Exponential Modeling

Pre-Calculus · Axiom Academy

One shape — y = a·bᵗ — runs growth, decay, and the long slow compounding that reshapes a planet. Put it in your hands. A bacterial colony, a buried bone, and the whole human population obey the same exponential rule — here are three things you can do with it. A colony that doubles on a clock 500 bacteria, doubling every few hours. Set the doubling time, hit Go, and watch the count race up the curve — that's P(t) = 500·2^(t/T), live. Run the model backward to date a bone Carbon-14 halves every 5,730 years. Drag the slider to how much C-14 is left in a sample, and the model runs in reverse to place its age on the timeline — that's solving 100·(½)^(t/5730) for t. A fraction of a percent, ninety years on Earth had ~3 billion people in 1960. Nudge the annual growth rate and watch where the model lands the population by 2050 — a tiny rate change, compounded for decades, is the whole ballgame. One shape, three jobs: it explodes (bacteria, viral spread, investment), it fades (C-14, medicine in your blood, a hot coffee cooling), and it compounds (population, savings). Whenever a quantity changes by a fixed percentage each step, you're looking at y = a·bᵗ — and these same three moves work.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.