Slope

Definition

The slope of the line through the points P = (x1, y1) and Q = (x2, y2) with x1 not equal to x2 is denoted by m where

The slope is sometimes referred to as or or . The slope gives the rate of change of y with respect to x .

Below is the graph of a line with slope of 3/2. The rise of 3 is shown in red while the run of 2 is in green.

Example 1

Find the slope of the line through ( -3, 5) and (2, 1).

Solution .

Since , we get or . Note that we would get the same result by computing

Below is a graph of the line through these two points.

Example 2

Here is a graph that shows lines through (1,2) with slopes of 3/2 and -2. Which line is increasing and which one is decreasing?

Example 3

Here are two lines through (0,0) with slopes of 3/2 and -2/3. What appears to be the geometric relationship between the lines?

Example 4

Suppose a company had 4.8 million dollars of sales in the first quarter of the year and 5.6 million dollars of sales in the second quarter. Find the slope of the linear relationship between sales in millions of dollars and time in quarters.

Solution .

or Expressing this in terms of the units given, we have that the rate of change of sales with respect to time is .8 million dollars per quarter.