3.05 Describing the slope of a line
Introduction
Previously, we used the slope of a line to graph proportional relationships in a coordinate plane. This time, we will derive the equation y=mx for a line through the origin. Using similar triangles, we will be able to explain why the slope m is the same between any two distinct points on a non-vertical line in a coordinate plane.
Identify slope from a graph
The slope of a line tells us how steep or slanted the line is. More specifically, it is the ratio of the change in the y-coordinates (vertical change) to the change in the x-coordinates (horizontal change).
Consider the two triangles and the points A, \, B, and C on the following coordinate plane:
We can see that from point A to Point B, we move 9 units down and 3 units to the right to create lines that form the sides of a triangle.
From point B to Point C, we also move 9 units down and 3 units to the right.
The triangle between A and B has a vertical height of 9 and a horizontal length of 3.
The triangle between B and C has a vertical height of 9 and a horizontal length of 3.
The simplified ratio of the vertical height to the horizontal length of both triangles is 3 to 1, which has a value of 3, indicating that the triangles are similar.
Now we can express this direction of movement as a simplified ratio, comparing vertical movement to horizontal movement.
From point A to Point B, the ratio \dfrac{-9}{3} can be simplified to \dfrac{-3}{1} or -3.
From point B to Point C, the ratio \dfrac{-9}{3} can be simplified to \dfrac{-3}{1} or -3.
This is the same as the ratio of the vertical change to the horizontal change, which is the slope of the line. Therefore, the slope is -3.
Substituting the value of m=-3 in the equation of the line which passes through the origin y=mx, we have y=-3x.
Examples
Example 1
Consider the following graph:
Derive the equation of the line in the form y=mx.
The slope of a line is the ratio of the change in the y-coordinates (vertical change) to the change in the x-coordinates (horizontal change).
Describe lines
Some lines have positive slopes, like these:
If we picture the line as a hill and imagine walking on the hill from left to right we will notice that with the lines above we would be walking uphill. This is why we consider the slope of these lines to be positive.
But some lines have decreasing slopes, like these:
If we imagine walking on a hill from left to right again, for these lines we would be walking downhill.
Exploration
Use the applet below to change the slope and create lines that have positive or negative slope. Make a note of anything interesting. Try to find or imagine types of slope other than positive or negative.
Notice that changing the slope doesn't move the line up or down, it simply adjusts the steepness. The closer to 0 the slope is, the flatter or less steep the line is. The farther from 0 the slope is (in the positive or negative direction) the steeper the line is. That is to say a line with a slope of 2 and a line with a slope of -2 are equally steep, just slanted in opposite directions and a line with a slope of 1 is steeper than a line with a slope of \dfrac{1}{4}.
Some lines have a slope that is neither positive nor negative.
The line shown below is horizontal. Imagine being at the beach and looking out at the horizon. The line where the ocean meets the sky is a horizontal line. The name makes a little more sense now.
Note the coordinates of points A, B, and C on the horizontal line above. A=(-4,4)\text{ }B=(2,4)\text{ }C=(4,4). Notice that all of the y-coordinates are the same. The line has a y-value of 4 no matter what the x-value is.
Horizontal lines have no vertical change. The only change is from left to right so these lines look flat. The slope of a horizontal line is 0 because the line isn't actually sloped at all so we can say that is has no slope or 0 slope.
The line shown below is vertical.
Note the coordinates of points A, B, and C on the horizontal line above. A=(5,-4)\text{ }B=(5,-2)\text{ }C=(5,4). Notice that all of the x-coordinates are the same. The line has an x-value of 5 no matter what the y-value is.
Vertical lines are infinitely sloped. If we imagine this line as a hill there is no way we could walk up it. We would need climbing equipment so this is really more of a cliff. For this reason, the slope of a vertical line is considered undefined.
Examples
Example 2
Answer the following.
Which line has a slope of 0?
Which line has a slope that is undefined?
A line slopes upwards when its slope is positive. A line slopes downwards when its slope is negative.
Horizontal lines have no vertical change and look flat. The slope of a horizontal line is 0 because the line isn't actually sloped at all so we can say that is has no slope or 0 slope.
Vertical lines are infinitely sloped. The slope of a vertical line is considered undefined.