Exponential Patterns

Algebra 1 · Axiom Academy

Watch a single bacterium become a crowd. Every step the colony doubles — and that one habit, multiplying by the same number again and again, is the seed of exponential growth. Drop a single bacterium in a warm petri dish and leave it alone. It doesn't grow by adding one neighbor at a time — it grows by splitting . Each cell becomes two, every two become four, and the dish fills faster than your intuition expects. That runaway is exponential growth, and it starts from one simple rule repeated over and over. Watch the dish below. Each generation, every cell divides in two — so the head count goes 1, then 2, then 4, then 8, doubling at each step. Keep an eye on the running total as the colony explodes. Every step the head count ×2 — not plus a fixed amount, but times a fixed amount. That single move is what makes growth exponential. Here is the move that matters. Slide through the generations and compare two colonies that both start at 1. One doubles every step (×2). The other just adds 2 every step. Early on they look close — then the doubling one leaves the other in the dust. Repeated multiplying grows wildly faster than repeated adding . That constant ×2 jump is the heartbeat of every exponential pattern. Doubling is just one choice of multiplier. What if each cell tripled, or barely changed? Slide the ratio r — the number you multiply by every step — and watch the growth curve answer. Crank the generations too, and read off the population P = r n .

This is the written version of the interactive lesson above. See the full Algebra 1 course.