Question

Use the table to answer the question.

Determining the starting value and rate of change from a given value table.
x y
1 4
3 8
Find the rate of change and the initial value of the linear function given 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 (slope) and the initial value (y-intercept) from the given table of values, we can use the points provided in the table.

The points given are:
- Point 1: (1, 4)
- Point 2: (3, 8)

### Step 1: Calculate the Rate of Change (m)

The formula for the rate of change (slope) \( m \) is:

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

Using the points:
- \((x_1, y_1) = (1, 4)\)
- \((x_2, y_2) = (3, 8)\)

Substituting the values:

\[
m = \frac{8 - 4}{3 - 1} = \frac{4}{2} = 2
\]

### Step 2: Determine the Initial Value (b)

We can use the slope-intercept form of a linear equation, \( y = mx + b \), to find the initial value (y-intercept) \( b \). We can rearrange this equation to solve for \( b \):

\[
b = y - mx
\]

We can use either of the points, let’s use (1, 4):

\[
b = 4 - (2 \cdot 1) = 4 - 2 = 2
\]

### Final Answer

- The rate of change is \( m = 2 \)
- The initial value is \( b = 2 \)

Thus, we have:
- The rate of change is \( m = 2 \), and the initial value is \( b = 2 \).