13_MMA0601_08AA_Exponential Models

Mathematical Models with Applications HS Mathematics Unit: 06 Lesson: 01 Exponential Models An exponential function is another type of non-linear function. Exponential functions have a common ratio that is raised to a power of x. a = initial value (0, a) b = common ratio Exponential growth Exponential decay b >1 0 < b <1 What was the common ratio for Population Explosion? What was the common ratio for Rebounding Ball? How did the common ratio affect the graphs of the exponential functions? What would be the common ratio of each of the following examples? o The population of a bacterial culture which doubles every hour o The value of investment property increases 5% per year o The value of a car which decreases by 10% per year The common ratio determines if exponential function represents growth or decay. Would the following examples represent exponential growth or decay? o The population of a bacterial culture which doubles every hour (C)2012, TESCCC 04/16/13 page 1 of 5 Mathematical Models with Applications HS Mathematics Unit: 06 Lesson: 01 o The value of investment property increases 5% per year o The value of a car which decreases by 10% per year Many problems can be modeled by exponential functions. Many biological systems exhibit exponential growth. Radioactive decay is represented by exponential decay. In business, percent increase and decrease are represented by exponential growth and decay models. (C)2012, TESCCC 04/16/13 page 2 of 5 Mathematical Models with Applications HS Mathematics Unit: 06 Lesson: 01 Exponential Models Draw a sketch of the following exponential functions using the graphing calculator. o What is the common ratio? o Does it represent exponential growth or decay? Be careful with problem 3. o Find the values at f(3), f(-1), and f(0). f ( x ) = 3(2)x 1. 2. 1 f ( x ) = ( )x - 1 2 f ( x ) = -(2)x + 3 3. (C)2012, TESCCC 04/16/13 page 3 of 5 Mathematical Models with Applications HS Mathematics Unit: 06 Lesson: 01 (C)2012, TESCCC 04/16/13 page 4 of 5 Mathematical Models with Applications HS Mathematics Unit: 06 Lesson: 01 Exponential Models Solve each of the following problems using exponential functions. 4. Suppose the future population of Smallville can be modeled by the formula f ( x ) = 12000(1.015)x , where x is the number of years since 2000. What is the initial population of Smallville in 2000? Use the function to predict the population in Smallville in the year 2020. 5. The future value of an investment at 6% compounded monthly can be modeled by the formula f ( x ) = 7500(1.005)x , where x represents the number of months that the money is invested. What is the initial amount invested according to the formula? What would be the value of the investment after 10 years? 6. Australia can be used for a study of population and effects of predators. In the 1800s, rabbits were brought into Australia which had no natural predators to keep the rabbit population in f ( x ) = 60(6.32)x check. Assume the number of rabbits increased exponentially by the function, , where x represents the number of years elapsed since 1865. How many rabbits were in Australia at time zero when the function model began in 1865? How many rabbits would be predicted at the end of 1875? Use the model to predict when the first pair of rabbits was introduced into Australia. 7. The number of hours milk can stay fresh decreases as temperature increases. A model for this f ( x ) = 192(0.9)x situation is a decreasing exponential function, , where x represents the temperature in oC. According to the formula, how long does milk stay fresh at 0oC? Predict how long the milk will stay fresh at 30oC and 60oC. (C)2012, TESCCC 04/16/13 page 5 of 5