I’m teaching a lot of calculus this term, and we just spent the last class period or two talking about straight lines. That makes sense. Calculus is especially concerned with measuring the slopes of functions, and straight lines are just about the simplest functions there are.
Now, the textbook we’re using this term, like pretty much all textbooks, defines a linear function as one that can be written in the form
I hate that!
It’s not that it’s wrong. It is perfectly true that straight lines, and only straight lines, can be expressed with equations of that form. But defining a straight line in terms of the form of its equation is like defining an even number as one whose final digit is 0, 2, 4, 6 or 8. It’s perfectly correct, but it doesn’t really get at what an even number is. Do we really want to suggest that cultures that use a different system for writing down numbers cannot understand the concept of an even number?
But it gets worse. The problem is not simply that y=mx+b fails to make clear what is special or interesting about linear functions. It is that even if you are going to define linear functions in terms of their equations, that is hardly the most natural one to use.
So let’s philosophize a bit. If I am thinking of a particular straight line, and I want you to be thinking about the same line, what information do I need to give you? The answer is that I need to give you two things: one point on the line, and the line’s slope. One point and the slope uniquely determine the line. Of course, I could also give you two points on the line, but that is because you can use the two points to determine the slope.
The key observation is that any point will do. There is no privileged point that plays a special role in defining the line. And when you appreciate that, you come to see the full horror of the y=mx+b form. It’s that b at the end. It represents the y-intercept of the line, which is to say the point where the line hits the y-axis. But why should we care about that point? Why should the y-intercept be considered the be-all end-all of linear functions? It’s madness! Heck, I could just move the axes, and suddenly the line has a different y-intercept. But I haven’t changed the line itself, now have I?
So how should we think about the equation of a straight line? Since the thing that makes a straight line special is that it has a constant slope, I would say the general equation is this:
Now that’s an equation I can get behind!
Let’s flesh it out a little bit more. I said before that you need to be given a point and a slope to determine a line. So let’s call the slope m and the given point
You probably remember that the slope of the line between two points is defined as rise over run. So, if (x,y) is any other point on the line then we have
And this gives us the equation
We could stop there, but fractions are annoying. So let’s cross-multiply to get
And we’re done.
Textbooks call this the point-slope form of the equation of a line. If the given point happens to be the y-intercept, so that it’s coordinates are (0,b), then our point-slope form quickly reduces to y=mx+b.
So, let’s wrap this up. The y=mx+b form certainly has its uses. For example, if you want to cook up an algebraic proof of the fact that two lines are parallel if and only if they have the same slope, (an exercise we did in class!), then this is the most convenient form to use. But please, let us stop the nonsense that this is the most appropriate way to define the notion of a linear function.
Of course, in this post I have been assuming that our goal is to describe a line in a two-dimensional plane. We could also ask how to describe a line that’s slashing through three-space. But that’s a subject for a different post…