Use the table to answer the question.

To determine the rate of change (m) and the initial value (b) of the linear function given the two points from the table, we will use the following formulas:

1. Rate of Change (slope) \( m \): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \((x_1, y_1) = (2, 22)\) and \((x_2, y_2) = (8, 25)\).

2. Initial Value (y-intercept) \( b \): For a linear function, if we know the slope \( m \) and one point, we can find \( b \) using the equation of the line \( y = mx + b \).

Step 1: Calculate the Rate of Change (m)

Using the values from the table:

• \( y_1 = 22 \), \( x_1 = 2 \)
• \( y_2 = 25 \), \( x_2 = 8 \)

Substituting these values into the formula for \( m \): \[ m = \frac{25 - 22}{8 - 2} = \frac{3}{6} = \frac{1}{2} \]

Step 2: Calculate the Initial Value (b)

We can use one of the points (we'll use (2, 22)) to find \( b \). Now we have \( m = \frac{1}{2} \).

The equation of the line is: \[ y = mx + b \] Substituting in \( m \) and the values for one of the points: \[ 22 = \frac{1}{2}(2) + b \] \[ 22 = 1 + b \] \[ b = 22 - 1 = 21 \]

Final Answers

The rate of change is \( m = \frac{1}{2} \), and the initial value is \( b = 21 \).

So, the answers are:

• The rate of change is \( m = \frac{1}{2} \)
• The initial value is \( b = 21 \)