Slope of a Line

Here we look at graphs of straight lines in more detail and study the concept of slope of a line. Slope If a highway has a 6% grade, then in 100 feet (measured horizontally) the road rises 6 feet (measured vertically). See the figure below. The ratio of 6 to 100 is 6%. If a roof rises 9 feet in a horizontal distance (or run) of 12 feet, then the roof has a 9–12 pitch. A roof with a 9–12 pitch is steeper than a roof with a 6–12 pitch. The grade of a road and the pitch of a roof are measurements of steepness. In each case the measurement is a ratio of rise (vertical change) to run (horizontal change).

We measure the steepness of a line in the same way that we measure steepness of a road or a roof. The slope of a line is the ratio of the change in y-coordinate, or the rise, to the change in x-coordinate, or the run, between two points on the line.

Slope

Consider the line in the following figure. In going from (0, 1) to (1, 3), there is a change of +1 in the x-coordinate and a change of +2 in the y-coordinate, or a run of 1 and a rise of 2. So the slope is or 2.

If we move from (1, 3) to (0, 1) as in the figure below, the rise is -2 and the run is -1. So the slope is or 2.

If we start at either point and move to the other point, we get the same slope.

Helpful Hint

Since the amount of run is arbitrary, we can choose the run to be 1. In this case slope

So the slope is the amount of change in y for a change of 1 in the x-coordinate.This is why rates like 50 miles per hour (mph), 8 hours per day, and two people per car are all slopes.

Example 1

Finding the slope from a graph

Find the slope of each line by going from point A to point B.

Solution

a) A is located at (0, 3) and B at (2, 0). In going from A to B, the change in y is -3 and the change in x is 2. So

b) In going from A(2, 1) to B(6, 3), we must rise 2 and run 4. So

c) In going from A(0, 0) to B(-6, -3), we find that the rise is -3 and the run is -6. So

Note that in Example 1(c) we found the slope of the line of Example 1(b) by using two different points. The slope is the ratio of the lengths of the two legs of a right triangle whose hypotenuse is on the line. See the following figure.

As long as one leg is vertical and the other leg is horizontal, all such triangles for a given line have the same shape: They are similar triangles. Because ratios of corresponding sides in similar triangles are equal, the slope has the same value no matter which two points of the line are used to find it.