Exponential Functions

An exponential function has the form $f(x) = a \cdot b^x$ where:

• $a$ = the starting value (the value when $x = 0$).
• $b$ = the growth factor — what you multiply by each time $x$ increases by 1.

$b > 1$: growth (function increases). $f(x) = 100 \cdot 2^x$ doubles every step.

$0 < b < 1$: decay (function decreases). $f(x) = 100 \cdot (0.5)^x$ halves every step.

The crucial difference from linear functions: linear adds a constant; exponential multiplies by a constant.

• Linear: $y = mx + b$ — $y$ increases by $m$ for every $+1$ in $x$.
• Exponential: $y = a \cdot b^x$ — $y$ multiplies by $b$ for every $+1$ in $x$.

Exponentials grow much faster than linear functions over time, even if they start slower.

Percent growth/decay translation:

• Growing by 5% per year: $b = 1 + 0.05 = 1.05$. So $f(t) = a \cdot (1.05)^t$.
• Decaying by 8% per year: $b = 1 - 0.08 = 0.92$. So $f(t) = a \cdot (0.92)^t$.
• Doubling every $k$ years: $f(t) = a \cdot 2^{t/k}$.
• Half-life of $h$ years: $f(t) = a \cdot (0.5)^{t/h}$.

SAT word-problem template:

"A bacteria population starts at 200 and doubles every 3 hours."

→ $P(t) = 200 \cdot 2^{t/3}$.

The $/3$ in the exponent says it takes 3 hours for the doubling to happen once.

Compound interest — a special case of exponential growth:

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

where $P$ is principal, $r$ is annual interest rate (as decimal), $n$ is times compounded per year, $t$ is years. For annual compounding, simplify to $A = P(1 + r)^t$.

Reading exponential graphs:

• $y$-intercept $= a$ (the starting value).
• If the curve rises sharply, $b > 1$ and $b$ is large.
• If the curve approaches but never touches the $x$-axis from above, $0 < b < 1$.