Exponential functions
The idea
An exponential function multiplies by the same factor over each equal step of input: f(x) = a × b^x, where a is the starting amount and b is the growth factor. Compare that with the linear functions you know, which add the same amount each step — exponential change compounds, so it starts deceptively slowly and then runs away. Populations, investments, radioactive decay, and viral spread all follow this multiply-each-period pattern, growing when b > 1 and decaying when 0 < b < 1.
The mental model is repeated multiplication wearing function notation. A quantity that doubles every 3 hours has been multiplied by 2 exactly t/3 times after t hours, so the model is N(t) = N₀ × 2^(t/3) — the exponent counts how many doubling periods have elapsed. The common misconception is reading 5% growth per year as adding a fixed amount annually; it actually multiplies by 1.05 each year, taking 5% of an ever-larger base, and over decades the gap between those two readings becomes enormous.
Worked example
A bacteria culture starts with 500 cells and doubles every 3 hours. Write a function for the population after t hours, find the population at t = 12 hours, and find the equivalent hourly growth rate.
- Each 3-hour block multiplies the population by 2, and t hours contain t/3 such blocks, so the model is N(t) = 500 × 2^(t/3).
- At t = 12 the exponent is 12/3 = 4 doublings: N(12) = 500 × 2⁴ = 500 × 16 = 8000 cells.
- For the hourly rate, one hour is a third of a doubling period, so the hourly factor is 2^(1/3) ≈ 1.26 — roughly 26% growth per hour, not 33%, because compounding does part of the work.
- Check the rate: 1.26³ ≈ 2.00, so three hours of 26% compounding really does double the count. Counting directly, 500 → 1000 → 2000 → 4000 → 8000 over four periods confirms N(12).
Answer. N(t) = 500 × 2^(t/3); the culture reaches 8000 cells at t = 12 hours, growing about 26% per hour.
Check your understanding
- How can you tell from a table of data whether a relationship is linear or exponential, and do you compare differences or ratios?
- Why does the same percentage growth rate produce larger and larger absolute increases as time passes?
- What does N(1.5) mean in the bacteria model, and how do you make sense of a fractional number of doublings?
- How would the model and its long-run behavior change if the culture instead lost half its cells every 3 hours?
Build the foundations first
Exponential functions builds on these concepts. If any feel shaky, start there.