### It’s Not The Cheat, It’s The Futility

The **transformations** section of the text begins, to my predictable chagrin, with a graphing-calculator “Exploration”: adding (or subtracting) a constant value at the end of a function (in the Function Editor [*i.e.*, the “Y=” screen] of the grapher) produces the by-now familiar (to any student in regular attendance) vertical shift. “We are led to the following conclusion:

If a real numberAnd this is about as clear as can be expected: this stuff is hard. But, doggone it, we’re adding to the right side of ankis added to the right side of a functiony = f(x), the graph of the new functiony = f(x) + kis the graph offshifted vertically upkunits (if ) ordown|k| units (if )."

*equation*, aren’t we. And shouldn’t we

*take a hint*from the doggone grapher and refer to and (the authors go on, in effect, to call by the name

*f*—but what’s the name for the

*new*graph?)?

As for the cases on the sign of *k* (and the use of absolute value), well, it’s not the way I’d handle it but it’s not *obviously* worse. I usually say something to the effect that the graph of [the equation] *y = f(x) + h* is obtained by shifting the graph of *y = f(x)* “up” by *k* units *with the understanding* that “shifting up by a negative number” is interpreted—in a very familiar way—as shifting *down*. It can be helpful to mention, say “*y = Mx + B*” here—the point being that we *don’t need to know* the sign of *B* for the formula to be valid. (Or the words: for the case of *y = f(x-h)* we *add h to each x co-ordinate* of the graph—*regardless* of the sign of *h*.)

Again, it’s not *obvious* that treating the “up” and “down” cases explicitly *right there in the display* that (effectively) defines “vertical shifting” was a bad idea … but I believe it *was* a bad idea (sort of): anyway, by the time a student reaches the Conic Sections stuff (next quarter) some of the “formula” displays have become *much* more complicated than they have any good reason to be (and goodness knows, Conics are hard enough already). The sign issues have to be discussed *somewhere*; using inequalites and absolute values to do it is probably very much the right idea … what I’m after here is that the definitions *be definitions*. Which would mean, first of all, *identifying* ’em as such; and then, as concise as you can be while getting the job done.

It would appear that somebody made an editorial decision *not* to discuss the kind of “understanding”s I mentioned a moment ago; the results are disastrous. This is particularly true for students inclined to the so-called “rote memory” strategy (learn the formulas by heart before even *beginning* to work at “seeing the big picture”). Of course, most of us probably consider it something of a duty to try to talk such students *out* of relying heavily on this strategy … but this hardly seems like the right way to do it. And yes, there sure does seem to be quite a bit of resistance (anyway at the level just *below* 148) to the idea that, say, “dividing by a number is just multiplying by its reciprocal”—this comes across as “mumblemumble” to many a 102 student (for example).

I call it “teacher talk”: the student somehow *knows for sure* that this technical stuff you’re always so careful about saying *cannot possibly* have anything to do with their existing ideas about how to solve equations; they’ll move heaven and earth to try to find some list of “rules” they can memorize if only they can avoid using the word “reciprocal” (or, of course, what’s worse, “multiplicative inverse”). And if you’d only stop trying to pretend any of this *means* anything and just *tell them what to do*, the scales would probably fall from their eyes in a minute flat. Of course it’s damn-right a duty to try to disabuse *these* poor souls of their misapprehensions as to the nature of our art. And this duty doesn’t fall on me alone.

Overlooking the power of the symbols until we end up with different formulas for, say, ellipses with vertical and horizontal major axes feels to me like pandering to the worst instincts of our weakest students and, I keep having to *apologize* (“Well, if they’d done this right, it’d be much easier … well, look: forget it. Here’s what they want you to *do* …”).

So. Let’s try and make this very difficult task as easy as it can (reasonably) be. Thinking of, say “stretching” and “compressing” graphs as two aspects of the *same* process (requiring only a *single* formula), is just a flat out good idea: the notation is beginning to do some of our thinking for us. It’s not *quite* the Heart of the Matter—an awful lot of people *have* learned about Transformations and Conics (for example) with the kind of overly-detailed treatment of cases I’m complaining of here, after all. But for me it’s pretty close. And, believe it or not, I don’t necessarily *want* to do constant battle with textbooks.

January 24, 2009 at 5:12 am

If they’re going to do like they’re doing in the definition, they shouldn’t conflate “up” and “down” in a way that forces them to say abs(k) units. If someone is trying to split things down to make it easier, they shouldn’t go half way.

Just calling it “vertical shift” without worry about the sign is likely better, but I’ve seen enough classroom confusion over f(x-h) I don’t mind listing out the four cases while they’re getting used to it. But certainly they should eventually have

f(x-h) + k

all as one thing, with k defined vertical and h horizontal and no mention of sign at all, right?

re: ellipses. I teach only one formula. Focus formula is now c^2=abs(a^2-b^2) with the extra abs().

If I did eccentricity I would have to tweak to make sure they used the major axis, but that’s still easy.

Don’t even get me started on hyperbolas. I think the need to avoid starting formulas with a negative is a holdover from the … 15th century? … before negative numbers were considered acceptable so expressions had to be written with “minus” instead?

January 25, 2009 at 5:26 am

This deep desire to simplify things makes a mess yet again.

Better to say it well and clearly, and then to provide clarifying examples and illustrations, as necessary.

Jonathan

January 26, 2009 at 4:14 pm

i consider it a *good idea* to avoid

starting a formula with a negative …

to the extent that sometimes i’ll prefer

“5 – 6x – x^2” to “-x^2 – 6x +5” …

but for an extramathematical reason

(namely, it’s easier to lose track of

the leading minus sign in the latter).

on a related issue, texts routinely claim that

Ax + By = C

is Standard Form for linear equations in two variables

(which is okay by me if that’s the way they want it);

quite often they’ll actually *want* A >= 0

and integer coefficients with no common factor

(which is *only* okay if it’s made explicit …

as it seldom is …).

of course they don’t dare *insist*

on integer coefficients since in principle

this isn’t always possible … one is led

to wonder why they made such a big fuss

in the beginning about “sets of numbers”

() when they plan

on *ignoring* the distinctions whenever

the opportunity to *use* them arises.