AP®MathsCollege BoardPrecalculusRevision NotesPolynomial & Rational FunctionsModeling with Polynomial & Rational FunctionsConstructing Models

Constructing Models (College Board AP® Precalculus): Revision Note

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 24 April 2026

Polynomial models

How can a model be constructed from a contextual scenario?

A model is a function that represents a real-world situation mathematically
A model can be constructed based on restrictions identified in a mathematical or contextual scenario
- These restrictions come from the given information, such as
  - specific data values
  - initial conditions
  - or known behaviors of the quantities involved
The general approach is
- Identify the appropriate function type
  - this is covered in the Selecting Polynomial Models study guide
- Use the given data or conditions to set up equations involving the unknown constants in the function
- Solve the system of equations to find the values of the constants

How do you construct a polynomial model from data points?

If you are given data points and told to use a polynomial model of a specific form (e.g. $D (t) = a t^{2} + b t + c$ )
- you can find the unknown constants by substituting each data point into the equation
- Each data point $(x, y)$ gives you one equation
- You need as many data points as there are unknown constants

E.g. a quadratic $a t^{2} + b t + c$ has 3 unknowns ( $a$ , $b$ , $c$ ),
- so you need 3 data points to set up 3 equations
A linear model $y = m x + b$ has 2 unknowns
- so you need 2 data points
Once you have the system of equations
- solve for the unknown constants
You can solve algebraically (substitution, elimination)
- or you can use a graphing calculator to solve the system

How can a model be constructed using transformations?

A model of a data set or a contextual scenario can be constructed using transformations of a parent function
If you recognize that a data set or scenario follows the shape of a known parent function (e.g. $x^{2}$ , $x^{3}$ ), you can apply transformations to fit the data
- Vertical and horizontal translations (shifts)
- Vertical and horizontal dilations (stretches/compressions)
E.g. if data appears to follow a parabola with its vertex at $(3, 5)$ , opening upward
- you might model it as $f (x) = a (x - 3)^{2} + 5$
- and use a data point to find $a$

How can technology be used to construct a model?

A model of a data set can be constructed using technology and regressions
A regression uses an algorithm to find the function of a given type that best fits the data
Available regression types for polynomial models include:
- Linear regression
  - fits a model of the form $y = a x + b$
- Quadratic regression
  - fits a model of the form $y = a x^{2} + b x + c$
- Cubic regression
  - fits a model of the form $y = a x^{3} + b x^{2} + c x + d$
- Quartic regression
  - fits a model of the form $y = a x^{4} + b x^{3} + c x^{2} + d x + e$

To perform a regression on a graphing calculator
- Enter the data into lists
- Select the appropriate regression type
- The calculator will output the values of the constants

Examiner Tips and Tricks

On the exam, if you are asked to write equations that can be used to find the constants in a model, make sure you show the equations before solving them.

This is often a separate scoring point from actually finding the values of the constants

On a calculator part of the exam, you can use your graphing calculator to solve systems of equations.

So even if you struggle with the algebra, you can still earn the point for finding the correct values

Worked Example

A town's population, in thousands, is recorded at three different times. At time $t = 0$ years, the population was 12 thousand. At $t = 3$ years, the population was 15.6 thousand. At $t = 6$ years, the population was 16.8 thousand.

The population can be modeled by the function $P$ given by $P (t) = a t^{2} + b t + c$ , where $P (t)$ is the population, in thousands, and $t$ is the number of years since the population was first recorded.

(a) Use the given data to write three equations that can be used to find the values for constants $a$ , $b$ , and $c$ in the expression for $P (t)$ .

(b) Find the values for $a$ , $b$ , and $c$ as decimal approximations.

Answer:

(a)

Substitute each data point into $P (t) = a t^{2} + b t + c$

$P (0) = 12$
$a (0)^{2} + b (0) + c = 12$
$c = 12$

$P (3) = 15.6$
$a (3)^{2} + b (3) + c = 15.6$
$9 a + 3 b + c = 15.6$

$P (6) = 16.8$
$a (6)^{2} + b (6) + c = 16.8$
$36 a + 6 b + c = 16.8$

(b)

From the first equation

$c = 12$

Substitute into the other two equations:

$9 a + 3 b + 12 = 15.6 \Rightarrow 9 a + 3 b = 3.6 1$

$36 a + 6 b + 12 = 16.8 \Rightarrow 36 a + 6 b = 4.8 2$

Multiplying equation 1 by 2

$2 \times (9 a + 3 b) = 2 \times 3.6$

$18 a + 6 b = 7.2 3$

Subtracting equation 3 from equation 2

$\begin{array}{l} 36 a + 6 b = 4.8 \\ - (18 a + 6 b = 7.2) \\ - - - - - - - - - \\ 18 a = - 2.4 \end{array}$

$a = - \frac{2.4}{18} = - \frac{2}{15} = - 0.133333 . . .$

Substitute that value of $a$ into one of the original equations and solve for $b$

E.g., substituting into equation 1 gives

$\begin{array}{rcl} 9 (- \frac{2}{15}) + 3 b & = & 3.6 \\ 3 b - 1.2 & = & 3.6 \\ 3 b & = & 4.8 \end{array}$

$b = \frac{4.8}{3} = 1.6$

Those are the three answers you are looking for

Round the answer for $a$ to 3 decimal places, to give the decimal approximation

$a = - 0.133 (3 d . p .)$ , $b = 1.6$ , $c = 12$

Piecewise-defined models

What is a piecewise-defined function model?

A piecewise-defined function model can be constructed through a combination of modeling techniques
A piecewise-defined model is appropriate when a data set or contextual scenario demonstrates different characteristics over different intervals
- E.g. a quantity might increase linearly for a period, then remain constant, then decrease
- Each 'piece' of the model captures the behavior over one interval

How do you construct a piecewise-defined model?

Identify the intervals where the behavior changes
- Look for points where the pattern or trend shifts
Choose an appropriate function type for each interval
- One interval might be best modeled by a linear function, another by a quadratic, etc.
Construct each piece separately
- Use the techniques for polynomial models, above
- I.e. substituting data points, using transformations, or regression
Combine the pieces into a single piecewise-defined function
- Be sure to specify the domain for each piece
  - Make sure the domain intervals do not overlap
  - Pay attention to whether endpoints use $\leq$ or $<$

Depending on the context, the pieces may or may not need to connect at the boundary points
- In many real-world models, the function value is expected to be the same from both sides at the join point

Worked Example

A city's water reservoir is being filled. For the first 5 hours ( $0 \leq t \leq 5$ ), water flows in at a constant rate, and the volume of water in the reservoir, in thousands of gallons, is given by $V (t) = 8 t + 10$ .

After 5 hours, the flow rate decreases. At $t = 5$ , $t = 7$ , and $t = 9$ , the volumes are 50, 58, and 62 thousand gallons respectively.

(a) Determine a quadratic model of the form $Q (t) = a t^{2} + b t + c$ for the volume of water in the reservoir for the interval $5 \leq t \leq 9$ .

(b) Write a piecewise-defined function $W (t)$ that models the volume of water in the reservoir for $0 \leq t \leq 9$ .

Answer:

(a)

Substitute each data point into $Q (t) = a t^{2} + b t + c$

$Q (5) = 50$ :

$25 a + 5 b + c = 50 1$

$Q (7) = 58$ :

$49 a + 7 b + c = 58 2$

$Q (9) = 62$ :

$81 a + 9 b + c = 62 3$

Subtracting equation 1 from equation 2 gives

$24 a + 2 b = 8 4$

Subtracting equation 2 from equation 3 gives

$32 a + 2 b = 4 5$

Equations 4 and 5 are simultaneous equations in $a$ and $b$ only

Subtracting equation 4 from equation 5 gives

$8 a = - 4$

$a = \frac{- 4}{8} = - \frac{1}{2} = - 0.5$

Substitute that back into equation 4 and solve for $b$

$\begin{array}{rcl} 24 (- 0.5) + 2 b & = & 8 \\ - 12 + 2 b & = & 8 \\ 2 b & = & 20 \end{array}$

$\begin{array}{rcl} b & = & \frac{20}{2} = 10 \end{array}$

Substitute those values into equation 1 and solve for $c$

$\begin{array}{rcl} 25 (- 0.5) + 5 (10) + c & = & 50 \\ - 12.5 + 50 + c & = & 50 \\ c + 37.5 & = & 50 \\ c & = & 12.5 \end{array}$

Therefore

$Q (t) = - 0.5 t^{2} + 10 t + 12.5$

(b)

Put the two pieces together

Be sure to indicate the correct, non-overlapping intervals for the two pieces

$W (t) = {\begin{cases} 8 t + 10, 0 \leq t \leq 5 \\ - 0.5 t^{2} + 10 t + 12.5, 5 < t \leq 9 \end{cases}$

Rational models

When is a rational function model appropriate?

Data sets and aspects of contextual scenarios involving quantities that are inversely proportional can often be modeled by rational functions
Two quantities are inversely proportional when
- one quantity increases as the other decreases
- and their product is constant
If $y$ is inversely proportional to $x$ ,
- then $y = \frac{k}{x}$ for some constant $k$
More generally, $y$ might be inversely proportional to a power of $x$
- E.g. $y$ is inversely proportional to $x^{2}$
  - in which case $y = \frac{k}{x^{2}}$
Examples of inverse proportionality in context:
- The time to complete a job is inversely proportional to the number of workers
  - assuming each worker works at the same rate
- The gravitational force between two objects is inversely proportional to the square of the distance between them
  - $F = \frac{k}{d^{2}}$
- The electromagnetic force between two charged particles is also inversely proportional to the square of the distance between them

How do you construct a rational function model?

Identify that the relationship involves inverse proportionality
- Look for language like "inversely proportional to"
- or data where one quantity decreases as another increases
  - with their product roughly constant
Set up the general form of the model
- E.g. $y = \frac{k}{x}$ or $y = \frac{k}{x^{2}}$ , depending on the relationship
- More complex rational models may take the form $y = \frac{a}{x - h} + k$ (a transformed reciprocal function)
Use given data to find the unknown constant(s)
- Substitute a known input-output pair and solve for the constant

Examiner Tips and Tricks

If a question describes a quantity that "varies inversely" with another, this is a direct signal to use a rational function model.

Remember that for an inversely proportional relationship $y = \frac{k}{x}$ , the product $x y = k$ is constant.

This is a quick way to verify that a data set is inversely proportional

Worked Example

A scientist is studying the intensity of a light source. The intensity $I$ , measured in lumens per square meter, at a distance $d$ meters from the source is inversely proportional to the square of the distance. At a distance of 2 meters, the intensity is measured to be 45 lumens per square meter.

(a) Construct a model for the intensity $I$ as a function of the distance $d$ .

(b) Use the model to predict the intensity at a distance of 5 meters from the source.

Answer:

(a)

Since $I$ is inversely proportional to $d^{2}$ , you can write

$I (d) = \frac{k}{d^{2}}$

Use the data point $I (2) = 45$ to find $k$

$\begin{array}{rcl} 45 & = & \frac{k}{2^{2}} \\ 45 & = & \frac{k}{4} \\ 4 \times 45 & = & k \end{array}$

$k = 180$

Therefore

$I (d) = \frac{180}{d^{2}}$

(b)

Substitute $d = 5$ into the answer from part (a)

$\begin{array}{rcl} I (5) & = & \frac{180}{5^{2}} \\ = & \frac{180}{25} \\ = & 7.2 \end{array}$

The intensity at a distance of 5 meters is 7.2 lumens per square meter

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Constructing Models (College Board AP® Precalculus): Revision Note

Polynomial models

How can a model be constructed from a contextual scenario?

How do you construct a polynomial model from data points?

How can a model be constructed using transformations?

How can technology be used to construct a model?

Examiner Tips and Tricks

Worked Example

Piecewise-defined models

What is a piecewise-defined function model?

How do you construct a piecewise-defined model?

Worked Example

Rational models

When is a rational function model appropriate?

How do you construct a rational function model?

Examiner Tips and Tricks

Worked Example

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis