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

Which of the following is a sinusoidal function?

  • space f open parentheses x close parentheses equals negative 2 cos open parentheses x minus pi over 3 close parentheses plus 5

  • space f open parentheses x close parentheses equals sin open parentheses x squared close parentheses

  • space f open parentheses x close parentheses equals tan open parentheses x close parentheses

  • space f open parentheses x close parentheses equals x sin open parentheses x close parentheses

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2
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1 mark

The graph of the function g is constructed by translating the graph of space f open parentheses theta close parentheses equals sin theta vertically up by 3 units and horizontally to the right by pi over 6 units. Which of the following could be an expression for g open parentheses theta close parentheses?

  • g open parentheses theta close parentheses equals sin open parentheses theta minus pi over 6 close parentheses minus 3

  • g open parentheses theta close parentheses equals sin open parentheses theta minus pi over 6 close parentheses plus 3

  • g open parentheses theta close parentheses equals sin open parentheses theta plus pi over 6 close parentheses minus 3

  • g open parentheses theta close parentheses equals sin open parentheses theta plus pi over 6 close parentheses plus 3

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3
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1 mark

The function g is given by g open parentheses theta close parentheses equals 5 sin open parentheses 3 theta close parentheses. What are the values of the amplitude and period of g?

  • The amplitude is 3 and the period is fraction numerator 2 pi over denominator 3 end fraction.

  • The amplitude is 5 and the period is fraction numerator 2 pi over denominator 3 end fraction.

  • The amplitude is 5 and the period is 2 pi.

  • The amplitude is 5 and the period is 6 pi.

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4
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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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5
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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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6
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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

Choose your answer

7
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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.

Choose your answer

8
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1 mark

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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9
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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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10
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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)

Choose your answer

11
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1 mark

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(t)?

  • H(t) = 40\sin\!\left(\dfrac{\pi t}{5}\right) + 60

  • H(t) = 40\cos\!\left(\dfrac{\pi t}{5}\right) + 20

  • H(t) = 40\cos\!\left(\dfrac{\pi t}{5}\right) + 60

  • H(t) = 40\cos\!\left(\dfrac{2\pi t}{5}\right) + 60

Choose your answer

12
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1 mark
A graph labelled "Graph of g" on the xy-plane. The horizontal axis runs from 0 to 12 with tick marks at each integer. The vertical axis runs from −4 to 4 with tick marks at each integer. The sinusoidal curve starts at y = 0 at x = 0, rises to a maximum of y = 4 at x = 1.5, returns to y = 0 at x = 3, falls to a minimum of y = −4 at x = 4.5, returns to y = 0 at x = 6, and then repeats this pattern, completing a second full cycle by x = 12.

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

  • The period is 3, and the amplitude is 4.

  • The period is 6, and the amplitude is 4.

  • The period is 3, and the amplitude is 8.

  • The period is 6, and the amplitude is 8.

Choose your answer

13
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1 mark

The figure shows the graph of a trigonometric function f.

Graph of a trigonometric function oscillating between −3 and 1 on the y-axis, with maxima at x = π/2 and x = 3π/2, and minima at x = 0, π, and 2π. The x-axis is labelled with π/2, π, 3π/2, and 2π. The y-axis shows tick marks at −3, −2, −1, 1, and 2.

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

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

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

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

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

Choose your answer

14
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1 mark

A sinusoidal function space f has a relative maximum at the point open parentheses 2 comma 5 close parentheses and a relative minimum at the point open parentheses 6 comma negative 3 close parentheses.

Which of the following could be an expression for space f open parentheses x close parentheses?

  • 4 \sin \left(\dfrac{\pi}{2} x\right) + 1

  • 4 \sin \left(\dfrac{\pi}{4} x\right) - 1

  • 4 \sin \left(\dfrac{\pi}{4} x\right) + 1

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

Choose your answer

15
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1 mark

The function space f is given by space f open parentheses x close parentheses equals 4 sin open parentheses pi over 6 open parentheses x minus 1 close parentheses close parentheses plus 3.

What is the value of space f open parentheses 8 close parentheses?

  • - 1

  • 3 - 2 \sqrt{3}

  • 1

  • 5

Choose your answer

16
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1 mark

The function space f is given by space f open parentheses x close parentheses equals 2 sin open parentheses pi over 3 x close parentheses plus 1.

On which of the following intervals of x is space f open parentheses x close parentheses both negative and increasing?

  • open parentheses 6 comma space 15 over 2 close parentheses

  • open parentheses 15 over 2 comma space 19 over 2 close parentheses

  • open parentheses 19 over 2 comma space 21 over 2 close parentheses

  • open parentheses 21 over 2 comma space 23 over 2 close parentheses

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17
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1 mark

A sinusoidal function can be written in the form space y equals a sin open parentheses b x close parentheses plus d, where a greater than 0 and b greater than 0. The function has maximum value 7, minimum value negative 3, and period 4. What is the value of a plus b plus d?

  • 7 + \dfrac{\pi}{2}

  • 7 + \pi

  • 9 + \dfrac{\pi}{2}

  • 11

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18
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1 mark

The graph of space f open parentheses x close parentheses equals sin x is transformed to obtain the graph of g open parentheses x close parentheses equals sin open parentheses 2 x plus pi over 3 close parentheses. Which of the following correctly describes the transformation?

  • Horizontal dilation by a factor of \dfrac{1}{2}, then horizontal shift left by \dfrac{\pi}{3}.

  • Horizontal dilation by a factor of \dfrac{1}{2}, then horizontal shift left by \dfrac{\pi}{6}.

  • Horizontal dilation by a factor of \dfrac{1}{2}, then horizontal shift right by \dfrac{\pi}{6}.

  • Horizontal dilation by a factor of 2, then horizontal shift left by \dfrac{\pi}{6}.

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19
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1 mark

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 equals 0. The rider's height h open parentheses t close parentheses, in meters, at time t (in minutes) is modeled by h open parentheses t close parentheses equals 17 minus 15 cos open parentheses pi t close parentheses. How many minutes elapse between the first and second times the rider reaches a height of 9.5 meters?

  • \dfrac{1}{3}

  • \dfrac{2}{3}

  • \dfrac{4}{3}

  • \dfrac{5}{3}

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