EXPONENTIAL GROWTH MODEL 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.

METHOD 1 SOLVE A SIMPLER PROBLEM 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 1 SOLVE 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

EXPONENTIAL GROWTH MODEL EXPONENTIAL GROWTH MODEL 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 2 USE A FORMULA Use the exponential growth model to find the account balance A. The growth rate is 0.08. The initial value is 500. EXPONENTIAL GROWTH MODEL 500 is the initial amount. 6 is the time period. (1 + 0.08) is the growth factor, 0.08 is the growth rate. A6 = 500(1.08) 6  793.437 Balance after 6 years A6 = 500 (1 + 0.08) 6 EXPONENTIAL GROWTH MODEL C is the initial amount. t is the time period. (1 + r) is the growth factor, r is the growth rate. The percent of increase is 100r. y = C (1 + r)t

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

Reminder: percent increase is 100r. Writing an Exponential Growth Model 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 the growth factor is 3. The population triples each year, so the growth factor is 3. 1 + r = 3 1 + r = 3 1 + r = 3 So, the growth rate r is 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%. So, the growth rate r is 2 and the percent of increase each year is 200%. Reminder: percent increase is 100r.

There will be about 4860 rabbits after 5 years. 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 Substitute C, r, and t. = 20 • 3 5 Simplify. = 4860 Evaluate. There will be about 4860 rabbits after 5 years.

GRAPHING EXPONENTIAL GROWTH MODELS A Model with a Large Growth Factor 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. t P 4860 60 180 540 1620 20 5 1 2 3 4 1000 2000 3000 4000 5000 6000 1 7 2 3 4 5 6 Time (years) Population Here, the large growth factor of 3 corresponds to a rapid increase P = 20 ( 3 ) t

EXPONENTIAL DECAY MODEL WRITING EXPONENTIAL DECAY MODELS A quantity is decreasing exponentially if it decreases by the same percent in each time period. EXPONENTIAL DECAY MODEL 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.

The purchasing power of a dollar in 1997 compared to 1982 was $0.59. Writing an Exponential Decay Model COMPOUND INTEREST From 1982 through 1997, the purchasing power of a dollar decreased by about 3.5% per year. Using 1982 as the base for comparison, what was the purchasing power of a dollar in 1997? SOLUTION Let y represent the purchasing power and let t = 0 represent the year 1982. The initial amount is $1. Use an exponential decay model. y = C (1 – r) t Exponential decay model = (1)(1 – 0.035) t Substitute 1 for C, 0.035 for r. = 0.965 t Simplify. Because 1997 is 15 years after 1982, substitute 15 for t. y = 0.96515 Substitute 15 for t. 0.59 The purchasing power of a dollar in 1997 compared to 1982 was $0.59.

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

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