To determine the equation of the periodic function based on the values provided in the table, we can analyze the information.
From the table, we see the following pairs of (x, y):
- \( (0, 0) \)
- \( \left(\frac{\pi}{2}, 34\right) \)
- \( (\pi, 0) \)
- \( \left(\frac{3\pi}{2}, -34\right) \)
- \( (2\pi, 0) \)
These points suggest that the function has a maximum value of 34 at \( x = \frac{\pi}{2} \) and a minimum value of -34 at \( x = \frac{3\pi}{2} \).
Since the function touches 0 at \( x = 0, \pi, \) and \( 2\pi \), a cosine function starting from 0 would work, but since the maximum occurs at \( \frac{\pi}{2} \), we can conclude it is a sine function starting from zero.
The amplitude of the function is half the distance from the maximum to the minimum:
\[ \text{Amplitude} = \frac{34 - (-34)}{2} = \frac{68}{2} = 34 \]
The period of the sine function is \( 2\pi \), and the sine function reaches its maximum at \( x = \frac{\pi}{2} \).
Thus, the function can be written as:
\[ y = 34 \sin x \]
Therefore, the correct response is:
y = 34 sin x
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