Lesson 3: Exponential Growth and Decay

Property

The function

• $P_0 = P(0)$ is the initial value of $P$;
• $b$ is the growth or decay factor.

(a) If $b > 1$, then $P(t)$ is increasing, and $b = 1 + r$, where $r$ represents percent increase.
(b) If $0 < b < 1$, then $P(t)$ is decreasing, and $b = 1 - r$, where $r$ represents percent decrease.

Examples

• A town with 1,000 people grows by 5% per year. Its population is modeled by $P(t) = 1000(1+0.05)^t = 1000(1.05)^t$.
• A 50mg dose of medicine decreases in the body by 20% each hour. The amount remaining is $A(t) = 50(1-0.20)^t = 50(0.8)^t$.
• The value of an investment doubles every 10 years. If the initial investment is 500 dollars, the growth factor is 2, and the value is $V(t) = 500(2)^{t/10}$.

Explanation

This function describes something that multiplies by the same amount over time. If the multiplier (b) is bigger than 1, it grows. If it's between 0 and 1, it shrinks. It's all about repeated multiplication!

Property

For exponential growth functions, the growth rate $r$ and growth factor $(1+r)$ are related by:

Examples

Property

For exponential decay functions $f(x) = a(1-r)^x$:

• Decay rate: $r$ (expressed as a decimal)
• Decay factor: $(1-r)$
• Conversion formulas: $r = 1 - \text{decay factor}$ and $\text{decay factor} = 1 - r$

Examples