{"id":867,"date":"2015-11-12T18:37:58","date_gmt":"2015-11-12T18:37:58","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=867"},"modified":"2017-03-31T20:11:59","modified_gmt":"2017-03-31T20:11:59","slug":"find-the-average-rate-of-change-of-a-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/find-the-average-rate-of-change-of-a-function\/","title":{"raw":"Find the average rate of change of a function","rendered":"Find the average rate of change of a function"},"content":{"raw":"<\/colgroup>\r\n\r\n\r\n\r\n
[latex]y[\/latex]<\/strong><\/td>\r\n2005<\/td>\r\n2006<\/td>\r\n2007<\/td>\r\n2008<\/td>\r\n2009<\/td>\r\n2010<\/td>\r\n2011<\/td>\r\n2012<\/td>\r\n<\/tr>\r\n
[latex]C\\left(y\\right)[\/latex]<\/strong><\/td>\r\n2.31<\/td>\r\n2.62<\/td>\r\n2.84<\/td>\r\n3.30<\/td>\r\n2.41<\/td>\r\n2.84<\/td>\r\n3.58<\/td>\r\n3.68<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

The price change per year is a rate of change<\/strong> because it describes how an output quantity changes relative to the change in the input quantity. We can see that the price of gasoline in\u00a0the table above\u00a0did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the average rate of change<\/strong> over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.<\/p>\r\n\r\n

Average rate of change=[latex]\\frac{\\text{Change in output}}{\\text{Change in input}}[\/latex]<\/div>\r\n
=[latex]\\frac{\\Delta y}{\\Delta x}[\/latex]<\/div>\r\n
=[latex]\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\r\n
=[latex]\\frac{f\\left({x}_{2}\\right)-f\\left({x}_{1}\\right)}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\r\n

The Greek letter [latex]\\Delta [\/latex] (delta) signifies the change in a quantity; we read the ratio as \"delta-y<\/em> over delta-x<\/em>\" or \"the change in [latex]y[\/latex] divided by the change in [latex]x[\/latex].\" Occasionally we write [latex]\\Delta f[\/latex] instead of [latex]\\Delta y[\/latex], which still represents the change in the function\u2019s output value resulting from a change to its input value. It does not mean we are changing the function into some other function.<\/p>\r\n

In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was<\/p>\r\n\r\n

[latex]\\frac{\\Delta y}{\\Delta x}=\\frac{{1.37}}{\\text{7 years}}\\approx 0.196\\text{ dollars per year}[\/latex]<\/div>\r\n

On average, the price of gas increased by about 19.6\u00a2 each year.<\/p>\r\n

Other examples of rates of change include:<\/p>\r\n\r\n