To identify the phase shift and write the equation of the sine function, we need to analyze the given table and the values provided for \( f(x) \).
The table provides:
- When \( x = 0 \), \( f(0) = 1 \)
- When \( x = \frac{\pi}{2} \), \( f\left(\frac{\pi}{2}\right) = 0 \)
- When \( x = \pi \), \( f(\pi) = -1 \)
- When \( x = \frac{3\pi}{2} \), \( f\left(\frac{3\pi}{2}\right) = 0 \)
- When \( x = 2\pi \), \( f(2\pi) = 1 \)
This behavior suggests that the function is a sine function, which typically has a maximum value of 1 at \( x = 0 \) without any phase shift.
However, the function appears to have values shifted one quarter of the way through its cycle. Specifically, a typical sine function, \( f(x) = \sin x \), would attain its maximum at \( x = 0 \). Since our function is also at \( f(0) = 1 \), this suggests it is a sine function, but we need to determine if there's a phase shift.
From the table, when \( x \) is at its maximum (\( f(x) = 1 \)), it results in a sine function that shifted to the left. The value of \( \frac{\pi}{2} \) would naturally give a 0 value from \( \sin x \), but here, it appears to be shifted to give a max at \( x = 0 \).
Recognizing this, the phase shift can be determined by observing that:
\[ f(x) = \sin(x + \frac{\pi}{2}) \]
Since \( \sin(x + \frac{\pi}{2}) \) would match the given values in the table,
The shifted sine function corresponds to:
- A phase shift to the left, which can be represented as "c" in the context of sine functions. Here, \( c = \frac{\pi}{2} \).
Thus, the equation of the function is:
\[ f(x) = \sin(x + \frac{\pi}{2}) \]
The correct response that reflects this fact is:
f(x) = sin(x + π/2).
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