I want to talk about drawing graphs, specifically, drawing curves on graphs. We know roughly what’s meant by that: it’s about wiggly shapes with a faint rectangular grid, usually in grey or maybe drawn in dotted lines, behind them. Sometimes the wiggly shapes will be in bright colors, to clarify a complicated figure or to justify printing the textbook in color. Those graphs.
I clarify because there is a type of math called graph theory in which, yes, you might draw graphs, but there what’s meant by a graph is just any sort of group of points, called vertices, connected by lines or curves. It makes great sense as a name, but it’s not what what someone who talks about drawing a graph means, up until graph theory gets into consideration. Those graphs are fun, particularly because they’re insensitive to exactly where the vertices are, so you get to exercise some artistic talent instead of figuring out whatever you were trying to prove in the problem.
The ordinary kind of graphs offer some wonderful advantages. The obvious one is that they’re pictures. People can very often understand a picture of something much faster than they can understand other sorts of descriptions. This probably doesn’t need any demonstration; if it does, try looking at a map of the boundaries of South Carolina versus reading a description of its boundaries. Some problems are much easier to work out if we can approach it as a geometric problem. (And I admit feeling a particular delight when I can prove a problem geometrically; it feels cleverer.)
But graphs have disadvantages, too. For one, we can’t work out everything interesting by eye. Eyes can mislead, as any optical illusion and a good number of Six Differences panels will show; reasoning can overcome that misleading impression, but that’s even harder than reasoning already is. Even when it doesn’t mislead we can’t get every answer we might want. If we’re interested in, say, whether the area inside one curve is greater or less than the area inside a different curve … maybe we can work that out, but I wouldn’t want to bet on it. We might be able to work something out experimentally, but that seems to lack a certain mathematical rigor, however practical it might be. (There is the story of Thomas Edison showing up a newly-hired college graduate with the problem of finding the volume of a light bulb; the graduate spent long hours working out exactly the shape of the bulb and doing the calculations from that, while Edison filled the bulb with water and poured that out into a measuring cup.)
So what we really want is to be able to shift a description into a drawing, and where possible, shift a drawing into a description, so that we can work out a problem by whatever method does better at getting whatever answer we want. This is a common trick of mathematics, I should mention; there is a decent section in any advanced course where one seems to just spend weeks going over ways to convert one kind of problem into a different kind, in the hopes that the different kind of problem will be easier to work out. If one’s lucky, one converts the hard problem into an easy one using steps that aren’t any harder than the original problem would have been. One spends most of those courses wondering why there aren’t any of those lucky problems assigned as homework. This is usually because the obvious problems are given as examples, in the book or in the class, and therefore can’t be given as homework; and it turns out all the problems about that hard are really just the same problem, so all that’s left are problems hard all the way through. At least that’s how it feels.
To talk about sliding between graphs and other descriptions, I should say what those other descriptions are, although since we all know we mean equations one might ask why we bother. Well, we have some reasons to bother; for example, the equation “2 = 2” is certainly true, but it’s not obvious how we’d make a graph of that. It’s also not clear why we’d want to, but whatever motivated us to write the equation is probably enough motivation to graph it, if we can think of a way to do that.
Another reason to think about the kinds of descriptions we might have, without just saying it’s equations, is that we might not be interested in equations. We might be interested in finding a whole region where some property applies, or fails to apply, and those are usually inequalities. But we can still draw graphs of them, or find interesting things about them well-described by graphs. But then we’re left saying we’re interested in things that are equations or inequalities, and that seems to cover everything. I think a well-rounded intellect and broad curiosity is a good thing, but it would be nice to be able to rule out something.