AP®MathsCollege BoardPrecalculusExam QuestionsTrigonometric & Polar FunctionsSinusoidal Functions & Modeling

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

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Sinusoidal Functions & Modeling

1
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1 mark

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

Graph of a sinusoidal function with amplitude 2, oscillating between y = 2 and y = -2. Peaks occur at x = 0 and x = 6, and troughs occur at x = 3 and x = 9. The function crosses zero at x = 1.5, x = 4.5, x = 7.5, etc.
Graph of f

What is the period of f ?

  • 2

  • 3

  • 4

  • 6

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2
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1 mark

The figure shows the graph of a trigonometric function f.

Graph of a sine or cosine wave, oscillating between -2 and 4 on the y-axis, with labels at π/2 (maximum), π (minimum), 3π/2 (maximum), and 2π (minimum) on the x-axis.

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

  • 3 \cos \left(2 \left(x - \frac{\pi}{8}\right)\right) + 1

  • 3 \sin \left(2 \left(x - \frac{\pi}{4}\right)\right) + 1

  • - 3 \cos \left(2 \left(x - \frac{\pi}{8}\right)\right) + 1

  • - 3 \sin \left(2 \left(x - \frac{\pi}{4}\right)\right) + 1

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3
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1 mark

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.

Diagram of a Ferris wheel with center 15 meters above the ground and radius 10 meters. The lowest point is 5 meters above the ground. A passenger starts at the bottom (lowest point) at t = 0 and the wheel rotates counterclockwise.

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?

  • h \left(t\right) = - 15 \cos \left(\frac{\pi}{20} t\right) + 10

  • h \left(t\right) = - 10 \sin \left(\frac{\pi}{20} t\right) + 15

  • h \left(t\right) = - 10 \cos \left(\frac{\pi}{10} t\right) + 15

  • h \left(t\right) = - 10 \cos \left(\frac{\pi}{20} t\right) + 15

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4
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1 mark

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 \leq t \leq 12 hours. Based on the model, which of the following is true?

  • The maximum depth of the water is 12.5 m.

  • The maximum depth of the water occurs at t = 6 hours.

  • The minimum depth of the water is 4 m.

  • The minimum depth of the water occurs at t = 12 hours.

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5
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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 ?

  • a = 4 and d = 1

  • a = 5 and d = 4

  • a = 8 and d = 5

  • a = 4 and d = 5

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6
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1 mark

The function g is given by g \left(x\right) = 3 \sin \left(6 x\right) + \cos \left(3 x\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 \leq x \leq 500 ?

  • 125

  • 238

  • 318

  • 477

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7
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1 mark

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 circle of radius 3 centered at the origin in the xy-plane. Point P is labeled at the Start position at (3, 0) on the positive x-axis. An arrow indicates the direction of rotation is counterclockwise.

  • f \left(t\right) = 3 \sin \left(\frac{\pi t}{10}\right)

  • f \left(t\right) = 3 \sin \left(\frac{\pi t}{3}\right)

  • f \left(t\right) = 3 \sin \left(6 t\right)

  • f \left(t\right) = 3 \sin \left(12 \pi t\right)

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