Unit 2 · Topic 1.14

Function Model Construction and Application

Turn conditions or data into a function, preserve the model's restrictions, and use values and rates to answer contextual questions with units.

Learning Goals

Translate points, zeros, extrema, rates, and restrictions into equations for unknown parameters.
Construct linear, quadratic, cubic, quartic, and degree-\(n\) polynomial models.
Build models by transforming a parent function.
Use technology to obtain and interpret linear, quadratic, cubic, or quartic regressions.
Combine rules into a piecewise-defined model with correct interval conditions.
Construct rational models for inverse and inverse-square relationships.
Use a model to predict outputs, solve for inputs, and analyze average and changing rates.
Report conclusions with appropriate units, rounding, domain, and reliability.

1. The Model-Construction Cycle

Define variables: identify input, output, and units.
Select a family: use the evidence developed in Topic 1.13.
Choose a revealing form: slope-intercept, vertex, factored, transformed, rational, or piecewise.
Translate conditions: substitute known points, zeros, rates, or extrema.
Solve parameters: determine every unknown coefficient.
Validate: check all original conditions, data patterns, and restrictions.
Apply: calculate the requested quantity and interpret it in context.

\[\text{conditions}\longrightarrow\text{parameters}\longrightarrow\text{model}\longrightarrow\text{contextual conclusion}.\]

2. Match Each Condition to an Equation

Given information	Equation or model feature
The graph contains \((p,q)\)	\(f(p)=q\)
\(r\) is a zero	\(f(r)=0\), often producing a factor \(x-r\)
Zero \(r\) has multiplicity \(m\)	Include the factor \((x-r)^m\)
The vertex is \((h,k)\)	Begin with \(f(x)=a(x-h)^2+k\)
Constant rate is \(m\)	Use linear slope \(m\)
Output is inversely proportional to \(x^n\)	Use \(f(x)=k/x^n\)

The best form exposes the supplied information. Convert to another equivalent form only when it helps answer the question.

3. Construct a Linear Model

A storage tank contains 140 liters at \(t=2\) minutes and 260 liters at \(t=6\) minutes. If the rate is constant,

\[m=\frac{260-140}{6-2}=30\text{ liters per minute}.\]

Using point-slope form,

\[V(t)-140=30(t-2)\quad\Longrightarrow\quad \boxed{V(t)=30t+80}.\]

Check both points. If the tank capacity is 500 liters and filling began at \(t=0\), a meaningful domain is \(0\le t\le14\).

4. Construct a Quadratic from a Vertex and a Point

A launched object's height \(h\), in meters, reaches a maximum of 20 meters at \(t=3\) seconds and begins at \(h(0)=2\). Start with vertex form:

\[h(t)=a(t-3)^2+20.\]

Use the initial condition to determine the vertical scale:

\[2=a(0-3)^2+20\quad\Longrightarrow\quad 9a=-18\quad\Longrightarrow\quad a=-2.\]

\[\boxed{h(t)=-2(t-3)^2+20}.\]

To find when the object reaches the ground, solve \(h(t)=0\):

\[t=3\pm\sqrt{10}.\]

The negative value is outside the time domain, so the contextual answer is \(t=3+\sqrt{10}\approx6.16\) seconds. The model's meaningful domain is \(0\le t\le3+\sqrt{10}\).

5. Construct a Polynomial from Zeros

Suppose a cubic model has zeros at \(x=-1\) and \(x=4\), touches rather than crosses the axis at \(x=4\), and contains \((1,18)\). The even multiplicity at 4 gives

\[P(x)=a(x+1)(x-4)^2.\]

Substitute the known point:

\[18=a(1+1)(1-4)^2=18a,\qquad a=1.\]

\[\boxed{P(x)=(x+1)(x-4)^2}.\]

The zeros determine factors, multiplicities determine local behavior, and one additional nonzero point determines the leading scale. Always verify the resulting degree and end behavior.

6. Construct by Transforming a Parent Function

If a parabola has the same basic shape as \(y=x^2\), is reflected across the x-axis, stretched vertically by 3, shifted right 5, and shifted up 12, then

\[f(x)=-3(x-5)^2+12.\]

The parameters are not decorative: the vertex is \((5,12)\), the maximum is 12, and the coefficient \(-3\) controls reflection and vertical scale. Translate each contextual feature into a transformation and check a point afterward.

7. Construct a Regression Model with Technology

When measurements do not lie exactly on one curve, regression estimates coefficients that best fit the selected family.

Enter inputs and outputs into separate lists and verify each ordered pair.
View a scatterplot before choosing linear, quadratic, cubic, or quartic regression.
Run the selected regression and record coefficients with enough precision.
Graph the regression over the data and inspect residual behavior.
Store the full-precision equation; round only the final requested result.
State the observed interval and any contextual domain restriction.

Technology output	Model to report
\(a=1.48,\ b=-3.07,\ c=6.12\)	\(Q(x)=1.48x^2-3.07x+6.12\)
Observed inputs \(0\le x\le8\)	Identify predictions within this interval as interpolation
Residual plot has no clear pattern	Use this as supporting evidence, not proof of causation

8. Construct a Piecewise-Defined Model

A rental service charges 18 dollars for up to 2 hours, then 7 dollars per additional hour until the 8-hour limit. Let \(t\) be rental time in hours.

\[C(t)=\begin{cases}18,&0<t\le2,\\18+7(t-2),&2<t\le8.\end{cases}\]

The second rule begins from the existing 18-dollar charge, so both pieces agree at \(t=2\). When constructing any piecewise model:

assign a formula to every intended interval;
avoid overlapping or missing input values;
use open and closed endpoints deliberately;
check whether the context requires continuity at each boundary.

9. Construct an Inverse-Variation Model

If travel time \(T\) for a fixed route is inversely proportional to constant speed \(v\), then \(T(v)=k/v\). A trip takes 3 hours at 80 kilometers per hour, so

\[3=\frac{k}{80}\quad\Longrightarrow\quad k=240.\]

\[\boxed{T(v)=\frac{240}{v}},\qquad v>0.\]

The constant 240 represents the route distance in kilometers. At 96 kilometers per hour, the model predicts \(T(96)=2.5\) hours.

10. Construct an Inverse-Square Model

Suppose signal intensity \(I\) is inversely proportional to the square of distance \(d\). Then

\[I(d)=\frac{k}{d^2},\qquad d>0.\]

If \(I(4)=45\), then \(45=k/16\), so \(k=720\) and

\[\boxed{I(d)=\frac{720}{d^2}}.\]

Doubling distance divides intensity by \(2^2=4\). The excluded input \(d=0\) is both mathematically undefined and physically inappropriate for this model.

11. Apply a Model in Both Directions

A model can predict an output from an input or solve for the input that produces a target output.

Question type	Mathematical action	Required interpretation
What is the output at \(x=a\)?	Evaluate \(f(a)\)	State output units and whether \(a\) is in the domain
When does the output equal \(b\)?	Solve \(f(x)=b\)	Reject solutions outside the contextual domain
What is the maximum or minimum?	Use the vertex, graph, or technology	Report both the input and output with units
When is one model larger than another?	Solve or graph \(f(x)>g(x)\)	Give the valid input interval

A calculator may produce several algebraic solutions. The context decides which solutions answer the question.

12. Interpret Average Rate of Change

For a model \(f\), the average rate of change from \(x=a\) to \(x=b\) is

\[\operatorname{AROC}_{[a,b]}=\frac{f(b)-f(a)}{b-a}.\]

Using \(h(t)=-2(t-3)^2+20\), \(h(1)=12\) and \(h(2)=18\). Thus

\[\frac{h(2)-h(1)}{2-1}=6\text{ meters per second}.\]

Interpret the sign and units: over this interval, height increases by an average of 6 meters for each second.

13. Interpret Changing Rates

For equal input steps, compare consecutive first differences. Their differences describe how the average rates themselves change.

For the quadratic \(Q(x)=3x^2+2x+1\), outputs at \(x=0,1,2,3\) are \(1,6,17,34\). First differences are \(5,11,17\), and second differences are \(6,6\).

\[\Delta^2Q=6=2a\quad\text{for unit input steps}.\]

If \(Q\) measures meters and \(x\) measures seconds, first differences have units of meters per second and second differences describe the change in that rate per second.

14. Units, Rounding, and Reliability

Quantity	Units	Reporting rule
\(f(x)\)	Output units	Round according to measurement or context
Average rate of change	Output units per input unit	Include direction through the sign
Input solving \(f(x)=c\)	Input units	Discard values outside the domain
Regression coefficients	Depend on powers of the input	Keep extra precision during calculations

Interpolation generally requires fewer assumptions than extrapolation. A good algebraic fit does not remove capacity limits, physical restrictions, changing conditions, or model breakdown.

15. Complete Modeling Response

A complete response should be readable without the original calculator screen.

Define the variables and their units.
Write the selected function form and explain why it matches the conditions.
Show the equations used to determine parameters.
State the final model and its contextual domain.
Perform the requested evaluation, equation solving, or rate calculation.
Interpret the result in a sentence with units and appropriate rounding.
Qualify the result if it is extrapolated or depends on a strong assumption.

Common errors: fitting the wrong variables, rounding coefficients too early, omitting interval conditions in a piecewise rule, using every algebraic solution, confusing output units with rate units, and reporting an equation without applying it.

Graph and Visual Model

The vertex encodes the maximum, the initial value determines the vertical scale, and only the nonnegative-time zero is meaningful.

Key Takeaways

Construct linear, polynomial, rational, regression, and piecewise-defined models from constraints or data, then use them to predict values and interpret rates with appropriate units and domains.
Core relationship: \(\text{constraints}\rightarrow\text{parameters}\rightarrow\text{model}\rightarrow\text{contextual conclusion}\)
Error check: Do not stop after finding an equation: preserve restrictions, show how each condition determines a parameter, and interpret predictions and rates with units.

Checkpoint · Topic 1.14

A small water display follows a parabolic height model. The stream leaves a nozzle 1.5 meters above the ground at \(x=0\), reaches a maximum height of 5.5 meters at \(x=2\), and lands on the ground to the right of the nozzle. Here \(x\) and height are measured in meters.

Begin with vertex form and use the nozzle point to show that \(H(x)=-(x-2)^2+5.5\).
Verify the maximum and initial height from the completed model.
Solve \(H(x)=0\), reject the noncontextual zero, and state the physical domain.
Predict the height at \(x=3\) and interpret the result with units.
Calculate the average rate of change from \(x=0\) to \(x=2\) and interpret its units.
Explain which model conditions would fail if wind substantially altered the path.
Describe how technology and residuals could be used if measured stream points did not lie exactly on the parabola.