Exploring Exponential Growth and Decay Models

Exploring Exponential Growth and Decay Models Sections 8.1 and 8.2

ExamplesDetermine if the function represents exponential growth or decay. 1. 2. 3. Exponential Growth Exponential Decay Exponential Decay

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

Example: Compound Interest You deposit $1500 in an account that pays 2.3% interest compounded yearly, • What was the initial principal (P) invested? • What is the growth rate (r)? The growth factor? • Using the equation A = P(1+r)t, how much money would you have after 2 years if you didn’t deposit any more money? • The initial principal (P) is $1500. • The growth rate (r) is 0.023. The growth factor is 1.023.

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

Example: Exponential Decay You buy a new car for $22,500. The car depreciates at the rate of 7% per year. • What was the initial amount invested? • What is the decay rate? The decay factor? • What will the car be worth after the first year? The second year? After 5 years? • When would the car be worth $10,000?

Writing an Exponential Decay Model From 1982 through 1997, the purchasing power of 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.

You Try It Your business had a profit of $25,000 in 2008. If the profit increased by 12% each year, what would your expected profit be in the year 2013? Identify P, t, r, and the growth factor. Write down the equation you would use and solve.

You deposit$5,000 in an account that pays 5% interest per year. If you leave the money in the account for 10 years, how much would you have in the account?

Iodine-131 is a radioactive isotope used in medicine. Its half-life or decay rate of 50% is 8 days. If a patient is given 25mg of iodine-131, how much would be left after 32 days or 4 half-lives. Identify P, t, r, and the decay factor. Write down the equation you would use and solve.

Solution P= 25 mg T = 4 R = 0.5 Decay factor = 0.5

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 = P (1 + r)t y = P (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

Exploring Exponential Growth and Decay Models