Unit 2 · Topic 1.13
Function Model Selection and Assumption Articulation
Select a function family from numerical, graphical, and contextual evidence, then explain the assumptions and restrictions that make the model defensible.
Learning Goals
- Distinguish a mathematical model from the real phenomenon it approximates.
- Use rates of change and finite differences to select linear, quadratic, or higher-degree polynomial models.
- Use symmetry, extrema, zeros, dimensions, and changes in behavior as model-selection evidence.
- Recognize when a piecewise-defined model is more appropriate than one global rule.
- Use residuals as supporting evidence without treating a numerical fit as proof.
- State assumptions about what remains consistent and how quantities vary together.
- Restrict domain and range using mathematical and contextual information.
- Communicate a model choice with a logical, evidence-based rationale.
1. A Model Is a Purposeful Approximation
A function model is a simplified rule that represents how an output quantity depends on an input quantity. A useful model preserves the features needed to answer a question, but it is not the phenomenon itself.
Before choosing a family, define the variables and units, identify the observed interval, and ask which behavior matters: constant change, curvature, turning points, zeros, geometric dimensions, or a change of rule.
2. Model-Selection Evidence
| Candidate family | Numerical or graphical evidence | Common contextual signal |
|---|---|---|
| Linear | Roughly constant rates or first differences | A quantity changes by about the same amount per input unit |
| Quadratic | Roughly linear first differences, constant second differences, or one symmetric maximum or minimum | Area, two changing dimensions, or accelerating change |
| Cubic or higher polynomial | Roughly constant nonzero higher differences, multiple zeros, or multiple extrema | Volume, three changing dimensions, or several changes in direction |
| Piecewise-defined | Different patterns on nonoverlapping input intervals | A threshold, policy, stage, or mechanism changes |
One clue is rarely enough. A strong choice combines at least two forms of evidence and explains why competing families are less suitable.
3. Equal Input Spacing Matters
Finite differences compare outputs at equally spaced inputs. If the input step is \(h\), first differences are
If the inputs are not equally spaced, raw output differences are not directly comparable. Compute rates by dividing by each input change or use a different representation.
4. Selecting a Linear Model
Suppose \(x\) is elapsed time in hours and \(L\) is a measured length in centimeters.
| \(x\) | 0 | 2 | 4 | 6 |
|---|---|---|---|---|
| \(L\) | 12 | 19 | 26 | 33 |
The output rises by 7 whenever the input rises by 2, so the rate is \(7/2=3.5\) centimeters per hour. A linear candidate is
This choice assumes the rate remains approximately constant over the interval where the model will be used.
5. Selecting a Quadratic Model from Differences
Consider the equally spaced data below.
| \(x\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \(y\) | 5 | 10 | 19 | 32 | 49 |
| First differences | 5 | 9 | 13 | 17 | |
| Second differences | 4 | 4 | 4 | ||
The constant nonzero second difference supports a quadratic model. For unit input spacing, a quadratic \(ax^2+bx+c\) has second difference \(2a\), so \(2a=4\) and \(a=2\). Since \(y(0)=5\), \(c=5\); using \(y(1)=10\) gives \(b=3\).
A roughly symmetric scatterplot with one clear maximum or minimum can also support a quadratic choice even when measurements are not exact.
6. Geometry Suggests Degree
If a rectangle has width \(x\) and length \(x+4\), its area is
Two varying dimensions often produce a quadratic model. Three varying dimensions often produce a cubic model; for a cube, \(V(x)=x^3\) with \(x>0\). Dimensional reasoning proposes a family, while the actual constraints determine the domain.
7. Higher-Degree Polynomial Evidence
For equally spaced inputs, roughly constant nonzero \(n\)th differences suggest a degree-\(n\) polynomial. Multiple real zeros or multiple maxima and minima can also indicate a polynomial beyond degree 2.
A polynomial of degree \(n\) or less can pass through \(n+1\) points having distinct inputs. However, exact interpolation alone does not show that the family describes the underlying process or predicts well outside those points.
8. When One Rule Is Not Enough
A local delivery service charges a flat amount for the first 5 kilometers and then adds a per-kilometer fee. One reasonable model is
The two formulas occupy nonoverlapping intervals and meet at the threshold. A single linear rule would miss the flat-fee stage, so the context supports a piecewise-defined model.
Piecewise modeling is also appropriate when a machine changes settings, a tax or fee bracket begins, or observed behavior shifts after a known event.
9. Residuals as Supporting Evidence
After fitting a candidate, compare each observation with its prediction.
- A positive residual means the model underpredicts that observation.
- A negative residual means the model overpredicts that observation.
- A residual of zero means the observation lies on the model.
Residuals measure vertical error in output units. They supplement the Topic 1.13 selection argument; they do not replace contextual reasoning.
10. Read the Residual Pattern
| Residual behavior | Interpretation |
|---|---|
| Small values scattered above and below 0 without structure | The candidate is supported over the observed interval |
| A curved or wave-like pattern | The selected family is missing systematic behavior |
| Residual spread grows as input grows | Prediction reliability changes across the domain |
| One unusually large residual | Check for an outlier, measurement error, or omitted condition |
For the quadratic \(Q(x)=2x^2+3x+5\), observations \(5.2,9.7,19.4,31.8,49.1\) at \(x=0,1,2,3,4\) produce residuals \(0.2,-0.3,0.4,-0.2,0.1\). Their small alternating signs support the model on this interval, but do not guarantee distant predictions.
11. Articulate the Assumptions
An assumption states what must remain true for the model to stay useful. Write assumptions as specific claims about the situation, not as a vague sentence that the pattern continues.
| Question | Possible assumption |
|---|---|
| What remains consistent? | The operating conditions and measurement method remain comparable |
| How do quantities vary together? | The output continues to respond to the input according to the selected rate pattern |
| Could the mechanism change? | No unmodeled threshold, policy, capacity limit, or external event occurs |
| Are the data representative? | The observations are sufficiently accurate for the intended prediction |
Example: For \(L(x)=3.5x+12\), state that the growth conditions remain stable and that length continues to increase by approximately 3.5 centimeters per hour over the modeled time interval.
12. Restrict Domain and Range
The algebraic domain is often wider than the meaningful domain. Restrictions can come from elapsed time, physical dimensions, capacity, observed extrema, or the interval represented by the data.
- Time and length may require nonnegative inputs or outputs.
- A finite experiment may justify only \(0\le x\le T\).
- Counts may require integer outputs or rounding.
- Percentages may require \(0\le y\le100\).
State rounding only after deciding what the output represents. Rounding a model input too early can change the prediction.
13. Interpolation, Extrapolation, and Limits
Interpolation predicts within the observed input interval. Extrapolation predicts outside it and relies on stronger assumptions.
A polynomial may eventually grow rapidly or become negative even when neither behavior is possible in context. Always identify whether a prediction is interpolated or extrapolated and explain why the model should or should not remain valid there.
14. A Complete AP-Style Justification
- Name the family: linear, quadratic, higher-degree polynomial, or piecewise-defined.
- Cite evidence: rates, differences, symmetry, extrema, zeros, dimensions, or changing intervals.
- Connect the context: explain why that mathematical behavior matches the quantities.
- State assumptions: identify what must remain consistent and how the variables continue to vary together.
- Give restrictions: specify a meaningful domain, range, discreteness, or rounding rule.
- Qualify predictions: distinguish interpolation from extrapolation and mention limits of the model.
Model response: A quadratic model is reasonable because equal input steps produce approximately constant nonzero second differences, and the context describes an area determined by two changing dimensions. This choice assumes the dimensional relationship remains unchanged. The domain is restricted to positive feasible lengths, and predictions far outside the measured interval should be treated cautiously.
Graph and Visual Model
Key Takeaways
- Select and justify linear, quadratic, polynomial, or piecewise models from rates, differences, graph features, geometry, and context; then state assumptions and meaningful domain and range restrictions.
- Core relationship: \(\Delta^n y\approx\text{constant and nonzero}\Rightarrow\text{degree-}n\text{ polynomial candidate}\)
- Error check: Do not select a model from appearance or exact fit alone; justify the family with numerical or contextual evidence and state its restrictions.
A measured output \(P\) is recorded at equally spaced inputs.
| \(x\) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| \(P\) | 7 | 12 | 23 | 40 | 63 |
- Compute the first and second differences and select a model family.
- Use the pattern to show that \(P(x)=3x^2+2x+7\) fits all five observations.
- Explain what numerical evidence makes a linear model less appropriate.
- If \(x\) counts completed production cycles, state suitable domain and range restrictions.
- Write two explicit assumptions needed to predict \(P(6)\).
- Classify the prediction at \(x=6\) as interpolation or extrapolation and discuss its reliability.
- Describe evidence that would make a piecewise model preferable in future data.