The idea
In an exponential function the variable is in the exponent: f(x) = a·bˣ, with b > 0 and b ≠ 1. The number a is the starting value (the y-intercept, since b⁰ = 1) and b is the growth factor per step.
- If b > 1 the function grows.
- If 0 < b < 1 it decays.
A growth rate r gives b = 1 + r; a decay rate r gives b = 1 − r. So 6% growth means b = 1.06 and 10% decay means b = 0.9.
The graph
For b > 1, the graph rises steeply to the right and flattens toward the x-axis on the left. The x-axis (y = 0) is a horizontal asymptote. The domain is all real numbers; the range is y > 0. Adding a constant shifts the asymptote: f(x) = 2ˣ + 3 has asymptote y = 3.
Solving simple exponential equations
If both sides can be written with the same base, set the exponents equal. For 5^(x + 1) = 125, write 125 = 5³, so x + 1 = 3 and x = 2. For 2ˣ = 32, note that 32 = 2⁵, so x = 5. If the bases cannot be matched, you need logarithms (next topic).
Compound interest
If P is invested at annual rate r, compounded n times per year, after t years:
A = P(1 + r/n)^(nt)
For yearly compounding this is A = P(1 + r)^t.
Worked example
You invest $2,000 at 5% compounded annually. What is it worth after 3 years?
- A = 2000(1.05)³
- 1.05³ = 1.157625
- A = 2000 × 1.157625 = $2,315.25
Compare with simple interest, which would give 2000 + 3(100) = $2,300. Compounding earns interest on interest, so it is larger.
Doubling and half-life
A quantity that doubles every 5 days from 4 units is 4 × 2^(t/5). At t = 15 that is 4 × 2³ = 32.
Decay works the same way with a factor of 1/2 per half-life. An 80 mg sample with a 3-day half-life has 40 mg after 3 days, 20 after 6, and 10 mg after 9 days. In general amount = 80 × (1/2)^(t/3).
Continuous growth
When compounding is continuous, A = Peʳᵗ, where e ≈ 2.718 is a special constant. It appears in many science models.
Common mistakes
- Using b = r instead of b = 1 + r. A 6% rise is 1.06, not 0.06.
- Rounding the factor too early. Keep extra digits until the end.
- Confusing a linear model (adds the same amount) with an exponential one (multiplies by the same factor).