Question

x y 2 22 8 25 Determine the rate of change and the initial value of the linear function given here as a table of values. (1 point) The rate of change is m= , and the initial value is b= .

Answer 1

To determine the rate of change (slope) and the initial value (y-intercept) from the given data points, we can first identify the points provided in the table:

1. Point 1: (2, 22)
2. Point 2: (8, 25)

### Step 1: Calculate 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 provided points:

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

Substituting the values into the formula:

\[
m = \frac{25 - 22}{8 - 2} = \frac{3}{6} = \frac{1}{2}
\]

### Step 2: Find the Initial Value (y-intercept)

The y-intercept \( b \) can be found using the formula of a linear equation:

\[
y = mx + b
\]

We can solve for \( b \) using one of the points. Let's use the point (2, 22):

\[
22 = \left(\frac{1}{2}\right)(2) + b
\]

This simplifies to:

\[
22 = 1 + b
\]

Solving for \( b \):

\[
b = 22 - 1 = 21
\]

### Conclusion

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

Thus, we can conclude:

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