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 (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 \( b = 21 \)