Dear Students Who Think Graphing is Stupid,
Right on! Graphs are stupid. Cosmically stupid. Deliberately stupid. In fact – and I hate to pull this rhetorical trick on you, but you leave me with no choice – that’s kind of the point.
Take the function f(x) = x2. This relationship includes a dizzying number of input-output pairs, more than you could ever hope to list. Even with every atom in the cosmos at your disposal, you could never create a “complete” graph of it: to fill in the details, you’d need particles smaller than quarks, and to reach the extremities, you’d have to extend far past the most distant galaxies.
So when you graph f(x) = x2, you’re not depicting the whole function. You’re drawing a simplified version, a pocket-sized map. “Function for Dummies,” you might call it, or “Idiot’s Guide to This Function.”
A world map shows the key features of the planet – oceans, continents, the more formidable mountain ranges – and leaves out the overwhelming mass of negligible details. No world map includes the chocolate shop near my apartment, even though it is emphatically – and deliciously – a part of the world.
So it is with graphing a function. A graph of f(x) = x2 does not usually illustrate the fact that when x = 3.2, f(x) = 10.24, even though any mathematician will tell you it’s true. (Actually, most mathematicians hate arithmetic, so if you ask them for the square of 3.2, they’ll grimace and refer you to the nearest engineer. In fact, the original version of this post had the number wrong. Bad at drawing and arithmetic!) Instead, a graph will illuminate a few key truths about a relationship, such as the following:
- The square of a real number can never be negative.
- When you square opposite numbers (like 2 and -2), you get the same result.
- When you square a number larger than 1, it grows.
- When you square a number between 0 and 1, it shrinks.
- As a number grows bigger (say, from 2 to 3), its square grows even faster (in this case, from 4 to 9).
I’m not trying to sell anyone on the charms of f(x) = x2 – although it is a rather enchanting little function. My only point is that a graph should be useful. It’s a tool of clarification, a lubricant for problem-solving, a map of truths.
Our math classes aren’t always great at conveying this. (Mine included.) Too often, we teach you guys how to graph, without teaching you why to graph. Fight against this! The last thing you want to be is an expert in mapmaking but an illiterate in map-reading. That’s like being able to walk into any room and sketch the floor plan, but when pressed, being unable even to find the door.
Anyway, you have my support: Graphs are stupid. Just as they should be.