Graphing Equations

In addition to its familiar meaning, the word "slope" has precise mathematical meaning. The slope of a line is the rise over the run, or the change in y divided by the change in x. To find the slope of a line, pick any two points on the line. Then subtract their x-coordinates and subtract their y-coordinates in the same order. Divide the difference of the y-coordinates by the difference of the x- coordinates:

Example 1. Find the slope of the line which passes through the points (2, 5) and (0, 1): m = = = 2. This means that every time x increases by 1 (anywhere on the line), y increase by 2, and whenever x decreases by 1, y decreases by 2.

If a line has a positive slope (i.e. m > 0), then y always increases when x increases and y always decreases when x decreases. Thus, the graph of the line starts at the bottom left and goes towards the top right.

Often, however, the slope of a line is negative. A negative slope implies that y always decreases when x increases and y always increases when x decreases. Here is an example of a graph with negative slope:

m = = = -
Thus, as x increases by 3, y decreases by 4, and as x decreases by 3, y increases by 4.

Sometimes, we will see equations whose graphs are horizontal lines. These are graphs in which y remains constant -- that is, in which y₁ - y₂ = 0 for any two points on the line:
m = = = 0.
The slope of any horizontal line is 0. In other words, as x increases or decreases, y does not change. x takes every possible value at a specific y value.

We will also see equations whose graphs are vertical lines. These are graphs in which x remains constant -- that is, in which x₁ - x₂ = 0 for any two points on the line:
m = = = undefined. We cannot divide a number by zero.
The slope of any vertical line is undefined. x does not increase or decrease; rather, y takes every possible value at a specific x value.

Intuitively, it makes sense that one cannot assign a slope to a vertical line; a line that is "almost" vertical (i.e. is very steeply inclined) could have a very large positive or negative slope. So there is no way to decide even whether or not a vertical line should have positive or negative slope (and it clearly cannot have zero slope).

Two lines are parallel if they have the same slope. Parallel lines, when extended, do not intersect at any point.

Two lines are perpendicular if their slopes are opposite reciprocals of each other. For example, if a line has a slope of , a perpendicular line has a slope of - . Perpendicular lines intersect each other at right angles.