Exponential Growth and Decay
Simple Explanation
When a quantity grows or shrinks by the same percentage every fixed period of time, it follows an exponential model: A = Aβ(1 + r)α΅ for growth, or A = Aβ(1 β r)α΅ for decay, where Aβ is the starting amount and r is the rate per period.
Why Do We Need It?
This is the real-world payoff of exponent rules β population growth, compound interest, and decay-style processes are all modeled with exactly this pattern, so understanding exponents means being able to reason about them quantitatively.
Formula
Exponential Growth/Decay Model
A = Aβ(1 + r)α΅ (growth, r > 0) A = Aβ(1 β r)α΅ (decay, 0 < r < 1)
A quantity that changes by the same percentage every fixed time period follows this pattern β starting amount times a growth/decay factor raised to the number of periods elapsed.
- A
- β the amount after t time periods
- Aβ
- β the initial amount, at t = 0
- r
- β the growth rate (or decay rate), as a decimal
- t
- β the number of time periods elapsed
When to use it: Whenever a quantity grows or shrinks by a fixed percentage each period β population, compound interest, or radioactive-style decay.
Worked Example
Apply the exponential growth model
A town has a population of 20,000 growing at 5% per year. Estimate the population after 3 years.
Why Does This Work?
Growing by the same percentage each period means multiplying by the same factor, (1 + r), every period β repeating that multiplication t times is exactly what raising the factor to the power t accomplishes.
Real-Life Example
Compound interest in a savings account
A bank account earns 4% interest per year, compounded annually.
The balance after t years is A = Aβ(1.04)α΅ β the same exponential growth model, with the interest rate as r.
Practice
A car worth $18,000 depreciates (loses value) at 10% per year. What is it worth after 2 years? (Round to the nearest whole number.)
MediumCommon mistake
Using (1 + r) for decay or (1 β r) for growth β growth always uses (1 + r) as the per-period multiplier, and decay always uses (1 β r), since decay must multiply by a factor less than 1.
Quick Review
- A = Aβ(1 + r)α΅ models exponential growth; A = Aβ(1 β r)α΅ models exponential decay.
- r is the rate per period as a decimal; t counts how many periods have elapsed.
- Compound interest, population growth, and depreciation are all real examples of this model.