To determine which table of values corresponds to the function \( y = \sin(x) + \frac{2}{3} \), we need to evaluate \( y \) for the given \( x \) values and see which table matches those results.
The function \( y = \sin(x) + \frac{2}{3} \) means that for each \( x \), we first calculate \( \sin(x) \) and then add \( \frac{2}{3} \) to it.
Let's do this calculation for the specified values of \( x \):
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For \( x = 0 \):
- \( y = \sin(0) + \frac{2}{3} = 0 + \frac{2}{3} = \frac{2}{3} \)
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For \( x = \frac{\pi}{2} \):
- \( y = \sin\left( \frac{\pi}{2} \right) + \frac{2}{3} = 1 + \frac{2}{3} = \frac{5}{3} \)
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For \( x = \pi \):
- \( y = \sin(\pi) + \frac{2}{3} = 0 + \frac{2}{3} = \frac{2}{3} \)
-
For \( x = \frac{3\pi}{2} \):
- \( y = \sin\left( \frac{3\pi}{2} \right) + \frac{2}{3} = -1 + \frac{2}{3} = -\frac{1}{3} \)
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For \( x = 2\pi \):
- \( y = \sin(2\pi) + \frac{2}{3} = 0 + \frac{2}{3} = \frac{2}{3} \)
Now, we summarize the resulting values:
- For \( x = 0 \), \( y = \frac{2}{3} \)
- For \( x = \frac{\pi}{2} \), \( y = \frac{5}{3} \)
- For \( x = \pi \), \( y = \frac{2}{3} \)
- For \( x = \frac{3\pi}{2} \), \( y = -\frac{1}{3} \)
- For \( x = 2\pi \), \( y = \frac{2}{3} \)
Now, let's look through the options provided and compare the \( y \) values from our calculations:
- Option 1:
- \( y = \frac{2}{3}, \frac{5}{3}, \frac{2}{3}, -\frac{1}{3}, \frac{2}{3} \) (matches!)
- Option 2:
- \( y = \frac{5}{3}, \frac{2}{3}, -\frac{1}{3}, \frac{2}{3}, \frac{5}{3} \) (does not match)
- Option 3:
- \( y = -\frac{2}{3}, \frac{1}{3}, -\frac{2}{3}, -\frac{5}{3}, -\frac{2}{3} \) (does not match)
- Option 4:
- \( y = 0, 1, 0, -1, 0 \) (does not match)
The correct table of values that corresponds to \( y = \sin(x) + \frac{2}{3} \) is Option 1:
\[ \begin{array}{|c|c|} \hline x & y \ \hline 0 & \frac{2}{3} \ \frac{\pi}{2} & \frac{5}{3} \ \pi & \frac{2}{3} \ \frac{3\pi}{2} & -\frac{1}{3} \ 2\pi & \frac{2}{3} \ \hline \end{array} \]