Question

Use the table to answer the 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) 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 \( b = 21 \)