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

The graph of the function $g$ is constructed by translating the graph of $f\left(\theta \right)=\sin \theta$ vertically up by $3$ units and horizontally to the right by $\frac{\pi }{6}$ units. Which of the following could be an expression for $g\left(\theta \right)$?

The function $g$ is given by $g\left(\theta \right)=5\sin \left(3\theta \right)$. What are the values of the amplitude and period of $g$?

Let $f$ be a sinusoidal function. The graph of $y=f\left(x\right)$ is given in the $xy$-plane.

What is the period of $f$ ?

The figure shows the graph of a trigonometric function $f$.

Which of the following could be an expression for $f\left(x\right)$ ?

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 is at the lowest point of the Ferris wheel. The wheel completes one full revolution every 40 seconds.

Which of the following could be an expression for $h\left(t\right)$, the height, in meters, of the passenger above the ground at time $t$ seconds?

At a harbor, the depth of water, in meters, is modeled by the function $d$, defined by $d\left(t\right)=4.5\cos \left(\frac{\pi t}{6}\right)+8$ for $0\le t\le 12$ hours. Based on the model, which of the following is true?

The function $f$ is defined by $f\left(x\right)=a\sin \left(b\left(x+c\right)\right)+d$, for constants $a$, $b$, $c$, and $d$. In the $xy$-plane, the points $\left(3,1\right)$ and $\left(7,9\right)$ represent a minimum value and a maximum value, respectively, on the graph of $f$. What are the values of $a$ and $d$ ?

The function $g$ is given by $g\left(x\right)=3\sin \left(6x\right)+\cos \left(3x\right)$. Using the period of $g$, which of the following is the number of complete cycles of the graph of $g$ in the $xy$-plane on the interval $0\le x\le 500$ ?

A carousel of radius 3 meters is rotated at a constant rate. The figure provides a representation of the carousel when viewed from above, in the $xy$-plane with the direction of rotation indicated. At time $t=0$ minutes, the carousel begins to rotate. Point $P$ on the carousel is at the starting position in the figure. At time $t=15$ minutes, 90 rotations of the carousel have been completed, and $P$ is in the same position as it was at time $t=0$. A sinusoidal function is used to model the $y$-coordinate of the position of $P$ as a function of time $t$ in minutes. Which of the following functions is an appropriate model for this situation?

A buoy bobs up and down in a harbour. The height of the buoy, in centimetres, above the seabed is modelled by the function $H$, where $t$ is measured in seconds. The buoy reaches a maximum height of 100 cm and a minimum height of 20 cm. It completes one full cycle every 10 seconds, and at $t=0$ the buoy is at its maximum height.

Which of the following defines $H\left(t\right)$?

The figure shows the graph of the sinusoidal function $g$. What are the values of the period and amplitude of $g$?

The figure shows the graph of a trigonometric function $f$.

Which of the following could be an expression for $f\left(x\right)$?

A sinusoidal function $f$ has a relative maximum at the point $\left(2,5\right)$ and a relative minimum at the point $\left(6,-3\right)$.

Which of the following could be an expression for $f\left(x\right)$?

The function $f$ is given by $f\left(x\right)=4\sin \left(\frac{\pi }{6}\left(x-1\right)\right)+3$.

What is the value of $f\left(8\right)$?

The function $f$ is given by $f\left(x\right)=2\sin \left(\frac{\pi }{3}x\right)+1$.

On which of the following intervals of $x$ is $f\left(x\right)$ both negative and increasing?

A sinusoidal function can be written in the form $y=a\sin \left(bx\right)+d$, where $a>0$ and $b>0$. The function has maximum value $7$, minimum value $-3$, and period $4$. What is the value of $a+b+d$?

The graph of $f\left(x\right)=\sin x$ is transformed to obtain the graph of $g\left(x\right)=\sin \left(2x+\frac{\pi }{3}\right)$. Which of the following correctly describes the transformation?

A Ferris wheel has diameter $30$ meters and rotates once every $2$ minutes. The bottom of the wheel is $2$ meters above the ground. A rider boards at the bottom at time $t=0$. The rider's height $h\left(t\right)$, in meters, at time $t$ (in minutes) is modeled by $h\left(t\right)=17-15\cos \left(\pi t\right)$. How many minutes elapse between the first and second times the rider reaches a height of $9.5$ meters?