The Lesson

The slope (or gradient) between two points measures the steepness of the line joining the points.

The Theory

The slope between two points can be found using the formula below:In the formula, (x1, y1) and (x2, y2) are the Cartesian coordinates of the two points. The image below shows what we mean by the slope between the two points:Note: (x1, y1) is the point on the left and (x2, y2) is the point on the right.

How to Find the Slope Between Two Points

Finding the slope between two points is easy.

Question

What is the slope between the points (1, 1) and (3, 5)?

Step-by-Step:

1

Start with the formula.:
$$Slope = \frac{y_2 - y_1}{x_2 - x_1}$$
Don't forget: / means ÷

2

Find the Cartesian coordinates of the points. In our example:
  • The first point is (1, 1), so x1 = 1 and y1 = 1.
  • The second point is (3, 5), so x2 = 3 and y2 = 5.

3

Substitute x1, y1, x2 and y2 into the formula.

$$Slope = \frac{5 - 1}{3 - 1}$$

$$\:\:\:\:\:\:\:\:\:\:\:\: = \frac{4}{2}$$

$$\:\:\:\:\:\:\:\:\:\:\:\: = 4 \div 2$$

$$\:\:\:\:\:\:\:\:\:\:\:\: = 2$$

Answer:

The slope between the points (1, 1) and (3, 5) is 2.

How to Visualize the Slope between Two Points

The slope between the points (1, 1) and (3, 5) is 2. By plotting the points, we can visualize what the slope means.To get from one point to the other (going left to right), you can see in the image above that you have to go 4 up and 2 across. The slope is simply how far up you go over how far across ("the change in y over the change in x" or "the rise over the run"). In this example it is 4/2 = 2. Another way of see this is by noticing that for each square you go across, you go 2 up.

Lesson Slides

The slider below gives another example of finding the slope between two points.

Positive and Negative Slopes

A positive slope means the line slopes up and to the right: A negative slope means the line slopes down and to the right:

Zero Slope

A line that goes straight across has zero slope:

Slope of 1

A slope of 1 is a 45° line going from bottom-left to top-right:

Fractional Slope

Slope can be a fraction, such as ½ and ¾. An improper fraction is positive, but less than 1. A slope of 1 gives a 45° line that splits the graph in 2. A fractional slope is less steep than this: Any slope greater than 1 is steeper than this.