To determine the rate of change \( m \) and the initial value \( b \) of the linear function from the given table of values, we need to identify the values of \( x \) and \( y \).
From the table, we interpret the values as follows:
- For \( x = 1 \), \( y = 4 \)
- For \( x = -3 \), \( y = 8 \)
- We are missing the value of \( y \) when \( x = 1 \).
However, we can calculate the rate of change \( m \) using the two points \((x_1, y_1) = (1, 4)\) and \((x_2, y_2) = (-3, 8)\):
The formula for the rate of change \( m \) (slope) between two points is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the values in:
\[ m = \frac{8 - 4}{-3 - 1} = \frac{4}{-4} = -1 \]
Now, the initial value \( b \) (y-intercept) can be found using the point-slope form of the linear equation \( y = mx + b \). We can plug one of the points into the equation to find \( b \).
Using the point \((1, 4)\):
\[ 4 = -1(1) + b \] \[ 4 = -1 + b \] \[ b = 4 + 1 = 5 \]
Thus, the rate of change \( m = -1 \) and the initial value \( b = 5 \).
In summary:
- The rate of change is \( m = -1 \)
- The initial value is \( b = 5 \)
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