8-3 Writing Equations for Exponential Functions

Property

To find an exponential function $f(x) = ab^x$ through two points:

1. Use the coordinates of the points to write two equations in $a$ and $b$.

2. Divide one equation by the other to eliminate $a$.

Property

A situation represents exponential growth when the quantity increases over time and exponential decay when the quantity decreases over time.

Use the clues below to identify the type and select the correct formula:

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!