Exponential Functions: Growth, Decay, and the Doubling Rule
When the rate of change is proportional to the amount, you have exponential. Master the y = a · b^x form, half-life, compound interest, and the visual difference between linear and exponential.
When change is proportional to amount
If a population doubles every year, or a substance loses half its mass every 5 years, or a savings account earns 3% interest annually — the rate of change depends on how much you currently have. That's the signature of an exponential function.
Linear: constant amount added each step (slope). Exponential: constant multiplier applied each step. Linear adds; exponential multiplies.
The standard form
Identify a growth graph
What does an exponential growth graph look like?
Open the question →Growth rate vs growth factor
Compound interest
Money in a savings account follows exponential growth. After t years at rate r:
Compound interest
$1000 at 3% annual interest compounded annually for 5 years. Find the amount.
Open the question →Half-life
For radioactive substances or any decay-by-half scenario, after each half-life period the amount is multiplied by ½.
Half-life calculation
A 100g substance has a 5-year half-life. How much remains after 15 years?
Open the question →Recognizing exponential from a table
Linear: each y-step differs by the same amount (1, 3, 5, 7, ... → +2 each). Exponential: each y-step multiplies by the same factor (2, 6, 18, 54, ... → ×3 each). Check the ratio between consecutive y-values; if it's constant, the function is exponential.
Exponential from a table
How can you tell if a function is exponential from a table of values?
Open the question →3-second recap
- y = a · bx: a = start, b = multiplier
- b > 1 → growth. 0 < b < 1 → decay.
- Growth rate r → b = 1 + r. Decay rate r → b = 1 − r.
- Linear adds the same amount; exponential multiplies by the same ratio.
- Half-life: divide elapsed time by half-life period → that's how many times you halve.