Exponential Functions and Growth/Decay

The General Exponential Function

$f(x) = a \cdot b^x$

• $a$ = initial value (when $x = 0$, $f(0) = a$)
• $b$ = growth/decay factor
• If $b > 1$: exponential growth
• If $0 < b < 1$: exponential decay

Growth Factor vs. Growth Rate

If a quantity increases by $r\%$ per period:

$b = 1 + \frac{r}{100}$

If it decreases by $r\%$:

$b = 1 - \frac{r}{100}$

Example: Population grows 5% per year: $b = 1.05$ Example: Car depreciates 15% per year: $b = 0.85$

Compound Interest

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

• $P$ = principal
• $r$ = annual rate (decimal)
• $n$ = compounding periods per year
• $t$ = years

Continuous Growth/Decay

$f(t) = a \cdot e^{kt}$

• $k > 0$: growth
• $k < 0$: decay

Key Properties

• Exponential functions have a horizontal asymptote (usually $y = 0$).
• They never reach zero (for decay) or become negative.
• They eventually outgrow any linear or polynomial function.

Doubling Time and Half-Life

Doubling time: how long until the quantity doubles. Set $a \cdot b^t = 2a$ → $b^t = 2$.

Half-life: how long until half remains. Set $a \cdot b^t = \frac{a}{2}$ → $b^t = \frac{1}{2}$.

Tip 1: Identify $a$ and $b$ from Context

$a$ is the starting amount. $b$ is $(1 + \text{rate})$ or $(1 - \text{rate})$.

Tip 2: Use the Table Pattern

In an exponential function, equal changes in $x$ produce equal ratios in $y$ (not equal differences — that's linear).

Tip 3: For "After How Many Years" Questions

Set up the equation and solve using logarithms or trial-and-error with the calculator.

Tip 4: Don't Confuse Linear and Exponential

Linear: constant amount added. Exponential: constant percentage multiplied.

Tip 5: Use the Calculator for Large Exponents

On the SAT, you can use the built-in calculator to evaluate expressions like $1.05^{20}$.