The big idea: Before fitting a model, choose the right type. Each model has a signature pattern you can spot from the context or from the shape of a scatter plot.
| Model | Equation | Signature pattern | Key word clues |
|---|---|---|---|
| Linear | y = mx + c | Straight line — constant rate of change | 'per unit', 'fixed rate', 'constant speed' |
| Quadratic | y = ax² + bx + c | Parabola — rises then falls (or vice versa) | 'projectile', 'profit vs price', 'maximum area' |
| Exponential | y = a·bˣ | Rapid growth or decay — multiplied each period | 'doubles', 'halves', 'percentage increase/decrease' |
| Power | y = axⁿ | Curved but not parabolic — no turning back | 'proportional to x²', 'cube root', 'inversely proportional' |
| Sinusoidal | y = a·sin(bx) + d | Regular repeating cycle | 'tides', 'hours of daylight', 'annual temperature cycle' |
Start with the context: Before looking at data, read the problem context. Words like 'doubles each year' → exponential; 'repeats every 12 hours' → sinusoidal; 'constant rate' → linear. This narrows your choice before you even graph the data.
The big idea: If you are given a scatter plot rather than a description, look at the overall shape of the data cloud. Each model type produces a distinctive curve (or line).
| Scatter plot shape | Most likely model |
|---|---|
| Points lie close to a straight line | Linear |
| Points form an upside-down U or U shape | Quadratic |
| Points rise steeply at first then level off (or vice versa) | Exponential or power |
| Points wave up and down in a regular cycle | Sinusoidal |
Justify your choice: IB may ask you to justify why you chose a particular model. State TWO things: (1) the shape of the scatter plot, and (2) the context. E.g. 'The data shows an increasing curve that levels off, consistent with a decay model. The context (bacteria dying) supports exponential decay.'
Worked example
Apply the key method from Choosing the right model type in a typical IB-style question.
Step by step
- Write the relevant formula or rule first.
- Substitute values carefully and show each step.
- State the final answer with correct units/context.
Final answer
Clear method and context-based interpretation secure most marks.
Memorize terms 3x faster
Smart flashcards show you cards right before you forget them. Perfect for definitions and key concepts.
Try Flashcards Free7-day free trial • No card required
The big idea: Exponential (y = a·bˣ) and power (y = axⁿ) models can look similar — both are curves. The key difference: exponential grows (or decays) faster than any power for large x. Use the context clue: if you're multiplying each period, it's exponential; if it's a physical formula (area, volume), it's likely a power model.
Exponential
- x is the exponent: y = a·bˣ
- Rate of change accelerates
- Grows faster than any power for large x
- Common: population, finance, radioactive decay
Power
- x is the base: y = axⁿ
- Rate of change also accelerates but more regularly
- Grows as a fixed power of x
- Common: area, volume, physical laws
The tell-tale phrase: 'Each year the value is multiplied by 1.5' → exponential (b = 1.5). 'The area is proportional to the square of the radius' → power (n = 2). Look for multiplication each period vs a power relationship.
The big idea: Model selection is a skill that takes practice. Work through a systematic check: Is it periodic? → sinusoidal. Does it multiply? → exponential. Does it have a turning point? → quadratic. Is it a power law? → power. Is the rate constant? → linear.
Model selection walkthrough
A scientist measures the number of bacteria in a sample every hour: 100, 200, 400, 800, 1600. What model is appropriate?
Step by step
- Check: is the change constant per hour?
- Check: is the ratio constant per hour?
- Conclusion: exponential growth with a = 100, b = 2.
Final answer
Exponential model: N = 100 · 2ᵗ. The constant ratio of 2 each hour confirms exponential growth.
Difference or ratio test: For linear: check first differences (constant?). For exponential: check ratios between successive values (constant?). For quadratic: check second differences (constant?). These quick checks work on data tables in IB questions.