Where should the type information be 205

When, as a student (and occasionally these days, when it's not worth throwing together a graphing toolchain on the computer), I graphed a function by hand, the process went something like:

1. Find the Y-intercept, or that it was undefined for the function. That gave me an idea of where my axes would be on the graph paper and what the scale would be like.

2. Look at the function and see what sort of X values would give me nice Y-values - ones that would be easy to graph, given my scales, the size of the paper, etc.

3. Pick some of those values and start making a table of coordinates.

4. Settle on scales and the position of the axes. Draw them on the graph paper and label them.

5. Plot the points from my table. For some graphs, this involved a ruler (particularly if there were a lot of points).

6. Is the shape of the curve clear? If not, compute some more coordinates. For something really unfamiliar (when we started to learn polar coordinates, for example), compute and plot quite a few to make sure I know where the curve is going.

7. Does it look like one or more points might be off? Go back and check algebra and arithmetic.

8. Draw the curve, connecting the points and extrapolating. For some graphs (eg in physics lab), this often meant using French curves and-or a flexible edge. Once in a while we'd have an buttignment to use some other construction method (compbutt, loops and pins, etc).

A hand-graphed curve like that is practically a work of art. It's a substantial effort that gives you a thorough feeling for what that particular function does. (Of course, I also did a lot of rough graphs in the margins of my notes and so on - those "I wonder what this would look like?" daydreams during clbutt. That's another sort of experience.)

Graphing with a calculator (or general-purpose computer), on the other hand:

1. Enter function.

2. Press button to graph.

3. Say "huh".

4. Return to function. Mess with graphing parameters or with function itself - coefficients or exponents or whatnot.

5. Graph again; say "huh".

6. Iterate until you have a sense of what this sort of function does.

Computer graphing gives you a sense of what a family of functions does, or how different functions compare. Done right, it's a fine way to get a good visual sense of various sorts of functions. It emphasizes breadth over depth, where graphing by hand emphasizes depth over breadth (because it takes longer, and because it's more work to vary parameters, so students typically won't do much exploring on their own).

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Shakespeare writes plantast and knows it; Mr Thomas writes plantast and doesn't. That is the difference. -- Geoffrey Johnson