Scatterplots and linear, quadratic and exponential models
Choose a model from a pattern, interpret its parameters and bound predictions to what the data support.
What you will tackle
- Interpret association and scatterplots
- Fit and use linear, quadratic and exponential models
- Interpret model coefficients, graphs and predictions
- Compare equal-difference linear growth with equal-ratio exponential growth
First inspect the shape
A scatterplot displays paired observations. A roughly straight trend suggests a linear model; a curved trend may need a quadratic or exponential model. A negative association means larger x tends to accompany smaller y, not that one variable has been shown to cause the other. A fitted line need not pass through every point.
A model is a claim that can be checked
For ŷ = 3x + 8, the predicted increase per unit of x is 3. A residual is observed y minus predicted y. Patterns in residuals can show that a straight line is inadequate. Equal time-step differences support linear growth; equal positive ratios support exponential growth. Predictions far beyond the data need stronger assumptions.
| x | B | C |
|---|---|---|
| 0 | 2 | 3 |
| 1 | 5 | 6 |
| 2 | 10 | 12 |
| 3 | 17 | 24 |
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
A model is ŷ = 3x + 8. Predict y at x = 4, and find the residual if the observed value is 23. Which table pattern is exponential: 4,7,10,13 or 4,8,16,32 at equal x steps?
Show a starting hint
A residual keeps the sign of observed minus predicted.
Compare with the explained answer
- The predicted value is 3(4) + 8 = 20.
- Residual = 23 − 20 = 3.
- The first sequence has constant difference 3; the second has constant ratio 2.
Answer / response criteria: Prediction 20; residual +3; the doubling sequence is exponential.
Guided 2
Read Scatterplot A, whose points include (0,2),(1,5),(2,8),(3,11). Derive a linear rule from the plotted points and predict at x = 5.
Compare with the explained answer
- The vertical change is 3 per unit horizontal change.
- The point (0,2) gives intercept 2, so y = 3x + 2.
- At x = 5 the model predicts 17; all displayed A points lie on this line.
Answer / response criteria: y = 3x + 2; prediction 17.
Independent practice
Independent 1
Values at x = 0,1,2 are 2,5,10. Does y = 3x + 2 fit all three? Test y = x² + 2x + 2 and predict at x = 3.
Compare with the explained answer
- The line gives 2,5,8, so it misses the third value.
- The quadratic gives 2,5,10 and fits these three points.
- At x = 3 it predicts 9 + 6 + 2 = 17; matching three points does not prove future validity.
Answer / response criteria: The line fails; the quadratic fits and predicts 17 under the model.
Independent 2
From Dataset B's points (0,2),(1,5),(2,10), determine a quadratic y = ax² + bx + c. From Dataset C's values 3,6,12,24 at x = 0,1,2,3, determine an exponential model.
Compare with the explained answer
- At x = 0, c = 2. The next two points give a + b = 3 and 4a + 2b = 8.
- Subtract twice the first relation: 2a = 2, so a = 1, b = 2.
- C starts at 3 and doubles each step, so y = 3·2^x.
Answer / response criteria: B: y = x² + 2x + 2. C: y = 3·2^x.
Application practice
Application 1
Plan A starts at 100 and adds 30 each year. Plan B starts at 100 and grows 20% per year. Compare years 1 and 3, and explain why a short comparison cannot establish which always stays larger.
Compare with the explained answer
- A(t) = 100 + 30t; B(t) = 100(1.2)^t.
- At year 1: A = 130, B = 120. At year 3: A = 190, B = 172.8.
- The models have different growth mechanisms; examining only those years is not a proof about all times.
Answer / response criteria: Year 1: 130 vs 120. Year 3: 190 vs 172.8. Long-run claims need further analysis.
Application 2
A study's scatterplot clusters near (0,2),(1,5),(2,8),(3,11), but one measured point is (2,8.4). Suggest a simple model, find that point's residual, and identify what evidence could show the line is inadequate.
Compare with the explained answer
- A simple approximate model is y = 3x + 2.
- At x = 2 its prediction is 8; the residual is 8.4 − 8 = 0.4.
- Systematic curved residual patterns or later values far from the line would suggest revising the model.
Answer / response criteria: For example y = 3x + 2; residual +0.4; inspect residual patterns and new observations rather than claiming a unique exact fit.
What can you explain now?
A study's scatterplot clusters near (0,2),(1,5),(2,8),(3,11), but one measured point is (2,8.4). Suggest a simple model, find that point's residual, and identify what evidence could show the line is inadequate.
Compare with the explanation
For example y = 3x + 2; residual +0.4; inspect residual patterns and new observations rather than claiming a unique exact fit.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
After trying it yourself, choose your next review. This is your self-assessment.
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Source and review status
Core coverage follows the detailed SAT skills in Appendix B of the September 2026 framework. The conditional School Day Essay is separate. The framework is maintained rather than issued as an annual subject syllabus.
Original teaching and explained self-check practice are available. Independent subject-specialist review and difficulty calibration remain pending; this is not an exam-board endorsement or a mastery certificate.
Official source specification / guide