To determine the correct equation for the function \( f(x) \) given the values in the table, we need to identify the pattern of the sine function.
The values given in the table appear to follow some shifts and transformations of the sine function at the specified intervals. Let's analyze it:
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Function Values at Specific Points:
- \( f(-\frac{\pi}{4}) = 0 \)
- \( f(\frac{\pi}{4}) = 1 \)
- \( f(\frac{3\pi}{4}) = 0 \)
- \( f(\frac{5\pi}{4}) = -1 \)
- \( f(\frac{7\pi}{4}) = 0 \)
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Sine Function Characteristics: The sine function \( f(x) = \sin(x) \) has zeros at integer multiples of \( \pi \) and has a maximum value of 1 at \( \frac{\pi}{2} + k\cdot2\pi \) and a minimum value of -1 at \( \frac{3\pi}{2} + k\cdot2\pi \).
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Identify the Shift: Observing the values:
- The function has a maximum of 1 at \( \frac{\pi}{4} \), which corresponds to \( \sin(\frac{\pi}{4} + \frac{\pi}{2}) \) = \( 1 \).
- If we transform to match this pattern, we can consider shifting the sine function to achieve matching values.
Possible Equation Forms:
- Shifted Sine Functions:
- \( f(x) = \sin(x + \frac{\pi}{4}) \)
- \( f(x) = \sin(x + \frac{\pi}{2}) \)
- \( f(x) = \sin(x - \frac{\pi}{4}) \)
Correct Function:
Given that the observed maximum occurs at \( \frac{\pi}{4} \), the correct transformation is:
The correct equation in the answer choices corresponds to:
Answer: \( f(x) = \sin(x + \frac{\pi}{4}) \)
Thus, the response is:
f(x) = sin(x + π/4) (which reflects the transformation of the function clearly).