Question

Select any two points on the graph and apply the slope formula, interpreting the result as a rate of change (units of measurement required); andUse rate of change (slope) to explain why your graph is linear (constant slope) or not linear (changing slopes).

Answer 1

Consider the following graph

Recall that the slope is given by

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

To find the slope, we need to select any two points on the graph.

Let us select the points (0, 4) and (2, 1)

$\begin{gathered} (x_1,y_1)=(0,4) \\ (x_2,y_2)=(2,1) \end{gathered}$

Let us substitute these points into the above slope formula

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{1-4}{2-0}=\frac{-3}{2}=-\frac{3}{2}$

So, the slope of the line is -3/2

Recall that a linear graph has a constant slope no matter which points we select on the graph, we will end up with the same slope.

Let us verify this by finding the slope again with two different points on the graph.

Let us select the points (4, -2) and (6, -5)

$\begin{gathered} (x_1,y_1)=(4,-2) \\ (x_2,y_2)=(6,-5) \end{gathered}$

Let us substitute these points into the slope formula

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{-5-(-2)}{6-4}=\frac{-5+2}{2}=-\frac{3}{2}$

As you can see, the slope of the line is still -3/2

This proves that our graph is linear since it has a constant slope.