Lesson 5 of 6 · Advanced math
Reading is free. Sign in to tick off lessons, keep your place and track your mastery.
Exponential functions are among the most dependable points in Advanced Math. Nearly every SAT form has a few, and the same small set of ideas decides them: constant ratio versus constant difference, what each number in means, and how to rewrite the exponent when the time unit changes. The hardest versions, late in a harder module 2, test percent language and equivalent forms rather than heavy algebra.
| Typical stem | What you must do |
|---|---|
| Which of the following is the best interpretation of (or of 1.04) in this context? | Name the initial value, or turn a factor into a percent change |
| Which equation could define ? (with a table or graph) | Find from and from the ratio |
3 questions on exponential functions and growth. Every option is explained once you answer.
Sign in to try the quick check
Answers are checked on our side, every option is explained, and your result feeds your mastery for this topic. It’s free.
The first 3 of 10 cards for this topic. Sign in and finish the lesson to review them with spaced repetition.
How do you tell linear from exponential in a table with equally spaced -values?
| Which situation is best modeled by an exponential function? | Separate constant percent change from constant amount change |
| Which equation models the population months after...? | Put the right fraction in the exponent |
| Which equivalent form shows the percent increase every 6 months as a constant or coefficient? | Rewrite the base and exponent so the exponent counts the period asked about |
| SPR: what is the value of , or ? | Solve from given values and type a plain number |
About 30% of SAT Math items are set in context. Exponential ones use populations, investments, depreciation, bacteria, medicine and radioactive decay.
| Linear | Exponential | |
|---|---|---|
| Equal steps in | add the same amount | multiply by the same factor |
| Table test | constant difference | constant ratio |
| Context words | “increases by 40 each year” | “increases by 4% each year”, “doubles every 3 hours” |
| Form | with , |
A percent of the current value means exponential. A fixed number of units per period means linear.
| Part | Meaning | Example |
|---|---|---|
| initial value, | : 850 at the start | |
| or | growth or decay factor per unit of | is +4%; is −15% |
| with | the quantity is multiplied by every units of | quadruples every 6 hours |
| with | half-life of | halves every 6 days |
A factor of means a percent change of .
| Factor | Change | Factor | Change |
|---|---|---|---|
| +7% | −7% | ||
| +100% (doubling) | −50% (halving) | ||
| +200% (tripling) | −75% | ||
| +300% | +44% |
When a model is written per year but you need months, change the exponent so that it counts years: months is years, so a 5% annual rate gives .
To show the factor for a different period, use the power rule :
For a half-year, : , which is 20% every six months, not 22%.
For , divide the outputs to cancel :
Take the root to find , then substitute back to find . If one of the points has , you can read directly.
y_1 ~ ab^{x_1} for an exact exponential model.