Growth vs. Linear

Algebra 1 · Axiom Academy

Two kinds of growth line up for a race. At first they look close — but one of them has a surprise that changes everything. Two ways to grow — and they are not in the same league A linear function grows by adding the same amount every step. An exponential function grows by multiplying by the same amount every step. That one-word difference — add versus multiply — turns out to be the difference between a steady climb and a takeoff. Watch the race. Both start at the same height. The blue line adds +3 each step; the red curve multiplies by ×1.6 each step. Early on the line is comfortably ahead — then the curve catches it, passes it, and leaves it far below. Same start, same number of steps — and yet the curve ends up in a different world. That gap is what exponential growth does to a head start. Step through it: +3 versus ×1.6 Walk forward one step at a time. Each step the blue total gets +3 added on — the same jump every time. The red total gets multiplied by 1.6 — and because it is multiplying its own ever-bigger value, its jumps keep getting larger. Drag to advance the step and watch the red jump swell. The linear jump is always the same size. The exponential jump grows every single step — that is the whole reason it eventually wins. Can the line ever keep its lead? Try to make it.

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