Topic 1.13 Notes – Function Model Selection and Assumption Articulation

Choosing a Function Model

A model is useful, not perfect. It captures the main pattern in a situation over a stated set of inputs.

A complete model choice has three parts:

1. Function type
   Linear, quadratic, polynomial, or piecewise-defined.

2. Evidence
   What in the table, graph, or context supports that type.

3. Assumptions and restrictions
   What must stay true, and what inputs/outputs actually make sense.

The main types here are:

• Linear: one steady rate of change
• Quadratic: rates change in a steady way; parabola shape
• Polynomial: more turning points or more zeros than a line/parabola can handle
• Piecewise-defined: different rules on different intervals

A common test mistake is choosing the fanciest model that can hit every data point. Usually you want the simplest model that matches the main behavior.

Evidence for Each Function Type

Linear

A linear model fits when the output changes by about the same amount for each unit of input.

Clues:

• Roughly constant rate of change
• For equally spaced inputs, first differences are roughly constant
• For unequally spaced inputs, compare average rates of change
  $\frac{f(b)-f(a)}{b-a}$
• On a graph, points lie about on a line
• In words, you may see “changes by about the same amount per unit”

Just increasing is not enough. Exponential and quadratic functions can increase too.

Quadratic

A quadratic model fits when the rates of change themselves change linearly.

Clues:

• For equally spaced inputs, second differences are roughly constant and nonzero
• Graph looks parabola-like
• Often roughly symmetric
• Has one clear maximum or minimum
• Geometry clue: area or a product of two linearly changing dimensions often gives a quadratic

Example: rectangle area $A(x)=x(24-x)$ is quadratic.

Cubic and higher-degree polynomial

These show up when the data has more bends or more zeros.

Clues:

• Constant nonzero third differences suggest a cubic
• Constant nonzero $n$th differences suggest degree $n$
• Multiple real zeros or multiple turning points suggest a polynomial of high enough degree
• A degree $n$ polynomial has:
  • at most $n$ real zeros
  • at most $n-1$ turning points
• Geometry clue: volume with three linearly changing dimensions often suggests a cubic

Example: $V(x)=x(20-2x)(14-2x)$ is cubic.

Piecewise-defined

A piecewise-defined function uses different rules on different nonoverlapping intervals.

Use it when behavior changes by:

• stage
• threshold
• slope or curvature
• price or policy
• rate

Its pieces can be linear, quadratic, polynomial, or even constant. Endpoint notation matters because it tells you which piece includes the transition value.

Finite Differences and Point-Based Clues

This is one of the most testable parts.

1. Check whether the input values are equally spaced.
2. If they are, compute:
   • first differences
   • second differences
   • third differences, if needed
3. Match the first roughly constant nonzero level:
   • first differences constant $\rightarrow$ linear
   • second differences constant $\rightarrow$ quadratic
   • third differences constant $\rightarrow$ cubic

If inputs are not equally spaced, raw differences are misleading. Use average rate of change instead.

Also remember the interpolation fact:

• 2 points with distinct inputs $\rightarrow$ polynomial of degree at most 1
• 3 points with distinct inputs $\rightarrow$ degree at most 2
• 4 points with distinct inputs $\rightarrow$ degree at most 3

The phrase “degree $n$ or less” matters. Three points might still lie on a line. And an exact fit through points does not automatically make that the best contextual model.

What a model is allowed to assume

A model only works under certain conditions.

Assumptions about what stays consistent

You may assume things like:

• conditions stay about the same
• dimensions remain fixed except for the stated change
• prices, procedures, or environment do not shift unexpectedly

Say this in context. “The cost per item stays constant” is better than just saying “it is linear.”

Assumptions about how quantities change together

Different models assume different behavior:

• Linear assumes equal input changes give about equal output changes
• Quadratic assumes rates of change vary about linearly
• Polynomial assumes the difference pattern reflects a real relationship, not random noise

Domain restrictions

The domain is the allowed input values.

Restrictions can come from:

• the formula itself
• the context, like nonnegative time or length
• the observed data interval
• practical endpoints such as capacity or when a process ends

Extrapolating beyond the data interval needs extra justification.

Range restrictions

The range is the allowed output values.

Common restrictions:

• outputs must be nonnegative
• percentages stay between $0$ and $100$
• counts may need whole numbers
• values may be capped by capacity

Round final interpreted answers when the context requires it, but keep intermediate calculations unrounded.

What Students Mix Up

• Increasing alone does not mean linear.
• Finite differences only work cleanly when inputs are equally spaced.
• Area often suggests quadratic and volume often suggests cubic, but only after checking how dimensions depend on the input.
• A higher-degree polynomial is not automatically better just because it passes through every point.
• A piecewise model can have a jump or just a change in slope.
• Naming a model type is incomplete unless you also give evidence, assumptions, and valid domain/range.
• Far extrapolation is weak unless the context clearly supports continuing the pattern.