The values you've provided correspond to specific angles and their sine values. The function \( f(x) \) takes on values of \( 0, 1, 0, -1, \) and \( 0 \) at the specified angles:
- \( f(-\frac{\pi}{4}) = 0 \)
- \( f(\frac{\pi}{4}) = 1 \)
- \( f(\frac{3\pi}{4}) = 0 \)
- \( f(\frac{5\pi}{4}) = -1 \)
- \( f(\frac{7\pi}{4}) = 0 \)
These points suggest a function that oscillates between these values.
A function that fits this description is \( f(x) = \sin(2x) \). To verify:
- Calculate \( f(-\frac{\pi}{4}) = \sin(2 \cdot -\frac{\pi}{4}) = \sin(-\frac{\pi}{2}) = 0 \).
- Calculate \( f(\frac{\pi}{4}) = \sin(2 \cdot \frac{\pi}{4}) = \sin(\frac{\pi}{2}) = 1 \).
- Calculate \( f(\frac{3\pi}{4}) = \sin(2 \cdot \frac{3\pi}{4}) = \sin(\frac{3\pi}{2}) = -1 \).
- Calculate \( f(\frac{5\pi}{4}) = \sin(2 \cdot \frac{5\pi}{4}) = \sin(\frac{5\pi}{2}) = 0 \).
- Calculate \( f(\frac{7\pi}{4}) = \sin(2 \cdot \frac{7\pi}{4}) = \sin(\frac{7\pi}{2}) = 1 \).
However, you have stated that \( f(\frac{5\pi}{4}) = -1 \) instead of moving forward in sine values.
Given the periodic nature and specific outputs of your data, a more suitable model could be a piecewise function defined specifically at those critical points, as any adjustment to the simple sine function or its transformations (like amplitude, phase shifts, vertical shifts, etc.) do not produce functions consistent with both the sine periodic waveform and the provided values elapsed straightforwardly.
Thus, based purely on the points given, you might want to represent \( f(x) \) as a piecewise function concluding specifically the outputs at those pivotal angles.
In any case, if you are looking for periodicity, then you might re-consider trigonometric adjustments such as:
- Adjustments about sine functions such as: \[ f(x) = \sin\left(2x\right), \text{ but ensure parameters fit clearly with outputs you listed. } \]
- Alternatively, if you seek more continuity you might consider composite functions that lead to those specific outputs respectively as confirmations.
Let me know if your focus is more linear or approximate modeling regarding traditional function forms on graphical frameworks or purely discrete selections. Please clarify if that’s sound or wish for more details.