To determine the periodic function represented in the table, we will first analyze the values provided.
From the table, we can observe the following:
- When \( x = 0 \), \( y = 0 \)
- When \( x = 2\pi \), \( y = 1 \)
- When \( x = 4\pi \), \( y = 0 \)
- When \( x = 6\pi \), \( y = -1 \)
- When \( x = 8\pi \), \( y = 0 \)
This pattern suggests that the function is sinusoidal, as we see oscillation through the values \( 0 \), \( 1 \), \( 0 \), \( -1 \), and back to \( 0 \).
The points indicate that:
- The function has a maximum value of \( 1 \) at \( x = 2\pi \).
- The function returns to the same maximum every \( 4\pi \) (from \( 0 \to 2\pi \to 4\pi \)).
- The period of the function is \( 4\pi \).
Since the sine function typically has a period of \( 2\pi \), we can adjust it to have a period of \( 4\pi \) by using the formula \( f(x) = \sin\left(\frac{x}{2}\right) \), which gives it a period of \( 4\pi \).
Thus, the equation of the periodic function represented in the table is:
\[ f(x) = \sin\left(\frac{x}{2}\right) \]
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