Question

Determine the rate of change and the initial value of this linear function as a table .. x/y 2/22. 8/25

Answer 1

To determine the rate of change and the initial value from the given points in the table:

Points:
- (2, 22)
- (8, 25)

### Step 1: Calculate the Rate of Change

The rate of change (slope) of a linear function can be found using the formula:

\[
\text{Rate of Change (m)} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's label the points:
- \( (x_1, y_1) = (2, 22) \)
- \( (x_2, y_2) = (8, 25) \)

Now substitute into the formula:

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

### Step 2: Determine the Initial Value

The linear function can be expressed in the slope-intercept form:

\[
y = mx + b
\]

Where \( m \) is the rate of change (slope) and \( b \) is the y-intercept (initial value when \( x = 0 \)).

We know \( m = \frac{1}{2} \). To find \( b \), we can use one of the points; let's use the point \( (2, 22) \):

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

### Summary:

- **Rate of Change**: \( \frac{1}{2} \)
- **Initial Value (y-intercept)**: \( 21 \)

### Result in Table Form:

| x | y |
|---|----|
| 2 | 22 |
| 8 | 25 |

- **Rate of Change**: \( \frac{1}{2} \)
- **Initial Value**: \( 21 \)