Modeling Data: Choosing a Model & Regression
When a quantity changes by a fixed percent each period, an exponential model captures it. Growth uses ; decay uses .
The whole task is translating a word problem into the right model: identify the starting amount, the rate, and whether it grows or shrinks.
Growth and decay factors
For growth of per period, each step multiplies by . For decay, it multiplies by . Convert the percent to a decimal first.
A 5% annual increase gives a factor of ; a 5% decrease gives . The base of the exponential is that factor.
Building the model
Start with the initial amount , attach the growth or decay factor as the base, and use time as the exponent.
A $2000 investment growing 5% yearly is modeled by . Read , the sign of the change, and straight from the problem.
Worked examples
Example 1: a growth model
A $2000 investment grows 5% each year. Which model gives its value after years?
Answer:
Example 2: a decay factor
What is the decay factor for a 12% yearly decrease?
Answer:
Try one yourself
Common questions
What is the difference between the growth and decay models?
Growth multiplies by each period; decay multiplies by . The sign in the base is the only difference.
How do I turn a percent into the base?
Convert to a decimal and add to or subtract from 1. A 7% increase gives base ; a 7% decrease gives .
What goes in the exponent?
The time — the number of periods the growth or decay runs.
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