To find the rate of change (\(m\)) of the linear function from the table, we use the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points from the table:
- Point 1: \((x_1, y_1) = (1, 4)\)
- Point 2: \((x_2, y_2) = (-3, 8)\)
Plugging in these values into the formula for \(m\):
\[ m = \frac{8 - 4}{-3 - 1} = \frac{4}{-4} = -1 \]
So, the rate of change is \(m = -1\).
Next, we find the initial value (\(b\)). The initial value refers to the value of \(y\) when \(x = 0\). We can organize the linear function in the slope-intercept form \(y = mx + b\) using one of the points. Using the point \((1, 4)\):
\[ 4 = -1 \cdot 1 + b \] \[ 4 = -1 + b \] \[ b = 4 + 1 = 5 \]
So, the initial value is \(b = 5\).
Finally, the answers are:
The rate of change is \(m = -1\), and the initial value is \(b = 5\).