Exponential Equations and Growth/Decay

Linear: grows by a constant amount (add the same each time)
$$f(x) = mx + b$$

Exponential: grows by a constant factor (multiply by the same each time)
$$f(x) = a \cdot b^x$$

where:
- $a$ = initial value ($f(0) = a$)
- $b$ = growth/decay factor
- $b > 1$: growth (increasing)
- $0 < b < 1$: decay (decreasing)
- $x$ = time or number of periods

Population growth: $P = P_0 \cdot (1 + r)^t$ (where $r$ = rate, e.g., $0.03$ for $3\%$)

Radioactive decay / half-life: $A = A_0 \cdot \left(\frac{1}{2}\right)^{t/h}$ (where $h$ = half-life)

Doubling: $A = A_0 \cdot 2^{t/d}$ (where $d$ = doubling time)

Compound interest: $A = P\left(1 + \frac{r}{n}\right)^{nt}$