Question

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 find the rate of change \( m \) and the initial value \( b \) of the linear function from the given table, we can use the points provided.

The points from the table are:
- \( (2, 22) \)
- \( (8, 25) \)

### Step 1: Calculate the rate of change (slope) \( m \)

The formula for the rate of change (slope) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

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

Using our points:
- \( (x_1, y_1) = (2, 22) \)
- \( (x_2, y_2) = (8, 25) \)

Substituting these values into the formula:

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

So, the rate of change \( m = \frac{1}{2} \).

### Step 2: Determine the initial value \( b \)

The linear function can be expressed in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

To find \( b \), we can use one of the points. Let's use the point \( (2, 22) \):

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

Now, solve for \( b \):

\[
22 = 1 + b
\]
\[
b = 22 - 1 = 21
\]

So, the initial value \( b = 21 \).

### Final Answer

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

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