Definition of Slope
One of the most important things to understand about lines is the definition of slope. Slope is the 'steepness' of the line, also commonly known as rise over run. We can calculate slope by dividing the change in the x-value between two points over the change in the y-value. In order to understand the importance of the definition of slope, one should understand how to interpret graphs and how to write an equation.
One of the most important ideas you're going to see in your whole study of algebra is the idea of slope. And slope just refers to how steep a line is. And that's kind of the basic definition. That's something that I help -- or helps me remember what slope means.
There's a whole lot of fancy notation that goes along with it. And we'll get into that in just a second. But if you can just remember in the back of your head that slope means how steep the line is, it will help you a lot.
So one thing you see is that anytime I draw a straight line, the slope ratio stays the same for that entire line. And here's what I mean.
Let's say those are my X and Y axes and I have this line right here. The line is equally steep the entire time. It has the same slope ratio. And we're going to get in a second what I mean by ratio. And here's what I'm talking about. When we talk about slope or steepness, the way it's defined is by change of Y over change of X. Like in a fraction. Change in Y on top of change in X. That's why we call it a ratio.
Remember ratio is like a fraction. So if I were to draw a little triangle here that represents how steep my line is, this would be my change in Y piece because Y is up and down. This would be my change in X piece, because X is horizontal. And whatever those numbers were on the graph, I would write as a fraction. That's one thing you want to keep in mind. Change in Y on top of change in X.
Sometimes we write it using this little triangle. This triangle is the Greek letter delta, which is tricky. Not only do you have to learn math but now you have to learn Greek. This means change in Y on top of change in X. That delta just represents the word change.
And a third way we write this is using the letter M. M stands for slope, and if I had two points, I'll use them up here. Let's say I had this point I'm going to call it X from my first point and then Y from my first point. Here's my second point. X from my second point I'm going to use that little 2 to show it's my second point. Y for my second point.
Then there's a formula I could use using those X and Y numbers to find M, or the slope. And the way I would write that is Y to take away Y1 on top of X2 take away X1. This is the same thing just written in a different way. I'm finding out how much did my Y values change and putting that on top of how much did my X values change in a fraction.
So this formula is really important anytime you have two points like this. These little numbers down here are tricky. It doesn't mean take your Y value and multiply it by 2 or take your Y value and multiply it by 1. What it means is we're just notating using what's called a subscript. That's my second Y value. I'm subtracting my first Y value. And we'll get into that more later as you guys start your homework problems.
A couple other things I want you to keep in mind with slope, and that is sometimes slopes are positive numbers and sometimes slopes are negative numbers. And here's how you can tell.
Anytime you're given a graph, there's my axis and here's my line. Think about if you were like skateboarding and you were coming along, and you hit this ramp. This is an uphill ramp. We call that a positive slope. As opposed to if you hit this line here, here you come along skateboarding, I don't know why that's a skateboarding noise. I just made that up. You're skateboarding. You hit that thing, you're going down. That's why we call it a negative slope. Anytime a line increases from left to right it's called a positive. Decreases from left to right, it's called a negative.
That's a whole lot to keep in mind when you guys approach your problems. This is all mathematical notation that you're going to see over and over and over again. So you'll get used to it.
The one last thing I want to leave you with is thinking about word problems, because a lot of times graphs, as we know, represent the real world. Like this graph might be growth per day, or how much something changes over time.
So slope becomes really important when you're looking at word problems. It can be growth per day. Like let's say my slope was 5. I would change like five inches per day or something like that.
It's really important when you're looking at word problems that you keep in mind the units that go with your X and Y values so you can relate the slope back to the real world. You guys are going to do all kinds of practice with this.
Again, slope is a really, really important idea that's going to show up over and over, but I think you'll be successful if you can just remember that slope means how steep the line is.