Exponential functions earn their place in algebra because the world is full of quantities that change by a percentage rather than a fixed amount. A savings account earns interest on its interest; a bacteria colony’s growth rate rises with its size; a radioactive sample loses the same fraction of what remains every hour. All of these are exponential growth and decay, and a single formula handles every one of them — the same one behind the half-life calculator and every compound-interest table you will ever see.
The Growth and Decay Formulas
Start from the general exponential function
| Symbol | Meaning |
|---|---|
| initial amount, at | |
| rate per period, as a decimal ( | |
| growth factor — what the amount is multiplied by each period | |
| decay factor — the fraction that remains each period | |
| number of periods elapsed |
The reason for the
Exponential Growth vs Decay
| Exponential growth | Exponential decay | |
|---|---|---|
| Factor | ||
| Graph | rises, steeper and steeper | falls, flatter and flatter |
| Long run | increases without bound | approaches |
| Examples | populations, compound interest, viral spread | radioactive decay, depreciation, medicine leaving the body |
Which equation represents exponential growth? Look only at the base.
Converting Between Rates and Factors
| Description | Rate | Factor |
|---|---|---|
| grows | ||
| grows | ||
| decays | ||
| decays | ||
| doubles each period | ||
| halves each period |
Reading a factor backwards is just as quick:
Doubling Time and Half-Life
Some quantities are described by how long they take to double or halve rather than by a percentage. Put the time scale into the exponent:
Here
Compound Interest
Money is the most common growth model. When an annual rate
and when it is compounded continuously, the base becomes
For
Logistic and Exponential Growth
Pure exponential growth cannot last. A population doubling every generation would outgrow its food, space or host. Logistic growth models that limit:
Early on, the logistic curve is almost identical to an exponential one. As
Example 1: Exponential Function Word Problems With Answers
| Situation | Model | Result |
|---|---|---|
Worked Example
A town of
Step 1 — identify the initial amount and the rate:
Step 2 — build the growth factor:
Step 3 — write the model:
Step 4 — evaluate at
Step 5 — evaluate at
The population grows by about
Common Mistakes to Avoid
- Using
as the factor. A growth rate gives , not . Multiplying by would wipe out of the amount each year. - Subtracting the percentage once instead of every period. Losing
a year for years is , a total loss — not . - Mismatching time units. If the rate is per month,
must be in months. A half-life in hours needs in hours. - Expecting decay to reach zero.
is always positive. In practice you round to zero when the amount becomes negligible, but the model never gets there.
Practice Problems
Work each one before opening the answer.
Problem 1. Classify each as growth or decay and give the percentage rate:
Show answer
Step 1 —
Step 2 —
Answer:
Problem 2. A town of
Show answer
Step 1 — the decay factor is
Step 2 — evaluate at
Answer: about
Problem 3.
Show answer
Step 1 — annual compounding means
Step 2 — evaluate:
Answer:
Problem 4. A
Show answer
Step 1 — count the half-lives:
Step 2 — halve three times:
Answer:
Problem 5. A culture of
Show answer
Step 1 — the number of tripling periods:
Step 2 — apply the factor
Answer:
Problem 6. A
Show answer
Step 1 — the model is
Step 2 — one more year:
Answer: after
Problem 6 was solved by trial because the unknown is in the exponent. To solve for time directly — “when will the balance double?” — you need the inverse of an exponential function, which is the subject of the logarithms lesson.