Hi, everyone. This is Anthony Varela. And this video is about determining slope. So what we're going to do is define slope mathematically. And then, we're going to calculate slope using points on a graph and using a table of values.
So let's first talk about the formula for slope. How do we calculate slope? Well, slope is defined as the change in y values divided by the change in x values. So you see we use the variable m to represent slope. And you might also hear this as the change in y divided by the change in x, or rise over run because y values would be a vertical displacement and the difference in x values would be a horizontal displacement.
So then what are y2, y1, x2, and x1? Well, these are coordinates for two coordinate points, x and y. Just because we have two of these coordinate points, we need to differentiate between the two. So one of them is x1, y1. And the other one is x2, y2.
And this is all we need to calculate slope. We just need to identify two coordinate points. And we can plug them into our formula and calculate slope. So let's write this down in our notes as the formula to calculate the slope. And we just need to know two coordinate points to find y2, y1, and x2, x1.
So let's use a graph to calculate slope. Here, I have a line plotted on a graph. And we need to identify any two coordinate points. It does not matter which two we identify. So here, I have plotted two points on our graph. We see this one is the point 3, 1. We go over 1, 2, 3, up 1, and we're at this point. And if we went over 12 and up 7, we would be at this point.
So now that we've identified two coordinate points on our graph, we can match this up with what we need to plug into our formula, which is x1, y1 and x2, y2. So I like to dis read my graph from left to right. And the first point I hit is going to be x1, y1. And the second point that I hit is going to be x2, y2.
So now that I have values that I can plug into my formula for slope, let's go ahead and calculate the slope. So m equals my difference in y-coordinates. So this would be 7 minus 1 divided by the difference in x-coordinates, 12 minus 3.
So let's go ahead and evaluate this then. 7 minus is 1 is 6. And 12 minus 3 is 9. So the slope is 6/9. Or I could say this is 2/3.
And so recall I said that slope is commonly referred to rise over run. So what that means is for every rise of 2 and run of 3, I'm going to hit another point on my line. So I'm going to take this coordinate and go up 2, over 3, and I'm on the line. I go up 2 and over 3, and I'm on the line. Go up 3, over 3, and I'm on the line, specifically that second coordinate point that we used to calculate slope.
So now, let's use a table of values to calculate slope of this linear relationship, y equals 3x minus 2. So what I like to do is create a couple of more columns in my table. So I'm going to identify a couple of x values. I'm going to plug those x values into this equation. And I've left this middle column space so I can do my calculations. And then whatever I get for y, I'm going to put in this third column.
So let's go ahead and identify a couple of x values. Once again, it doesn't matter what you choose. I like starting at zero, because it's easy to multiply things by zero. And then just to some other points, 1, 2, and 3. Make it easy on us.
So now, what I'm going to do is for each of these x values plug that into my equation. So for this first row here, I'm going to have 3 times 0 minus 2. For the next row, I'll have 3 times 1 minus 2. And then I'll have 3 times 2 minus 2. And then I'll have 3 times 3 minus 2.
So let's go ahead and find out what y is at each of these values of x. Well, 3 times 0 is 0. And then I take away 2, so I get negative 2. 3 times 1 is 3. Taking away 2, gives me 1. 3 times 2 equals 6. Take away 2 gives me 4. And finally, 3 times 3 is 9. Take away 2, I get 7.
So I filled up my table of values. What do I do? Well, I notice that this table of values then gives me x-coordinates and y-coordinates. And these are paired up in the rows here. So I can just choose any two rows. It doesn't matter which ones you choose. So I'm just going to highlight two rows. And these form my coordinate points that I need to put into my formula for slope.
So what is x1, y1? Well, that can be the first row that I highlighted. So it's going to give me x is 0 and y is negative 2. And then the other coordinate point gives me x is 2 and y is 4. So let's plug these into our formula for slope.
So slope is the difference in the y-coordinate-- so this would be 4 minus negative 2. So I wrote that as 4 plus 2 because subtracting a negative is the same as adding a positive. So be careful about our negative numbers. Then, we put this over our difference in x values, which is 2 minus 0. So for plus 2 is 6, and 2 minus 0 is 2. So my slope is 6/2, which I can simplify to 3.
Now, the cool thing here-- and this is actually true for all linear relationships-- is that the slope is the same as the coefficient in front of the x term. So we see that this is a 3 here. And we've calculated that the slope is 3. And you would get this number 3 for whatever values you took from this table. And it's always going to match up to this coefficient in front of x.
Well, we can use any two points to calculate the slope. So here, I've highlighted two other points. That's the points 1, 1 and 3, 7. So if we plug in our difference in y-coordinate 7 minus 1, we'll still get 6. And our difference in x-coordinates, 3 minus 1, we'll still get 2. So our slope will still be 6 over 2, or 3.
So let's review determining slope. What did we talk about today? Well, here is the formula that we use to calculate slope. m is the slope. And then we have y2 minus y1, difference in y-coordinates, over x2 minus x1, difference in x-coordinates. So all we need then are two coordinate points. And we can get these coordinate points from points on a line. Or if we have the equation of the line, we can use a table of values to calculate slope.
Well, thanks for watching this tutorial on determining slope. Hope to see you next time.
