Identify the Correct Equation: Linking X and Y with Table Data

To determine the correct function, we first observe the changes in $X$ and their corresponding changes in $Y$. This can help us establish the slope of the function.

Let's calculate the slope $m$ using two points from the table. From $X = -4$ to $X = 0$, $Y$ changes from -3 to -2, resulting in a change of 1 in $Y$ and 4 in $X$. Hence the slope $m = \frac{1}{4}$.

Now, let's use the point $(0, -2)$ to find the y-intercept $c$ in the equation $y = mx + c$. Substituting $m = \frac{1}{4}$ and point $(0, -2)$ in, we solve for $c$:

$-2 = \frac{1}{4}(0) + c$
$c = -2$

This gives us the equation $y = \frac{1}{4}x - 2$.

Let's verify this equation against other points in the table:

• For $X = 4$: $y = \frac{1}{4}(4) - 2 = 1 - 2 = -1$. ✓
• For $X = 8$: $y = \frac{1}{4}(8) - 2 = 2 - 2 = 0$. ✓
• For $X = 12$: $y = \frac{1}{4}(12) - 2 = 3 - 2 = 1$. ✓

All table values satisfy the equation $y = \frac{1}{4}x - 2$.

Therefore, the function that corresponds to the table is: $y = \frac{1}{4}x - 2$.