To find the rate of change \( m \) and the initial value \( b \) of the linear function from the given table, we can use the points provided.
The points from the table are:
- \( (2, 22) \)
- \( (8, 25) \)
Step 1: Calculate the rate of change (slope) \( m \)
The formula for the rate of change (slope) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using our points:
- \( (x_1, y_1) = (2, 22) \)
- \( (x_2, y_2) = (8, 25) \)
Substituting these values into the formula:
\[ m = \frac{25 - 22}{8 - 2} = \frac{3}{6} = \frac{1}{2} \]
So, the rate of change \( m = \frac{1}{2} \).
Step 2: Determine the initial value \( b \)
The linear function can be expressed in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
To find \( b \), we can use one of the points. Let's use the point \( (2, 22) \):
\[ 22 = \frac{1}{2}(2) + b \]
Now, solve for \( b \):
\[ 22 = 1 + b \] \[ b = 22 - 1 = 21 \]
So, the initial value \( b = 21 \).
Final Answer
The rate of change is \( m = \frac{1}{2} \), and the initial value is \( b = 21 \).
\[ \text{The rate of change is } m = \frac{1}{2}, \text{ and the initial value is } b = 21. \]
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