Hi, and welcome back. My name is Anthony Varela. And today I'm going to be introducing slope. So we're going to define slope mathematically. We'll be calculating positive and negative slopes, take a look at the slopes of vertical and horizontal lines, and also the slopes of parallel and perpendicular lines.
So let's take a look at this line on a graph, and slope is commonly referred to as rise over run. Now what rise over run means is that we consider rise being vertical movement or vertical displacement and run being horizontal movement or horizontal displacement. So rise over run is vertical movement divided by horizontal movement.
Now when we're talking about movement, we're talking about between two points on the graph of a line. So here, I have identified two points that are on this line. And rise is the vertical movement from one point to another, and run is the horizontal movement from one point to another.
So more specifically, we can say that the rise is a change in y. So what that means is a difference in y-coordinates. So if we're calling this point number one and this point number two, the y-coordinate of this point is y2, and the y-coordinate of this point is y1. So if we take the difference of those two coordinates, we've found our change in y, which is that vertical movement, rise.
Now run, then, would be a change in x or the difference of x-coordinates, x2 minus x1. So then we can say that the slope is y2 minus y1 over x2 minus x1, a difference in y-coordinates divided by the difference in x-coordinates. And we use this variable m to define slope. So we have this formula then that m equals y2 minus y1 divided by x2 minus x1. So slope is the steepness of a line, and it can be found by dividing the change in y-coordinates by the change and x-coordinates from any two points on a line.
So returning to this graph, we're going to calculate the slope of the line that we see. So I need to first identify these coordinate points. So if we're assuming then that this graph has units of 1, then this coordinate right here on the x-axis is 1, 2, 3. Y-axis, 1, 2, 3, 4, 5. So this is the point 3, 5, and this has the point 7, 1. So we're going to call this point 1 and this point 2, which makes this x1, y1 and this, x2, y2. And we can substitute those into our formula for slope.
So plugging in our y-coordinates, we have 11 minus 5. And then our difference in x-coordinates, we have 7 minus 3. Let's go ahead and evaluate this. Well, 11 minus 5 is 6, and 7 minus 3 is 4. And I can further simplify 6/4 as 3/2. So the slope of this line is 3/2.
Now, notice that 3/2 is a positive number. So lines that have positive slopes-- we read the graph left to right, so we're looking at x values approaching positive infinity. Our y value also approaches positive infinity. So that's what a positive slope looks like. Well here's another line, which I think then might have a negative slope because I'm noticing, as I read the graph left to right, the line is decreasing, heading towards negative infinity in the y direction.
So let's go ahead and confirm this assumption. So I'm going to pick out two points on my line. I have the point 2, 5, and I have the point 6, 2. So this is x1, y1, x2, y2. Let's plug those into our formula for slope. So our difference in y-coordinates is 2 minus 5, and our difference in x-coordinates is 6 minus 2. So evaluating this, I have negative 3/4. And that is my negative slope.
So next, I'd like to talk about the slopes of horizontal and vertical lines. So here we have a horizontal line in blue and a vertical line in purple here. And we're going to calculate the slopes of these lines. So taking a look at our horizontal line first, I've identified the points 5, 7, and 10, 7. Let's go ahead and plug in our x and y-coordinates.
So our difference in y-coordinates, that would be y2 minus y1, is 7 minus 7. And the difference then in x-coordinates is 10 minus 5. Well notice that my numerator is 0. So it doesn't really matter what my denominator is because anything-- a 0 divided by anything as long as we're not dividing by 0 evaluates to 0. And this is true for all horizontal lines.
Notice, if I were to choose any other two points, they would all share the same y values. So my change in y would be 0. There is no change in that rise. The line doesn't go up or down. So horizontal lines have a slope of 0.
What about the vertical line that we see So let's go ahead and pull up two points on our vertical line, and let's find the difference in our y-coordinates. So this would be 8 minus 2 and the difference then in our x-coordinates would be 4 minus 4. Now whereas our horizontal line shared all y-coordinates, vertical lines share the same x-coordinates.
So our denominator is always going to have one value minus itself because that x value is always the same. So we get a fraction that has 0 in the denominator no matter what the y values are. And we cannot divide by 0. So what does this mean then for the slope of our vertical line? The slope is undefined. So horizontal lines have a slope of 0, and vertical lines have a slope that is undefined because we cannot divide by 0.
Lastly, I'd like to talk about the slopes of parallel and perpendicular lines. So here we have a set of two parallel lines. Now parallel lines never intersect. So what does this mean then about their slope? Well this means that we have the same change in y for the same change in x.
So let's illustrate this idea here. Here we have a rise of 2 and a run of 3. And if we go over to our line that is parallel to it, we also have a rise of 2 and a run of 3. So we have the same change in y for the same change in x. So they have identical slopes. Two parallel lines have the same slope.
So parallel lines are lines that never intersect, and they have identical slopes. How about perpendicular lines? Well perpendicular lines intersect at a 90 degree angle. So here, we have this square shows us that there's a 90-degree angle between are blue and purple lines.
And one thing that I noticed right away is that the slopes certainly have opposite signs. One of them is a positive slope, and the other has a negative slope. But more specifically, their slopes are opposite reciprocals.
Now what does that mean? So let's take a look at the slope of this line. We know that it has a rise of 2 for every run of 3. So we can say the slope of this line is 2/3. Well, I'm going to take this picture for the slope. I'm going to rotate it a bit and line it up against our purple line. And what do we see?
Well, from one point to another, we have a run of 2 and a rise of negative 3. So our rise and our run flipped, but then they also went from positive to negative or negative to positive. So that's what we mean, then, by opposite reciprocal slopes. So perpendicular lines are lines that intersect at a right angle, and they have opposite reciprocal slopes.
So here's an example then of the opposite reciprocal. We go from 2/3, and to take the reciprocal, we flip that fraction around, so 2 is now in our denominator, and then we would have 3 in the numerator. But the opposite reciprocal means we go from positive to negative or negative to positive, so the opposite reciprocal of 2/3 is negative 3/2. And 2/3 is the slope of this line, and negative 3/2 is the slope of this line.
So let's review our introduction to slope. The formula to calculate the slope of a line as m equals y2 minus y1, a difference in y-coordinates, divided by x2 minus x1, the difference in x-coordinates. So all we need are two points on a line. If we can identify those x and y-coordinates, we can calculate the slope.
Horizontal lines have a slope of 0. There is no rise between two points in a horizontal line. And vertical slopes-- vertical lines, their slopes are undefined because that change in x, our denominator of this fraction here that defines slope, is 0, and we cannot divide by 0.
Parallel lines never intersect, so their slopes are identical. And perpendicular lines intersect at a right angle, and their slopes are opposite reciprocals. So thanks for watching this tutorial, an introduction to slope. Hope to see you next time.