There are many real life situations that can be modeled using a linear relationship. Within the context of these situations there needs to be a constant rate of change. But what if there isn’t? (To determine if a linear relationship or exponential relationship exists refer to Exponential Functions Target C). In some instances, the rate of change is not constant, but increases or decreases by a growth factor. When this occurs, we use an exponential growth/decay model to represent the situation.
The exponential growth/decay model: \( y = a ( 1 \pm r )^t \)
The symbol \( \pm \) is used to refer to both addition and subtraction. If a situation represented exponential growth, use addition. If the situation represents exponential decay, use subtraction. So what do all of those variables represent? The “\( a \)” represents the initial amount, the “\( r \)” represents the rate at which something is growing (increasing) or decaying (decreasing), “\( t \)” represents the time that has or will pass and the "\( y \)" represents the ending amount or value at time "\( t \)". There are certain key terms that can help you differentiate between exponential growth and exponential decay. For growth, you may see terms such as increases, earns, accumulates, and so on. For decay some key terms may be decrease, depreciates, loses, etc.
Here are just a few examples of when you would use the exponential growth/decay model:
Example 1:
A college graduate accepts a job at an advertising agency. The job has a salary of \( \$ 40,000 \) per year, plus a pay increase of \( 2 \% \) per year. How much will the graduate be making in \( 10 \) years?
Here are just a few examples of when you would use the exponential growth/decay model:
- finding the balance of a savings account after a given period of time
- determining the value of a car after so many years
- estimating/calculating a city’s population
Example 1:
A college graduate accepts a job at an advertising agency. The job has a salary of \( \$ 40,000 \) per year, plus a pay increase of \( 2 \% \) per year. How much will the graduate be making in \( 10 \) years?
Example 2:
Francine just bought a new car for \( \$ 32,000 \). She plans on keeping it until it is worth \( \$ 18,000 \) when she finally decides to sell it. If the average depreciation rate for cars is \( 15 \% \) per year, how many years will she own the car before she needs to sell it?
Francine just bought a new car for \( \$ 32,000 \). She plans on keeping it until it is worth \( \$ 18,000 \) when she finally decides to sell it. If the average depreciation rate for cars is \( 15 \% \) per year, how many years will she own the car before she needs to sell it?
Quick Check
1) You deposit \( \$ 125 \) into a savings account that earns \( 5 \% \) annual interest, compounded yearly. Your friend deposits \( \$ 200 \) into a savings account that earns \( 3.75 \% \) interest compounded yearly. Who will have a higher balance in their account when they reach \( 17 \) years old if no other deposits are made?
1) You deposit \( \$ 125 \) into a savings account that earns \( 5 \% \) annual interest, compounded yearly. Your friend deposits \( \$ 200 \) into a savings account that earns \( 3.75 \% \) interest compounded yearly. Who will have a higher balance in their account when they reach \( 17 \) years old if no other deposits are made?