Question

Which of the following tables of values would you use to graph the function y=sinx+2/3
?(1 point)
Responses

x 0 π2
π
3π2
2π
y 23
53
23
−13
23
x 0 Start Fraction pi over 2 End Fraction pi Start Fraction 3 pi over 2 End Fraction 2 pi y Start Fraction 2 over 3 End Fraction Start Fraction 5 over 3 End Fraction Start Fraction 2 over 3 End Fraction negative Start Fraction 1 over 3 End Fraction Start Fraction 2 over 3 End Fraction

x 0 π2
π
3π2
2π
y 53
23
−13
23
53
x 0 Start Fraction pi over 2 End Fraction pi Start Fraction 3 pi over 2 End Fraction 2 pi y Start Fraction 5 over 3 End Fraction Start Fraction 2 over 3 End Fraction negative Start Fraction 1 over 3 End Fraction Start Fraction 2 over 3 End Fraction Start Fraction 5 over 3 End Fraction

x 0 π2
π
3π2
2π
y −23
13
−23
−53
−23
x 0 Start Fraction pi over 2 End Fraction pi Start Fraction 3 pi over 2 End Fraction 2 pi y negative Start Fraction 2 over 3 End Fraction Start Fraction 1 over 3 End Fraction negative Start Fraction 2 over 3 End Fraction negative Start Fraction 5 over 3 End Fraction negative Start Fraction 2 over 3 End Fraction

x 0 π2
π
3π2
2π
y 0 1 0 −1
0
x 0 Start Fraction pi over 2 End Fraction pi Start Fraction 3 pi over 2 End Fraction 2 pi y 0 1 0 negative 1 0

Answer 1

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 \( x \):

1. For \( x = 0 \):
- \( y = \sin(0) + \frac{2}{3} = 0 + \frac{2}{3} = \frac{2}{3} \)

2. For \( x = \frac{\pi}{2} \):
- \( y = \sin\left( \frac{\pi}{2} \right) + \frac{2}{3} = 1 + \frac{2}{3} = \frac{5}{3} \)

3. For \( x = \pi \):
- \( y = \sin(\pi) + \frac{2}{3} = 0 + \frac{2}{3} = \frac{2}{3} \)

4. For \( x = \frac{3\pi}{2} \):
- \( y = \sin\left( \frac{3\pi}{2} \right) + \frac{2}{3} = -1 + \frac{2}{3} = -\frac{1}{3} \)

5. 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:

1. **Option 1**:
- \( y = \frac{2}{3}, \frac{5}{3}, \frac{2}{3}, -\frac{1}{3}, \frac{2}{3} \) (matches!)
2. **Option 2**:
- \( y = \frac{5}{3}, \frac{2}{3}, -\frac{1}{3}, \frac{2}{3}, \frac{5}{3} \) (does not match)
3. **Option 3**:
- \( y = -\frac{2}{3}, \frac{1}{3}, -\frac{2}{3}, -\frac{5}{3}, -\frac{2}{3} \) (does not match)
4. **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**:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & \frac{2}{3} \\
\frac{\pi}{2} & \frac{5}{3} \\
\pi & \frac{2}{3} \\
\frac{3\pi}{2} & -\frac{1}{3} \\
2\pi & \frac{2}{3} \\
\hline
\end{array}
\]