E XPONENTIAL G ROWTH M ODEL

1. WRITING EXPONENTIAL GROWTH MODELS A quantity is growing exponentially if it increases by the same percent in each time period. EXPONENTIAL GROWTH MODEL C is the initial amount. t is the time period. y = C (1 + r)t (1 + r) is the growth factor,r is the growth rate. The percent of increase is 100r.

2. Finding the Balance in an Account • • • • • • COMPOUND INTEREST You deposit $500 in an account that pays 8% annual interest compounded yearly. What is the account balance after 6 years? SOLUTION METHOD 1SOLVE A SIMPLER PROBLEM Find the account balance A1 after 1 year and multiply by the growth factor to find the balance for each of the following years. The growth rate is 0.08, so the growth factor is 1 + 0.08 = 1.08. A1 = 500(1.08) = 540 Balance after one year A2 = 500(1.08)(1.08) = 583.20 Balance after two years A3 = 500(1.08)(1.08)(1.08) = 629.856 Balance after three years A6 = 500(1.08)6793.437 Balance after six years

3. Finding the Balance in an Account EXPONENTIAL GROWTH MODEL EXPONENTIAL GROWTH MODEL C is the initial amount. t is the time period. 500 is the initial amount. 6 is the time period. y = C (1 + r)t A6 = 500 (1 + 0.08)6 (1 + r) is the growth factor,r is the growth rate. (1 + 0.08) is the growth factor,0.08 is the growth rate. The percent of increase is 100r. A6 = 500(1.08)6793.437Balance after6years COMPOUND INTEREST You deposit $500 in an account that pays 8% annual interest compounded yearly. What is the account balance after 6 years? SOLUTION METHOD 2USE A FORMULA Use the exponential growth model to find the account balance A. The growthrate is 0.08. The initial value is 500.

4. Writing an Exponential Growth Model A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years.

5. Writing an Exponential Growth Model The population triples each year, so thegrowth factoris 3. 1 + r = 3 So, the growth rate ris 2 and the percent of increase each year is 200%. So, the growth rate r is 2 and the percent of increase each year is 200%. A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years. a. What is the percent of increase each year? SOLUTION The population triples each year, so thegrowth factoris 3. 1 + r = 3 1 + r = 3 So, the growth rate r is 2 and the percent of increase each year is 200%. Reminder: percent increase is 100r.

6. Writing an Exponential Growth Model A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years. b.What is the population after 5 years? Help SOLUTION After 5 years, the population is P = C(1 + r)t Exponential growth model = 20(1 + 2)5 SubstituteC,r, andt. = 20 • 35 Simplify. = 4860 Evaluate. There will be about 4860 rabbits after 5 years.

7. A Model with a Large Growth Factor t 0 1 2 3 4 5 6000 P 20 60 180 540 1620 4860 5000 4000 Population 3000 2000 1000 0 1 2 3 4 5 6 7 Time (years) GRAPHING EXPONENTIAL GROWTH MODELS Graph the growth of the rabbit population. SOLUTION Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points. Here, the large growth factor of 3 corresponds to a rapid increase P = 20(3)t

8. WRITING EXPONENTIAL DECAY MODELS EXPONENTIAL DECAY MODEL A quantity is decreasing exponentially if it decreases by the same percent in each time period. C is the initial amount. t is the time period. y = C (1 – r)t (1 – r) is the decay factor,r is the decay rate. The percent of decrease is 100r.

9. Writing an Exponential Decay Model COMPOUND INTERESTFrom 1982 through 1997, the purchasing powerof a dollar decreased by about3.5% per year. Using 1982 as the base for comparison, what was the purchasing power of a dollar in 1997? Let y represent the purchasing power and lett = 0 represent the year 1982. The initial amount is $1. Use an exponential decay model. SOLUTION y = C(1 – r)t Exponential decay model = (1)(1 – 0.035)t Substitute1forC,0.035forr. = 0.965t Simplify. Because 1997 is 15 years after 1982, substitute 15 for t. y = 0.96515 Substitute 15fort. 0.59 The purchasing power of a dollar in 1997 compared to 1982 was $0.59.

10. Graphing the Decay of Purchasing Power t 0 1 2 3 4 5 6 7 8 9 10 y 1.00 0.965 0.931 0.899 0.867 0.837 0.808 0.779 0.752 0.726 0.7 1.0 0.8 0.6 Purchasing Power (dollars) 0.4 0.2 0 1 2 3 4 5 6 7 8 9 10 11 12 Years From Now GRAPHING EXPONENTIAL DECAY MODELS Help Graph the exponential decay model in the previous example. Use the graph to estimate the value of a dollar in ten years. Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points. SOLUTION y = 0.965t Your dollar of today will be worth about 70 cents in ten years.

11. CONCEPT EXPONENTIAL GROWTH AND DECAY MODELS SUMMARY (1 + r) is the growth factor,r is the growth rate. (1 – r) is the decay factor,r is the decay rate. (0,C) (0,C) GRAPHING EXPONENTIAL DECAY MODELS EXPONENTIAL GROWTH MODEL EXPONENTIAL DECAY MODEL y = C (1 + r)t y = C (1 – r)t An exponential model y = a•btrepresents exponential growth ifb > 1 and exponential decay if 0 < b < 1. C is the initial amount. t is the time period. 0 < 1 – r < 1 1 + r > 1