To determine the rate of change (m) and the initial value (b) of the linear function given the table of values, we use the following information from the table:
- Point 1: (2, 22)
- Point 2: (8, 25)
Step 1: Calculate the rate of change (m)
The rate of change (slope) \( m \) is calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the values from the points:
\[ m = \frac{25 - 22}{8 - 2} = \frac{3}{6} = \frac{1}{2} \]
So, the rate of change \( m = \frac{1}{2} \).
Step 2: Determine the initial value (b)
The initial value (b) is the value of \( y \) when \( x = 0 \). We can use the slope-intercept form of the linear equation:
\[ y = mx + b \]
We can use one of the points to solve for \( b \). Let's use the point (2, 22):
\[ 22 = \frac{1}{2}(2) + b \]
Calculating this gives:
\[ 22 = 1 + b \] \[ b = 22 - 1 = 21 \]
So, the initial value \( b = 21 \).
Final Answer:
The rate of change is \( m = \frac{1}{2} \), and the initial value is \( b = 21 \).
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