The final’s soon. Somewhat to my surprise, the co-ordinators haven’t prepared a Practice Test (they recently took over the *whole* final exam; in the old regime one had to write half the Final for one’s own sections). Which, believe me, makes perfect sense: lots of good teachers consider this kind of thing a *bad* idea (the whole point of a final exam on their model being to cover the entire course).

So it’s on me and I don’t *care* if it’s a bad idea just now: my students have grown to expect such things and I hate to disappoint them. At the same time, I’ll preserve a little of the roll-your-own flavor: part of the point is to get students to *read the book*, not just to consider certain problem types. Anywhow, from top to bottom, we did, and will soon do again, problems like these.

**Reflections.** Given a graph, sketch its various reflections (origin, *x*-axis, *y*-axis); keep track of one or more points (given as ordered pairs); show algebraically that a function is “even” or “odd”.

**Information From A Graph.**Given the graph of a function, write down its Domain, its Range, its local Maxima or Minima (collectively Extrema; know the singular forms of the three plural nouns just mentioned… and in general, for heck sake, learn these *words* I keep slinging around… the point isn’t just to be able to *do* the calculations, but to *talk about them*…), the sets (subsets of the Reals, natch—specifically, of the *domain*… so we’re talking about *x* co-ordinates… behave appropriately here I beg of you) on which the given function is *increasing* or *decreasing* (or, I suppose, *constant*, though this strikes me as somewhat affected and I typically distain to give its definition), the sets on which it’s *positive* or *negative*, its various *intercepts*, and whatever I’ve forgotten to mention. This is stuff you mostly all knew before you even took 148 so just get the bloody notations right: what’s a point, what’s a set of *x* co-ordinates, what have you. Because if you can’t *write this stuff down* just by looking at the graph and be essentially correct about almost all of it, you aren’t *ready* to “talk about” graphs of functions…

**Piecewise-defined Functions.** Graph it if we give you the algebra; write down the code if we give you the graph.

**Transformations.** We did a lot of this and if I had my way we’d’ve done even more. Be able to *name* and *calculate* “shifts” (I actually prefer to call these “translations”), “shrinks” and “stretches” (“scalings”) as applied to algebraic “formulas” and to graphs (and their individual points).

**Quadratic Functions.** We looked in some detail at some Optimize-the-Area problems and some Optimize-the-Revenue problems, so prepare for these: in either case, the point is to set up the right quadratic and look at its vertex. We did more review in here than the syllabus called for… complete the square, *QF* (the Quadratic Formula), what have you. You can thank me later.

**Zeros of a Polynomial.** With real coefficients. In the easiest cases, one can read the factored form of a polynomial directly from its graph; in the more interesting cases we’ll have to “divide away” the factors—*(x-R)*, say—associated to certain roots (*R* such that *f(R)=0*, also known as zeros and, loosely, as *x* intercepts [strictly speaking, the root is a *number* and the intercept is a *point* (*i.e.*, an *ordered pair* of numbers); some students appear to feel that clinging to the belief that details like this are *never* important is some higher moral duty… as if thinking clearly is some sort of sacrilege…]); typically one finally cleans up by applying *QF* to get the last two roots. These can be real or complex. One should also be prepared to *construct* polynomials given certain roots (and their multiplicities): remember here that complex roots of real polynomials occur in *conjugate pairs*.

**Rational Functions.** A single problem of the form “graph *y = R(x)*“, where *R* is a rational function, involves *lots* of sub-problems. Find zeros and vertical asymptotes (factor numerator and denominator first if necessary; we reviewed this *very little*, alas); put appropriate features in the graph. Determine the “end behavior” (horizontal, oblique, or no asymptote); sketch. Find a “new” point & make the appropriate connections. There’s an *inequality* to solve under this heading too: I recommend solving these *graphically* even though the exam itself says “solve algebraically”. If they give you ruled paper, write the other way.

**Composite Functions.** Compute ’em. But also, evidently *because* it’s slightly tricky, give their *domains*. The point here is to consider, not only the computed “formula”, but the domain of the right-hand “factor” in the composite (*g* in , for example).

**Inverse Functions.** Be able to compute the formula for the inverse of a function (given by a formula). First of all, in textbook fashion: to invert *f*, put *x = f(y)* and solve for *y*; the result is (a “formula” for) (so write down the appropriate equation). But, to thrill your instructor, use “shoes and socks” to calculate *verbally* (appropriate wigglings of chalk or pencil are optional here).

And **exponential** stuff. But that’s recent. Ongoing, even. And the coffeeshop’s ready and waiting. Gotta go.

March 10, 2009 at 4:57 pm

also, duh, circles. also ongoing, i shoulda said.

x^2 + y^2 = 1 and all that. amazing stuff.