Question

Use the table to answer the question.
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & 0 & $(\pi)/(2)$ & $\pi$ & $(3 \pi)/(2)$ & $2 \pi$ \\
\hline
$y$ & 0 & $(3)/(4)$ & 0 & $-(3)/(4)$ & 0 \\
\hline
\end{tabular}

Write the equation of the periodic function based on the table.

A. $y=(3)/(4) \cos x$
B. $y=-(3)/(4) \sin x$
C. $y=(3)/(2) \sin x$
D. $y=(3)/(4) \sin x$

Answer 1

Are you sure

Answer 2

Is that the correct option

Answer 3

To determine the equation of the periodic function based on the provided table, we can analyze the values of \( y \) corresponding to the values of \( x \):

1. At \( x = 0 \), \( y = 0 \).
2. At \( x = \frac{\pi}{2} \), \( y = \frac{3}{4} \).
3. At \( x = \pi \), \( y = 0 \).
4. At \( x = \frac{3\pi}{2} \), \( y = -\frac{3}{4} \).
5. At \( x = 2\pi \), \( y = 0 \).

From these observations, we can recognize the following points:

- The function starts at \( (0, 0) \), rises to \( \frac{3}{4} \) at \( x = \frac{\pi}{2} \), returns to 0 at \( x = \pi \), drops to \( -\frac{3}{4} \) at \( x = \frac{3\pi}{2} \), and returns to 0 at \( x = 2\pi \).

Thus, the function displays a characteristic sinusoidal behavior, which strongly resembles the sine function transposed and scaled.

To determine the correct equation, we note the following:

- The maximum value at its peak \( y = \frac{3}{4} \) suggests amplitude of \( \frac{3}{4} \).
- The sine function \( \sin(x) \) has a value of 0 at \( x = 0 \) and \( x = \pi \), and reaches its minimum at \( x = \frac{3\pi}{2} \).
- The output is positive for \( x = \frac{\pi}{2} \) and negative for \( x = \frac{3\pi}{2} \), which matches the behavior of a negative sine function.

Considering all these factors, the function that fits these values is most appropriately given by the equation:

\[
y = -\frac{3}{4} \sin x
\]

So the correct answer is:

**B. \( y = -\frac{3}{4} \sin x \)**.

Answer 4

Let's carefully analyze the data again step by step to verify the function that fits the given values.

The table shows the points:

- \( x = 0 \): \( y = 0 \)
- \( x = \frac{\pi}{2} \): \( y = \frac{3}{4} \)
- \( x = \pi \): \( y = 0 \)
- \( x = \frac{3\pi}{2} \): \( y = -\frac{3}{4} \)
- \( x = 2\pi \): \( y = 0 \)

### Analyzing the Points:
1. **At \( x = 0 \)**: \( y = 0 \)
2. **At \( x = \frac{\pi}{2} \)**: \( y = \frac{3}{4} \) (maximum)
3. **At \( x = \pi \)**: \( y = 0 \)
4. **At \( x = \frac{3\pi}{2} \)**: \( y = -\frac{3}{4} \) (minimum)
5. **At \( x = 2\pi \)**: \( y = 0 \)

### Observing the Pattern:
- The function starts from zero, goes up to \(\frac{3}{4}\), back to zero, down to \(-\frac{3}{4}\), and back to zero again. This indicates a sine function that is shifted down.

### Investigating Possible Equations:
- A sine function has a maximum at \( \frac{\pi}{2} \) and is zero at \( 0 \) and \( \pi \), and negative at \( \frac{3\pi}{2} \).
- The behavior can be modeled by a sine function with a negative amplitude:

\[
y = A \sin(x)
\]

### Determining the Amplitude:
We need the amplitude to be \(\frac{3}{4}\), so we'll adjust the amplitude:

Since the sine function starts from 0 and goes to a maximum, our equation must account for the vertically shifted behavior that matches:

\[
y = \frac{3}{4} \sin(x)
\]

If we check:
- It goes to zero at \( x=0 \) and \( x=\pi \).
- It reaches \(\frac{3}{4}\) at \( x=\frac{\pi}{2}\).
- It goes to \(-\frac{3}{4}\) at \( x = \frac{3\pi}{2}\).

### Conclusion:
The correct equation that fits the given data is:

**D. \( y = \frac{3}{4} \sin x \)**

I appreciate your patience, and I apologize for the confusion in my previous response. The right answer is indeed **D. \( y = \frac{3}{4} \sin x \)**.

Answer 5

Let's review the options and analyze the table values again to find the correct option for the periodic function matching the provided \( x \) and \( y \) values.

The table provided is:

\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 0 & \frac{\pi}{2} & \pi & \frac{3\pi}{2} & 2\pi \\
\hline
y & 0 & \frac{3}{4} & 0 & -\frac{3}{4} & 0 \\
\hline
\end{array}
\]

### Analyzing Behavior:

- **At \( x = 0 \)**: \( y = 0 \) (starting point)
- **At \( x = \frac{\pi}{2} \)**: \( y = \frac{3}{4} \) (maximum point)
- **At \( x = \pi \)**: \( y = 0 \) (returns to zero)
- **At \( x = \frac{3\pi}{2} \)**: \( y = -\frac{3}{4} \) (minimum point)
- **At \( x = 2\pi \)**: \( y = 0 \) (returns to zero)

This pattern corresponds to the behavior of a sine function, but it has a specific amplitude of \(\frac{3}{4}\) and is reflected through the x-axis when it reaches a minimum.

### Testing the Options:

1. **Option A**: \( y = \frac{3}{4} \cos x \)
- At \( x=0 \), \( y = \frac{3}{4} \) — not \(0\).
- This is incorrect.

2. **Option B**: \( y = -\frac{3}{4} \sin x \)
- At \( x=0 \), \( y = 0\) (correct).
- At \( x=\frac{\pi}{2} \), \( y = -\frac{3}{4} \) (not matching).
- This is incorrect.

3. **Option C**: \( y = \frac{3}{2} \sin x \)
- At \( x=0 \), \( y = 0\) (correct).
- At \( x=\frac{\pi}{2} \), \( y = \frac{3}{2} \) (not matching).
- This is incorrect.

4. **Option D**: \( y = \frac{3}{4} \sin x \)
- At \( x=0 \), \( y = 0\) (correct).
- At \( x = \frac{\pi}{2} \), \( y = \frac{3}{4}\) (correct).
- At \( x = \pi \), \( y = 0 \) (correct).
- At \( x = \frac{3\pi}{2} \), \( y = -\frac{3}{4}\) (correct).
- At \( x = 2\pi \), \( y = 0 \) (correct).
- This matches all the points.

### Conclusion:
The correct answer based on the table values is:

**D. \( y = \frac{3}{4} \sin x \)**

Thank you for your patience, and I confirm that **Option D** is indeed the correct option.