\\dfrac{1}{2}\\cdot\\dfrac{1}{2}=\\dfrac{1}{4}[\/latex] when\u00a0[latex]x=-2[\/latex]. Recall that dividing by 2 is equivalent to multiplying by [latex]\\dfrac{1}{2}[\/latex]. This pattern shows that the value of\u00a0[latex]y[\/latex] is a fraction where\u00a0[latex]y=\\dfrac{1}{2^{|x|}}[\/latex] when the exponent\u00a0[latex]x[\/latex] is a negative number (See Table 2). For example,\u00a0[latex]y=2^{-3}=\\dfrac{1}{2^{|-3|}}=\\dfrac{1}{2^3}[\/latex]. In fact, the value of\u00a0[latex]y=a^x[\/latex] will alway be [latex]\\dfrac{1}{a^{|x|}}[\/latex] when the exponent [latex]x[\/latex] is a negative number and the base [latex]a \u2265 0[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Negative Exponents<\/h3>\n<p style=\"text-align: center;\">[latex]y=a^x=\\dfrac{1}{a^{|x|}}[\/latex] when the exponent [latex]x[\/latex] is a negative number and [latex]a \u2265 0[\/latex].<\/p>\n<\/div>\n<p>Table 3 shows how the values of [latex]y=2^x[\/latex] for the exponents\u00a0[latex]x=0, -1, -2, -3...[\/latex] are obtained following the reverse pattern starting at [latex]x=0[\/latex].<\/p>\n<table style=\"border-collapse: collapse; width: 126.373%; height: 189px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<th style=\"width: 15.0718%; text-align: center; height: 12px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 15.3369%; text-align: center; height: 12px;\">[latex]y[\/latex]<\/th>\n<th style=\"width: 15.6378%; height: 12px; text-align: left;\">Method for Obtaining\u00a0[latex]y[\/latex]<\/th>\n<th style=\"width: 12.87%; height: 12px; text-align: left;\">Equation<\/th>\n<\/tr>\n<tr style=\"height: 28px;\">\n<td style=\"width: 15.071770334928232%; text-align: center; height: 28px;\"><span style=\"color: #0000ff;\">-3<\/span><\/td>\n<td style=\"width: 15.336945869602815%; text-align: center; height: 28px;\"><span style=\"color: #0000ff;\">[latex]\\dfrac{1}{8}[\/latex]<\/span><\/td>\n<td style=\"width: 15.637757011498225%; height: 28px; text-align: left;\"><span style=\"color: #0000ff;\">[latex]=\\dfrac{1}{2} \\cdot \\dfrac{1}{2} \\cdot \\dfrac{1}{2}[\/latex]<\/span><\/td>\n<td style=\"width: 12.870007312340563%; height: 28px; text-align: left;\"><span style=\"color: #0000ff;\">[latex]2^{-3}[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.071770334928232%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">-2<\/span><\/td>\n<td style=\"width: 15.336945869602815%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">[latex]\\dfrac{1}{4}[\/latex]<\/span><\/td>\n<td style=\"width: 15.637757011498225%; height: 12px; text-align: left;\"><span style=\"color: #0000ff;\">[latex]=\\dfrac{1}{2} \\cdot \\dfrac{1}{2}[\/latex]<\/span><\/td>\n<td style=\"width: 12.870007312340563%; height: 12px; text-align: left;\"><span style=\"color: #0000ff;\">[latex]2^{-2}[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.071770334928232%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">-1<\/span><\/td>\n<td style=\"width: 15.336945869602815%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">[latex]\\dfrac{1}{2}[\/latex]<\/span><\/td>\n<td style=\"width: 15.6378%; height: 12px; text-align: left;\"><span style=\"color: #0000ff;\">[latex]=\\dfrac{1}{2}[\/latex]<\/span><\/td>\n<td style=\"width: 12.870007312340563%; height: 12px; text-align: left;\"><span style=\"color: #0000ff;\">[latex]2^{-1}[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.071770334928232%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">0<\/span><\/td>\n<td style=\"width: 15.336945869602815%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">1<\/span><\/td>\n<td style=\"width: 15.6378%; height: 12px; text-align: left;\"><span style=\"color: #0000ff;\">[latex]= 2 \\cdot \\dfrac{1}{2}[\/latex] = 1<\/span><\/td>\n<td style=\"width: 12.870007312340563%; height: 12px; text-align: left;\"><span style=\"color: #0000ff;\">[latex]2^0[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.071770334928232%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">1<\/span><\/td>\n<td style=\"width: 15.336945869602815%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">2<\/span><\/td>\n<td style=\"width: 15.637757011498225%; height: 12px; text-align: left;\"><span style=\"color: #ff0000;\">[latex]=2[\/latex]<\/span><\/td>\n<td style=\"width: 12.870007312340563%; height: 12px; text-align: left;\"><span style=\"color: #ff0000;\">[latex]2^1[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.071770334928232%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">2<\/span><\/td>\n<td style=\"width: 15.336945869602815%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">4<\/span><\/td>\n<td style=\"width: 15.637757011498225%; height: 12px; text-align: left;\"><span style=\"color: #ff0000;\">[latex]= 2 \\cdot 2[\/latex]<\/span><\/td>\n<td style=\"width: 12.870007312340563%; height: 12px; text-align: left;\"><span style=\"color: #ff0000;\">[latex]2^2[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 25px;\">\n<td style=\"width: 15.071770334928232%; text-align: center; height: 25px;\"><span style=\"color: #ff0000;\">3<\/span><\/td>\n<td style=\"width: 15.336945869602815%; text-align: center; height: 25px;\"><span style=\"color: #ff0000;\">8<\/span><\/td>\n<td style=\"width: 15.637757011498225%; height: 25px; text-align: left;\"><span style=\"color: #ff0000;\">[latex]=2 \\cdot 2 \\cdot 2[\/latex]<\/span><\/td>\n<td style=\"width: 12.870007312340563%; height: 25px; text-align: left;\"><span style=\"color: #ff0000;\">[latex]2^3[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 25px;\">\n<td style=\"width: 15.071770334928232%; text-align: center; height: 25px;\"><span style=\"color: #ff0000;\">4<\/span><\/td>\n<td style=\"width: 15.336945869602815%; text-align: center; height: 25px;\"><span style=\"color: #ff0000;\">16<\/span><\/td>\n<td style=\"width: 15.637757011498225%; height: 25px; text-align: left;\"><span style=\"color: #ff0000;\">[latex]=2 \\cdot 2 \\cdot 2 \\cdot 2[\/latex]<\/span><\/td>\n<td style=\"width: 12.870007312340563%; height: 25px; text-align: left;\"><span style=\"color: #ff0000;\">[latex]2^4[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 25px;\">\n<td style=\"width: 15.071770334928232%; text-align: center; height: 25px;\"><span style=\"color: #ff0000;\">5<\/span><\/td>\n<td style=\"width: 15.336945869602815%; text-align: center; height: 25px;\"><span style=\"color: #ff0000;\">32<\/span><\/td>\n<td style=\"width: 15.637757011498225%; height: 25px; text-align: left;\"><span style=\"color: #ff0000;\">[latex]=2 \\cdot 2 \\cdot 2 \\cdot 2 \\cdot 2[\/latex]<\/span><\/td>\n<td style=\"width: 12.870007312340563%; height: 25px; text-align: left;\"><span style=\"color: #ff0000;\">[latex]2^5[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 15.071770334928232%; text-align: left; height: 14px;\" colspan=\"4\">Table 2. Exponential growth with base 2<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Zero and negative exponents will be discussed again in section 5.3.2 using the Product and Quotient Rule for Exponents.<\/p>\n<div class=\"textbox examples\">\n<h3>Example 1<\/h3>\n<p>Complete the table for the equation [latex]y=10^x[\/latex].<\/p>\n<table style=\"border-collapse: collapse; width: 126.373%; height: 189px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<th style=\"width: 15.0718%; text-align: center; height: 12px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 15.3369%; text-align: center; height: 12px;\">[latex]y[\/latex]<\/th>\n<th style=\"width: 15.6378%; text-align: center; height: 12px;\">Method for Obtaining\u00a0[latex]y[\/latex]<\/th>\n<th style=\"width: 12.87%; text-align: center; height: 12px;\">Equation<\/th>\n<\/tr>\n<tr style=\"height: 28px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 28px;\"><span style=\"color: #0000ff;\">-3<\/span><\/td>\n<td style=\"width: 15.3369%; text-align: center; height: 28px;\"><\/td>\n<td style=\"width: 15.6378%; height: 28px; text-align: left;\"><\/td>\n<td style=\"width: 12.87%; height: 28px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=10^{-3}[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">-2<\/span><\/td>\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\"><\/td>\n<td style=\"width: 15.6378%; height: 12px; text-align: left;\"><\/td>\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=10^{-2}[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">-1<\/span><\/td>\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\"><\/td>\n<td style=\"width: 15.6378%; height: 12px; text-align: left;\"><\/td>\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=10^{-1}[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">0<\/span><\/td>\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\"><\/td>\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\"><\/td>\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=10^0[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">1<\/span><\/td>\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">10<\/span><\/td>\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\"><span style=\"color: #ff0000;\">[latex]= 10[\/latex]<\/span><\/td>\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #ff0000;\">[latex]y=10^1[\/latex]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Solution<\/h4>\n<p>Since the base of the function is 10, we divide the [latex]y[\/latex]-value by 10 as [latex]x[\/latex] decreases by 1.<\/p>\n<table style=\"border-collapse: collapse; width: 126.373%; height: 189px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<th style=\"width: 15.0718%; text-align: center; height: 12px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 15.3369%; text-align: center; height: 12px;\">[latex]y[\/latex]<\/th>\n<th style=\"width: 15.6378%; text-align: center; height: 12px;\">Method for Obtaining\u00a0[latex]y[\/latex]<\/th>\n<th style=\"width: 12.87%; text-align: center; height: 12px;\">Equation<\/th>\n<\/tr>\n<tr style=\"height: 28px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 28px;\"><span style=\"color: #0000ff;\">-3<\/span><\/td>\n<td style=\"width: 15.3369%; text-align: center; height: 28px;\">[latex]\\dfrac{1}{1000}[\/latex]<\/td>\n<td style=\"width: 15.6378%; height: 28px; text-align: center;\">[latex]\\dfrac{1}{100}\\div10=\\dfrac{1}{1000}[\/latex]<\/td>\n<td style=\"width: 12.87%; height: 28px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=10^{-3}[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">-2<\/span><\/td>\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\">[latex]\\dfrac{1}{100}[\/latex]<\/td>\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\">[latex]\\dfrac{1}{10}\\div10=\\dfrac{1}{100}[\/latex]<\/td>\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=10^{-2}[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">-1<\/span><\/td>\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\">[latex]\\dfrac{1}{10}[\/latex]<\/td>\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\">[latex]1\\div10=\\dfrac{1}{10}[\/latex]<\/td>\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=10^{-1}[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">0<\/span><\/td>\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\">1<\/td>\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\">[latex]10\\div10=1[\/latex]<\/td>\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=10^0[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">1<\/span><\/td>\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">10<\/span><\/td>\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\"><span style=\"color: #ff0000;\">[latex]= 10[\/latex]<\/span><\/td>\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #ff0000;\">[latex]y=10^1[\/latex]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 1<\/h3>\n<p>Complete the table for the equation [latex]y=3^x[\/latex].<\/p>\n<table style=\"border-collapse: collapse; width: 126.373%; height: 189px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<th style=\"width: 15.0718%; text-align: center; height: 12px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 15.3369%; text-align: center; height: 12px;\">[latex]y[\/latex]<\/th>\n<th style=\"width: 15.6378%; text-align: center; height: 12px;\">Method for Obtaining\u00a0[latex]y[\/latex]<\/th>\n<th style=\"width: 12.87%; text-align: center; height: 12px;\">Equation<\/th>\n<\/tr>\n<tr style=\"height: 28px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 28px;\"><span style=\"color: #0000ff;\">-3<\/span><\/td>\n<td style=\"width: 15.3369%; text-align: center; height: 28px;\"><\/td>\n<td style=\"width: 15.6378%; height: 28px; text-align: left;\"><\/td>\n<td style=\"width: 12.87%; height: 28px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=3^{-3}[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">-2<\/span><\/td>\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\"><\/td>\n<td style=\"width: 15.6378%; height: 12px; text-align: left;\"><\/td>\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=3^{-2}[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">-1<\/span><\/td>\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\"><\/td>\n<td style=\"width: 15.6378%; height: 12px; text-align: left;\"><\/td>\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=3^{-1}[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">0<\/span><\/td>\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\"><\/td>\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\"><\/td>\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=3^0[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">1<\/span><\/td>\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">3<\/span><\/td>\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\"><span style=\"color: #ff0000;\">[latex]3[\/latex]<\/span><\/td>\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #ff0000;\">[latex]y=3^1[\/latex]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Solution<\/h4>\n<p>Since the base of the function is 3, we divide the [latex]y[\/latex]-value by 3 as [latex]x[\/latex] decreases by 1.<\/p>\n<table style=\"border-collapse: collapse; width: 126.373%; height: 189px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<th style=\"width: 15.0718%; text-align: center; height: 12px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 15.3369%; text-align: center; height: 12px;\">[latex]y[\/latex]<\/th>\n<th style=\"width: 15.6378%; text-align: center; height: 12px;\">Method for Obtaining\u00a0[latex]y[\/latex]<\/th>\n<th style=\"width: 12.87%; text-align: center; height: 12px;\">Equation<\/th>\n<\/tr>\n<tr style=\"height: 28px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 28px;\"><span style=\"color: #0000ff;\">-3<\/span><\/td>\n<td style=\"width: 15.3369%; text-align: center; height: 28px;\">[latex]\\dfrac{1}{27}[\/latex]<\/td>\n<td style=\"width: 15.6378%; height: 28px; text-align: center;\">[latex]\\dfrac{1}{9}\\div3=\\dfrac{1}{27}[\/latex]<\/td>\n<td style=\"width: 12.87%; height: 28px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=3^{-3}[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">-2<\/span><\/td>\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\">[latex]\\dfrac{1}{9}[\/latex]<\/td>\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\">[latex]\\dfrac{1}{3}\\div3=\\dfrac{1}{9}[\/latex]<\/td>\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=3^{-2}[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">-1<\/span><\/td>\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\">[latex]\\dfrac{1}{3}[\/latex]<\/td>\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\">[latex]1\\div3=\\dfrac{1}{3}[\/latex]<\/td>\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=3^{-1}[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #0000ff;\">0<\/span><\/td>\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\">1<\/td>\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\">[latex]3\\div3=1[\/latex]<\/td>\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #0000ff;\">[latex]y=3^0[\/latex]<\/span><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 15.0718%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">1<\/span><\/td>\n<td style=\"width: 15.3369%; text-align: center; height: 12px;\"><span style=\"color: #ff0000;\">3<\/span><\/td>\n<td style=\"width: 15.6378%; height: 12px; text-align: center;\"><span style=\"color: #ff0000;\">[latex]=3[\/latex]<\/span><\/td>\n<td style=\"width: 12.87%; height: 12px; text-align: center;\"><span style=\"color: #ff0000;\">[latex]y=3^1[\/latex]<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 2<\/h3>\n<p>Evaluate [latex]y=4^x[\/latex] when,<\/p>\n<ul>\n<li>a) [latex]x=1[\/latex]<\/li>\n<li>b) [latex]x=3[\/latex]<\/li>\n<li>c) [latex]x=0[\/latex]<\/li>\n<li>d) [latex]x=-2[\/latex]<\/li>\n<li>e) [latex]x=-3[\/latex]<\/li>\n<\/ul>\n<h4>Solution<\/h4>\n<ul>\n<li>a) [latex]4^1=4[\/latex]<\/li>\n<li>b) [latex]4^3=4\\cdot4\\cdot4=64[\/latex]<\/li>\n<li>c) [latex]4^0=1[\/latex]<\/li>\n<li>d) [latex]4^{-2}=\\dfrac{1}{4^2}=\\dfrac{1}{16}[\/latex]<\/li>\n<li>e) [latex]4^{-3}=\\dfrac{1}{4^3}=\\dfrac{1}{64}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 2<\/h3>\n<p>Evaluate [latex]y=7^x[\/latex] when,<\/p>\n<ul>\n<li>a) [latex]x=1[\/latex]<\/li>\n<li>b) [latex]x=2[\/latex]<\/li>\n<li>c) [latex]x=0[\/latex]<\/li>\n<li>d) [latex]x=-2[\/latex]<\/li>\n<li>e) [latex]x=-3[\/latex]<\/li>\n<\/ul>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm725\">Show Answer<\/span><\/p>\n<div id=\"qhjm725\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>a) [latex]7^1=7[\/latex]<\/li>\n<li>b) [latex]7^2=49[\/latex]<\/li>\n<li>c) [latex]7^0=1[\/latex]<\/li>\n<li>d) [latex]7^{-2}=\\dfrac{1}{49}[\/latex]<\/li>\n<li>e) [latex]7^{-3}=\\dfrac{1}{343}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<h3>The Difference between Exponential Growth and Power Growth<\/h3>\n<p>In exponential growth, the variable is the exponent (e.g., [latex]2^x[\/latex]). In power growth, the variable is the base (e.g., [latex]x^2[\/latex]). Exponential growth grows faster than power growth. For example, table 3 shows that the exponential growth [latex]2^x[\/latex] grows much faster than the power growth [latex]x^2[\/latex] while table 4 shows that the exponential growth [latex]3^x[\/latex] grows much faster than the power growth [latex]x^3[\/latex].<\/p>\n<table style=\"border-collapse: collapse; width: 100%; height: 154px;\">\n<tbody>\n<tr style=\"height: 14px;\">\n<th style=\"width: 14.2857%; height: 14px;\"><span style=\"caret-color: #339966;\">[latex]x[\/latex]<\/span><\/th>\n<th style=\"width: 14.2857%; height: 14px;\"><span style=\"caret-color: #339966;\">[latex]x^2[\/latex]<\/span><\/th>\n<th style=\"width: 14.2857%; height: 14px;\"><span style=\"caret-color: #339966;\">[latex]2^x[\/latex]<\/span><\/th>\n<th style=\"width: 14.2857%; height: 14px;\">____<\/th>\n<th style=\"width: 14.2857%; height: 14px;\"><span style=\"caret-color: #339966;\">[latex]x[\/latex]<\/span><\/th>\n<th style=\"width: 14.2857%; height: 14px;\"><span style=\"caret-color: #339966;\">[latex]x^3[\/latex]<\/span><\/th>\n<th style=\"width: 14.2857%; height: 14px;\"><span style=\"caret-color: #339966;\">[latex]3^x[\/latex]<\/span><\/th>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 14.2857%; height: 14px;\">0<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">0<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">1<\/td>\n<td style=\"width: 14.2857%; height: 14px;\"><\/td>\n<td style=\"width: 14.2857%; height: 14px;\">0<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">0<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">1<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 14.2857%; height: 14px;\">1<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">1<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">2<\/td>\n<td style=\"width: 14.2857%; height: 14px;\"><\/td>\n<td style=\"width: 14.2857%; height: 14px;\">1<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">1<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">3<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 14.2857%; height: 14px;\">2<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">4<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">4<\/td>\n<td style=\"width: 14.2857%; height: 14px;\"><\/td>\n<td style=\"width: 14.2857%; height: 14px;\">2<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">8<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">9<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 14.2857%; height: 14px;\">3<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">9<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">8<\/td>\n<td style=\"width: 14.2857%; height: 14px;\"><\/td>\n<td style=\"width: 14.2857%; height: 14px;\">3<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">27<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">27<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 14.2857%; height: 14px;\">4<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">16<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">16<\/td>\n<td style=\"width: 14.2857%; height: 14px;\"><\/td>\n<td style=\"width: 14.2857%; height: 14px;\">4<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">64<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">81<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 14.2857%; height: 14px;\">5<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">25<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">32<\/td>\n<td style=\"width: 14.2857%; height: 14px;\"><\/td>\n<td style=\"width: 14.2857%; height: 14px;\">5<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">125<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">243<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 14.2857%; height: 14px;\">6<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">36<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">64<\/td>\n<td style=\"width: 14.2857%; height: 14px;\"><\/td>\n<td style=\"width: 14.2857%; height: 14px;\">6<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">216<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">729<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 14.2857%; height: 14px;\">7<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">49<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">128<\/td>\n<td style=\"width: 14.2857%; height: 14px;\"><\/td>\n<td style=\"width: 14.2857%; height: 14px;\">7<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">343<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">\u00a02187<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 14.2857%; height: 14px;\">8<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">64<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">256<\/td>\n<td style=\"width: 14.2857%; height: 14px;\"><\/td>\n<td style=\"width: 14.2857%; height: 14px;\">8<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">512<\/td>\n<td style=\"width: 14.2857%; height: 14px;\">6561<\/td>\n<\/tr>\n<tr style=\"height: 14px;\">\n<td style=\"width: 14.2857%; height: 14px;\" colspan=\"3\">Table 3. Exponential versus power growth<\/td>\n<td style=\"width: 14.2857%; height: 14px;\"><\/td>\n<td style=\"width: 14.2857%; height: 14px;\" colspan=\"3\">Table 4.\u00a0Exponential versus power growth<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Initial Value<\/h3>\n<p>The initial value of exponential growth occurs at [latex]x=0[\/latex].\u00a0So far we have considered only initial values of 1. Tables 1 and 2 show a pattern where [latex]y=1\\cdot 2^x[\/latex]. However, the initial value can be any real number.<\/p>\n<p>For example, let&#8217;s consider an initial value of 8 and an exponential growth rate of 2. We can create a table that illustrates this scenario starting with [latex]y=8[\/latex] when [latex]x=0[\/latex], then multiplying each [latex]y[\/latex]-value by 2:<\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 33.3333%;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 33.3333%;\">[latex]y[\/latex]<\/th>\n<th style=\"width: 33.3333%;\">Equation<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">-2<\/td>\n<td style=\"width: 33.3333%;\">[latex]4\\div 2=2[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]8\\cdot2^{-2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">-1<\/td>\n<td style=\"width: 33.3333%;\">[latex]8\\div 2=4[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]8\\cdot2^{-1}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">0<\/td>\n<td style=\"width: 33.3333%;\">[latex]8[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]y=8\\cdot2^0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">1<\/td>\n<td style=\"width: 33.3333%;\">[latex]8\\cdot 2=16[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]8\\cdot2^1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">2<\/td>\n<td style=\"width: 33.3333%;\">[latex]16\\cdot 2=32[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]8\\cdot2^2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 33.3333%;\">3<\/td>\n<td style=\"width: 33.3333%;\">[latex]32\\cdot 2=64[\/latex]<\/td>\n<td style=\"width: 33.3333%;\">[latex]8\\cdot2^3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The equation that models an exponential growth rate of 2 with an initial value of 8 is [latex]y=8\\cdot2^x[\/latex].<\/p>\n<div class=\"textbox shaded\">\n<h3>Exponential Growth<\/h3>\n<p>The equation that models an exponential growth rate of [latex]r[\/latex] with an initial value of [latex]a[\/latex] is [latex]y=a\\cdot r^x[\/latex].<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 3<\/h3>\n<p>Complete the table for an exponential growth of 4 and an initial value of 3.\u00a0 Then write an equation for the exponential growth pattern.<\/p>\n<table style=\"border-collapse: collapse; width: 0%; height: 166px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<th style=\"width: 52.6328%; text-align: right; height: 12px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 143.5%; height: 12px; text-align: left;\">[latex]y[\/latex]<\/th>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 52.6328%; text-align: right; height: 12px;\">[latex]\u20132[\/latex]<\/td>\n<td style=\"width: 143.5%; height: 12px;\"><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 52.6328%; text-align: right; height: 12px;\">[latex]\u20131[\/latex]<\/td>\n<td style=\"width: 143.5%; height: 12px;\"><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 52.6328%; text-align: right; height: 12px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 143.5%; height: 12px;\"><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 52.6328%; text-align: right; height: 12px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 143.5%; height: 12px;\"><\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 52.6328%; text-align: right; height: 12px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 143.5%; height: 12px;\"><\/td>\n<\/tr>\n<tr>\n<td style=\"width: 52.6328%; text-align: right;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 143.5%;\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Solution<\/h4>\n<p>The initial value of 3 occurs when [latex]x=0[\/latex] so we can add that to the table. Then we can multiply by 4 each time to determine the [latex]y[\/latex]-values for [latex]x=1,\\;2[\/latex]. To find the\u00a0[latex]y[\/latex]-values for [latex]x=-1,\\;-2[\/latex], we work backwards by dividing the initial value by 4.<\/p>\n<table style=\"border-collapse: collapse; width: 0%; height: 72px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<th style=\"width: 52.2644%; text-align: right; height: 12px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 47.7356%; height: 12px; text-align: left;\">[latex]y[\/latex]<\/th>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 52.2644%; text-align: right; height: 12px;\">[latex]\u20132[\/latex]<\/td>\n<td style=\"width: 47.7356%; height: 12px;\">[latex]3\u00f74\u00f74=\\dfrac{3}{16}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 52.2644%; text-align: right; height: 12px;\">[latex]\u20131[\/latex]<\/td>\n<td style=\"width: 47.7356%; height: 12px;\">[latex]3\u00f74=\\dfrac{3}{4}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 52.2644%; text-align: right; height: 12px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 47.7356%; height: 12px;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 52.2644%; text-align: right; height: 12px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 47.7356%; height: 12px;\">[latex]3\\cdot 4=12[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 52.2644%; text-align: right; height: 12px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 47.7356%; height: 12px;\">[latex]3\\cdot 4\\cdot 4=48[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 52.2644%; text-align: right;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 47.7356%;\">[latex]3\\cdot 4\\cdot 4\\cdot 4=192[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The pattern is [latex]y=3\\cdot 4^x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 3<\/h3>\n<p>Complete the table for an exponential growth of 3 and an initial value of 5.\u00a0 Then write an equation for the exponential growth pattern.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm585\">Show Answer<\/span><\/p>\n<div id=\"qhjm585\" class=\"hidden-answer\" style=\"display: none\">\n<table style=\"border-collapse: collapse; width: 0%; height: 72px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<th style=\"width: 46.5554%; text-align: right; height: 12px;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 53.4446%; text-align: right; height: 12px;\">[latex]y[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 46.5554%; text-align: right;\">[latex]\u20133[\/latex]<\/td>\n<td style=\"width: 53.4446%;\">[latex]\\dfrac{5}{27}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 46.5554%; text-align: right; height: 12px;\">[latex]\u20132[\/latex]<\/td>\n<td style=\"width: 53.4446%; height: 12px;\">[latex]\\dfrac{5}{9}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 46.5554%; text-align: right; height: 12px;\">[latex]\u20131[\/latex]<\/td>\n<td style=\"width: 53.4446%; height: 12px;\">[latex]\\dfrac{5}{3}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 46.5554%; text-align: right; height: 12px;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 53.4446%; height: 12px;\">[latex]5[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 46.5554%; text-align: right; height: 12px;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 53.4446%; height: 12px;\">[latex]15[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"width: 46.5554%; text-align: right; height: 12px;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 53.4446%; height: 12px;\">[latex]45[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The pattern is [latex]y=5\\cdot 3^x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 4<\/h3>\n<p>Write an equation for the following exponential growth patterns:<\/p>\n<ol>\n<li>growth = 6; initial value = 4<\/li>\n<li>growth = 2; initial value = 9<\/li>\n<li>growth = [latex]\\dfrac{2}{3}[\/latex]; initial value = 5<\/li>\n<li>growth = 0.77; initial value = \u20135.4<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm118\">Show Answer<\/span><\/p>\n<div id=\"qhjm118\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]y=4\\left(6^x\\right)[\/latex]<\/li>\n<li>[latex]y=9\\left(2^x\\right)[\/latex]<\/li>\n<li>[latex]y=5\\left(\\dfrac{2}{3}\\right)^x[\/latex]<\/li>\n<li>[latex]y=-5.4(0.77)^x[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<h2>Exponential Functions and Their Graphs<\/h2>\n<p>Exponential growth has an <strong><em>initial value<\/em><\/strong> and an <strong><em>exponential rate of change<\/em><\/strong>. <span style=\"color: #0000ff;\"><span style=\"color: #000000;\">The initial value occurs at [latex]x=0[\/latex].<\/span>\u00a0<\/span>In table 1, the initial value is 1 (when [latex]x=0[\/latex]), and the exponential rate of change is 2. This creates a pattern where [latex]y=1\\cdot 2^x[\/latex]. Consequently, the exponential growth in table 1 may be modeled or represented by the function [latex]f(x) = 2^x[\/latex].<\/p>\n<p>If we graph the values [latex](x, y)[\/latex] from table 1, we can then connect the points to draw the graph the exponential function [latex]f(x)=2^x[\/latex] (figure 1).<\/p>\n<div id=\"attachment_2730\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2730\" class=\"wp-image-2730 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-16T164852.907-300x300.png\" alt=\"Curve increasing gradually from left to right until reaching (0,1), then steeply increasing\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-16T164852.907-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-16T164852.907-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-16T164852.907-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-16T164852.907-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-16T164852.907-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-16T164852.907-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-16T164852.907.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-2730\" class=\"wp-caption-text\">Figure 1. The graph of the function [latex]f(x)=2^x[\/latex].<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example 4<\/h3>\n<p>Create a table of values then graph the function [latex]f(x)=\\left(\\dfrac{1}{2}\\right)^x[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>We can choose any [latex]x[\/latex]-values to create a table for [latex]y=f(x)[\/latex]:<\/p>\n<table style=\"border-collapse: collapse; width: 50%;\">\n<tbody>\n<tr>\n<th style=\"width: 15.8986%;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 26.4977%;\">[latex]y[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 15.8986%;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 26.4977%;\">[latex]\\left(\\dfrac{1}{2}\\right)^0=1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15.8986%;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 26.4977%;\">[latex]\\left(\\dfrac{1}{2}\\right)^1=\\dfrac{1}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15.8986%;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 26.4977%;\">[latex]\\left(\\dfrac{1}{2}\\right)^2=\\dfrac{1}{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15.8986%;\">[latex]-1[\/latex]<\/td>\n<td style=\"width: 26.4977%;\">[latex]\\left(\\dfrac{1}{2}\\right)^{-1}=2^1=2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15.8986%;\">[latex]-2[\/latex]<\/td>\n<td style=\"width: 26.4977%;\">[latex]\\left(\\dfrac{1}{2}\\right)^{-2}=2^2=4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 15.8986%;\">[latex]-3[\/latex]<\/td>\n<td style=\"width: 26.4977%;\">\u00a0[latex]\\left(\\dfrac{1}{2}\\right)^{-3}=2^3=8[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We plot the [latex](x,y)[\/latex] points from the table:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-2614\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/10184730\/desmos-graph-2022-06-10T124619.629-300x300.png\" alt=\"Plotted points\" width=\"300\" height=\"300\" \/><\/p>\n<p>Then join the points with a smooth curve:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2615 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T124636.600-300x300.png\" alt=\"Exponential graph passing through points, decreasing by half as it moves left to right\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T124636.600-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T124636.600-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T124636.600-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T124636.600-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T124636.600-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T124636.600-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T124636.600.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It 5<\/h3>\n<p>Create a table of values then graph the function [latex]f(x)=4^x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm469\">Show Answer<\/span><\/p>\n<div id=\"qhjm469\" class=\"hidden-answer\" style=\"display: none\">\n<table style=\"border-collapse: collapse; width: 24.2396%; height: 118px;\">\n<tbody>\n<tr>\n<th style=\"width: 48.1255%; text-align: right;\">[latex]x[\/latex]<\/th>\n<th style=\"width: 51.8745%;\">[latex]y[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 48.1255%; text-align: right;\">[latex]\u20133[\/latex]<\/td>\n<td style=\"width: 51.8745%;\">[latex]\\dfrac{1}{64}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 48.1255%; text-align: right;\">[latex]\u20132[\/latex]<\/td>\n<td style=\"width: 51.8745%;\">[latex]\\dfrac{1}{16}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 48.1255%; text-align: right;\">[latex]\u20131[\/latex]<\/td>\n<td style=\"width: 51.8745%;\">[latex]\\dfrac{1}{4}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 48.1255%; text-align: right;\">[latex]0[\/latex]<\/td>\n<td style=\"width: 51.8745%;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 48.1255%; text-align: right;\">[latex]1[\/latex]<\/td>\n<td style=\"width: 51.8745%;\">[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 48.1255%; text-align: right;\">[latex]2[\/latex]<\/td>\n<td style=\"width: 51.8745%;\">[latex]16[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 48.1255%; text-align: right;\">[latex]3[\/latex]<\/td>\n<td style=\"width: 51.8745%;\">[latex]64[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-2538\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5774\/2022\/05\/06180547\/desmos-graph-2022-06-06T120534.071-300x300.png\" alt=\"Exponential graph\" width=\"300\" height=\"300\" \/><\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>The Definition of an Exponential Function<\/h3>\n<p>An exponential function has the form\u00a0[latex]f(x) = r^x[\/latex], where [latex]r[\/latex] is a real number with [latex]r >0[\/latex] and [latex]r \\neq 1[\/latex].<\/p>\n<\/div>\n<p>Figure 2 illustrates how the graph changes as the value of [latex]r[\/latex] changes. Move the red circle up or down to change the value of [latex]r[\/latex] and watch what happens to the function.<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" style=\"border: 1px solid #ccc;\" title=\"Interactive graph\" src=\"https:\/\/www.desmos.com\/calculator\/jgj3wtjxxx?embed\" width=\"500\" height=\"500\" frameborder=\"0\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<p style=\"text-align: center;\">Figure 2. Interactive graph [latex]f(x)=r^x[\/latex]<\/p>\n<p>Manipulate the graph of [latex]f(x)=r^x[\/latex] in figure 2 to answer the following questions:<\/p>\n<p>1. What happens to the point (0, 1) as [latex]r[\/latex]changes?<\/p>\n<p style=\"padding-left: 30px;\">The point (0, 1) never changes. The point (0, 1) is always on the graph of [latex]f(x)=r^x[\/latex].<\/p>\n<p><span style=\"font-size: 1em;\">2. What happens to the point on the graph at [latex]x=1[\/latex] as\u00a0[latex]r[\/latex]changes?<\/span><\/p>\n<p style=\"padding-left: 30px;\">At\u00a0[latex]x=1[\/latex], the point on the graph will always be\u00a0[latex](1, r)[\/latex], because [latex]f(1)=r^1=r[\/latex].<\/p>\n<p>3. What happens to the graph when [latex]r=1[\/latex]?<\/p>\n<p style=\"padding-left: 30px;\"><span style=\"font-size: 1em;\">If [latex]r=1[\/latex] we get a flat line; a linear equation [latex]y=1[\/latex]. This is why [latex]r[\/latex] is never allowed to equal 1.<\/span><\/p>\n<p>4. What happens to the graph when [latex]r=0[\/latex]?<\/p>\n<p style=\"padding-left: 30px;\">If [latex]r=0[\/latex] we get a flat line starting at [latex]x>0[\/latex]. [latex]f(0)=0^0[\/latex] which is undefined.\u00a0 For any [latex]x<0[\/latex], [latex]0^{\\text{negative number}}=\\dfrac{1}{0^{\\text{positive number}}}=\\dfrac{1}{0}[\/latex], which is undefined. This is why [latex]r[\/latex] is never allowed to equal 0.<\/p>\n<p>5. What happens to the graph when [latex]r<0[\/latex]?\n\n\n<p style=\"padding-left: 30px;\">When [latex]r<0[\/latex], the graph disappears!! This is why [latex]r[\/latex] is a positive real number \u2260 0, 1.<\/p>\n<p>6. What happens to the graph when [latex]r>1[\/latex]?<\/p>\n<p style=\"padding-left: 30px;\">The graph comes up from [latex]y=0[\/latex], passes through (0, 1) and (1, r), then quickly moves towards [latex]+\\infty[\/latex].<\/p>\n<p>7. What happens to the graph when [latex]0<r<1[\/latex]?\n\n\n<p style=\"padding-left: 30px;\">When [latex]r[\/latex] is a proper fraction, the graph comes in from\u00a0[latex]+\\infty[\/latex], passes through (0, 1) and (1, r), then moves slowly towards\u00a0[latex]y=0[\/latex].<\/p>\n<h2>The Asymptote and Intercepts<\/h2>\n<p>A significant feature on the graph of any exponential function\u00a0[latex]f(x) = r^x[\/latex] ([latex]r>0,\\;r\\neq1[\/latex]) is that the graph never crosses the [latex]x[\/latex]-axis. It continually approaches the [latex]x[\/latex], getting closer and closer, but the graph never meets the\u00a0[latex]x[\/latex]-axis. Figure 2 illustrates that\u00a0<span style=\"font-size: 1em;\">when [latex]0<r<1,[\/latex] the graph gets closer and closer to the [latex]x[\/latex]-axis when [latex]x[\/latex] gets closer and closer to positive infinity (figure 3).\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">However,\u00a0<\/span><span style=\"font-size: 1em;\">when [latex]r>1[\/latex], the graph gets closer and closer to the [latex]x[\/latex]-axis when [latex]x[\/latex] gets closer and closer to negative infinity (figure 4).<\/span><\/p>\n<table style=\"border-collapse: collapse; width: 100%;\">\n<tbody>\n<tr>\n<th style=\"width: 50%;\" colspan=\"2\">Graphs with horizontal asymptotes<\/th>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2620 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T131850.316-300x300.png\" alt=\"r^x with 0, a decreasing exponential graph passing through (0,1)\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T131850.316-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T131850.316-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T131850.316-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T131850.316-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T131850.316-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T131850.316-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T131850.316.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/td>\n<td style=\"width: 50%;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2619 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T131830.3471-300x300.png\" alt=\"r^x with 0&lt;r&lt;1, an exponentially increasing curve passing through (0,1)\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T131830.3471-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T131830.3471-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T131830.3471-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T131830.3471-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T131830.3471-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T131830.3471-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-10T131830.3471.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/td>\n<\/tr>\n<tr>\n<td class=\"shaded\" style=\"width: 50%; text-align: center;\">Figure 3. [latex]f(x)=r^x[\/latex] with [latex]0<r<1[\/latex]<\/td>\n<td class=\"shaded\" style=\"width: 50%; text-align: center;\">Figure 4. [latex]f(x)=r^x[\/latex] with [latex]r>1[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Why does the graph never meet the [latex]x[\/latex]-axis? Consider the following examples using the function [latex]f(x)=2^x[\/latex] that was graphed in figure 1:<\/p>\n<p style=\"text-align: center;\">[latex]f(-10)=2^{-10}=\\dfrac{1}{2^{10}}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]f(-100)=2^{-100}=\\dfrac{1}{2^{100}}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]f(-100000)=2^{-100000}=\\dfrac{1}{2^{100000}}[\/latex]<\/p>\n<p>As the the value of [latex]x[\/latex] gets closer to negative infinity, the value of the function [latex]y[\/latex] is a fraction with a numerator of 1 and a denominator that is a very large positive number. The value of [latex]x[\/latex] gets more and more negative as it gets closer to negative infinity, so the value of the function will get smaller and smaller. It will get close to zero but will never be zero because [latex]\\dfrac{1}{\\text{very large positive number}}[\/latex] is always positive and therefore greater than zero.<\/p>\n<p>Figure 2 shows that for all values of [latex]r>0[\/latex] and [latex]r\\neq1[\/latex], the graph gets close to but never crosses the [latex]x[\/latex]-axis, it is a <em><strong>horizontal asymptote<\/strong><\/em> of the function [latex]f(x)=r^x[\/latex]. Also, s<span style=\"font-size: 1rem; text-align: initial;\">ince the graph never meets the\u00a0[latex]x[\/latex]-axis, there is no\u00a0[latex]x[\/latex]-intercept for the function. The\u00a0[latex]y[\/latex]-intercept of the function\u00a0[latex]f(x)=r^x[\/latex] is always (0, 1).<\/span><\/p>\n<p>We use a dotted line to show that a graph has a horizontal asymptote (figure 5).<\/p>\n<div id=\"attachment_2732\" style=\"width: 310px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2732\" class=\"wp-image-2732 size-medium\" src=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-16T171836.699-300x300.png\" alt=\"Exponential function with asymptote at y=0\" width=\"300\" height=\"300\" srcset=\"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-16T171836.699-300x300.png 300w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-16T171836.699-150x150.png 150w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-16T171836.699-768x768.png 768w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-16T171836.699-65x65.png 65w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-16T171836.699-225x225.png 225w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-16T171836.699-350x350.png 350w, https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-content\/uploads\/sites\/5774\/2022\/05\/desmos-graph-2022-06-16T171836.699.png 800w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p id=\"caption-attachment-2732\" class=\"wp-caption-text\">Figure 5. Exponential function with horizontal asymptote.<\/p>\n<\/div>\n<h2>Domain and Range<\/h2>\n<p>Figure 5 shows the graph of [latex]f(x)=2^x[\/latex]. The domain of the function is the set of all possible [latex]x[\/latex]-values, so domain = [latex]\\{x\\;|\\;x\\in \\mathbb{R}\\}[\/latex]. Any [latex]x[\/latex]-value from [latex]-\\infty[\/latex] to\u00a0[latex]+\\infty[\/latex] has a corresponding function value. It&#8217;s range, the set of all function values, lies above the line [latex]y=0[\/latex]. Consequently, the range =[latex]\\{f(x)\\;|\\;f(x)\\in\\mathbb{R}^+\\}[\/latex], where\u00a0[latex]\\mathbb{R}^+[\/latex] is the set of all positive real numbers.<\/p>\n<div><\/div>\n<div class=\"textbox shaded\">\n<h3>DOMAIN and RANGE<\/h3>\n<p>The domain of any exponential function [latex]f(x)=r^x[\/latex] is all real numbers, or [latex]\\{x | x \\in \\mathbb{R}\\}[\/latex], or [latex](-\\infty, \\infty)[\/latex]. The range of any exponential function[latex]f(x)=r^x[\/latex] is all real numbers that are above the horizontal asymptote. Range = [latex]\\{f(x)\\;|\\;f(x)\\in\\mathbb{R}^+\\}[\/latex], or [latex](0, \\infty)[\/latex].<\/p>\n<\/div>\n<p>The exponential function\u00a0[latex]f(x)=r^x[\/latex] is the parent function of all exponential functions. In the next section, we will see what happens to the graph of the function when we transform the parent function.<\/p>\n<p><script>\nwindow.embeddedChatbotConfig = {\nchatbotId: \"ejVb5sgc1-w972OOCgl5x\",\ndomain: \"www.chatbase.co\"\n}\n<\/script><br \/>\n<script src=\"https:\/\/www.chatbase.co\/embed.min.js\" defer=\"defer\">\n<\/script><\/p>\n<p><iframe style=\"height: 100%; min-height: 700px;\" src=\"https:\/\/www.chatbase.co\/chatbot-iframe\/ejVb5sgc1-w972OOCgl5x\" width=\"100%\" frameborder=\"0\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-2121\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Exponential functions and Their Graphs. <strong>Authored by<\/strong>: Hazel McKenna and Leo Chang. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>All graphs created using desmos graphing calculator. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>All examples and Try its: hjm585; hjm469; hjm118. <strong>Authored by<\/strong>: Hazel McKenna. <strong>Provided by<\/strong>: Utah Valley University. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":422608,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Exponential functions and Their Graphs\",\"author\":\"Hazel McKenna and Leo Chang\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All graphs created using desmos graphing calculator\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"All examples and Try its: hjm585; hjm469; hjm118\",\"author\":\"Hazel McKenna\",\"organization\":\"Utah Valley University\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2121","chapter","type-chapter","status-publish","hentry"],"part":2116,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2121","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/users\/422608"}],"version-history":[{"count":72,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2121\/revisions"}],"predecessor-version":[{"id":4757,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2121\/revisions\/4757"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/parts\/2116"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapters\/2121\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/media?parent=2121"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2121"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/contributor?post=2121"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-combinedalgebra\/wp-json\/wp\/v2\/license?post=2121"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}