Which transformations are needed to change the parent cosine function to [tex]y=3 \cos (10(x-\pi))[/tex]?
A. vertical compression of 3, horizontal stretch to a period of [tex]5 \pi[/tex], phase shift of [tex] \pi[/tex] units to the left
B. vertical stretch of 3, horizontal compression to a period of [tex]\frac{\pi}{5}[/tex], phase shift of [tex]\pi[/tex] units to the right
C. vertical compression of 3, horizontal stretch to a period of [tex]10 \pi[/tex], phase shift of [tex]\pi[/tex] units to the right
D. vertical stretch of 3, horizontal compression to a period of [tex]\frac{\pi}{5}[/tex], phase shift of [tex]\pi[/tex] units to the left
Answer (2)
Identify the vertical stretch: The amplitude is 3, indicating a vertical stretch by a factor of 3.
Determine the horizontal compression: The period is 5 π , indicating a horizontal compression by a factor of 10.
Find the phase shift: The phase shift is π to the right.
Combine the transformations: The parent cosine function undergoes a vertical stretch of 3, a horizontal compression to a period of 5 π , and a phase shift of π units to the right. vertical stretch of 3, horizontal compression to a period of 5 π , phase shift of π units to the right
Explanation
Analyze the problem We are given the function y = 3"." cos ( 10 ( x − π )) and we want to determine the transformations needed to obtain this function from the parent cosine function y = cos ( x ) . We need to identify the vertical stretch/compression, horizontal stretch/compression, and phase shift.
General form of cosine function The general form of a cosine function with transformations is y = A cos ( B ( x − C )) , where:
∣ A ∣ is the vertical stretch/compression factor.
B affects the period of the function. The period is given by B 2 π .
C is the phase shift (horizontal shift).
Identify the parameters In our given function, y = 3 cos ( 10 ( x − π )) , we can identify the following:
A = 3 , so there is a vertical stretch by a factor of 3.
B = 10 , so the period is 10 2 π = 5 π . This means there is a horizontal compression by a factor of 10, resulting in a period of 5 π .
C = π , so there is a phase shift of π units to the right.
List the transformations Therefore, the transformations needed are:
Vertical stretch by a factor of 3.
Horizontal compression to a period of 5 π .
Phase shift of π units to the right.
Final Answer Comparing our findings with the given options, the correct answer is:
Vertical stretch of 3, horizontal compression to a period of 5 π , phase shift of π units to the right.
Examples
Understanding transformations of trigonometric functions is crucial in fields like signal processing and physics. For instance, when analyzing sound waves, the amplitude (vertical stretch) determines the loudness, the frequency (horizontal compression) determines the pitch, and the phase shift helps align different waves for interference studies. Imagine designing an audio equalizer; you'd use these transformations to boost or suppress certain frequencies, shaping the sound to your liking. Similarly, in optics, understanding these transformations helps in designing lenses and understanding how light waves behave.
The transformations needed to change the parent cosine function to y = 3 cos ( 10 ( x − π )) include a vertical stretch of 3, a horizontal compression to a period of 5 π , and a phase shift of π units to the right.
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