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.
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.
This page was written by Stephen Clarke.
You might also like...
graphs and coordinate geometryfinding the distance between pointsfinding the midpoint between pointstranslating shapes
Help Us Improve Mathematics Monster
- Do you disagree with something on this page?
- Did you spot a typo?
Find Us Quicker!
- When using a search engine (e.g., Google, Bing), you will find Mathematics Monster quicker if you add #mm to your search term.
Share This Page
If you like Mathematics Monster (or this page in particular), please link to it or share it with others.
If you do, please tell us. It helps us a lot!
Create a QR Code
Use our handy widget to create a QR code for this page...or any page.
