Adding versus multiplying
A line grows by the same *amount* every step: start at 100, add 30, get 130, add 30 again, 160. The change is constant. An exponential function grows by the same *factor* every step: start at 100, multiply by 1.3, get 130, multiply by 1.3 again, 169. The change itself keeps getting bigger, because you are taking a percentage of a number that is itself growing.
We write the exponential as f(x) = a·b^x. The number a is the starting value (what you have when x = 0, since b^0 = 1). The number b is the base — the factor you multiply by each time x goes up by one. This is exactly the rule behind a geometric sequence, where b plays the role of the common ratio; an exponential function is that same multiplying pattern made continuous.
Linear y = 100 + 30x Exponential y = 100·(1.3)^x x | linear x | exponential 0 | 100 0 | 100 1 | 130 1 | 130 2 | 160 2 | 169 3 | 190 3 | 219.7 4 | 220 4 | 285.6 Linear: each step adds 30 (constant gap). Exponential: each step multiplies by 1.3 (constant ratio).
The base decides everything
For an exponential f(x) = a·b^x we always require b > 0 and b ≠ 1. If b > 1, multiplying by b each step makes the values climb — this is exponential growth. If 0 < b < 1, multiplying by a number smaller than one shrinks the values toward zero — this is exponential decay. The base b = 1 is banned because multiplying by one never changes anything; the function would be the flat constant a.
Notice what makes an exponential different from a power like x^2. In a power the variable sits in the base and the exponent is fixed; in an exponential the variable sits in the [[exponent|exponent]] and the base is fixed. That swap is the whole difference: x^2 grows like a parabola, but 2^x eventually outruns it and every other polynomial, no matter how high its degree.
What the graph looks like
Every exponential a·b^x with a > 0 stays strictly positive — its graph never touches the x-axis. As x runs off to one side the curve hugs the axis without ever reaching it (a horizontal asymptote at y = 0), and to the other side it shoots upward. Growth curves rise as you read left to right; decay curves fall. The y-intercept is always a, because setting x = 0 gives a·b^0 = a·1 = a.
Growth: y = 2^x Decay: y = (1/2)^x x | y x | y -2 | 1/4 = 0.25 -2 | 4 -1 | 1/2 = 0.5 -1 | 2 0 | 1 0 | 1 1 | 2 1 | 1/2 2 | 4 2 | 1/4 3 | 8 3 | 1/8 Note: (1/2)^x = 2^(-x), so decay is just growth read backwards — a mirror across the y-axis.