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

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

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 9 April 2026

Applying models

How can a function model be used to draw conclusions?

Once a model has been constructed for a data set or contextual scenario, it can be used to
- Predict values
  - Evaluate the model at a specific input to estimate the corresponding output
- Calculate rates of change
  - Find the average rate of change over an interval
  - or approximate the rate of change at a point
- Make estimates using average rates of change
  - Use a known average rate of change to estimate function values at points within or near the interval
- Describe changing rates of change
  - Explain how the rate of change itself is increasing or decreasing
    - This is connected to the concavity of the function

How do I use an average rate of change to estimate a function value?

If you know the value of a function $f (x)$ at one point, $a$
- and the average rate of change over an interval, $[a, b]$
- then you can estimate the function's value at another point within that interval
The following formula can be used to work out the estimate
- $Estimate = f (a) + (average rate of change) \times (c - a)$
  - where $f (a)$ is the known function value at $x = a$
  - and $c$ is the input value where you want the estimate
This estimate assumes the function changes at a constant rate equal to the average rate of change
- It is the value corresponding to $x = c$
  - on the secant line connecting the endpoints of the interval

When is an estimate using average rate of change an overestimate or underestimate?

The estimate using the average rate of change lies on the secant line between two points on the graph
Whether this estimate is above or below the actual function value depends on the concavity of the graph
- If the graph is concave up on the interval, the secant line lies above the graph
  - so the estimate is an overestimate
- If the graph is concave down on the interval, the secant line lies below the graph
  - so the estimate is an underestimate

Graph showing curve y = f(x) with points a, t, and b marked on the x-axis. Secant line connects points on graph between a and b. Labels for estimated and actual values of f(t). Secant line is above the curve. — **The secant line lies above the function on a concave up interval**

What about units?

When applying a model in context, always include appropriate units in your answer
Units for the average rate of change are
- the units of the output
- divided by the units of the input
E.g. if the output is "thousands of units sold" and the input is "days"
- then the average rate of change has units of "thousands of units per day"

Examiner Tips and Tricks

On the exam, units are not always required to earn full credit, but including them demonstrates strong understanding.

Examiner Tips and Tricks

Explaining whether an estimate based on an average rate of change is an overestimate or an underestimate is a recurring feature of the FRQ questions in the AP® Precalculus exam. To earn the point, your explanation must include two things

that the graph of the model is concave down or concave up (as appropriate)
and a reference to the secant line or the linear estimate and its position relative to the graph

Simply stating, for example, "the estimate is too low" without explaining why is not sufficient.

Worked Example

The temperature, in degrees Celsius (°C), at a scientific research station on a particular day is modeled by the function $T$ defined by $T (t) = \frac{71 t^{3} - 715 t^{2} + 3000 t - 4315}{17 t^{2} + 9 t - 20}$ , where $t$ is measured in hours from 6 A.M. for $2 \leq t \leq 10$ . Based on the model, how many hours did it take for the temperature to increase from 0°C to 10°C?

(A) 2.744

(B) 4.295

(D) 8.411

Answer:

Graph the function on your graphing calculator

and use the solving feature to find the coordinates of the point where graph crosses the horizontal axis $(T = 0)$
That gives the value of $t$ when the temperature is 0°C

Graph the horizontal line $T = 10$ on the same set of axes

and use the solving feature to find the coordinates of the point where the two graphs intersect
That gives the value of $t$ when the temperature is 10°C

Graph of a curve intersecting the x-axis at (2.74354, 0) and crossing y=10 at (8.41123, 10) on a grid, with axis labels.

So $T = 0$ when $t = 2.74354$ and $T = 10$ when $t = 8.41123$

Subtract those two times to find how many hours it took for the temperature to increase from 0°C to 10°C

$8.41123 - 2.74354 = 5.66769$

Rounded to 3 decimal places that is 5.668 hours

Worked Example

The total number of books, in thousands, sold by an online retailer can be modeled by a quadratic function $B$ . At time $t = 0$ months, the total number of books sold was 12 thousand. Over the interval from $t = 0$ to $t = 6$ months, the average rate of change of $B$ was 3.5 thousand books per month. The graph of $B$ is concave down on the interval $0 < t < 6$ .

(a) Use the average rate of change to estimate the total number of books sold, in thousands, at $t = 4$ months. Show the work that leads to your answer.

Answer

You know the value of $B$ when $t = 0$

and you want to estimate the value where $t = 4$

$\begin{array}{rcl} Estimate & = & B (0) + (average rate of change) \times (4 - 0) \\ = & 12 + 3.5 \times 4 \\ = & 12 + 14 \\ = & 26 \end{array}$

The estimated total number of books sold at $t = 4$ months is 26 thousand

(b) Is the estimate found in part (a) less than or greater than the value predicted by the model $B$ ? Explain your reasoning.

Answer:

The graph is concave down, so the estimate is going to be an underestimate

Explain this in a way that shows your understanding of the situation

The estimate of 26 thousand is the $y$ -coordinate of a point on the secant line passing through $(0, B (0))$ and $(6, B (6))$ . Because the graph of $B$ is concave down on the interval $0 < t < 6$ , the secant line lies below the graph of $B$ on this interval.

Therefore, the estimate using the average rate of change is less than the actual value of $B (4)$ predicted by the model.

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

Applying models

How can a function model be used to draw conclusions?

How do I use an average rate of change to estimate a function value?

When is an estimate using average rate of change an overestimate or underestimate?

What about units?

Examiner Tips and Tricks

Examiner Tips and Tricks

Worked Example

Worked Example

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Author: Roger B

Reviewer: Mark Curtis