Exponential Functions and Modeling

Exponential Functions

$f(x) = ab^x$

• $a$ = initial value
• $b$ = base (growth/decay factor)
• Domain: all reals; Range: $y > 0$ (if $a > 0$)

Natural Exponential Function

$f(x) = e^x \quad (e \approx 2.71828...)$

Growth and Decay Models

Continuous growth/decay

$f(t) = ae^{kt}$

• $k > 0$: growth
• $k < 0$: decay

Half-life

$A(t) = A_0 \left(\frac{1}{2}\right)^{t/h}$

where $h$ = half-life

Doubling time

$A(t) = A_0 \cdot 2^{t/d}$

where $d$ = doubling time

Constructing Exponential Models

Given two points $(x_1, y_1)$ and $(x_2, y_2)$:

$b = \left(\frac{y_2}{y_1}\right)^{\frac{1}{x_2 - x_1}}$

Then $a = \frac{y_1}{b^{x_1}}$

Comparing Linear and Exponential

Over equal intervals:

• Linear: Constant rate of change (differences)
• Exponential: Constant ratio (multiplicative change)

| Feature | Linear | Exponential | |---------|--------|-------------| | Equal intervals | Add constant | Multiply by constant | | Equation | $f(x) = mx + b$ | $f(x) = ab^x$ | | Graph | Line | Curve | | Long-term | Constant growth | Accelerating growth |

Logistic Growth

$f(t) = \frac{L}{1 + Ce^{-kt}}$

• $L$ = carrying capacity
• Starts exponential, levels off at $L$

AP Precalculus Tip: The College Board expects you to model with exponential functions given contextual data and interpret parameters in context.