Question

Use the table to answer the question.

x
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=
m
=
, and the initial value is b=
b
=
.

Answer 1

To find the rate of change (\(m\)) of the linear function from the table, we use the formula:

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

Using the points from the table:
- Point 1: \((x_1, y_1) = (1, 4)\)
- Point 2: \((x_2, y_2) = (-3, 8)\)

Plugging in these values into the formula for \(m\):

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

So, the rate of change is \(m = -1\).

Next, we find the initial value (\(b\)). The initial value refers to the value of \(y\) when \(x = 0\). We can organize the linear function in the slope-intercept form \(y = mx + b\) using one of the points. Using the point \((1, 4)\):

\[
4 = -1 \cdot 1 + b
\]
\[
4 = -1 + b
\]
\[
b = 4 + 1 = 5
\]

So, the initial value is \(b = 5\).

Finally, the answers are:

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