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.
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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.'
Picking a model from a scatter plot
A scatter plot shows data that rises slowly at first, then increases more steeply, and never crosses the x-axis.
The context: bacteria population over time.
Which model type best fits?
Step by step
- Look at the shape: rising curve, increasing rate, never crosses x-axis. This rules out linear (no straight line) and rules out quadratic (no turning point).
- Bacteria multiply each period (a constant ratio of cells per generation) — that is the signature of exponential growth.
- Power y = axⁿ would also rise but would typically pass through (0, 0). Bacteria start with a > 0 population, so y-intercept is not zero — fits exponential better.
Final answer
Exponential growth (y = a·bˣ with b > 1). The combination of a continuously accelerating rise AND the population context (constant ratio per period) is the decisive signal.
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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.