The graph of $y=\sin \left(x-\frac{3 \pi}{2}\right)$ is the graph of the $y=\sin (x)$ shifted in which direction?
A. $\frac{3 \pi}{2}$ units to the left
B. $\frac{3 \pi}{2}$ units to the right
C. $\frac{3 \pi}{2}$ units up
D. $\frac{3 \pi}{2}$ units down
Answer (2)
The given function is y = sin ( x − 2 3 π ) .
Comparing with the general form y = sin ( x − c ) , we identify c = 2 3 π .
Since 0"> c > 0 , the shift is to the right.
The graph of y = sin ( x − 2 3 π ) is the graph of y = sin ( x ) shifted 2 3 π units to the right. 2 3 π units to the right
Explanation
Understanding the Problem We are given the function y = sin ( x − 2 3 π ) and asked to describe how its graph is related to the graph of y = sin ( x ) . This involves understanding horizontal shifts of trigonometric functions.
Identifying the Shift The general form of a horizontally shifted sine function is y = sin ( x − c ) , where c represents the amount of the horizontal shift. If 0"> c > 0 , the graph is shifted to the right by c units. If c < 0 , the graph is shifted to the left by ∣ c ∣ units.
Determining the Direction and Magnitude In our case, we have y = sin ( x − 2 3 π ) . Comparing this to y = sin ( x − c ) , we see that c = 2 3 π . Since c is positive, the graph is shifted to the right.
Conclusion Therefore, the graph of y = sin ( x − 2 3 π ) is the graph of y = sin ( x ) shifted 2 3 π units to the right.
Examples
Understanding horizontal shifts is crucial in various fields, such as signal processing and physics. For instance, when analyzing sound waves, a horizontal shift can represent a time delay in the signal. Similarly, in physics, understanding phase shifts in wave phenomena is essential for analyzing interference patterns. Knowing how to interpret and apply these transformations allows engineers and scientists to accurately model and predict the behavior of systems involving periodic functions.
The graph of y = sin ( x − 2 3 π ) is shifted 2 3 π units to the right compared to the graph of y = sin ( x ) . The correct answer is option B. 2 3 π units to the right.
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