Explosive Growth

Pre-Calculus · Axiom Academy

Keep multiplying by the same factor and a quantity doesn't just grow — it erupts, leaving any steady-rate growth in the dust. One bacterium, one rule: every hour, double Start with a single bacterium in a dish. Every hour, each one splits in two. That rule sounds harmless — and for a while it looks harmless. Then it doesn't. Watch how a quantity that only multiplies behaves, and you'll meet the shape behind population booms, compound interest, and viral shares. Press play and follow the sweep line across the hours. The bar over each hour is the colony size — 1, then 2, 4, 8 — and the smooth curve through their tops is 2 raised to the hour . Notice it barely lifts off the floor early on, then erupts toward the ceiling. That sudden launch is exponential growth. Each step doesn't add the same amount — it multiplies by the same amount. That single difference is what makes the curve explode. A race: steady growth vs. doubling Two quantities both start at 1. The steady one adds 10 every hour (1, 11, 21, 31, …). The doubling one only multiplies by 2 (1, 2, 4, 8, …). Drag the time slider and watch: for the first few hours steady is winning — by a lot. Keep dragging. Somewhere past hour 5 doubling catches up, passes it, and then it isn't close. Doubling starts behind, then overtakes near hour 6 (2⁶ = 64 passes 61) and never looks back — at hour 10 it's 1,024 against the steady 101.

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