Fitting and refining a function model
| English | 中文 | Pinyin |
|---|---|---|
| residual/rɪˈsɪdʒuːəl/ | 残差 | cán chà |
Two different curve models fit the same three measurements. Which extra observation can help choose between them?
- Two different curve models fit the same three measurements. Which extra observation can help choose between them?
- This lesson studies residual 残差: An observed output minus the output predicted by a model.
Choose the mathematical structure
- State inputs, outputs, units and the modelling domain. Fit parameters using the stated observations, then compare predictions with independent measurements. An exact fit to the calibration points does not prove the chosen function is correct everywhere. A residual records observed minus predicted output; model refinement needs evidence and a reasoned change.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines residual?
An observed output minus the output predicted by a model.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For a proposed arch, x is horizontal distance and h its height, both in metres. Use the illustrative points (0,0),(3,9),(6,0), with 0≤x≤6. Let h=ax²+bx+c. Then c=0, 9a+3b=9 and 36a+6b=0 give a=−1,b=6. Thus h=−x²+6x=9−(x−3)², with maximum 9 m at x=3. A piecewise-linear alternative h=3x for 0≤x≤3 and h=18−3x for 3≤x≤6 fits the same three points. At x=1, an illustrative extra measurement 5.2 m gives residual 0.2 m for the quadratic but 2.2 m for the linear alternative. This favours the quadratic at that point, not a universal verdict.
Fitting and refining a function model
State inputs, outputs, units and the modelling domain
Check the evidence and conditions behind a transformed or fitted function.
Find a in the fitted quadratic h=ax²+bx+c.
Subtract twice 9a+3b=9 from 36a+6b=0: 18a=−18, hence a=−1.
Test a tempting shortcut
- Do not use a fitted calibration point as independent validation. Outside 0≤x≤6, the quadratic can give negative height and no longer describes the proposed arch. A smaller residual at one point does not establish accuracy everywhere. The simpler model can be easier to interpret but still miss curvature.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A function that fits three calibration points is proven correct at every input. This claim is false. Explain which definition or assumption it violates.
Find its maximum modelled height in metres.
Complete the square: h=9−(x−3)², with maximum 9.
A function that fits three calibration points is proven correct at every input.
Do not use a fitted calibration point as independent validation. Outside 0≤x≤6, the quadratic can give negative height and no longer describes the proposed arch. A smaller residual at one point does not establish accuracy everywhere. The simpler model can be easier to interpret but still miss curvature.
Interpret a new situation
- For height at least 8 m, 9−(x−3)²≥8 gives 2≤x≤4: modelled horizontal width 2 m. Collect more intermediate measurements and estimate measurement uncertainty before relying on this clearance. Thickness and changes in shape can require a refined domain or formula; explain which observation motivates the refinement.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the modelled horizontal width where h≥8, in metres.
The allowed interval is [2,4], with width 4−2=2 metres.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 7357 · A-level · B. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
An observed output minus the output predicted by a model. Choose the relationship, show the method, check its assumptions and interpret the result.