Choosing the right function
A function model describes a real relationship so you can understand, predict and decide. The first step is choosing the right type.
| Pattern | Function type | Example |
|---|---|---|
| constant rate of change | linear | hourly wages |
| constant second differences, rises and falls | quadratic | projectile height |
| constant ratio (percent change) | exponential | population, compound interest |
| rapid change then levelling off | logarithmic | learning curves, sound perception |
| repeating cycles | sinusoidal | tides, daylight hours |
| inverse relationship | rational | time vs speed for a fixed distance |
Clues from tables
| Check | Suggests |
|---|---|
| First differences constant | linear |
| Second differences constant | quadratic |
| Ratios constant | exponential |
| Values repeat in a cycle | sinusoidal |
Clues from context
- "Increases by $5 each month" → linear.
- "Increases by 5% each year" → exponential.
- "Thrown into the air" → quadratic.
- "Repeats every 12 hours" → sinusoidal.
Regression with technology
Graphing calculators, spreadsheets and Desmos can perform regression: finding the best-fitting function for data.
The coefficient of determination, R², measures fit: R² = 1 is perfect; values close to 1 indicate a strong fit.
Example data: World population (billions)
| Year | 1960 | 1980 | 2000 | 2020 |
|---|---|---|---|---|
| Population | 3.0 | 4.4 | 6.1 | 7.8 |
A linear model fits well over this range, but over longer periods population growth isn’t purely linear or exponential, so a model must be chosen carefully.
Residuals
A residual = actual value − predicted value. Small, randomly scattered residuals indicate a good model. A pattern in residuals suggests a different function type would fit better.
Making predictions
- Interpolation (inside the data range): usually reliable.
- Extrapolation (outside it): risky. Conditions may change.
Example: cooling
A cup of coffee cools: 90 °C, 75 °C, 64 °C, 56 °C at 0, 5, 10, 15 min. Differences shrink, approaching room temperature: an exponential decay model fits, T = 20 + 70(0.955)ᵗ (approximately).
Example: sinusoidal
Daylight hours in a Canadian city vary from about 8.5 h in December to about 15.5 h in June. Model:
D(t) = 3.5 sin(30(t − 3)) + 12, with t in months and the angle in degrees.
Worked examples
Example 1: A gym charges $40 to join and $25/month. Model? C = 25m + 40 (linear).
Example 2: Data with ratios 1.5, 1.5, 1.5. Type? Exponential.
Example 3: A model has R² = 0.98. Good fit? Yes, a strong fit.
Common mistakes
- Choosing a model without checking differences or ratios.
- Trusting extrapolations far beyond the data.
- Ignoring context: negative heights or populations make no sense.
Try it yourself
- Data: 3, 6, 12, 24. Model type?
- Data: 5, 8, 11, 14. Model type?
- A ball’s height over time. Model type?
- Why is predicting a city’s population 200 years from now risky?
Answers: 1) exponential 2) linear 3) quadratic 4) It’s extreme extrapolation; growth conditions change.
Quick recap
- Choose a function model from data patterns and context.
- Use regression and check R² and residuals.
- Interpolation is safer than extrapolation.