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 table of values, we can use the points given: (2, 22) and (8, 25).

1. **Calculate the rate of change (slope, m)**: The formula for the slope \( m \) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

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

Plugging in the values:

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

2. **Find the initial value (y-intercept, b)**: We can use one of the points and the slope to find the equation of the line in the form \( y = mx + b \). Using the point (2, 22):

\[
22 = \frac{1}{2}(2) + b
\]
\[
22 = 1 + b
\]
\[
b = 21
\]

So, the values are:
- The rate of change \( m = \frac{1}{2} \)
- The initial value \( b = 21 \)

Thus, the final answer is:
- The rate of change is \( m = \frac{1}{2} \)
- The initial value is \( b = 21 \)