Slope Concept 3 – Slopes of Vertical and Horizontal Lines

Note: This is the third part of the Slope Concept Series.

Part I: Basic Understanding of Slope

Part II:  Slope of the Graph of a Linear Function

In the Understanding the Basic Concepts of Slope post, we have discussed that the slope of a line in the coordinate plane is described as the change in y over the change in x. When we say change in x, or change in y, we talk about the ‘change distance.’ To determine a distance we need two points.  If we are in the coordinate plane, and we have two points with coordinates (x_1,y_1) and (x_2,y_2), then the rise is y_2 - y_1 and the run is x_2 - x_1. Thus, the slope of the line containing the two points is \displaystyle\frac{y_2-y_1}{x_2-x_1}.

Fiigure 1 - Slope of a line containing points with coordinates (x1,y1) and (x2,y2).

We also have discussed that the slope of a horizontal line is 0 by rotation. Here, we will show the same fact using coordinates as shown.

A horizontal line has the same y-coordinates everywhere. Let us consider line l in Figure 2 with equation y = a.   Let us pick two points with coordinates (x_1,a) and (x_2,a). Using the formula above, calculating for the slope we have \displaystyle\frac{a -a}{x_2 - x_1} = \frac{0}{x_2 - x_1 }= 0. Hence, the slope of a horizontal line l is 0.

Figure 2 - A horizontal line containing points with coordinates (x_1,a) and (x_2,a)..

Similarly, a vertical line has the same x-coordinate everywhere. Let us consider line m in Figure 3 with equation x = b.   Let us pick two points with coordinates (b,y_1) and (b,y_2). Using the formula above, calculating the slope, we have \displaystyle\frac{y_2 - y_1}{b - b} = \frac{ y_2 - y_1}{0} which is undefined.

Figure 3 - A horizontal line containing points with coordinates (b,y2) and (b,y1).

In the sequel of this post, we will discuss the lines with positive and negative slopes.

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