Linear things ADD the same amount each step. Exponential things MULTIPLY by the same factor each step — and that one difference is why a savings account, a rumour and a petri dish all behave alike. The shape is $A = A_0 b^{t}$: read aloud, A equals A-nought b to the t. $A_0$ is said “A-nought” and means the amount you started with; $b$ is the base, the factor applied once per period; $t$ counts the periods.

Everything about the story lives in $b$. Above 1 and the quantity climbs; below 1 and it fades; exactly 1 and nothing ever happens. Two names are worth keeping apart from the first day. The growth rate $r$ is what you GAIN — 8% a year is $r = 0.08$. The growth factor is what you MULTIPLY by — $1 + r = 1.08$, because the 1 carries forward everything you already had. Decay flips the sign, not the idea: lose 20% and the factor is $1 - 0.20 = 0.80$. Mixing up the rate and the factor is the single most expensive slip in this chapter, and it is why this lesson does no arithmetic at all.