Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for exponential models algebra 2? Then this Algebra 2 lesson on Exponential Models: Growth, Decay, and Interest explains each idea step by step, with worked examples.
Quick answer: An exponential model y = a · bt grows if b > 1 and decays if 0 < b < 1. Compound interest uses A = P(1 + r/n)nt, continuous growth uses A = Pert, and half-life problems raise 1/2 to the power t over the half-life.
Standards: F-IF.C.8b, F-LE.A.4, F-LE.B.5 · Level: Grade 11 Algebra 2
What does an exponential model look like?
An exponential function has the form y = a · bt, where a is the initial amount and b is the growth or decay factor. If b > 1 the quantity grows; if 0 < b < 1 it decays. A percent change of r per period gives b = 1 + r for growth and b = 1 − r for decay.
| Situation | Factor b | Model |
|---|---|---|
| Grows 6.5% per year | 1.065 | y = a(1.065)t |
| Loses 12% per year | 0.88 | y = a(0.88)t |
| Doubles every d periods | 21/d | y = a · 2t/d |
| Half-life h | (1/2)1/h | y = a(1/2)t/h |
How does compound interest work?
Compounded n times per year: A = P(1 + rn)nt.
Compounded continuously: A = Pert, where e ≈ 2.71828.
Here P is the principal, r is the annual rate as a decimal, and t is time in years. More frequent compounding earns slightly more, and continuous compounding is the limit.
Find the value of $4,000 at 3.6% compounded monthly for 5 years. Here n = 12 and nt = 60: A = 4,000(1 + 0.036/12)60 = 4,000(1.003)60 ≈ $4,787.58.
What are half-life and doubling time?
A half-life is the time for a quantity to fall to half its amount; a doubling time is the time for it to double. In both cases the factor 2 or 1/2 is raised to t divided by the period.
- Find the initial amount a.
- Find the factor b from the percent rate, doubling time, or half-life.
- Write y = a · bt and check one known value.
- Substitute to answer the question and attach units.
A 12% decrease uses b = 0.88, not 0.12. Also, do not multiply the rate by t: that describes linear change, not exponential change.
Check the factor first: if it is above 1 the graph must rise, and if it is below 1 it must fall.
- y = a · bt: a is the initial value and b the growth or decay factor.
- A percent increase of r gives b = 1 + r; a percent decrease gives b = 1 − r.
- Compounded n times a year: A = P(1 + r/n)nt. Continuous: A = Pert.
- Half-life models use (1/2)t/h; doubling models use 2t/d.
- More frequent compounding earns more, but continuous compounding sets an upper limit.
Exponential Models: Growth, Decay, and Interest: frequently asked questions
How do you tell growth from decay in y = a · bt?
Look at b. If b is greater than 1 the quantity grows, and if b is between 0 and 1 it decays. The percent change per period is |b − 1| written as a percent.
What is the number e?
The number e ≈ 2.71828 is the base that arises when interest is compounded more and more often. Continuous growth is modeled by A = Pert, where r is the annual rate as a decimal.
How do you write a half-life model?
Use N(t) = N0(1/2)t/h, where N0 is the starting amount and h is the half-life. After each period of length h the amount is cut in half.
Why does compounding more often earn more?
Interest earned in each period is added to the balance and then earns interest itself. With more compounding periods this happens sooner, so the balance is slightly higher, but the gain shrinks toward the continuous limit.
What do the parameters of an exponential model mean?
The parameter a is the value at t = 0. The factor b tells how much the value is multiplied by during one period, so b − 1 is the percent rate of change per period.
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