Exponential functions describe anything that doubles, halves, or multiplies by a constant — population growth, compound interest, radioactive decay. The pattern: the variable lives in the exponent.
Identify $a$ (starting value) and $b$ (growth factor). Convert percent rates to factors: $1 + r$ for growth, $1 - r$ for decay.
A quantity starts at 1000 and increases by 10% each year. Which function models the quantity Q after t years?
Worked examples
Example 1
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Example 2
A radioactive isotope has a half-life of 8 years. If the initial amount is 80 grams, how many grams remain after 24 years?
Common pitfalls
Confusing exponential with linear
Decreases by 4% per year is exponential (multiplicative). Decreases by 4 per year is linear (additive). The word percent almost always signals exponential.
Using the wrong growth factor
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Forgetting to divide by the period in the exponent
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Plugging in $b$ when you should plug in $b - 1$
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Key takeaways
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Linear adds; exponential multiplies. Look for percent or each year to know which model fits.
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Try it yourself
5 practice questions on Exponential Functions, drawn from the question bank. The tutor is one click away if you get stuck.
An exponential function has the form f(x)=a⋅bx where:
a = the starting value (the value when x=0).
b = the growth factor — what you multiply by each time x increases by 1.
b>1: growth (function increases). f(x)=100⋅2x doubles every step.
0<b<1: decay (function decreases). f(x)=100⋅(0.5)x halves every step.
The crucial difference from linear functions: linear adds a constant; exponential multiplies by a constant.
Linear: y=mx+b — y increases by m for every +1 in x.
Exponential: y=a⋅bx — y multiplies by b for every +1 in x.
Exponentials grow much faster than linear functions over time, even if they start slower.
Percent growth/decay translation:
Growing by 5% per year: b=1+0.05=1.05. So f(t)=a⋅(1.05)t.
Decaying by 8% per year: b=1−0.08=0.92. So f(t)=a⋅(0.92)t.
Doubling every k years: f(t)=a⋅2t/k.
Half-life of h years: f(t)=a⋅(0.5)t/h.
SAT word-problem template:
"A bacteria population starts at 200 and doubles every 3 hours."
→ P(t)=200⋅2t/3.
The /3 in the exponent says it takes 3 hours for the doubling to happen once.
Compound interest — a special case of exponential growth:
A=P(1+nr)nt
where P is principal, r is annual interest rate (as decimal), n is times compounded per year, t is years. For annual compounding, simplify to A=P(1+r)t.
Reading exponential graphs:
y-intercept =a (the starting value).
If the curve rises sharply, b>1 and b is large.
If the curve approaches but never touches the x-axis from above, 0<b<1.
A town's population was 12,000 in 2020 and is decreasing by 4% per year. Which function models the population t years after 2020?
Growing by 5% → factor is 1.05, not 0.05. Decaying by 5% → factor is 0.95, not −0.05. Always: b=1± (rate as decimal).
Doubles every 3 years → f(t)=a⋅2t/3, NOT a⋅2t or a⋅2⋅3. The /3 scales time so the doubling happens once per 3 years.
If the function is a⋅(1.07)t, the growth rate is 7% per year (not 107%). The factor is 1.07. The rate is 0.07. Don't conflate them.
Exponential function: f(x)=a⋅bx. a = starting value. b = growth factor.
b>1: growth. 0<b<1: decay.
Percent translation: growing by r → b=1+r. Decaying by r → b=1−r.
Doubling every k years: a⋅2t/k. Half-life h: a⋅(0.5)t/h.