Question

x: -pi/4 pi/4 3pi/4 5pi/4 7pi/2
F(x): 0 1 0 -1 0

The table lists values of a sine function at intervals of π/2. Given the values, identify the equation of the function.

Answer 1

Given the values of \( F(x) \) at specific intervals of \( x \), we can analyze the behavior of the function to determine the equation of the sinusoidal function.

The values given are:

- \( F(-\frac{\pi}{4}) = 0 \)
- \( F(\frac{\pi}{4}) = 1 \)
- \( F(\frac{3\pi}{4}) = 0 \)
- \( F(\frac{5\pi}{4}) = -1 \)
- \( F(\frac{7\pi}{4}) = 0 \)

From the values given, we can observe the following:

1. **Periodicity:** The function returns to zero at multiple points, indicating it is periodic.
2. **Amplitude:** The maximum value is 1 (at \( \frac{\pi}{4} \) and minimum is -1 (at \( \frac{5\pi}{4} \)), which suggests an amplitude of 1.
3. **Function Behavior:**
- It passes through zero at \( -\frac{\pi}{4} \), \( \frac{3\pi}{4} \), and \( \frac{7\pi}{4} \).
- It reaches 1 at \( \frac{\pi}{4} \) and -1 at \( \frac{5\pi}{4} \).

Based on this behavior, we can identify that the function is likely a sine function modified for phase shift.

To find a possible function:
- The sine function normally has roots (zero crossings) at multiples of \( \pi \). However, since the first zero occurs at \( -\frac{\pi}{4} \), we can use a phase shift to account for this.
- Noticing that it goes from zero to positive to zero to negative suggests the use of the sine function, which generally has the form:

\[
F(x) = A \sin(B(x - C)) + D
\]

Where:
- \( A \) is the amplitude (1),
- \( B \) is related to the period,
- \( C \) is the horizontal shift (phase shift),
- \( D \) is the vertical shift (which is 0 here since values range from -1 to 1).

The normal period of \( \sin(x) \) is \( 2\pi \). The values appear to indicate a sine function with a 0 mean (vertical shift \( D = 0 \)).

Given that it oscillates, the behavior and zeros can align with a sine function if we shift it by \( \frac{\pi}{2} \) to the right. Thus, we can form:

\[
F(x) = \sin(x + \frac{\pi}{4})
\]

This function has the following properties:
- At \( x = -\frac{\pi}{4} \): \( F(-\frac{\pi}{4}) = \sin(0) = 0 \)
- At \( x = \frac{\pi}{4} \): \( F(\frac{\pi}{4}) = \sin(\frac{\pi}{2}) = 1 \)
- At \( x = \frac{3\pi}{4} \): \( F(\frac{3\pi}{4}) = \sin(\frac{5\pi}{4}) = 0 \)
- At \( x = \frac{5\pi}{4} \): \( F(\frac{5\pi}{4}) = \sin(\frac{9\pi}{4}) = -1 \)
- At \( x = \frac{7\pi}{4} \): \( F(\frac{7\pi}{4}) = \sin(\frac{11\pi}{4}) = 0 \)

Thus, the equation of the function is:

\[
F(x) = \sin(x + \frac{\pi}{4})
\]