Based on the values presented in the table, we have the points for the function \( f(x) \):
- \( f\left(-\frac{\pi}{4}\right) = 0 \)
- \( f\left(\frac{\pi}{4}\right) = 1 \)
- \( f\left(\frac{3\pi}{4}\right) = 0 \)
- \( f\left(\frac{5\pi}{4}\right) = -1 \)
- \( f\left(\frac{7\pi}{4}\right) = 0 \)
The values suggest that the function has the following key features:
- The function equals 0 at \( x = -\frac{\pi}{4} \), \( x = \frac{3\pi}{4} \), and \( x = \frac{7\pi}{4} \).
- The function reaches its maximum value of 1 at \( x = \frac{\pi}{4} \).
- The function reaches its minimum value of -1 at \( x = \frac{5\pi}{4} \).
These points and their behavior align closely with a sine function, particularly the sine function shifted and scaled.
Let's express these observations in terms of the sine function. The function \( f(x) \) can be represented as:
\[ f(x) = \sin\left(x + \frac{\pi}{4}\right) \]
To validate this, let's analyze the adjustments:
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The sine function, \( \sin(x) \), has zeros at \( x = n\pi \) for \( n \) being any integer. The function \( \sin\left(x + \frac{\pi}{4}\right) \) effectively shifts these points.
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Setting \(\sin\left(x + \frac{\pi}{4}\right) = 0\) gives \(x + \frac{\pi}{4} = n\pi\), or \(x = n\pi - \frac{\pi}{4}\), which matches the zeros we have.
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The maximum value of sine occurs at \( \frac{\pi}{2} \), and translating it gives a peak at:
\[ x + \frac{\pi}{4} = \frac{\pi}{2} \implies x = \frac{\pi}{4} \]
And the minimum occurs similarly, confirming the behavior at \( \frac{5\pi}{4} \).
Thus, the equation of the function is:
\[ f(x) = \sin\left( x + \frac{\pi}{4} \right) \]
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