Exponential Functions: Growth and Decay

The anatomy of $f(x)=ab^x$

With $a\neq0$, $b>0$, $b\neq1$: the value $a=f(0)$ is the initial amount, and $b$ is the growth factor per unit of $x$. If $b>1$ the function models growth; if $0<b<1$, decay.

The defining property — the sentence AP quotes verbatim — is: over equal-length input intervals, the output changes proportionally. Whenever $x$ increases by $\Delta x$, the output is multiplied by the same factor $b^{\Delta x}$, no matter where the interval sits. Linear functions add the same amount over equal intervals; exponential functions multiply by the same amount.

Consequences worth memorizing: $f$ is always increasing or always decreasing (no extrema), always concave up when $a>0$, has horizontal asymptote $y=0$, and $\lim_{x\to\infty}ab^x=\infty$ (growth) or $0$ (decay). Given two points $(x_1,y_1)$, $(x_2,y_2)$, the base is recovered from $b^{\,x_2-x_1}=y_2/y_1$.