Write exponential equations from contexts

A context with a constant multiplicative change is represented by an exponential model such as $y=a b^t$ , where $a$ is the initial value, $b$ is the growth or decay factor, and $t$ measures elapsed time or repeated periods. The model connects percent increase to $b=1+r$ and percent decrease to $b=1-r$ , while distinguishing multiplicative change from constant additive change and preserving meaningful units and parameter interpretations. This scope excludes variable rates, regression-based model selection, and more advanced generalized exponential models.

Detailed Explanation: Write exponential equations from contexts

A situation has exponential change when a quantity is multiplied by the same factor during each time period. Use the model

y=ab^t

where:

$a$ is the initial value, when $t=0$ ;
$b$ is the multiplicative change factor;
$t$ is the number of elapsed time periods;
$y$ is the amount after $t$ periods.

For a percent increase of $r$ ,

b=1+r.

For a percent decrease of $r$ ,

b=1-r.

Remember to write the percent as a decimal before using it.

Example

A town has a population of $12{,}000$ . Its population increases by $3\%$ each year. Write an exponential equation for the population $P$ after $t$ years.

Step 1: Identify the initial value.

The population starts at $12{,}000$ , so

a=12{,}000.

Step 2: Find the growth factor.

A $3\%$ increase means

r=0.03.

Therefore,

b=1+0.03=1.03.

This means the population is multiplied by $1.03$ each year.

Step 3: Identify the time variable.

Since the change happens each year, let $t$ represent the number of years.

Step 4: Substitute into the exponential model.

\boxed{P=12{,}000(1.03)^t}

Here, $P$ is the population and $t$ is measured in years.

For example, after $5$ years,

P=12{,}000(1.03)^5.

The expression is exponential because the population is multiplied by the same factor, $1.03$ , each year. It is not modeled by $12{,}000+0.03t$ , because that would represent a constant additive change rather than a constant percent change.

Learn by doing: Write exponential equations from contexts

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Exponential Function Growth (Discrete) - Scenario to Equation

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