Determine the growth or decay factor

In an exponential model $y=ab^x$ , the growth or decay factor $b$ is the constant multiplier applied when $x$ increases by one unit; it can be identified from an equation, successive values in a table, or a graph. A factor $b>1$ indicates growth, $0<b<1$ indicates decay, and $b=1$ indicates a constant relationship; the factor differs from the initial value $a$ and from the percent rate, with $b=1+r$ for growth and $b=1-r$ for decay. Continuous compounding and logarithmic methods are outside this scope.

Detailed Explanation: 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 $1.04=1+0.04$ .

Learn by doing: Determine the growth or decay factor

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Exponential Function Growth (Discrete) - Scenario to Equation

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