Use the table to answer the question.

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

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 2

Are you sure

Answer 3

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 4

Is that the correct option

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.