## Remarks Toward The Dissolution Of This Blog

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 $f\circ g$, 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) $f^{-1}(x)$ (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.