Modeling with Polynomial & Rational Functions (College Board AP® Precalculus): Exam Questions

The table gives values of $f\left(x\right)$ at selected values of $x$.

Based on this data, which type of polynomial function is most appropriate to model $f$?

The piecewise-defined function $f$ is given by

$f\left(x\right)=\{\begin{array}{l}{x}^2+1,x<1\\ 4-x,x\ge 1\end{array}$

What is the value of $f\left(0\right)+f\left(1\right)+f\left(3\right)$ ?

The depth, in metres, of water at a sensor in a tidal estuary relative to the mean water level is modelled by the function $W$ defined by

$W\left(t\right)=\frac{84t^3-950t^2+3556t-4620}{16t^2+12t-45}$

where $t$ is measured in hours after 6 a.m., for $2\le t\le 9$. Based on the model, how many hours did it take for the water depth to increase from 0 metres to 4 metres above the mean level?

A car is traveling on a highway. The total distance, $D\left(t\right)$ (in miles), the car has traveled $t$ hours after entering the highway is modeled by the linear function $D\left(t\right)=65t+5$. Which of the following best describes an underlying assumption of this model?

The quadratic function $f$, given by $f\left(x\right)=a{x}^2+bx+c$, models a set of data. The graph of $f$ passes through the points $\left(0,5\right)$, $\left(2,9\right)$, and $\left(4,21\right)$. What is the value of $a$?

The profit (in thousands of dollars) earned by a small business $t$ months after launch is modeled by $P\left(t\right)=-0.4t^3+4.8t^2-12t+15$ for $0\le t\le 8$. What is the maximum value of $P\left(t\right)$ on this interval?

A parking garage charges a flat fee of $5 for parking up to 1 hour. Beyond 1 hour, an additional $3 per hour is charged at a continuous rate. Let $C\left(t\right)$ represent the total parking charge, in dollars, for parking $t$ hours, where $t>0$. Which of the following piecewise functions correctly models $C\left(t\right)$ ?

The intensity $I$ (in watts per square meter) of light from a particular source is inversely proportional to the square of the distance $d$ (in meters) from the source. When $d=1.5$ meters, the intensity is 240 watts per square meter. To 3 decimal places, what is the intensity when $d=4.7$ meters?

A model for the squirrel population (in hundreds) in a small park is given by $N\left(t\right)=-0.3t^2+6t+5$ for $0\le t\le 10$, where $t$ is the number of years since the park opened. What is the value of $t$, to 3 decimal places, at which $N\left(t\right)=25$?

A community garden plot is being designed in the shape of a rectangle. One side will run along an existing wall, so fencing is needed only for the other three sides. The gardener has 40 feet of fencing material.

The area of the plot, in square feet, can be modeled by the function $A$ given by

$A\left(x\right)=x\left(40-2x\right)$

where $x$ is the length of the side perpendicular to the wall, in feet, and $0<x<20$.

What is the contextually appropriate range of $A$?

A telecommunications company's monthly cell phone bill, in dollars, depends on the number of gigabytes (GB) of data used during the month. Customers using less than 5 GB pay a flat $30. Customers using at least 5 GB but less than 20 GB pay a flat $45. Customers using 20 GB or more pay a flat $65.

Let $C$ be the function such that $C\left(d\right)$ gives the monthly bill, in dollars, for a customer using $d$ GB of data, where $d\ge 0$.

Which of the following correctly defines $C\left(d\right)$?

A scatterplot of a data set in the $xy$-plane shows a sequence of points that switches between increasing and decreasing three times.

Which of the following polynomial function types best models the data?

The force, in Newtons, between two electrically charged objects is inversely proportional to the square of the distance, in meters, between them. When the distance between the objects is $2$ meters, the force is $25$ Newtons.

Based on the model, what is the distance between the objects, in meters, when the force is $4$ Newtons?

A polynomial model for the population of a colony (in thousands) is given by $P\left(t\right)=-t^2+20t+100$ for $t\ge 0$, where $t$ is measured in years from the start of observation. According to this model, after how many years does the population first reach zero?

Let $S\left(t\right)$ represent the total amount of snow, in inches, that has fallen during a snowstorm, where $t$ is the number of hours since the start of the storm. The total snowfall is modeled by the function $S\left(t\right)=-\frac{1}{4}{t}^2+3t$ for $0\le t\le T$, where $T$ is the latest time at which the model is appropriate. What is the largest value of $T$ for which the model is appropriate in this context?

The cost per unit of producing $x$ items of a product, in dollars, is modeled by the function $C$ given by $C\left(x\right)=\frac{a}{x-h}$ for some positive constants $a$ and $h$. When $60$ items are produced, the cost per unit is $50 When 80 items are produced, the cost per unit is $25.

Based on the model, what is the cost per unit, in dollars, when $100$ items are produced?

The height, in meters, of a model rocket $t$ seconds after launch is modeled by the function $h$ given by $h\left(t\right)=-4.9t^2+30t$ for $0\le t\le 6$.

Based on the model, for how many seconds is the height of the rocket greater than $30$ meters?

The function $f$ is defined by $f\left(x\right)=2x+a$ for $x<3$, and by $f\left(x\right)=x^2-4x+b$ for $x\ge 3$, where $a$ and $b$ are constants. The two pieces of $f$ agree in value at $x=3$, and $f\left(0\right)=-1$. What is the value of $a+b$?

A polynomial function $f$ has the following two properties:

• $f$ has exactly $1$ distinct real zero.

• $f$ has exactly $2$ local maxima and exactly $2$ local minima.

What is the smallest possible degree of $f$?