AnswersAlgebra 2 MP 234 CHSModeling with Periodic Functions

Modeling with Periodic Functions Answers

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Mark sits on a swing that is 2.5 feet off the ground. Swinging as hard as he can, the swing reaches a height of 6 feet. Assuming Mark holds a steady pace, the height of the swing could be represented by a cosine curve.If Mark makes a full swing forward and backward in 3 seconds, which graph below could represent the height of the swing, in feet, for any time, t?

A
On a coordinate plane, seconds is on the x-axis and feet is on the y-axis. A curve starts at (0, 3). It has a maximum of y = 6 and a minimum of y = 3. It goes through 1 cycle in 1.5 seconds.
Option A
B
On a coordinate plane, seconds is on the x-axis and feet is on the y-axis. A curve starts at (0, 3). It has a maximum of y = 8 and a minimum of y = 4. It goes through 1 cycle in 1.5 seconds.
Option B
C
On a coordinate plane, seconds is on the x-axis and feet is on the y-axis. A curve starts at (0, 3). It has a maximum of y = 6 and a minimum of y = 3. It goes through 1 cycle in 3 seconds.
Option C
D
On a coordinate plane, seconds is on the x-axis and feet is on the y-axis. A curve starts at (0, negative 2). It has a maximum of y = 2 and a minimum of y = negative 2. It goes through 1 cycle in 1.5 seconds.
Option D
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The height, d, in feet of a ball suspended from a spring as a function of time, t, in seconds can be modeled by the equation . The ball is released from its lowest point at seconds. Using your knowledge of the general form of sine and cosine functions, which of the following equations can also model this situation?

Question illustration
A
d = negative 2 cosine (pi t) + 5
Option A
B
d = negative 2 cosine (pi (t + one-half)) + 5
Option B
C
d = 2 cosine (pi t) + 5
Option C
D
d = 2 cosine (pi (t + one-half)) + 5
Option D
3

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
4

Suppose that you want to model the height of a rider on a Ferris wheel as a function of time, t, in minutes. If the rider is at the bottom of the Ferris wheel at t = 0 minutes, which of the following would be easiest to use?

A
the cosine function unreflected
B
the sine function unreflected
C
the cosine function reflected about its midline
D
the sine function reflected about its midline
5

The height, h, in feet of a buoy in relation to sea level as a function of time, t, in seconds can be modeled by the graph below. How long does it take for the buoy to go from its highest point to its lowest point and then back to its highest point?

Question illustration
A
1 second
B
2 seconds
C
3 seconds
D
4 seconds
6

The distance, d, in inches of a weight attached to a spring from its equilibrium as a function of time, t, in seconds can be modeled by the graph below. Which equation is represented in the graph below?

Question illustration
A
d = negative 10 cosine (StartFraction pi Over 2 EndFraction t)
Option A
B
d = negative 10 cosine (pi t)
Option B
C
d = negative 5 cosine (StartFraction pi Over 2 EndFraction t)
Option C
D
d = negative 5 cosine (pi t)
Option D
7

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
8

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
9

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
10

The height, h, in feet of a flag on one blade of a windmill as a function of time, t, in seconds can be modeled by the equation . What is the minimum height of the flag?

Question illustration
A
3 feet
B
9 feet
C
12 feet
D
15 feet

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