This post continues my series of mathematical explanations with another popular topic. In keeping with the theme from my previous post, where I discussed the Cartesian coordinate system and ordered pairs, this post is going to focus on another rudimentary graphing concept: finding the slope of a line. Before you can move on to drawing parabolas and other fancy curves, you have to understand how to graph straight lines first. I am going to tell you everything you need to know about drawing lines, finding their slope, and and then coming up with the equation that describes the line.
What is a Line Segment and a Line?
Drawing a line connecting two points on a graph is simple. If you know where point A is, and also where point B is, it is just a matter of using your ruler to connect the dots. However, this simple act could mean a few things. First, by connecting your two points, this results in a line segment, which is defined by having end points. Alternatively, the more general form of a line refers to a line segment that has no ends – that is, it extends in both directions to infinity. On a graph, you represent this by including arrowheads on either end of a representative line segment, which indicates that the line continues beyond the limits of your drawing. This may seem like a trivial difference, but in math, it is an important to know if a line continues on forever, or is restricted to a defined length.
One additional point I’d like to make about… points… is how you denote them on a graph. If you draw the point as a small, solid, filled-in circle, then that indicates that you are referring to that precise point, as in the ordered pair (1,2) is exactly at that coordinate. However, when you are drawing points to use for lines, if you leave it as a small, open, unfilled circle, then that indicates that you are including all points on the line EXCEPT exactly at the point you have circled. This will be more clear when I discuss domains and ranges in a later post, but it is something that I thought was worth mentioning now – that is, solid circle points and unfilled circle points mean different things on a graph!
Finding the Slope
So, now that you know how to represent points on a line, and how to differentiate between a line and a line segment, let’s move on to one of the most important parts of line drawing – finding the slope of the line!
So, possibly your first question is: what is slope in math?! A slope in math is exactly what you think it is, as if you were talking about a hill or a driveway. No tricks here. It is a measure of how steep your hill or line is. A larger slope means that you have a steeper driveway. Or maybe you can think of it in terms of skiing in the winter – you want to find a larger slope, because that is the steeper hill and will mean it’s more fun to ski down.
To calculate slope, you don’t need a lot of information. Basically, all you need to know is the location of any two points on the line. Any points will do; they can be super close together or way on opposite ends of the line. This is because for a straight line, the slope is always the same at every point on it, and so if you calculate the slope with two points over here, you will end up with the same slope as if you used two points over there. The slope is a property of the line itself, and the line is made up of all of the individual points on the line.
Thankfully, the slope formula is one of the easiest equations you will probably ever learn in math. “The slope is equal to the ratio of the difference in height of two points on a line to the difference in horizontal location of those same two points.” Easy, right? Just kidding! While that IS the slope formula, the far more widely used and memorable form of the equation is: “slope equals rise over run.” THAT is much easier to memorize.
Now, let me break it down a little bit so that the terms actually make sense. The “rise” part refers to the change in height, or the difference of the y-values of two points on the line. Conversely, the “run” part can be thought of as running down the flat street, moving from one side to another, or the difference in the x-values of your same two points.
The usual convention to represent the slope term is to use the lowercase letter m. There are several potential reasons for this, but I won’t get into those here. It’s enough if you just remember that m = slope.
S0, now let’s put these words into the form of a mathematical equation that you can more easily use in your work. Let’s assume that we have one generic point at the coordinate (x1,y1), and a second point represented by the ordered pair (x2,y2). All of this together gives us the generic form of the slope equation:
The 1’s and 2’s are just reference notes to help you keep track of what point you are talking about. They could be A’s and B’s, or squares and triangles – it doesn’t matter. The key point that you need to remember is that it is VERY important that you keep track of your points! Don’t add extra points into the mix, and definitely don’t mix up your x-values and y-values. It is very easy to do, and you will probably do it several times. I remember always making that mistake. The only trick I can suggest is that you pay attention! When you’re asked to find the slope of a line with two points, just take your time and make sure your x’s and y’s are in the right places!
Once you have determined your slope of your line, one final easy check you can do is to look at which direction is it sloping? If your line is rising to the right, that is a positive slope. Conversely, a line that is dropping to the right has a negative slope. You can very easily see by looking at the graph, and then comparing to the sign of your determined slope value if you are at least correct as far as the sign of the slope is concerned. If the sign doesn’t agree with what your drawing is doing, you likely have a sign error, quite possibly because you mixed up the y’s and x’s in the formula. That’s easy to fix, and quite honestly, happens to everyone a lot more frequently than you’d think.
So, that’s it. Now you know about lines, line segments, and finding the slope. With this information, you can move on to do slightly more complex things, such as find the equation that describes the line, not just the slope. With that equation, you have all the information you need to perfectly place and draw a line. In my next post, I will go into more detail about how to actually do this.