Exponential Growth and Decay

Exponential Functions

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

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

Exponential Growth

$y = a(1 + r)^t$

• $a$ = initial amount
• $r$ = growth rate (as a decimal)
• $t$ = time
• $(1 + r)$ = growth factor

Example: A population of 500 grows at 3% per year: $y = 500(1.03)^t$

After 10 years: $y = 500(1.03)^{10} \approx 672$

Exponential Decay

$y = a(1 - r)^t$

• $(1 - r)$ = decay factor

Example: A car worth $25,000 depreciates at 15% per year: $y = 25000(0.85)^t$

After 5 years: y = 25000(0.85)^5 \approx \11,093$

Compound Interest

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

• $P$ = principal (initial investment)
• $r$ = annual interest rate
• $n$ = times compounded per year
• $t$ = years

Graphs of Exponential Functions

Growth ($b > 1$): Curve goes up, gets steeper Decay ($0 < b < 1$): Curve goes down, flattens out

Both have:

• Horizontal asymptote at $y = 0$
• Y-intercept at $(0, a)$
• Domain: All real numbers
• Range: $y > 0$ (if $a > 0$)

Comparing Linear vs. Exponential

| Feature | Linear | Exponential | |---------|--------|-------------| | Rate of change | Constant | Increasing/decreasing | | Equation | $y = mx + b$ | $y = ab^x$ | | Graph | Straight line | Curve | | Eventually wins? | No | Always grows faster |

Key insight: Exponential functions eventually grow faster than ANY linear function, no matter how steep the line.