1. 6.1 Exponential Growth and Decay
With Applications

2. Exponential Expression
An expression where the exponent is the variable and the base is a fixed number

3. Multiplier
The base of an exponential expression

4. Growth vs. Decay When b>1, f(x) = bx represents GROWTH
When 0<b<1, f(x) = bx represents DECAY

5. Applications
Exponential Growth and Decay can be found in many applications
Ex: population growth, stocks, science studies, compound interest, and effective yield

6. Basic Growth/Decay Applications:
When dealing with most growth and decay apps, you have an equation such as:

7. Base is your multiplier
Growth: multiplier = 100% + rate
Decay: multiplier = 100% - rate

8. Growth/Decay App. WS Problem 1:
The population of the United States was 248,718,301 in 1990 and was projected to grow at a rate of about 8% per decade. Predict the population, to the nearest hundred thousand, for the years 2010 and 2025.

9. Growth/Decay App. WS Problem 1:
Growth Application
Initial Population = 248,718,301
Multiplier = 100% + 8% = 108% = 1.08
Expression to model the problem:

10. Growth/Decay App. WS Problem 1:
2010: 2 decades after 1990
n = 2

11. Growth/Decay App. WS Problem 1:
Round to the nearest hundred thousand:
290,100,000 = Population in 2010

12. Growth/Decay App. WS Problem 1:
2025: 3.5 decades after 1990
n = 3.5

13. Growth/Decay App. WS Problem 1:
Round to the nearest hundred thousand:
325,600,000 = Population in 2025

14. Population Formula WS
This is for homework, but it is for you to practice writing population formulas.
YOU DO NOT HAVE TO SOLVE ANYTHING….JUST WRITE THE FORMULAS

15. Compound Interest Formula
Another application of exponential growth
The total amount of an investment A, earning compound interest is:
P = Principal, r = annual interest rate, n = # of times interest is compounded per year, t = time in years

16. Example
Find the final amount of a $500 investment after 8 years, at 7% interest compounded annually, quarterly, monthly, daily.

17. Example (cont)
P = $500
r = 7% = .07
t = 8 years
Annually, n = 1

18. Example (cont): Annually

19. Example (cont): Quarterly

20. Example (cont): Monthly

21. Example (cont): Daily
n = 365

22. Effective Yield
The annually compounded interest rate that yields the final amount of an investment.
Determine the effective yield by fitting an exponential regression equation to 2 points.
Effective Yield = b - 1

23. Example
A collector buys a painting for $100,000 at the beginning of 1995 and sells it for $150,000 at the beginning of Write an equation to model this situation and then find the effective yield.

24. Example (Cont)
When modeling the situation, you use the compounded interest formula, and you let n = 1 for compounded annually.
A(t) = ending value = $150,000
P = initial = $100,000
n = 1 and t = 5 years

25. Example (Cont)

26. Example (Cont) Now we need to find the effective yield
First we need 2 points that would model the data:
(0, ) and (5, )

27. Example (Cont)
Plug these points into your calc
STAT, EDIT

28. Example (Cont)
Then generate the Exponential Regression: STAT, CALC, 0:ExpReg
y = abx
a =
b = 1.084
Effective yield = – 1 = .084 = 8.4% annual interest rate

29. Homework: Finish BOTH WS Pg 358 #15, 18, 21, 37, 42, 47, 48
Pg. 367 #17-23odd, 29-33odd, 47, 49