What is the phase shift of a periodic function?
A. a horizontal translation of the function
B. the horizontal length of one cycle of the function
C. the number of cycles of the function that occur in one horizontal unit
D. a vertical translation of the function
Answer (2)
Phase shift is defined as the horizontal translation of a periodic function.
The horizontal length of one cycle is the period, and the number of cycles per unit is the frequency.
A vertical translation is not a phase shift.
Therefore, the phase shift is a horizontal translation of the function. a horizontal translation of the function
Explanation
Understanding the Question The question asks for the definition of the phase shift of a periodic function. We need to choose the correct option from the given choices.
Recalling the Definition of Phase Shift The phase shift of a periodic function refers to the horizontal shift or translation of the function. It indicates how much the function is shifted horizontally from its original position.
Comparing with the Options Now, let's examine the given options:
a horizontal translation of the function: This matches the definition of phase shift.
the horizontal length of one cycle of the function: This describes the period of the function.
the number of cycles of the function that occur in one horizontal unit: This describes the frequency of the function.
a vertical translation of the function: This describes a vertical shift, not a phase shift.
Identifying the Correct Option The correct definition of the phase shift of a periodic function is a horizontal translation of the function.
Final Answer The phase shift of a periodic function is a horizontal translation of the function.
Examples
In music, the phase shift is like adjusting when a note starts in a melody. Imagine two identical melodies, but one starts slightly later than the other. That delay is similar to a phase shift. In electrical engineering, phase shifts are crucial in alternating current (AC) circuits, affecting how power is delivered and used. Understanding phase shifts helps engineers design efficient and stable electrical systems.
The phase shift of a periodic function is the horizontal translation of the function, indicating how far the graph is shifted left or right along the x-axis. This is different from concepts like period or frequency, which relate to the length and number of cycles, respectively. The correct choice from the options given is A: a horizontal translation of the function.
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