Sinusoidal Functions & Modeling (College Board AP® Precalculus): Exam Questions

The tire of a car has a radius of 9 inches, and a person rolls the tire forward at a constant rate on level ground, as shown in the figure. Point $W$ on the edge of the tire touches the ground at time $t=\frac{1}{2}$ second. The tire completes a full rotation, and the next time $W$ touches the ground is at time $t=\frac{5}{2}$ seconds. The maximum height of $W$ above the ground is 18 inches. As the tire rolls, the height of $W$ above the ground periodically increases and decreases.

The sinusoidal function $h$ models the height of point $W$ above the ground, in inches, as a function of time $t$, in seconds.

The graph of $h$ and its dashed midline for two full cycles is shown. Five points, $F$, $G$, $J$, $K$, and $P$, are labeled on the graph. No scale is indicated, and no axes are presented.

Determine possible coordinates $\left(t,h\left(t\right)\right)$ for the five points: $F$, $G$, $J$, $K$, and $P$.

The function $h$ can be written in the form $h\left(t\right)=a\sin \left(b\left(t+c\right)\right)+d$. Find values of constants $a$, $b$, $c$, and $d$.

A sinusoidal function $f$ has a relative maximum of $12$ at $x=2$, and its next relative minimum, of $0$, occurs at $x=6$.

The function $f$ can be written in the form $f\left(x\right)=a\sin \left(b\left(x-h\right)\right)+k$, where $a$, $b$, $h$, and $k$ are real numbers.

Find the value of $b$. Show the work that leads to your answer.

The function $f$ is given by $f\left(x\right)=-3\sin \left(\frac{\pi }{6}x\right)+5$.

Find the amplitude of $f$.

The depth of water at a coastal location is modeled by a sinusoidal function $g$ of time $t$ (in hours after midnight). The graph of $g$ is shown above for $0\le t\le 10$ hours. The graph has a relative maximum at the point $\left(3,8\right)$ and a relative minimum at the point $\left(7,2\right)$.

Find the coordinates $\left(t,g\left(t\right)\right)$ of the next relative maximum of $g$ that occurs after $t=10$.

The depth of water at a harbor, in meters, is modeled by the function $D$, where $D\left(t\right)=8\cos \left(\frac{\pi t}{6}\right)+12$ for $0\le t\le 24$. $D\left(t\right)$ is the depth in meters, and $t$ is the number of hours since midnight on a particular day.

On the interval $\left(3,6\right)$, is $D$ increasing or decreasing?

On the interval $\left(3,6\right)$, describe the concavity of the graph of $D$ and determine whether the rate of change of $D$ is increasing or decreasing.

The average daily temperature in a particular town varies sinusoidally throughout the year. The function $T$ models the temperature, in degrees Fahrenheit, on month $t$ of the year, with $0\le t\le 12$. The temperature reaches its maximum of $80°\mathrm{F}$ at $t=7$ and its minimum of $40°\mathrm{F}$ at $t=1$.

The function $T$ can be written in the form $T\left(t\right)=a\cos \left(b\left(t+c\right)\right)+d$. Find the values of $a$, $b$, $c$, and $d$.

Find the value of $T\left(9\right)$. Show the work that leads to your answer.

Interpret the meaning of your answer to part (b) in the context of the problem.

The depth of water at a certain coastal point varies sinusoidally with a period of 24 hours due to the daily tidal cycle. The table shows the depth $D\left(t\right)$, in meters, measured at selected times $t$ in hours after midnight on a particular day.

Construct a sinusoidal function $D\left(t\right)$ in the form $a\sin \left(b\left(t+c\right)\right)+d$ that fits the data in the table.

Use your model to find the value of $D\left(8\right)$. Express your answer as an exact value. Show the work that leads to your answer.

Interpret the meaning of your answer to part (b) in the context of the problem.

Let $f\left(\theta \right)=\sin \theta$. The graph of $g$ is constructed by applying the following transformations to the graph of $f$, in order:

• A vertical dilation by a factor of $3$

• A horizontal dilation by a factor of $\frac{1}{2}$

• A vertical translation up by $4$ units

Find an expression for $g\left(\theta \right)$.

State the amplitude, period, and midline of $g$.

The temperature, in degrees Fahrenheit, in a particular city on a certain day is modeled by the function $T$, where $T\left(t\right)=12\sin \left(\frac{\pi t}{12}\right)+65$ for $0\le t\le 24$. $T\left(t\right)$ is the temperature in degrees Fahrenheit, and $t$ is the number of hours since 6:00 a.m. on that day.

Find the maximum temperature reached during the 24-hour period and the time at which this maximum occurs.

Find all values of $t$, as decimal approximations, in the interval $0\le t\le 24$ for which $T\left(t\right)=70$, or indicate that there are no such values.

Interpret your answer to part (b) in the context of the problem.

A surfboard floating in the sea moves up and down as waves pass beneath it. The height of the surfboard above the sea floor varies periodically.

The surfboard reaches a maximum height of 11 feet above the sea floor and a minimum height of 3 feet above the sea floor. One full cycle takes 8 seconds. At time $t=0$, the surfboard is at its midline height and is rising.

The function $h$ models the height of the surfboard above the sea floor, in feet, as a function of time $t$, in seconds.

A graph of $h$ and its dashed midline for two full cycles is shown. Five points $F$, $G$, $J$, $K$ and $P$ are labeled on the graph. No scale is indicated and no axes are presented. Determine possible coordinates $\left(t,h\left(t\right)\right)$ for the five points: $F$, $G$, $J$, $K$ and $P$.

The function $h$ can be written in the form $h\left(t\right)=a\cos \left(b\left(t+c\right)\right)+d$. Find the values of constants $a$, $b$, $c$ and $d$.

A Ferris wheel at an amusement park has a radius of 10 meters. The center of the Ferris wheel is 15 meters above the ground. At time $t=0$ seconds, a passenger at point $P$ is at the lowest point of the Ferris wheel, as indicated in the figure. The Ferris wheel rotates at a constant speed in a counterclockwise direction and completes one full revolution every 40 seconds. As the wheel rotates, the height of point $P$ above the ground periodically increases and decreases.

The sinusoidal function $h$ models the height of point $P$ above the ground, in meters, as a function of time $t$, in seconds.

A graph of $h$ and its dashed midline for two full cycles is shown. Five points $F$, $G$, $J$, $K$ and $P$ are labeled on the graph. No scale is indicated and no axes are presented.

Determine possible coordinates $\left(t,h\left(t\right)\right)$ for the five points: $F$, $G$, $J$, $K$ and $P$.

The function $h$ can be written in the form $h\left(t\right)=a\sin \left(b\left(t+c\right)\right)+d$. Find the values of constants $a$, $b$, $c$ and $d$.

The function $h$ is given by $h\left(t\right)=4\sin \left(\frac{\pi t}{6}\right)+5$.

Find all values of $t$ in the interval $0\le t\le 12$ for which $h\left(t\right)=7$, or indicate that there are no such values. Show the work that leads to your answer.

Find all real values of $t$ for which $h\left(t\right)=9$. Show the work that leads to your answer.

The function $g$ can be written in the form $g\left(x\right)=a\sin \left(b\left(x+c\right)\right)+d$, where $a$, $b$, $c$, and $d$ are real numbers. The graph of $g$ has a maximum value of $7$ and a minimum value of $1$. The graph crosses its midline going downward at $x=5$ and has its next relative maximum at $x=8$.

Find values of $a$, $b$, $c$, and $d$.

A wind turbine blade tip rotates so that the height of a marked point on the tip above the ground is modeled by a sinusoidal function $h$ of time $t$, where $t$ is measured in seconds and $h\left(t\right)$ is in meters. The function $h$ has a maximum value of $80$ meters and a minimum value of $20$ meters. The marked point first reaches its maximum height at $t=1$, and the period of $h$ is $6$ seconds.

The function $h$ can be written in the form $h\left(t\right)=a\sin \left(b\left(t+c\right)\right)+d$, where $a$, $b$, $c$, and $d$ are real numbers.

Find values of constants $a$, $b$, $c$, and $d$.

On the interval $\left(2.5,4\right)$, describe the concavity of the graph of $h$ and determine whether the rate of change of $h$ is increasing or decreasing.