Exponential Functions

An exponential function has the form:

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

Where:

• $a$ = initial value (y-intercept when $x = 0$)
• $b$ = base (growth/decay factor)
• $b > 1$: exponential growth
• $0 < b < 1$: exponential decay

The Natural Exponential

The most important base is $e \approx 2.71828...$

$f(x) = e^x$

$e$ appears naturally in compound interest, population growth, and calculus.

Properties

$b^0 = 1$ (always) $b^1 = b$ $b^{-x} = \frac{1}{b^x}$ $b^{x+y} = b^x \cdot b^y$ $b^{x-y} = \frac{b^x}{b^y}$ $(b^x)^y = b^{xy}$

Graphing

For $f(x) = b^x$ where $b > 1$:

• Always passes through $(0, 1)$
• Increases from left to right
• Horizontal asymptote at $y = 0$
• Domain: all real numbers
• Range: $(0, \infty)$

For $f(x) = b^x$ where $0 < b < 1$:

• Always passes through $(0, 1)$
• Decreases from left to right
• Horizontal asymptote at $y = 0$

Applications

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

Continuous Compounding: $A = Pe^{rt}$

Population Growth: $P(t) = P_0 e^{kt}$