To determine the rate of change (slope) and the initial value (y-intercept) of the linear function represented by the given table of values, we can use the two points provided: (2, 22) and (8, 25).
Finding the Rate of Change (Slope)
The formula for the slope \( m \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points (2, 22) and (8, 25):
- \( x_1 = 2 \), \( y_1 = 22 \)
- \( x_2 = 8 \), \( y_2 = 25 \)
Substituting these values into the slope formula:
\[ m = \frac{25 - 22}{8 - 2} = \frac{3}{6} = \frac{1}{2} \]
Finding the Initial Value (y-intercept)
To find the initial value (y-intercept) \( b \), we can use the slope-intercept form of a linear equation \( y = mx + b \). We can substitute one of the points and the slope we've just calculated.
Using point (2, 22):
\[ 22 = \frac{1}{2}(2) + b \]
Calculating:
\[ 22 = 1 + b \] \[ b = 22 - 1 = 21 \]
Summary
The rate of change is \( m = \frac{1}{2} \) and the initial value is \( b = 21 \).
Thus:
- The rate of change is \( m = \frac{1}{2} \)
- The initial value is \( b = 21 \)