Exponential growth and decay · AP Precalculus

Equal input increments produce equal multiplicative output factors in an exponential model. The initial value and growth factor play different roles.

A reliable method

Express each percentage change as a multiplier. Identify the length of one period, then count periods. Check the first two modeled values against the context.

The common trap

A growth factor of 1.08 means an 8 percent increase, not a 108 percent increase. Equal percentages in opposite directions do not cancel.

Make the connection

Compare additive and multiplicative changes in a data table before selecting a model. Do not extrapolate beyond a sensible contextual domain without explanation.

Multiplying periods versus adding percentages

A model starts at 80 and grows 25% each hour. After one hour it gives 100 and after two hours 125. Adding 20 in every hour would produce 120 at hour two and describes a linear model instead. The exponential increment changes because it is a percentage of the current amount. For a half-life of two hours, four hours contains two halvings, not four; 80 becomes 20. State the duration of a model period before choosing the exponent. Reversing a percentage change uses division by its multiplier, not an equal percentage subtraction.

Transfer, explain and check

An amount grows 10% and then falls 10%. Starting from 100 gives 110 and then 99. Explain why the final value differs from 100: the second percentage uses a different base.

WORKED EXAMPLE

A culture begins with 30 cells and triples each hour in a model. How many cells does the model predict after 3 hours?

There are three one-hour growth intervals.
Multiply by 3 in each interval.
30×3×3×3=810.

Answer: 810

Compare the other answer choices

Add three growth factors instead of multiplying.

Multiply the initial count by elapsed hours alone.

2430

Apply one extra tripling interval.

Try it, then explain it.

Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.

Practice this skill

Sources & authorship

Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.

AP Precalculus course and syllabus