To determine the rate of change (slope) and the initial value (y-intercept) of the linear function from the given table of values, we can use the following information from the table:
- When \( x = 2 \), \( y = 22 \)
- When \( x = 8 \), \( y = 25 \)
Step 1: Calculate the Rate of Change (Slope)
The formula for the rate of change (slope \( m \)) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the values:
- Let \( (x_1, y_1) = (2, 22) \)
- Let \( (x_2, y_2) = (8, 25) \)
\[ m = \frac{25 - 22}{8 - 2} = \frac{3}{6} = \frac{1}{2} \]
Step 2: Determine the Initial Value (y-intercept)
To write the equation of the linear function in the form \( y = mx + b \), we can substitute one of the points and the slope into the equation to find \( b \).
Using the slope \( m = \frac{1}{2} \) and the point \( (2, 22) \):
\[ 22 = \left(\frac{1}{2}\right)(2) + b \]
Calculating \( b \):
\[ 22 = 1 + b \ b = 22 - 1 \ b = 21 \]
Final Values
- The rate of change is \( m = \frac{1}{2} \)
- The initial value is \( b = 21 \)
Thus, the answer is:
- The rate of change is \( m = \frac{1}{2} \)
- The initial value is \( b = 21 \)