Many of my students have trouble defining and finding slope, especially after not seeing it for some time, so I’d like to go over a solid definition and some examples to help you solidify the concept.
Slope is just a number that tells us how steep a line is and what direction it is going in. Every line (besides vertical lines) has a slope. Before I show you the formula, I’m going to give you a brief description:
When we’re looking at a graph, we’re talking about the vertical and horizontal change between two points. Below, I chose the points (-1, -1) and (0,1). So, let’s see what the slope between them is. This will tell us the slope of the entire line! Assume each grid has a length of 1 unit.
Starting from the point (-1, -1) we had to move up 2 and right 1. If we started at (0, 1) we would have had to go down 2 and left 1.
The graph shows us that the vertical change is 2 and the horizontal change is 1, so if we plug that into the formula we get:
We just found out that our slope is 2. This tells us that the slope is positive and for every 2 units I move up on the graph, I must move right 1 unit. I’m going to pick another point on the same graph to show you that the slope will still be the same. Let’s pick (-2, -3) and (1,3):
As you can see, our vertical change is 6 and our horizontal change is 3, so let’s find our slope:
Our slope is still 2, even though we chose two new coordinates. Remember, the slope is always the same, no matter what two points of the line you choose! In this way, we call slope proportional (like above, we found 6 over 3 was equal to 2 over 1.)
Let’s try another example: Find the slope of the following line
Once you’ve got your answer, move on below.
I chose the points (-1,-4) and (1, -1) but you could have chosen others as well:
You can see the vertical change from (-1,-4) to (1, -1) is 3 and the horizontal change is 2 so our slope is:
What if we had decided to go from (1, -1) to (-1, -4) instead? Notice how the arrows change:
We still get the same answer! It doesn’t matter what order you choose; you will end up with the same slope 😊
Let’s try one more:
Find the slope of the following line:
Once you get an answer, check it below:
If we start from the point (-1, 4) to (0, 1) (you can choose other points as well) our slope formula gives us:
Notice how a line with a negative slope has a different direction than a line with a positive slope.🧐
I hope this article helped a bit to find the slope from graphs, as slope is an important concept in Algebra and in the next post I’ll show you the proper formulas to find the slope between two points (and then from a table 😉)