Question

Use the table to answer the question.

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=
.
Remaining Attempts : 2View previous attempt
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page 26 of 28

Answer 1

To find the rate of change (m) and the initial value (b) of the linear function represented by the table of values, we first need to identify the points from the table.

From the information provided, we have the following points:

- Point 1: (1, 4)
- Point 2: (-3, 8)

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

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

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

Using the points we have:

- \((x_1, y_1) = (1, 4)\)
- \((x_2, y_2) = (-3, 8)\)

Substituting these values into the formula gives:

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

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

The initial value corresponds to the y-intercept of the linear function. We can use the slope-intercept form of a linear equation, which is:

\[
y = mx + b
\]

We can substitute one of the points (let's use (1, 4)) and the slope (m = -1) into this equation to solve for b:

\[
4 = -1(1) + b
\]
\[
4 = -1 + b
\]
\[
b = 4 + 1 = 5
\]

### Summary

So, to summarize:

- The rate of change \( m = -1 \)
- The initial value \( b = 5 \)

Therefore:

The rate of change is m = -1, and the initial value is b = 5.