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.

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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 vs exponential

Linear: constant amount added each step (slope). Exponential: constant multiplier applied each step. Linear adds; exponential multiplies.

The standard form

y = a · b^x a = starting value (when x = 0) b = base / multiplier each step If b > 1: exponential growth. If 0 < b < 1: exponential decay. If b = 1: just a horizontal line.

Growth curves up and right. Decay curves down. Both pass through (0, a) — the starting value.

Identify a growth graph

Exponential growth graph is:

Identify decay

Which shows exponential decay?

Growth rate vs growth factor

growth: b = 1 + r (r = growth rate) decay: b = 1 − r (r = decay rate) A 3% interest rate means b = 1.03. A 5% loss per year means b = 0.95.

Compound interest

Money in a savings account follows exponential growth. After t years at rate r:

A = P (1 + r)^t $1000 at 3% for 5 years: 1000 · (1.03)⁵ = 1000 · 1.159 ≈ $1159

Compound interest

$1000 at 3% annual interest compounded annually for 5 years?

Half-life

For radioactive substances or any decay-by-half scenario, after each half-life period the amount is multiplied by ½.

100g, half-life 5 years, after 15 years 15 / 5 = 3 half-lives elapsed 100 · (½)³ = 100 · ⅛ = 12.5g

Half-life calculation

Half-life: 100g substance with 5-year half-life. Amount after 15 years?

Recognizing exponential from a table

Constant ratio = exponential

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

Linear vs exponential: constant rate of change means:

3-second recap

y = a · b^x: 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.

Check yourself

Quick check #1

A bacterial culture doubles every hour. Starting with 100 cells, how many cells after 4 hours?

Quick check #2

Which function represents exponential DECAY?