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Exponential Growth and Exponential Decay Section 8.1 and 8.2 WHAT YOU WILL LEARN: 1. How to graph exponential growth functions. 2. How to graph exponential decay functions. Exponential Growth • This is demonstrated by the classic riddle in which a child is offered two choices for an increasing weekly allowance: the first option begins at 1 cent and doubles each week, while the second option begins at $1 and increases by $1 each week. Exponential Growth • This is demonstrated by the classic riddle in which a child is offered two choices for an increasing weekly allowance: the first option begins at 1 cent and doubles each week, while the second option begins at $1 and increases by $1 each week. Although the second option, growing at a constant rate of $1/week, pays more in the short run, the first option eventually grows much larger: W 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 .01 .02 .04 .08 .16 .32 .64 1.28 2.56 5.12 10.2 4 20.4 8 40. 96 81. 92 163 .84 327 .68 655 .36 131 0.7 2 2 $1 $2 $3 $4 $5 $6 $7 $8 $9 $10 $11 $12 $13 $14 $15 $16 $17 $18 Why! Exponential Growth! The equation for option 1 is: y = 2n where n is the number of weeks. The equation for option 2 is y = 1 + n where n is the number of weeks. Oh Boy! Vocabulary An exponential function involves the expression bx where the base “b” is a positive number other than 1. The variable is going to be in the “position” of the exponent. Let’s Graph an Example = Question: Will the graph ever pass below y of 0? y 2 10 5 -10 -5 5 -5 -10 10 x Let’s Graph an Example Question: Will the graph ever pass below y of 0? y 10 We say that there is an asymptote at y = 0. 5 -10 -5 5 -5 -10 10 x Let’s Graph an Example Question: Will the graph ever pass below y of 0? y 10 We say that there is an asymptote at y = 0. 5 -10 -5 An asymptote is a line that a graph approaches as you move away from the origin. 5 -5 -10 10 x Try the following on your graphing calculator Group 1: y Group 2: 1 2 x 3 2 x 5 y 32 y 2 y 1 x x How does “a” in the function y 5 2 y2 y ab x x x affect the graph? A Definition y = abx is an exponential growth function. When a is greater than 0 and b is greater than 1. Graphing Examples • Graph y 1 3 x 2 y 10 5 -10 -5 5 -5 -10 10 x Another Example Graph 3 y ( ) 2 x y 10 5 -10 -5 5 -5 -10 10 x Graphing by Translation The generic form of an exponential function is: y = abx-h + k Where h is movement along the x axis and k is movement along the y axis. An Example of Graphing by Translation Graph y 3 2 x 1 4 y 10 5 -10 -5 5 -5 -10 10 x You Try • Graph y 23 x2 1 y 10 5 -10 -5 5 -5 -10 10 x Exponential Growth Model • We will use the formula: y = a(1 + r)t a is the initial amount, r is the percent increase expressed as a decimal and t is the number of years. The term 1 + r is called the growth factor. An Example Problem • In January 1993, there were about 1,313,000 Internet hosts. During the next five years, the number of hosts increased by about 100% per year. • Write a model. • How many hosts were there in 1996? • Graph the model. • When will there be 30 million hosts? Section 8.2 – Exponential Decay • These functions will have the form y = abx where a is greater than zero and b is between 0 and 1. 19 Example 1 • State whether the function is an exponential growth or exponential decay function. 2 x 3 x 1. f ( x ) 5( ) 3 2. f ( x ) 8( ) 2 3 . f ( x ) 10 ( 3 ) x 20 You Try • State whether the function is an exponential decay or growth function. 1. f ( x) 1 (2) x 3 5 2. f ( x) 4( ) 8 x A Basic Graph • A graph of 1 y 2 x y 10 5 -10 -5 5 -5 -10 10 x Graphing Exponential Functions…again • Graph: 1 y 3 4 x y 10 5 -10 -5 5 -5 -10 10 x Another Example • Graph: 2 y 5 3 x y 10 5 -10 -5 5 -5 -10 10 x Graphing by Translation The generic form of an exponential function is: y = abx-h + k Where h is movement along the x axis and k is movement along the y axis. Graphing by Translation • Graph: 1 y 3 2 x2 1 y 10 5 -10 -5 5 -5 -10 10 x An Exponential Decay Word Problem • We will use the formula: y = a(1 - r)t (1-r) is called the decay factor. The Word Problem • You buy a new car for $24,000. The value y of the car decreases by 16% each year. 1. Write an exponential decay model for the value of the car. 2. Use the model to estimate the value after 2 years. 3. Graph the model. 4. When will the car have a value of $12,000. Homework : Page 469, 14-18 even, 19-24 all, 34, 36, 38, 43-45 all Page 477, 12, 16, 18, 19-24 all, 36, 40, 42, 47-49 all