{"id":903,"date":"2021-09-25T21:11:53","date_gmt":"2021-09-25T21:11:53","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&#038;p=903"},"modified":"2021-12-06T23:56:20","modified_gmt":"2021-12-06T23:56:20","slug":"6-4-1-rate-of-change","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/6-4-1-rate-of-change\/","title":{"raw":"6.4.1: Linear Rate of Change","rendered":"6.4.1: Linear Rate of Change"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Determine the rate of change between two data points.<\/li>\r\n \t<li>Determine if the rate of change is linear.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Key words<\/h3>\r\n<ul>\r\n \t<li><strong>Rate of change<\/strong>:\u00a0the ratio of the change in one variable with respect to the change in another variable<\/li>\r\n \t<li><strong>Linear rate of change<\/strong>: a constant rate of change between any two data points<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Linear Rate of Change<\/h2>\r\n<em><strong>Rate of change<\/strong><\/em> is defined as the ratio of the change in one variable with respect to the change in another variable. The following table shows values of two variables,\u00a0 [latex]x[\/latex] and [latex]y[\/latex].\r\n<table style=\"border-collapse: collapse; width: 50%; height: 30px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 16px;\">\r\n<th class=\"border\" style=\"width: 3.05192%; height: 16px;\">[latex]x[\/latex]<\/th>\r\n<th class=\"border\" style=\"width: 4.64221%; height: 16px;\">-2<\/th>\r\n<th class=\"border\" style=\"width: 4.82929%; height: 16px;\">-1<\/th>\r\n<th class=\"border\" style=\"width: 4.73572%; height: 16px;\">0<\/th>\r\n<th class=\"border\" style=\"width: 4.08094%; height: 16px;\">1<\/th>\r\n<th class=\"border\" style=\"width: 4.45514%; height: 16px;\">2<\/th>\r\n<th class=\"border\" style=\"width: 4.26804%; height: 16px;\">3<\/th>\r\n<th class=\"border\" style=\"width: 4.08096%; height: 16px;\">4<\/th>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<th class=\"border\" style=\"width: 3.05192%; height: 14px;\">[latex]y[\/latex]<\/th>\r\n<th class=\"border\" style=\"width: 4.64221%; height: 14px;\">-7<\/th>\r\n<th class=\"border\" style=\"width: 4.82929%; height: 14px;\">-5<\/th>\r\n<th class=\"border\" style=\"width: 4.73572%; height: 14px;\">-3<\/th>\r\n<th class=\"border\" style=\"width: 4.08094%; height: 14px;\">-1<\/th>\r\n<th class=\"border\" style=\"width: 4.45514%; height: 14px;\">1<\/th>\r\n<th class=\"border\" style=\"width: 4.26804%; height: 14px;\">3<\/th>\r\n<th class=\"border\" style=\"width: 4.08096%; height: 14px;\">5<\/th>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\nNotice that as the values of [latex]x[\/latex] increase so do the values of [latex]y[\/latex]. As the value of\u00a0[latex]x[\/latex] increases by 1 unit, the value of [latex]y[\/latex] increases by 2 units. The ratio of change in\u00a0[latex]y[\/latex] per change in [latex]x =\\frac{\\text{change in }y}{\\text{change in } x} = \\frac{2}{1} = 2[\/latex]. This rate of change is constant. That is what makes it linear.\r\n\r\n<em><strong>Linear rate of change<\/strong><\/em> means the rate of change ratio for any two data points is always the same. To illustrate, let's use the values in the following table to find the rate of change for any two points. For example, for the two data points (-2, -7) and (1, -1), the rate of change ratio of change in [latex]y[\/latex] over change in [latex]x[\/latex] is:\r\n<p style=\"text-align: center;\">[latex] \\frac{\\text{change in }y}{\\text{change in }x} = \\frac{-1-(-7)}{1-(-2)} = \\frac{6}{3} = 2[\/latex].<\/p>\r\nFor another two data points (12, 21) and (25, 47), the rate of change ratio of change in\u00a0[latex]y[\/latex] over change in\u00a0[latex]x[\/latex] is:\r\n<p style=\"text-align: center;\">[latex]\\frac{\\text{change in }y}{\\text{change in }x} = \\frac{47-21}{25-12} = \\frac{26}{13} = 2[\/latex]<\/p>\r\n\r\n<table style=\"border-collapse: collapse; width: 41.16%;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 16px;\">\r\n<th class=\"border\" style=\"width: 3.05192%; height: 16px; text-align: center;\">[latex]x[\/latex]<\/th>\r\n<th class=\"border\" style=\"width: 5.2035%; text-align: center;\">-2<\/th>\r\n<th class=\"border\" style=\"width: 5.10991%; text-align: center;\">1<\/th>\r\n<th class=\"border\" style=\"width: 5.6712%; text-align: center;\">4<\/th>\r\n<th class=\"border\" style=\"width: 5.10992%; text-align: center;\">6<\/th>\r\n<th class=\"border\" style=\"width: 5.57766%; text-align: center;\">12<\/th>\r\n<th class=\"border\" style=\"width: 5.95178%; text-align: center;\">25<\/th>\r\n<th class=\"border\" style=\"width: 5.48408%; text-align: center;\">35<\/th>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<th class=\"border\" style=\"width: 3.05192%; height: 14px; text-align: center;\">[latex]y[\/latex]<\/th>\r\n<th class=\"border\" style=\"width: 5.2035%; text-align: center;\">-7<\/th>\r\n<th class=\"border\" style=\"width: 5.10991%; text-align: center;\">-1<\/th>\r\n<th class=\"border\" style=\"width: 5.6712%; text-align: center;\">5<\/th>\r\n<th class=\"border\" style=\"width: 5.10992%; text-align: center;\">9<\/th>\r\n<th class=\"border\" style=\"width: 5.57766%; text-align: center;\">21<\/th>\r\n<th class=\"border\" style=\"width: 5.95178%; text-align: center;\">47<\/th>\r\n<th class=\"border\" style=\"width: 5.48408%; text-align: center;\">67<\/th>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe table has a linear rate of change ratio because the ratio is always 2 between any two data points.\r\n<div class=\"textbox shaded\">\r\n<h3>Rate of change<\/h3>\r\n<p style=\"text-align: center;\">Rate of Change = [latex] \\frac{y_{2}-y_{1}}{x_{2}-x_{1}} [\/latex],\u00a0where\u00a0 [latex]\\left ( x_1, y_1\\right )[\/latex] and [latex]\\left (x_2, y_2\\right )[\/latex] are\u00a0 two data points.<\/p>\r\n&nbsp;\r\n<p style=\"text-align: center;\">The rate of change is linear when it is constant for any two data points.<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nDetermine if the data in the table represents a linear rate of change.\r\n<table style=\"border-collapse: collapse; width: 26.3742%; height: 41px;\" border=\"1\">\r\n<tbody>\r\n<tr class=\"border\" style=\"height: 16px;\">\r\n<th class=\"border\" style=\"width: 8.53293%; height: 16px; text-align: center;\">[latex]x[\/latex]<\/th>\r\n<th class=\"border\" style=\"width: 12.026%; text-align: center;\">-3<\/th>\r\n<th class=\"border\" style=\"width: 13.6227%; text-align: center;\">-1<\/th>\r\n<th class=\"border\" style=\"width: 11.2525%; text-align: center;\">1<\/th>\r\n<th class=\"border\" style=\"width: 14.0469%; text-align: center;\">3<\/th>\r\n<\/tr>\r\n<tr class=\"border\" style=\"height: 14px;\">\r\n<th class=\"border\" style=\"width: 8.53293%; height: 14px; text-align: center;\">[latex]y[\/latex]<\/th>\r\n<th class=\"border\" style=\"width: 12.026%; text-align: center;\">-4<\/th>\r\n<th class=\"border\" style=\"width: 13.6227%; text-align: center;\">0<\/th>\r\n<th class=\"border\" style=\"width: 11.2525%; text-align: center;\">4<\/th>\r\n<th class=\"border\" style=\"width: 14.0469%; text-align: center;\">8<\/th>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Solution<\/h4>\r\nMoving from left to right,\u00a0[latex]x[\/latex] increases by 2 units between cells, and\u00a0[latex]y[\/latex] increases by 4 units between cells.\r\n\r\nThe rate of change between points is constant at [latex]\\frac{2}{4}=\\frac{1}{2}[\/latex], which is a linear rate of change.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nDetermine if the data in the table represents a linear rate of change.\r\n<table style=\"border-collapse: collapse; width: 26.3742%; height: 41px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 16px;\">\r\n<th class=\"border\" style=\"width: 8.53293%; height: 16px; text-align: center;\">[latex]x[\/latex]<\/th>\r\n<th class=\"border\" style=\"width: 12.026%; text-align: center;\">-5<\/th>\r\n<th class=\"border\" style=\"width: 13.6227%; text-align: center;\">-1<\/th>\r\n<th class=\"border\" style=\"width: 11.2525%; text-align: center;\">3<\/th>\r\n<th class=\"border\" style=\"width: 14.0469%; text-align: center;\">7<\/th>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<th class=\"border\" style=\"width: 8.53293%; height: 14px; text-align: center;\">[latex]y[\/latex]<\/th>\r\n<th class=\"border\" style=\"width: 12.026%; text-align: center;\">2<\/th>\r\n<th class=\"border\" style=\"width: 13.6227%; text-align: center;\">0<\/th>\r\n<th class=\"border\" style=\"width: 11.2525%; text-align: center;\">-4<\/th>\r\n<th class=\"border\" style=\"width: 14.0469%; text-align: center;\">-8<\/th>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h4>Solution<\/h4>\r\nMoving from left to right,\u00a0[latex]x[\/latex] increases by 4 units between cells, but [latex]y[\/latex] decreases by 2 units, then 4 units, then 4 units between cells.\r\n\r\nThe rate of change between points is not constant so this is not a linear rate of change.\r\n\r\n<\/div>\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nDetermine if the data in the table represents a linear rate of change.\r\n<table style=\"border-collapse: collapse; width: 26.3742%; height: 41px;\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 16px;\">\r\n<th class=\"border\" style=\"width: 8.53293%; height: 16px;\">[latex]x[\/latex]<\/th>\r\n<th class=\"border\" style=\"width: 12.026%; text-align: center;\">1<\/th>\r\n<th class=\"border\" style=\"width: 13.6227%; text-align: center;\">5<\/th>\r\n<th class=\"border\" style=\"width: 11.2525%; text-align: center;\">9<\/th>\r\n<th class=\"border\" style=\"width: 13.0489%; text-align: center;\">11<\/th>\r\n<\/tr>\r\n<tr style=\"height: 14px;\">\r\n<th class=\"border\" style=\"width: 8.53293%; height: 14px;\">[latex]y[\/latex]<\/th>\r\n<th class=\"border\" style=\"width: 12.026%; text-align: center;\">2<\/th>\r\n<th class=\"border\" style=\"width: 13.6227%; text-align: center;\">4<\/th>\r\n<th class=\"border\" style=\"width: 11.2525%; text-align: center;\">6<\/th>\r\n<th class=\"border\" style=\"width: 13.0489%; text-align: center;\">7<\/th>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"hjm590\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm590\"]\r\n\r\nThe rate of change is linear. Rate of change [latex]=\\frac{1}{2}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nFind the rate of change between the points [latex]\\left (4, -7\\right )[\/latex] and [latex]\\left (-3, 4\\right )[\/latex].\r\n<h4>Solution<\/h4>\r\nIf we let [latex]\\left (4, -7\\right )=\\left ( x_1, y_1\\right )[\/latex] and\u00a0[latex]\\left (-3, 4\\right )=\\left ( x_2, y_2\\right )[\/latex],\r\n\r\n[latex]\\begin{equation}\\begin{aligned}\\text{Rate of Change} &amp; =\\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\\\ &amp; = \\frac{4-(-7)}{-3-4} \\\\ &amp; = \\frac{11}{-7} \\\\ &amp; = -\\frac{11}{7} \\end{aligned}\\end{equation}[\/latex]\r\n\r\n&nbsp;\r\n\r\nOn the other hand, if we let [latex]\\left (-3, 4\\right )=\\left ( x_1, y_1\\right )[\/latex] and\u00a0[latex]\\left (4, -7\\right )=\\left ( x_2, y_2\\right )[\/latex],\r\n\r\n[latex]\\begin{equation}\\begin{aligned}\\text{Rate of Change} &amp; =\\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\\\ &amp; = \\frac{-7-4}{4-(-3)} \\\\ &amp; = \\frac{-11}{7} \\\\ &amp; = -\\frac{11}{7} \\end{aligned}\\end{equation}[\/latex]\r\n\r\n<\/div>\r\nNotice that we get the same rate of change irrespective of the designation of the points.\r\n<div class=\"textbox tryit\">\r\n<h3>Try It<\/h3>\r\nFind the rate of change between the points [latex]\\left (0, -3\\right )[\/latex] and [latex]\\left (5, -4\\right )[\/latex].\r\n\r\n[reveal-answer q=\"hjm842\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"hjm842\"]\r\n\r\n[latex]-\\frac{1}{5}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Determine the rate of change between two data points.<\/li>\n<li>Determine if the rate of change is linear.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key words<\/h3>\n<ul>\n<li><strong>Rate of change<\/strong>:\u00a0the ratio of the change in one variable with respect to the change in another variable<\/li>\n<li><strong>Linear rate of change<\/strong>: a constant rate of change between any two data points<\/li>\n<\/ul>\n<\/div>\n<h2>Linear Rate of Change<\/h2>\n<p><em><strong>Rate of change<\/strong><\/em> is defined as the ratio of the change in one variable with respect to the change in another variable. The following table shows values of two variables,\u00a0 [latex]x[\/latex] and [latex]y[\/latex].<\/p>\n<table style=\"border-collapse: collapse; width: 50%; height: 30px;\">\n<tbody>\n<tr style=\"height: 16px;\">\n<th class=\"border\" style=\"width: 3.05192%; height: 16px;\">[latex]x[\/latex]<\/th>\n<th class=\"border\" style=\"width: 4.64221%; height: 16px;\">-2<\/th>\n<th class=\"border\" style=\"width: 4.82929%; height: 16px;\">-1<\/th>\n<th class=\"border\" style=\"width: 4.73572%; height: 16px;\">0<\/th>\n<th class=\"border\" style=\"width: 4.08094%; height: 16px;\">1<\/th>\n<th class=\"border\" style=\"width: 4.45514%; height: 16px;\">2<\/th>\n<th class=\"border\" style=\"width: 4.26804%; height: 16px;\">3<\/th>\n<th class=\"border\" style=\"width: 4.08096%; height: 16px;\">4<\/th>\n<\/tr>\n<tr style=\"height: 14px;\">\n<th class=\"border\" style=\"width: 3.05192%; height: 14px;\">[latex]y[\/latex]<\/th>\n<th class=\"border\" style=\"width: 4.64221%; height: 14px;\">-7<\/th>\n<th class=\"border\" style=\"width: 4.82929%; height: 14px;\">-5<\/th>\n<th class=\"border\" style=\"width: 4.73572%; height: 14px;\">-3<\/th>\n<th class=\"border\" style=\"width: 4.08094%; height: 14px;\">-1<\/th>\n<th class=\"border\" style=\"width: 4.45514%; height: 14px;\">1<\/th>\n<th class=\"border\" style=\"width: 4.26804%; height: 14px;\">3<\/th>\n<th class=\"border\" style=\"width: 4.08096%; height: 14px;\">5<\/th>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>Notice that as the values of [latex]x[\/latex] increase so do the values of [latex]y[\/latex]. As the value of\u00a0[latex]x[\/latex] increases by 1 unit, the value of [latex]y[\/latex] increases by 2 units. The ratio of change in\u00a0[latex]y[\/latex] per change in [latex]x =\\frac{\\text{change in }y}{\\text{change in } x} = \\frac{2}{1} = 2[\/latex]. This rate of change is constant. That is what makes it linear.<\/p>\n<p><em><strong>Linear rate of change<\/strong><\/em> means the rate of change ratio for any two data points is always the same. To illustrate, let&#8217;s use the values in the following table to find the rate of change for any two points. For example, for the two data points (-2, -7) and (1, -1), the rate of change ratio of change in [latex]y[\/latex] over change in [latex]x[\/latex] is:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\text{change in }y}{\\text{change in }x} = \\frac{-1-(-7)}{1-(-2)} = \\frac{6}{3} = 2[\/latex].<\/p>\n<p>For another two data points (12, 21) and (25, 47), the rate of change ratio of change in\u00a0[latex]y[\/latex] over change in\u00a0[latex]x[\/latex] is:<\/p>\n<p style=\"text-align: center;\">[latex]\\frac{\\text{change in }y}{\\text{change in }x} = \\frac{47-21}{25-12} = \\frac{26}{13} = 2[\/latex]<\/p>\n<table style=\"border-collapse: collapse; width: 41.16%;\">\n<tbody>\n<tr style=\"height: 16px;\">\n<th class=\"border\" style=\"width: 3.05192%; height: 16px; text-align: center;\">[latex]x[\/latex]<\/th>\n<th class=\"border\" style=\"width: 5.2035%; text-align: center;\">-2<\/th>\n<th class=\"border\" style=\"width: 5.10991%; text-align: center;\">1<\/th>\n<th class=\"border\" style=\"width: 5.6712%; text-align: center;\">4<\/th>\n<th class=\"border\" style=\"width: 5.10992%; text-align: center;\">6<\/th>\n<th class=\"border\" style=\"width: 5.57766%; text-align: center;\">12<\/th>\n<th class=\"border\" style=\"width: 5.95178%; text-align: center;\">25<\/th>\n<th class=\"border\" style=\"width: 5.48408%; text-align: center;\">35<\/th>\n<\/tr>\n<tr style=\"height: 14px;\">\n<th class=\"border\" style=\"width: 3.05192%; height: 14px; text-align: center;\">[latex]y[\/latex]<\/th>\n<th class=\"border\" style=\"width: 5.2035%; text-align: center;\">-7<\/th>\n<th class=\"border\" style=\"width: 5.10991%; text-align: center;\">-1<\/th>\n<th class=\"border\" style=\"width: 5.6712%; text-align: center;\">5<\/th>\n<th class=\"border\" style=\"width: 5.10992%; text-align: center;\">9<\/th>\n<th class=\"border\" style=\"width: 5.57766%; text-align: center;\">21<\/th>\n<th class=\"border\" style=\"width: 5.95178%; text-align: center;\">47<\/th>\n<th class=\"border\" style=\"width: 5.48408%; text-align: center;\">67<\/th>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The table has a linear rate of change ratio because the ratio is always 2 between any two data points.<\/p>\n<div class=\"textbox shaded\">\n<h3>Rate of change<\/h3>\n<p style=\"text-align: center;\">Rate of Change = [latex]\\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[\/latex],\u00a0where\u00a0 [latex]\\left ( x_1, y_1\\right )[\/latex] and [latex]\\left (x_2, y_2\\right )[\/latex] are\u00a0 two data points.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">The rate of change is linear when it is constant for any two data points.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Determine if the data in the table represents a linear rate of change.<\/p>\n<table style=\"border-collapse: collapse; width: 26.3742%; height: 41px;\">\n<tbody>\n<tr class=\"border\" style=\"height: 16px;\">\n<th class=\"border\" style=\"width: 8.53293%; height: 16px; text-align: center;\">[latex]x[\/latex]<\/th>\n<th class=\"border\" style=\"width: 12.026%; text-align: center;\">-3<\/th>\n<th class=\"border\" style=\"width: 13.6227%; text-align: center;\">-1<\/th>\n<th class=\"border\" style=\"width: 11.2525%; text-align: center;\">1<\/th>\n<th class=\"border\" style=\"width: 14.0469%; text-align: center;\">3<\/th>\n<\/tr>\n<tr class=\"border\" style=\"height: 14px;\">\n<th class=\"border\" style=\"width: 8.53293%; height: 14px; text-align: center;\">[latex]y[\/latex]<\/th>\n<th class=\"border\" style=\"width: 12.026%; text-align: center;\">-4<\/th>\n<th class=\"border\" style=\"width: 13.6227%; text-align: center;\">0<\/th>\n<th class=\"border\" style=\"width: 11.2525%; text-align: center;\">4<\/th>\n<th class=\"border\" style=\"width: 14.0469%; text-align: center;\">8<\/th>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Solution<\/h4>\n<p>Moving from left to right,\u00a0[latex]x[\/latex] increases by 2 units between cells, and\u00a0[latex]y[\/latex] increases by 4 units between cells.<\/p>\n<p>The rate of change between points is constant at [latex]\\frac{2}{4}=\\frac{1}{2}[\/latex], which is a linear rate of change.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Determine if the data in the table represents a linear rate of change.<\/p>\n<table style=\"border-collapse: collapse; width: 26.3742%; height: 41px;\">\n<tbody>\n<tr style=\"height: 16px;\">\n<th class=\"border\" style=\"width: 8.53293%; height: 16px; text-align: center;\">[latex]x[\/latex]<\/th>\n<th class=\"border\" style=\"width: 12.026%; text-align: center;\">-5<\/th>\n<th class=\"border\" style=\"width: 13.6227%; text-align: center;\">-1<\/th>\n<th class=\"border\" style=\"width: 11.2525%; text-align: center;\">3<\/th>\n<th class=\"border\" style=\"width: 14.0469%; text-align: center;\">7<\/th>\n<\/tr>\n<tr style=\"height: 14px;\">\n<th class=\"border\" style=\"width: 8.53293%; height: 14px; text-align: center;\">[latex]y[\/latex]<\/th>\n<th class=\"border\" style=\"width: 12.026%; text-align: center;\">2<\/th>\n<th class=\"border\" style=\"width: 13.6227%; text-align: center;\">0<\/th>\n<th class=\"border\" style=\"width: 11.2525%; text-align: center;\">-4<\/th>\n<th class=\"border\" style=\"width: 14.0469%; text-align: center;\">-8<\/th>\n<\/tr>\n<\/tbody>\n<\/table>\n<h4>Solution<\/h4>\n<p>Moving from left to right,\u00a0[latex]x[\/latex] increases by 4 units between cells, but [latex]y[\/latex] decreases by 2 units, then 4 units, then 4 units between cells.<\/p>\n<p>The rate of change between points is not constant so this is not a linear rate of change.<\/p>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Determine if the data in the table represents a linear rate of change.<\/p>\n<table style=\"border-collapse: collapse; width: 26.3742%; height: 41px;\">\n<tbody>\n<tr style=\"height: 16px;\">\n<th class=\"border\" style=\"width: 8.53293%; height: 16px;\">[latex]x[\/latex]<\/th>\n<th class=\"border\" style=\"width: 12.026%; text-align: center;\">1<\/th>\n<th class=\"border\" style=\"width: 13.6227%; text-align: center;\">5<\/th>\n<th class=\"border\" style=\"width: 11.2525%; text-align: center;\">9<\/th>\n<th class=\"border\" style=\"width: 13.0489%; text-align: center;\">11<\/th>\n<\/tr>\n<tr style=\"height: 14px;\">\n<th class=\"border\" style=\"width: 8.53293%; height: 14px;\">[latex]y[\/latex]<\/th>\n<th class=\"border\" style=\"width: 12.026%; text-align: center;\">2<\/th>\n<th class=\"border\" style=\"width: 13.6227%; text-align: center;\">4<\/th>\n<th class=\"border\" style=\"width: 11.2525%; text-align: center;\">6<\/th>\n<th class=\"border\" style=\"width: 13.0489%; text-align: center;\">7<\/th>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm590\">Show Answer<\/span><\/p>\n<div id=\"qhjm590\" class=\"hidden-answer\" style=\"display: none\">\n<p>The rate of change is linear. Rate of change [latex]=\\frac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Find the rate of change between the points [latex]\\left (4, -7\\right )[\/latex] and [latex]\\left (-3, 4\\right )[\/latex].<\/p>\n<h4>Solution<\/h4>\n<p>If we let [latex]\\left (4, -7\\right )=\\left ( x_1, y_1\\right )[\/latex] and\u00a0[latex]\\left (-3, 4\\right )=\\left ( x_2, y_2\\right )[\/latex],<\/p>\n<p>[latex]\\begin{equation}\\begin{aligned}\\text{Rate of Change} & =\\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\\\ & = \\frac{4-(-7)}{-3-4} \\\\ & = \\frac{11}{-7} \\\\ & = -\\frac{11}{7} \\end{aligned}\\end{equation}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>On the other hand, if we let [latex]\\left (-3, 4\\right )=\\left ( x_1, y_1\\right )[\/latex] and\u00a0[latex]\\left (4, -7\\right )=\\left ( x_2, y_2\\right )[\/latex],<\/p>\n<p>[latex]\\begin{equation}\\begin{aligned}\\text{Rate of Change} & =\\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\\\ & = \\frac{-7-4}{4-(-3)} \\\\ & = \\frac{-11}{7} \\\\ & = -\\frac{11}{7} \\end{aligned}\\end{equation}[\/latex]<\/p>\n<\/div>\n<p>Notice that we get the same rate of change irrespective of the designation of the points.<\/p>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<p>Find the rate of change between the points [latex]\\left (0, -3\\right )[\/latex] and [latex]\\left (5, -4\\right )[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qhjm842\">Show Answer<\/span><\/p>\n<div id=\"qhjm842\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]-\\frac{1}{5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-903\">\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>Linear rate of change. <strong>Authored by<\/strong>: Leo Chang and 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":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Linear rate of change\",\"author\":\"Leo Chang and 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-903","chapter","type-chapter","status-publish","hentry"],"part":659,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/903","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/users\/422608"}],"version-history":[{"count":15,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/903\/revisions"}],"predecessor-version":[{"id":1958,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/903\/revisions\/1958"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/parts\/659"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapters\/903\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/media?parent=903"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=903"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/contributor?post=903"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/wp-json\/wp\/v2\/license?post=903"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}