Slope | Definition, Formula & Examples

Definition of Slope

The slope of a line is the ratio of the amount that y increases as x increases some amount. Slope tells you how steep a line is, or how much y increases as x increases. The slope is constant (the same) anywhere on the line.

Steepness

One way to think of the slope of a line is by imagining a roof or a ski slope. Both roofs and ski slopes can be very steep or quite flat. In fact, both ski slopes and roofs, like lines, can be perfectly flat (horizontal). You would never find a ski slope or a roof that was perfectly vertical, but a line might be.

We can usually visually tell which ski slope is steeper than another. Clearly, the three ski slopes get gradually steeper.

In mathematics, we often want to measure the steepness. You can tell that slopes B and C are higher than slope A. They are both seven units high, while slope A is only four units high. So, it appears that height has something to do with steepness.

Slope C, however, is clearly steeper than slope B, even though both are seven high. So, there must be more to steepness than height. If you look at the width of slopes B and C, you see that slope B is ten units, while slope C is only six units. The narrower ski slope is steeper.

It is not height alone or width alone that determines how steep the ski slope is. It is the combination of the two. In fact, the ratio of the height to the width (the height divided by the width) tells you the slope.

Think of it this way: suppose you need to change seven feet in height to get from the bottom of the ski slope to the top. For the moment we will pretend you are trying to climb up to the top of the slope. We will discuss going downward later. If you have only four feet straight ahead of you (the width in the picture) in which to get to the top, you have to climb up at a very steep angle. If, on the other hand, you have six feet ahead of you in which to ascend those seven feet, the angle is less steep. It is the relationship between height and width that matter.

You could write the relationship like this:

Slope = (Change in height)/(Change in width)

Or:

Slope = rise/run

If y represents the vertical direction on a graph, and x represents the horizontal direction, then this formula becomes:

Slope = (Change in y)/(Change in x)

Or:

In this equation, m represents the slope. The small triangles are read 'delta' and they are Greek letters that mean 'change.'

For the first ski slope example, the skier travels four units vertically and ten units horizontally. So, the first slope is m = 4/10.

The second ski slope involves a seven unit change vertically and ten units horizontally. So, the slope is m = 7/10. The second slope is steeper than the first because 7/10 is greater 4/10.

The third ski slope involves a seven unit change vertically and six units horizontally. So, the slope is m = 7/6. The third slope is the steepest of all.

Slope on the Cartesian Plane

The Cartesian plane is a two-dimensional mathematical graph. When graphing on it, a line may not start at zero as in the ski slope examples. In fact, a line goes on forever at both ends. The slope of a line, however, is exactly the same everywhere on the line. So, you can choose any starting and ending point on the line to help you find its slope. It is also possible that you might be given a line segment, which is a section of a line that has a beginning and an end. Or, you might be given two points and you are expected to draw (or imagine) the line segment between them. In all these situations, finding the slope works the same way.