Exponential Functions and Data Modeling
Exponential Functions and Data Modeling
This strand tests exponential growth and decay models, geometric sequences, and the statistics of fitting a model to data: scatter plots, lines and curves of best fit, correlation, and residuals. Expect percent-change contexts such as population, depreciation, and compound interest, plus items asking you to choose between a linear and an exponential model.
Core rules
- Exponential form f(x) = a·bˣ, where a is the initial value and b the growth factor. Growth: b = 1 + r; decay: b = 1 − r, with r written as a decimal. The graph has y = 0 as a horizontal asymptote.
- Compound interest: A = P(1 + r/n)^(nt). Continuous compounding: A = Pe^(rt).
- Geometric sequence: aₙ = a₁r^(n−1). An arithmetic sequence has a constant difference; a geometric one has a constant ratio.
- A trend line summarizes a scatter plot. A correlation coefficient near 1 or −1 indicates a strong linear association; near 0 indicates a weak one. Correlation does not establish causation.
- Interpolation inside the data range is safer than extrapolation far beyond it.
Worked example
$2,000 is invested at 6% annual interest compounded quarterly. Find the balance after 3 years.
A = 2000(1 + 0.06/4)^(4·3) = 2000(1.015)^12 ≈ 2000(1.1956) ≈ $2,391.24. The rate per period is 0.06/4 = 0.015 and the number of periods is 12, not 3.
Traps that cost marks
- Using r instead of r/n, or multiplying n·t incorrectly.
- Adding 6 to a quantity for a 6% increase instead of multiplying by 1.06.
- Reading a decay factor of 0.94 as a 94% decrease; it is a 6% decrease.
- Treating a strong correlation as proof of causation.
- Extrapolating a fitted model far outside the observed data.