From the table provided, we can observe the values of the sine function at specific intervals:
- \( F(0) = 1 \)
- \( F(\frac{\pi}{2}) = 0 \)
- \( F(\pi) = -1 \)
- \( F(\frac{3\pi}{2}) = 0 \)
- \( F(2\pi) = 1 \)
These correspond to the standard sine function, but the values suggest a cosine function due to the fact that the maximum occurs at \( x = 0 \).
We know that the sine function has the general form:
\[ F(x) = A \sin(B(x - c)) + D \]
Where:
- \( A \) is the amplitude,
- \( B \) affects the period,
- \( c \) is the phase shift,
- \( D \) is the vertical shift.
From the table, we can see a complete cycle of the sine function occurring from \( 0 \) to \( 2\pi \), which indicates a regular period, suggesting \( B = 1 \).
Since the maximum value occurs at \( x = 0 \), it is actually equivalent to the cosine function shifted. The cosine function can be expressed as:
\[ F(x) = A \cos(B(x - c)) + D \]
The maximum of the cosine function is \( A \) when \( x = 0 \), and since \( F(0) = 1 \), we have:
- \( A = 1 \),
- \( D = 0 \) (no vertical shift).
The cosine function has a maximum value at \( x = 0 \) when there is no phase shift, indicating:
\[ F(x) = \cos(x) \]
Thus, comparing to the standard sine function, we recognize that this can be reinterpreted using the sine function with a phase shift. The cosine function can also be expressed in terms of the sine function:
\[ \cos(x) = \sin\left(x + \frac{\pi}{2}\right) \]
Thus, the phase shift \( c \) is \( -\frac{\pi}{2} \) to rewrite the equation as:
\[ F(x) = \sin\left(x + \frac{\pi}{2}\right) \]
Putting this all together, we have:
- Phase Shift (c): \( -\frac{\pi}{2} \)
- Equation of the Function: \( F(x) = \sin\left(x + \frac{\pi}{2}\right) \)
This matches the values provided in the table.