AnswersAZ-Common Core Algebra II BChanges in Period and Phase Shift of Sine and Cosine Functions

Modeling with Periodic Functions Answers

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Suppose that you want to model the height of a rider on a Ferris wheel as a function of time. The amplitude of the function you use as a model should be equal to which of the following?

A
the area of the Ferris wheel
B
the circumference of the Ferris wheel
C
the diameter of the Ferris wheel
D
the radius of the Ferris wheel
2
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Tides in a specific location can be approximated using the periodic function shown on the graph.What is the interpretation of the amplitude in this application?

Question illustration
A
the time between high tides
B
the time between high tide and low tide
C
the sum of the heights at high tide and low tide
D
the distance halfway between the height at high tide and low tide
3

The height, h, in feet of a piece of cloth tied to a waterwheel in relation to sea level as a function of time, t, in seconds can be modeled by the equation . What is the period of the function?

Question illustration
A
seconds
Option A
B
seconds
Option B
C
8 seconds
D
20 seconds
4

Throughout the day, the depth of water at the end of a dock varies with the tides. The function represents the height, in feet, of the water t hours after midnight.Which graph shows the height of the water at the dock at any time after midnight?

Question illustration
A
On a coordinate plane, feet is on the x-axis and depth is on the y-axis. A curve starts at (0, 5). It decreases through the y-axis. It has a minimum of y = 2 and a maximum of y =12. It goes through 1 cycle in 13 feet.
Option A
B
On a coordinate plane, feet is on the x-axis and depth is on the y-axis. A curve starts at (0, 5). It increases through the y-axis. It has a minimum of y = 2 and a maximum of y =12. It goes through 1 cycle in 13 feet.
Option B
C
On a coordinate plane, feet is on the x-axis and depth is on the y-axis. A curve starts at (0, 5). It increases through the y-axis. It has a minimum of y = 2 and a maximum of y =12. It goes through 1 cycle in 7 feet.
Option C
D
On a coordinate plane, feet is on the x-axis and depth is on the y-axis. A curve starts at (0, negative 2). It increases through the y-axis. It has a minimum of y = negative 5 and a maximum of y =5. It goes through 1 cycle in 13 feet.
Option D
5

The displacement, d, in millimeters of a tuning fork as a function of time, t, in seconds can be modeled with the equation . What is the maximum displacement of the tuning fork?

Question illustration
A
0.2 mm
B
0.4 mm
C
0.8 mm
D
2.5 mm
6

Which of the following situations can be modeled with a periodic function?

A
the height of a football after it has been thrown
B
the height of a person riding on an escalator
C
the height of a building after it has been constructed
D
the height of a pebble stuck in the tread of a tire
7

A weight attached to a spring is at its lowest point, 9 inches below equilibrium, at time t = 0 seconds. When the weight it released, it oscillates and returns to its original position at t = 3 seconds. Which of the following equations models the distance, d, of the weight from its equilibrium after t seconds?

A
d = negative 9 cosine (StartFraction pi Over 3 EndFraction t)
Option A
B
d = negative 9 cosine (StartFraction 2 pi Over 3 EndFraction t)
Option B
C
d = negative 3 cosine (StartFraction pi Over 9 EndFraction t)
Option C
D
d = negative 3 cosine (StartFraction 2 pi Over 9 EndFraction t)
Option D
8

Blood pressure can be modeled by a sinusoidal curve, where the maximum and minimum on the curve correlate to the person’s blood pressure reading. Henry’s blood pressure is modeled by the function , where t is time in seconds.Which graph accurately depicts this model?

Question illustration
A
On a coordinate plane, seconds is on the x-axis and blood pressure is on the y-axis. A curve starts at (0, 100). It has a maximum of y = 130 and a minimum of y = 70. It goes through one cycle in 1 second.
Option A
B
On a coordinate plane, seconds is on the x-axis and blood pressure is on the y-axis. A curve starts at (0, 100). It has a maximum of y = 130 and a minimum of y = 70. It goes through one cycle in 2 seconds.
Option B
C
On a coordinate plane, seconds is on the x-axis and blood pressure is on the y-axis. A curve starts at (0, 0). It has a maximum of y = 30 and a minimum of y = negative 30. It goes through one cycle in 1 second.
Option C
D
On a coordinate plane, seconds is on the x-axis and blood pressure is on the y-axis. A curve starts at (0, 30). It has a maximum of y = 60 and a minimum of y = negative 0. It goes through one cycle in 1 second.
Option D
9

The height, h, in feet of a piece of cloth tied to a waterwheel in relation to sea level as a function of time, t, in seconds can be modeled by the equation . How long does it take for the waterwheel to complete one turn?

Question illustration
A
5 seconds
B
10 seconds
C
20 seconds
D
40 seconds
10

The equation models the horizontal distance, d, in inches of the pendulum of a grandfather clock from the center as it swings from right to left and left to right as a function of time, t, in seconds. According to the model, how long does it take for the pendulum to swing from its rightmost position to its leftmost position and back again? Assume that right of center is a positive distance and left of center is a negative distance.

Question illustration
A
0.625 seconds
B
0.8 seconds
C
1.25 seconds
D
1.6 seconds

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