Download Exponential Growth and Decay: Modeling and Analyzing Exponential Functions and more Exercises Painting in PDF only on Docsity! © H ou g ht on M if fl in H ar co ur t P ub lis hi n g C om p an y • I m ag e C re d it s: © Te tr a Im ag es /C or b is Explain 1 Modeling Exponential Growth Recall that a function of the form y = a b x represents exponential growth when a > 0 and b > 1. If b is replaced by 1 + r and x is replaced by t, then the function is the exponential growth model y = a (1 + r) t , where a is the initial amount, the base (1 + r) is the growth factor, r is the growth rate, and t is the time interval. The value of the model increases with time. Example 1 Write an exponential growth function for each situation. Graph each function and state its domain, range and an asymptote. What does the y-intercept represent in the context of the problem? A A painting is sold for $1800, and its value increases by 11% each year after it is sold. Find the value of the painting in 30 years. Write the exponential growth function for this situation. y = a (1 + r) t = 1800 (1 + 0.11) t = 1800 (1.11) t Find the value in 30 years. y = 1800 (1.11) t = 1800 (1.11) 30 ≈ 41,206.13 After 30 years, the painting will be worth approximately $41,206. Create a table of values to graph the function. t y (t, y) 0 1800 (0, 1800) 8 4148 (8, 4148) 16 9560 (16, 9560) 24 22,030 (24, 22,030) 32 50,770 (32, 50,770) Determine the domain, range and an asymptote of the function. The domain is the set of real numbers t such that t ≥ 0. The range is the set of real numbers y such that y ≥ 1800. An asymptote for the function is y = 0. The y-intercept is the value of y when t = 0, which is the value of the painting when it was sold. (0, 1800) (8, 4148) (16, 9560) (24, 22,030) (32, 50,770) 11,000 22,000 33,000 44,000 y 16 Time (years) Va lu e (d ol la rs ) 248 32 t 0 Module 15 723 Lesson 2 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-D;CA-D © H ou g ht on M if fl in H ar co ur t P ub lis hi n g C om p an y Your Turn 10. Write and graph an exponential growth function, and state the domain and range. Tell what the y-intercept represents. Sara sold a coin for $3, and its value increases by 2% each year after it is sold. Find the value of the coin in 8 years. Explain 2 Modeling Exponential Decay Recall that a function of the form y = a b x represents exponential decay when a > 0 and 0 < b < 1. If b is replaced by 1 - r and x is replaced by t, then the function is the exponential decay model y = a (1 - r) t , where a is the initial amount, the base (1 - r) is the decay factor, r is the decay rate, and t is the time interval. Example 2 Write an exponential decay function for each situation. Graph each function and state its domain and range. What does the y-intercept represent in the context of the problem? A The population of a town is decreasing at a rate of 3% per year. In 2005, there were 1600 people. Find the population in 2013. Write the exponential decay function for this situation. y = a (1 - r) t = 1600 (1 - 0.03) t = 1600 (0.97) t Find the value in 8 years. y = 1600 (0.97) t = 1600 (0.97) 8 ≈ 1254 After 8 years, the town’s population will be about 1254 people. 1 2 3 4 y 4 6 Time (years) Va lu e (d ol la rs ) 2 8 t 0 Module 15 725 Lesson 2 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-D;CA-D
