Nonlinear Functions • Topic 2 of 3

Exponential Functions

An exponential function has the form f(x) = a · bˣ, where the variable is in the exponent. The base b controls behavior: b > 1 gives growth (values rise quickly), and 0 < b < 1 gives decay (values shrink toward zero). To evaluate, raise the base to the input power first, then multiply by a — for f(x) = 3 · 2ˣ, f(4) = 3 · 2⁴ = 3 · 16 = 48. Note f(0) = a, since b⁰ = 1. Exponential functions grow far faster than linear or quadratic ones for large x. The SAT tests evaluating them and identifying growth versus decay from the base.

sat30t2 graphO(0, 1)Exponential functiony = 2 to the power x — exponential growth

✅ Solved examples

1. If f(x) = 3 · 2ˣ, find f(4).
3 · 2⁴ = 3 · 16 = 48.
2. If f(x) = 5 · 3ˣ, find f(2).
5 · 9 = 45.
3. Does y = 100 · (1.2)ˣ show growth or decay?
Base 1.2 > 1, so growth.
4. Does y = 80 · (0.5)ˣ show growth or decay?
Base 0.5 < 1, so decay.

✏️ Practice — try these, take hints as needed

1. If f(x) = 2 · 3ˣ, find f(2).
2 · 3².
2 · 9.
—
18.
2. If f(x) = 4 · 2ˣ, find f(3).
4 · 8.
—
—
32.
3. If f(x) = 5 · 2ˣ, find f(0).
2⁰ = 1.
5 · 1.
—
5.
4. Does y = 50 · (3)ˣ represent growth or decay?
Base > 1?
—
—
Growth.
5. Does y = 200 · (0.8)ˣ represent growth or decay?
Base between 0 and 1.
—
—
Decay.

📝 Topic test — 8 questions

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