Use the table to answer the question.

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 $b=5$.