To determine the rate of change (slope) and the initial value (y-intercept) of the linear function from the given table of values, we will follow these steps:
-
Calculate the rate of change (m) using the formula for the slope of a line, which is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] For the values provided:
- \( (x_1, y_1) = (2, 22) \)
- \( (x_2, y_2) = (8, 25) \)
Plugging into the formula: \[ m = \frac{25 - 22}{8 - 2} = \frac{3}{6} = \frac{1}{2} \]
-
Determine the initial value (b), which is the y-intercept. We can use the slope-intercept form of a linear equation, \( y = mx + b \). We can substitute one of the points to solve for \( b \).
Using the point \( (2, 22) \): \[ 22 = \frac{1}{2}(2) + b \] \[ 22 = 1 + b \] \[ b = 22 - 1 = 21 \]
Thus, we have:
- The rate of change \( m = \frac{1}{2} \)
- The initial value \( b = 21 \)
In summary:
- The rate of change is \( m = \frac{1}{2} \)
- The initial value is \( b = 21 \)