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Introduction to Slope

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Today we're going to talk about slope. Remember the slope of a line is basically just how steep it is. We're going to do some examples calculating the slopes of positive and negative lines and then we'll look at some special cases, including the slopes of horizontal and vertical lines and looking at the slopes of lines that are both parallel and perpendicular.

So let's look at how to calculate the slope of a line. The slope of the line is basically just how steep a line is. When you're calculating the slope, you need at least two points on a line, so for this example I've got negative 2, negative 2, and two, four, which are two points on this line. The slope mathematically can be thought of as the rise over the run, where the rise is the vertical displacement between our two points and the run is the horizontal displacement between two points.

We can think the coordinates for this point as x1 and Y1 and the coordinates for this point as x2 and y2. When we're looking at a formula for slope, we can think of the rise as the difference between our y-coordinates, which would be y2 minus y2, and we can think of the run as the difference between our x-coordinates, so x2 minus x1. Now when we're talking about slope, we often use the variable m to mean slope, so our formula becomes m is equal to y2 minus y1 over x2 minus x1.

Again, we can think of our slope as the change in our y-coordinates over the change in our x-coordinates, so let's see how to calculate the slope of this line. My two y values, my y2 value is four and my y1 value is negative 2, so in my formula, I've got 4 minus negative 2 and my x2 value is 2, my x1 value is negative 2, so now I've got 2 minus negative 2. Simplifying this, I've got 4 minus negative 2 in my numerator, which is 6, and 2 minus negative 2 in my denominator, which is 4. Six over 4 reduces to 3 over 2, so the slope of my line is 3 over 2.

So we can see that this line has a positive slope, but we also can verify that by again looking at our graph and noting that as we read this graph from left to right, our line is heading towards positive infinity on our y-axis, which again is characteristic of lines that have a positive slope.

So let's do another example. Here I've got a line and two points on the line are 0, 4 and 3, negative 2, so I'm going to use these two points to calculate the slope. So I'm going to label this first point with the coordinates of x1 and y1 and I'm going to label this point as x2 y2. So using my formula, I've got y2 minus y1 in my numerator, so that's going to give me negative 2 minus 4 and in my denominator, I've got x2 minus x1, so 3 minus 0. Simplifying this, negative 2 minus 4 gives me negative 6 and 3 minus 0 gives me three. Dividing negative 6 by 3, I get negative 2, so my slope of this line is negative 2. We see that this has a negative slope and again, we can verify that by looking at our line. Reading it from left to right, we see that our line heads to negative infinity on the y-axis, which is characteristic of lines that have a negative slope.

So let's look at the slopes of horizontal and vertical lines. So in this example of a vertical line, I've got two points, negative 2, 3 and 2, 3, so I'm going to substitute that into our formula for slope. So m is going to be equal to my rise, which is going to be the difference in my y-coordinates, 3 and 3, so in my numerator I'll have 3 minus 3 and my denominator will be the difference in my x-coordinates, so 2 minus negative 2. Simplifying this, I have 0 over 4, which is the same as just 0. So whenever we have a horizontal line, our run could be any value, depending on the points that we choose on the line, but our rise is always going to be zero. So the slope of any horizontal line is always going to be 0.

Let's look at an example of a vertical line. For this example I've got a vertical line and two points on it are negative 2, 4 and negative 2, negative 2. So substituting these values into our formula, in my numerator for my rise, I'm going to have the difference in my y-coordinates 4 and negative 2, so 4 minus negative 2. In my denominator, my run will be the difference in my x-coordinates, so negative 2 and negative 2-- negative 2 minus negative 2. This in the numerator will simplify to be 6 and negative 2 minus negative 2 is going to be zero, which is undefined because you can't divide a number by zero. A fraction, 6 over 0 or any number over 0 is going to be undefined, so for any vertical line, the slope is going to be undefined because the rise can be any value, depending on which points you choose on your line, but the run is always going to be 0, which is always going to lead, again, to a slope which is undefined.

So let's look at parallel lines. So here on my graph I've got two parallel lines, and parallel lines are lines that do not intersect. Because they don't intersect, their rise over run from one point to the next must be the same, which means that their slopes are also going to be the same. So parallel lines do not intersect and they have the same slope. Finally, let's look at perpendicular lines. Perpendicular lines are lines that cross at a 90 degree angle, so they're going to cross at a 90 degree or right angle. There are two things you need to know about perpendicular lines. The first is that you're going to have one line with a positive slope and another line with a negative slope, and the second thing to know is that their slopes are also going to be reciprocals of each other. So I'll show you with an example of what that means.

Let's say we have one line with a slope of 3 over 2. Since this is positive, we know that the slope of the perpendicular line is going to be negative and because it's reciprocal, the slope is going to be 2 over 3. So reciprocal just means you're flipping the numerator and the denominator. So if we have a line with a slope of 3 over 2, the slope of the line perpendicular to it is going to be negative 2 over 3.

Let's go over our key points from today. The slope of a line is a measure of how steep it is. When reading a graph from left to right, lines with a negative slope head towards negative infinity on the y-axis and lines with a positive slope head towards positive infinity on the y-axis. Horizontal lines have a slope of zero and vertical lines have a slope that is undefined. Finally, parallel lines have the same slope and perpendicular lines have slopes that are opposite reciprocals of each other.

I hope that these key points and examples helped you understand a little bit more about calculating the slope. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.