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
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide