To determine the linear function \( y = mx + b \) that represents the relationship, we need to find the slope (\( m \)) and the y-intercept (\( b \)).
First, let's find the slope \( m \). The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
We can use two of the given points. Let's use the points \((-4, 10)\) and \((-2, 7)\):
\[ x_1 = -4, \quad y_1 = 10 \] \[ x_2 = -2, \quad y_2 = 7 \]
Calculating the slope:
\[ m = \frac{7 - 10}{-2 - (-4)} = \frac{-3}{2} = -\frac{3}{2} \]
Now, we can use the slope-intercept form of the equation to solve for \( b \) by substituting one of the points into the equation \( y = mx + b \). Let's use the point \((-2, 7)\):
\[ 7 = -\frac{3}{2}(-2) + b \]
Calculating \( -\frac{3}{2}(-2) \):
\[ 7 = 3 + b \]
To solve for \( b \), we subtract 3 from both sides:
\[ b = 7 - 3 = 4 \]
Now we have both \( m \) and \( b \):
\[ m = -\frac{3}{2}, \quad b = 4 \]
The equation of the linear function is:
\[ y = -\frac{3}{2}x + 4 \]
Thus, the function that represents the relationship is:
\[ y = -\frac{3}{2}x + 4 \]