Describe how the graph of the function $ y = \sin(x) $ changes when it is transformed to $ y = \sin(x - \frac{\pi}{2}) $. - 3.6 Sinusoidal Function Transformations - Review - Use this assignment in your class!

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3.6 Sinusoidal Function Transformations - Review

Describe how the graph of the function $y = \sin (x)$ changes when it is transformed to $y = \sin (x - \frac{π}{2})$ .

Explain the effect of the transformation $y = 2 \sin (x)$ on the graph of $y = \sin (x)$ .

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How does the graph of

y = \cos (x)

change when it is reflected over the x-axis to become

y = - \cos (x)

What is the effect of the transformation

y = \sin (2 x)

on the period of the function

y = \sin (x)

Describe the combined effect of the transformations

y = - 3 \sin (x) + 2

on the graph of

y = \sin (x)

Describe how the amplitude of the function

y = 3 \sin (x)

compares to the amplitude of the function

y = \sin (x)

. Explain your reasoning.

How does the period of the function

y = \sin (2 x)

compare to the period of the function

y = \sin (x)

? Provide a detailed explanation.

Explain how a horizontal shift affects the graph of the function

y = \sin (x)

. Use the function

y = \sin (x - \frac{π}{2})

as an example.

How does the vertical shift affect the function

y = \sin (x)

? Illustrate your answer using the function

y = \sin (x) + 2

Compare the effects of a horizontal stretch and a horizontal compression on the function

y = \cos (x)

. Use the functions

y = \cos (\frac{1}{2} x)

and

y = \cos (2 x)

as examples.

The average temperature in a city varies sinusoidally over the course of a year. The highest temperature is 30°C in July, and the lowest is -10°C in January. Write a sinusoidal function to model the temperature

T (t)

as a function of time

t

in months, where

t = 0

corresponds to January.

A sound wave can be modeled by the function

y (t) = 3 \sin (440 π t)

. If the sound wave is transformed to have an amplitude of 5 and a frequency of 880 Hz, write the new sinusoidal function.

A spring oscillates vertically with a maximum displacement of 15 cm and a period of 2 seconds. Write a sinusoidal function that models the displacement

d (t)

of the spring from its equilibrium position as a function of time

t

The depth of water at a pier varies sinusoidally with time. At low tide, the depth is 2 meters, and at high tide, it is 10 meters. The time between consecutive high tides is 12 hours. Write a sinusoidal function that models the depth

D (t)

of the water as a function of time

t

in hours.

Describe how the graph of the function

y = \sin (x)

changes when it is transformed to

y = \sin (x - \frac{π}{2})

The graph of

y = \sin (x)

is shifted to the right by

\frac{π}{2}

units. This is because the transformation

y = \sin (x - \frac{π}{2})

represents a horizontal translation to the right by

\frac{π}{2}

units. The shape and amplitude of the sine wave remain unchanged.

Which of the following transformations correctly describes the change from

y = \sin (x)

y = \sin (x - \frac{π}{2})

A horizontal shift to the left by

\frac{π}{2}

units.

A horizontal shift to the right by

\frac{π}{2}

units.

A vertical shift up by

\frac{π}{2}

units.

A vertical shift down by

\frac{π}{2}

units.

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