AP®MathsCollege BoardPrecalculusExam QuestionsTrigonometric & Polar FunctionsSinusoidal Functions & Modeling

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

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Sinusoidal Functions & Modeling

1a
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2 marks

For a guitar to make a sound, the strings need to vibrate, or move up and down or back and forth, in a motion that can be modeled by a periodic function.

At time t equals 0 seconds, point X on one vibrating guitar string starts at its highest position, 2 millimeters above its resting position. Then it passes through its resting position and moves to its lowest position, 2 millimeters below the resting position. Point X then passes through its resting position and returns to 2 millimeters above the resting position. This motion occurs 200 times in 1 second.

The sinusoidal function h models how far point X is from its resting position, in millimeters, as a function of time t, in seconds. A positive value of h open parentheses t close parentheses indicates the point is above the resting position; a negative value of h open parentheses t close parentheses indicates the point is below the resting position.

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 open parentheses t comma space h open parentheses t close parentheses close parentheses for the five points: F, G, J, K, and P.

Graph of a sinusoidal wave showing peaks labelled F and P, a valley labelled J, and with points G and K on the midline. Horizontal lines indicate maximum and minimum levels, and a dashed line shows the midline.

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1b
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2 marks

The function h can be written in the form h open parentheses t close parentheses equals a sin open parentheses b open parentheses t plus c close parentheses close parentheses plus d. Find values of constants a, b, c, and d.

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2a
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2 marks
Woman pushing a tire, indicated by her posture and an arrow indicating the tyre is rolling forwards. A point W is marked on the circumference of the tire. Note reads: "Figure not drawn to scale."

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 equals 1 half second. The tire completes a full rotation, and the next time W touches the ground is at time t equals 5 over 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 open parentheses t comma space h open parentheses t close parentheses close parentheses for the five points: F, G, J, K, and P.

Graph of a sinusoidal wave showing peaks labelled F and P, a valley labelled J, and with points G and K on the midline. Horizontal lines indicate maximum and minimum levels, and a dashed line shows the midline.

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2b
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2 marks

The function h can be written in the form h open parentheses t close parentheses equals a sin open parentheses b open parentheses t plus c close parentheses close parentheses plus d. Find values of constants a, b, c, and d.

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3a
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1 mark

The depth of water at a harbor, in meters, is modeled by the function D, where D open parentheses t close parentheses equals 8 cos open parentheses fraction numerator pi t over denominator 6 end fraction close parentheses plus 12 for 0 less or equal than t less or equal than 24. D open parentheses t close parentheses is the depth in meters, and t is the number of hours since midnight on a particular day.

On the interval open parentheses 3 comma 6 close parentheses, is D increasing or decreasing?

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3b
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1 mark

On the interval open parentheses 3 comma 6 close parentheses, describe the concavity of the graph of D and determine whether the rate of change of D is increasing or decreasing.

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4a
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2 marks

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 less or equal than t less or equal than 12. The temperature reaches its maximum of 80 degree straight F at t equals 7 and its minimum of 40 degree straight F at t equals 1.

The function T can be written in the form T open parentheses t close parentheses equals a cos open parentheses b open parentheses t plus c close parentheses close parentheses plus d. Find the values of a, b, c, and d.

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4b
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1 mark

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

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4c
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1 mark

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

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5a
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2 marks

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.

t (hours)

0

6

12

18

24

D \left(t\right) (meters)

10

14

10

6

10

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.

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5b
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1 mark

Use your model to find the value of D open parentheses 8 close parentheses. Express your answer as an exact value. Show the work that leads to your answer.

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5c
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1 mark

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

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6a
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2 marks

Let space f open parentheses theta close parentheses equals sin theta. The graph of g is constructed by applying the following transformations to the graph of space f, in order:

  • A vertical dilation by a factor of 3

  • A horizontal dilation by a factor of 1 half

  • A vertical translation up by 4 units

Find an expression for g open parentheses theta close parentheses.

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6b
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1 mark

State the amplitude, period, and midline of g.

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7a
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1 mark

The temperature, in degrees Fahrenheit, in a particular city on a certain day is modeled by the function T, where T open parentheses t close parentheses equals 12 sin open parentheses fraction numerator pi t over denominator 12 end fraction close parentheses plus 65 for 0 less or equal than t less or equal than 24. T open parentheses t close parentheses 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.

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7b
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1 mark

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

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7c
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1 mark

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

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8a
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2 marks

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 equals 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 open parentheses t comma space h open parentheses t close parentheses close parentheses for the five points: F, G, J, K and P.

A black-and-white sketch of a cosine wave showing two full cycles. the graph has a horizontal dashed midline and solid horizontal lines at the maximum and minimum values. Five points are labeled in italics: F at a peak, G where the graph crosses the midline while decreasing, J at a trough, K where the graph crosses the midline while increasing, and P at the next peak. the axes are not shown.

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8b
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2 marks

The function h can be written in the form h open parentheses t close parentheses equals a cos open parentheses b open parentheses t plus c close parentheses close parentheses plus d. Find the values of constants a, b, c and d.

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9a
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2 marks
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.

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 equals 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 open parentheses t comma space h open parentheses t close parentheses close parentheses for the five points: F, G, J, K and P.

A black-and-white sketch of a sine wave showing two full cycles. the graph has a horizontal dashed midline and solid horizontal lines at the maximum and minimum values. Five points are labeled in italics: F at a trough, G where the graph crosses the midline while increasing, J at a peak, K where the graph crosses the midline while decreasing, and P at the next trough. The axes are not shown.

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9b
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2 marks

The function h can be written in the form h open parentheses t close parentheses equals a sin open parentheses b open parentheses t plus c close parentheses close parentheses plus d. Find the values of constants a, b, c and d.

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10a
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1 mark

The function h is given by h open parentheses t close parentheses equals 4 sin open parentheses fraction numerator pi t over denominator 6 end fraction close parentheses plus 5.

Find all values of t in the interval 0 less or equal than t less or equal than 12 for which h open parentheses t close parentheses equals 7, or indicate that there are no such values. Show the work that leads to your answer.

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10b
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2 marks

Find all real values of t for which h open parentheses t close parentheses equals 9. Show the work that leads to your answer.

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