Describe how the amplitude of a sine or cosine function affects its graph. Provide an example with a specific function.

3.4 Sine and Cosine Function Graphs - Review

Explain how the period of a sine or cosine function is determined and provide an example with a specific function.

How does a phase shift affect the graph of a sine or cosine function? Provide an example with a specific function.

Describe the effect of a vertical shift on the graph of a sine or cosine function. Provide an example with a specific function.

Explain how the shape of the sine and cosine graphs differ and how they are similar. Provide an example of each.

Describe how the amplitude of the function $y = A \sin (x)$ affects its graph. Provide an example with a specific value for $A$ .

Explain how the period of the function $y = \cos (B x)$ changes with different values of $B$ . Provide an example with a specific value for $B$ .

How does a phase shift affect the graph of the function $y = \sin (x - C)$ ? Provide an example with a specific value for $C$ .

Compare the graphs of $y = \sin (x)$ and $y = \sin (x - \frac{π}{3})$ . How does the phase shift affect the position of the graph?

10.

How does the graph of $y = 2 \cos (2 x - π)$ differ from the graph of $y = \cos (x)$ ? Discuss the changes in amplitude, period, and phase shift.

11.

A coastal town experiences high tides and low tides every day. The height of the tide can be modeled by the function $h (t) = 3 \sin (\frac{π}{6} t) + 5$ , where $h (t)$ is the height of the tide in feet and $t$ is the time in hours. Explain what the amplitude, period, and vertical shift of this function represent in the context of the tides.

12.

A sound wave can be modeled by the function $y (t) = 0.5 \cos (440 π t)$ , where $y (t)$ is the displacement and $t$ is the time in seconds. Determine the frequency and period of this sound wave.

13.

A Ferris wheel has a diameter of 40 meters and completes one full rotation every 5 minutes. If the height of a seat on the Ferris wheel above the ground can be modeled by a cosine function, write the equation for the height $h (t)$ in terms of time $t$ in minutes, assuming the lowest point of the Ferris wheel is 2 meters above the ground.

14.

A temperature sensor records the temperature in a room over time and finds that it can be modeled by the function $T (t) = 10 \cos (\frac{π}{12} t) + 25$ , where $T (t)$ is the temperature in degrees Celsius and $t$ is the time in hours. Describe the temperature fluctuations in the room over a 24-hour period.

15.

A pendulum swings back and forth, and its displacement from the equilibrium position can be modeled by the function $d (t) = 5 \sin (2 π t)$ , where $d (t)$ is the displacement in meters and $t$ is the time in seconds. Determine the amplitude, period, and frequency of the pendulum's motion.

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AI InstructorAUG 12, 2024, 11:36:57 AM

The amplitude of a sine or cosine function determines the height of the peaks and the depth of the troughs from the midline (usually the x-axis). For example, in the function $y = 3 \sin (x)$ , the amplitude is 3, meaning the graph will oscillate between 3 and -3. This is because the amplitude is the coefficient in front of the sine or cosine function, which stretches or compresses the graph vertically.

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Given the function $y = 2 \cos (x)$ , what is the amplitude of the graph?

Select one of the following options:

A.
2
B.
0.5
C.
$π$
D.
1

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