Exponential Growth / Decay (Appreciation / Depreciation) Formulas and Example

Exponential growth/appreciation per time period 't' by a percentage (or rate) 'r' and an initial amount 'a' is given by the function f, where f(t)=a*(1+r)^t. That is:



Exponential decay/depreciation per time period 't' by a percentage 'r' and an initial amount 'a' is given by the function 'f', where f(t)=a*(1-r)^t. That is:
 

Example:
In 1967, you could by an Austin Healy 3000 Mark III sports car for $4000.00. Since then, the car has appreciated in value by 6.12% per year.

What function models the value, f, of the car t years after 1967?
Solution: a=4000, r-.0612. Thus,


What was the value of the car in 2004?
Solution: t=37. Thus,


In what year will the car's value appreciate to over $50,000? Explain the process you used to find the answer.
Solution: 
Solve 50000=4000*(1.0612)^t
by graphing
y1=50000
y2=4000*(1.0612)^x
and find the x-value of the intersection of y1 and y2.
x=42.5 year

2009.5, or 2010 by rounding up.