Topic 1.14 Notes – Function Model Construction and Application

What a Complete Function Model Is

A function model connects an input variable to an output variable in a way that matches a scenario, graph, or data set.

A complete model includes all of these:

• Variables
  Say what each variable means and include units.
  Example: $x$ is cut size in centimeters, $V(x)$ is volume in cubic centimeters.
• Function rule plus domain
  The equation alone is incomplete. A box model like $V(x)=x(20-2x)(14-2x)$ only makes sense for certain $x$-values.
• Range restrictions, rounding, assumptions
  Money may need cents, population may need whole numbers, and a model may assume constant rate, fixed dimensions, or negligible thickness.

One easy-to-miss point is that the function family is already chosen here. You are not deciding whether it should be linear or polynomial. You are building the specific model in that family.

Clues can come from:

• a scenario
• a graph
• a table
• stated features like points, zeros, extrema, asymptotes, and restrictions

Building the Model from Conditions or Structure

This is where conditions become algebra.

Linear models

For a linear model,

$$L(x)=mx+b$$

Use two points to find slope:

$$m=\frac{y_2-y_1}{x_2-x_1}$$

Then plug one point in to find $b$.

Example with points $(3,14)$ and $(7,26)$:

$$m=\frac{26-14}{7-3}=3$$

So $14=3(3)+b$, which gives $b=5$.
Model: $L(x)=3x+5$

Here, slope means constant rate of change, with units like dollars per hour.

Polynomial models from zeros

If a polynomial has zeros $r_1,r_2,\dots$ with multiplicities $m_1,m_2,\dots$, then

$$P(x)=a(x-r_1)^{m_1}(x-r_2)^{m_2}\cdots$$

• The sum of multiplicities matches the degree.
• Use one extra point to find $a$.
• Odd multiplicity means the graph crosses the axis.
• Even multiplicity means it touches and turns.

If you build from standard form, a degree $n$ polynomial needs enough conditions to solve for its coefficients. In general, $n+1$ distinct points determine one polynomial of degree at most $n$.

Geometric or contextual structure

Sometimes the structure gives the model directly.
Open-box example:

$$V(x)=x(20-2x)(14-2x)$$

The factors come from height, length, and width. Restrictions matter here too. Positive dimensions give $0<x<7$.

Building the Model with Transformations, Regression, and Pieces

Three common routes show up a lot.

Transformations

A transformed parent function has form

$$g(x)=af(b(x-h))+k$$

For quadratics, vertex form is the usual tool:

$$Q(x)=a(x-h)^2+k$$

If the vertex is known, or a center/repeated zero is obvious, transformations are usually fastest.

Regression

Regression uses technology to fit data approximately.

In scope:

• linear
• quadratic
• cubic
• quartic

Make sure paired data is entered correctly. The regression curve does not need to hit every point. A residual is:

$$\text{observed} - \text{predicted}$$

Keep calculator precision in the model and round only the final answer.

Piecewise-defined models

A piecewise model uses different rules on different intervals.

• Intervals must cover the domain
• Intervals must not overlap
• Continuity only matters if the context says it should
• Whole-hour charges or round-up rules must come from the scenario

Rational Models and Inverse Proportionality

A rational model is a quotient of polynomials. In context, it often shows inverse variation.

$$y=\frac{k}{x} \qquad y=\frac{k}{x^2}$$

Use one known point to find $k$.

If $(3,20)$ fits an inverse-square model:

$$20=\frac{k}{3^2} \Rightarrow k=180$$

So

$$y=\frac{180}{x^2}$$

Multiplicative behavior matters:

• multiply $x$ by $c$ in $y=\frac{k}{x}$ $\Rightarrow$ divide $y$ by $c$
• multiply $x$ by $c$ in $y=\frac{k}{x^2}$ $\Rightarrow$ divide $y$ by $c^2$

So doubling distance makes force one-fourth as large in an inverse-square model.

Illustrative examples:

• gravitational force
• electromagnetic force

Also watch domain restrictions. Denominator cannot be zero, and distance is often positive.

Using the Model and Avoiding Common Errors

You may use a model to predict outputs, solve for inputs, or answer threshold questions.

Average rate of change is

$$\frac{f(b)-f(a)}{b-a}$$

Include units. If you want rate of change at a point in this course, approximate it with a small interval around that input.

Comparing rates on consecutive intervals helps describe whether the rate itself is increasing or decreasing.

Interpolation is inside the data interval.
Extrapolation is outside it, so it is less reliable.

Common exam mistakes:

• giving only the equation and forgetting domain, units, or assumptions
• rounding regression coefficients too early
• keeping algebraic solutions that do not fit the context
• treating a regression model as exact when it is only approximate