Skill: Determine the growth or decay factor

In an exponential model,

$$y=ab^x,$$

the growth or decay factor is the number $b$ being raised to the power $x$. It tells how the output changes when $x$ increases by $1$.

• If $b>1$, the model shows growth.
• If $0<b<1$, the model shows decay.
• If $b=1$, the model is constant.

Example

Suppose the population of a town is modeled by

$$P=12{,}000(1.04)^t,$$

where $t$ is the number of years.

Step 1: Compare the equation with $y=ab^x$.

$$P=12{,}000(1.04)^t$$

The initial value is

$$a=12{,}000,$$

and the base is

$$b=1.04.$$

Step 2: Identify the factor.

The growth factor is

$$\boxed{1.04}.$$

Step 3: Interpret the factor.

Because $1.04>1$, the population is growing. Each year, the population is multiplied by $1.04$, or increased by $4\%$.

The factor is not the initial value $12{,}000$. Also, the factor is $1.04$, while the percent growth rate is $4\%$. For growth,

$$b=1+r,$$

so here $1.04=1+0.04$.