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Exponential Growth and Decay for the PSAT/NMSQT: Complete Study Guide + Free Practice Problems
Everything the PSAT/NMSQT tests about exponential growth and decay in one place: the exponential function formula, growth vs. decay factors, percent change over time, and comparing exponential to linear growth — with worked examples and a free practice quiz.
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Free · No signupExponential Growth and Decay
Growth and decay factors, percent change over time, and word problems.
What is exponential growth and decay?
An exponential quantity changes by a constant percentage or factor for every fixed unit of time, rather than by a constant amount. This means the quantity grows (or shrinks) faster and faster over time — the classic curved shape, distinct from the straight line of linear growth.
Common real-world exponential scenarios on the PSAT: population growth, investment/compound interest, radioactive decay, and depreciation of an asset's value.
The exponential function formula
a is always the value at the very start (t = 0) — it's what you get when you substitute 0 for t, since b⁰ = 1 for any nonzero b.
Growth vs. decay factors
| Factor b | Behavior |
|---|---|
| b > 1 | Exponential growth |
| 0 < b < 1 | Exponential decay |
A population starts at 800 and increases by 6% each year. Write a function for the population P after t years.
For decay, subtract the percentage from 100% — a value decreasing by 6% per year uses b = 1 − 0.06 = 0.94, not b = 0.06.
Percent change over time
A car's value is modeled by V(t) = 24000(0.88)ᵀ. What is the annual percent decrease in value?
Whenever a question gives you the equation and asks for the percent rate, isolate b first, then compare it to 1: subtract from 1 for decay, subtract 1 from b for growth.
Exponential vs. linear growth
| Type | Change pattern | Graph shape |
|---|---|---|
| Linear | Adds the same amount every step | Straight line |
| Exponential | Multiplies by the same factor every step | Curves faster and faster |
Even an exponential function with a small growth factor will eventually overtake a linear function with a much steeper starting rate — it just takes long enough for the multiplicative effect to compound.
Word problems
A bacteria colony starts with 200 cells and doubles every 3 hours. Write a function for the population P after h hours.
When growth or decay happens over a period other than one unit of time (like "doubles every 3 hours" instead of "every hour"), the exponent must be adjusted to a fraction of the elapsed time divided by that period — not left as a plain t.
Common PSAT traps
- Using the percentage directly as b: b is always 1 plus or minus the rate, never the raw percentage itself.
- Mixing up growth and decay: b > 1 means growth; 0 < b < 1 means decay.
- Forgetting to adjust the exponent for a non-yearly period: "every 3 hours" or "every 5 years" requires dividing t by that period in the exponent.
- Assuming exponential and linear models look the same short-term: over a short interval they may seem similar, but their long-run behavior is fundamentally different.
Test day strategy for exponential growth and decay
| Question signal | Fastest approach |
|---|---|
| "Increases/decreases by ___% each ___" | Growth factor = 1 + rate; decay factor = 1 − rate |
| Equation given, percent rate asked | Isolate b, then compare to 1 |
| "Doubles/triples every ___" language | Adjust the exponent to (elapsed time) / (period) |
| Comparing which model grows faster long-term | Exponential eventually outgrows linear, regardless of starting rate |
| "What is the initial value" question | Substitute t = 0 into the function, or read a directly |
Now put it to work
One comprehensive quiz covering the full topic — growth and decay factors, percent change over time, and word problems.
Exponential Growth and Decay
Growth and decay factors, percent change over time, and word problems.