Exponential functions · SAT Maths

Concept

An exponential function has the form y = a · b^x where a is the starting value (the y-intercept at x = 0) and b is the growth/decay factor per period.

Growth — b > 1 (e.g. b = 1.05 means 5% growth per period).
Decay — 0 < b < 1 (e.g. b = 0.85 means 15% decay per period).
Compound interest — A = P(1 + r)^t for t annual periods.
Half-life decay — y = a · (1/2)^(t/T) where T is the half-life.

Whatever the period, exponential change multiplies rather than adds. That's why even small growth rates compound to large quantities over time.

Worked example 1

A car loses 15% of its value every year. If it's worth $20,000 today, what is it worth in 4 years?

Solution

Setup. Decay: b = 1 − 0.15 = 0.85. V = 20000 · (0.85)^t

Plug t = 4. V = 20000 · (0.85)⁴ ≈ 20000 · 0.5220

Approximately $10,440.

Worked example 2

A bacterial culture triples every 2 hours. Starting from 100 cells, how many are there after 6 hours?

Solution

Periods. 6 / 2 = 3 tripling periods.

Compute. N = 100 · 3³ = 100 · 27

2,700 cells.

Practice test

8 questions on growth/decay rates, compound interest, half-life, and reading exponential models.

Practice test · 8 questions Question 1 of 8 · Score 0