Use the table to answer the question.

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=

Answer 1

To determine the rate of change $m$ and the initial value $b$ of the linear function from the given table of values, we need to identify the values of $x$ and $y$.

From the table, we interpret the values as follows:

• For $x=1$, $y=4$
• For $x=-3$, $y=8$
• We are missing the value of $y$ when $x=1$.

However, we can calculate the rate of change $m$ using the two points $(x_1,y_1)=(1,4)$ and $(x_2,y_2)=(-3,8)$:

The formula for the rate of change $m$ (slope) between two points is given by:

$$m=\frac{y_2-y_1}{x_2-x_1}$$

Substituting the values in:

$$m=\frac{8-4}{-3-1}=\frac{4}{-4}=-1$$

Now, the initial value $b$ (y-intercept) can be found using the point-slope form of the linear equation $y=mx+b$. We can plug one of the points into the equation to find $b$.

Using the point $(1,4)$:

$$4=-1(1)+b$$ $$4=-1+b$$ $$b=4+1=5$$

Thus, the rate of change $m=-1$ and the initial value $b=5$.

In summary:

• The rate of change is $m=-1$
• The initial value is $b=5$