To determine which of the given equations satisfies the data in the table, we will substitute the [tex]\( x \)[/tex] values into each equation and compare the result with the corresponding [tex]\( y \)[/tex] values.
The table is as follows:
[tex]\[
\begin{array}{c|c|c|c}
x & -3 & 0 & 3 \\
\hline
y & 0 & 2 & 4
\end{array}
\][/tex]
Let’s examine each equation one by one.
### Option A: [tex]\( y = \frac{2}{3} x + 2 \)[/tex]
Substitute [tex]\( x = -3 \)[/tex]:
[tex]\[
y = \frac{2}{3} \times (-3) + 2 = -2 + 2 = 0
\][/tex]
Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[
y = \frac{2}{3} \times 0 + 2 = 0 + 2 = 2
\][/tex]
Substitute [tex]\( x = 3 \)[/tex]:
[tex]\[
y = \frac{2}{3} \times 3 + 2 = 2 + 2 = 4
\][/tex]
The [tex]\( y \)[/tex] values match the given table.
### Option B: [tex]\( y = \frac{3}{2} x + 4 \)[/tex]
Substitute [tex]\( x = -3 \)[/tex]:
[tex]\[
y = \frac{3}{2} \times (-3) + 4 = -4.5 + 4 = -0.5
\][/tex]
This does not match the [tex]\( y \)[/tex] value 0 for [tex]\( x = -3 \)[/tex], so Option B is incorrect.
### Option C: [tex]\( y = -\frac{3}{2} x + 4 \)[/tex]
Substitute [tex]\( x = -3 \)[/tex]:
[tex]\[
y = -\frac{3}{2} \times (-3) + 4 = 4.5 + 4 = 8.5
\][/tex]
This does not match the [tex]\( y \)[/tex] value 0 for [tex]\( x = -3 \)[/tex], so Option C is incorrect.
### Option D: [tex]\( y = -\frac{2}{3} x + 2 \)[/tex]
Substitute [tex]\( x = -3 \)[/tex]:
[tex]\[
y = -\frac{2}{3} \times (-3) + 2 = 2 + 2 = 4
\][/tex]
This does not match the [tex]\( y \)[/tex] value 0 for [tex]\( x = -3 \)[/tex], so Option D is incorrect.
Therefore, the correct equation that satisfies the given data is:
[tex]\[
\boxed{y = \frac{2}{3} x + 2}
\][/tex]
Thus, the correct choice is [tex]\( \boxed{A} \)[/tex].
