To determine the rate of change (slope) and the initial value (y-intercept) of the linear function represented by the table of values, we can use the points given: (2, 22) and (8, 25).
- Calculate the rate of change (slope, m): The formula for the slope \( m \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Where:
- \( (x_1, y_1) = (2, 22) \)
- \( (x_2, y_2) = (8, 25) \)
Plugging in the values:
\[ m = \frac{25 - 22}{8 - 2} = \frac{3}{6} = \frac{1}{2} \]
- Find the initial value (y-intercept, b): We can use one of the points and the slope to find the equation of the line in the form \( y = mx + b \). Using the point (2, 22):
\[ 22 = \frac{1}{2}(2) + b \] \[ 22 = 1 + b \] \[ b = 21 \]
So, the values are:
- The rate of change \( m = \frac{1}{2} \)
- The initial value \( b = 21 \)
Thus, the final answer is:
- The rate of change is \( m = \frac{1}{2} \)
- The initial value is \( b = 21 \)
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