Review Problems

Somebody sez Brutus is absent today for good reason & she’s been asked to take the notes. Which is a bunch of exercises. And Brutus reads the blog. So here goes, quick and dirty.

1. The order of transformations matters. Demonstrate this by graphing the “original” graph y = | x| , then (i) Shift up 2, then reflect in the x axis; and (ii) Reflect in x axis, then shift up 2.

2.Write a definition for the piecewise function on the blackboard. Sorry. Not ready to try to draw it here. Piecewise-linear if that’s any help. (Part of the point here is that both directions — graphics to algebra and algebra to graphics — make good exercises. We had the other one on the quiz.)

3. Give (exactly — e.g. 32/113, not .3274) both co-ordinates for the vertex of $y = -17x^2 +11x + 5$.

4.The Demand function for a certain product is p = 100 – .2 x. (a) Determine the Revenue function. (b)Find the maximum revenue. (c) What are the quantity and price that give this revenue?

Also covered (though not here): anything from the first 2 quizzes (domain & range; increasing & decreasing; intercepts; maxes and mins [nonquadratic]; symmetries …)

1. ellie

1) will the step functions be on the test?

2)to find the vertex with fractions instead of decimal points [you’re killing me here btw] can we find it with decimal points first and then switch it to fractions via the calculator?

2. brutus12

Well i’m just getting in and figured i would see what i missed and that pretty much explains it. Thanks so much for the update and posting that information!! :-)

3. @ellie

1) step functions are supposed to be
“fair game” … but we’ve had very little
actual practice with ’em …
so *if* i put one on the exam
it’ll be in some inessential way
(like “what transformation changes
int(x) to -int(x)”; the student answers
“reflection in the x axis”; this is a
not the “int” function … any other
function would have done just as well).

2) the “Frac” feature of the calculator
is indeed the tool of choice for
“exact co-ordinates”.

“plug in” the x co-ordinate of the vertex
(found via “H = -B/(2A)”) on the formula
(previously loaded into the Y_1 function
of the calculator, say); evalaluate; use
the “Frac”.

probably one should compute by hand
and use the calculator to check just for practice
(when working such problems at leisure;
under exam conditions, verifying everything
right on the calculator is faster (and exams
are no time for “practice”!).

anyway, that’s what i like to do in class.
most algebra students can sure *use* the practice
(in rational number arithmetic — “manipulating fractions”).

in one-to-one work in the 10:00
and i realized we’d never done
any in-class work with the Frac feature.
it’s still a good problem but might
call for a hint …

@ brutus
see you in a couple hours … well prepared i hope!

4. brutus12

pheww soo much for the well prepared on the revenue problem, number 4 i think. Dang it!! For some reason i drew a blank, decided i’d go back to it at the end and with just minutes left before my next class, that i had to give a speech in, i had to throw in the towl. Let’s just say i’m pissed, i never leave problems blank or give up!! Anyway the review helped alot to prepare for the test and the clarity of the questions were great. Between all the madness going on with the 19credit hours im taking this quarter, im actually learning some math, and you can bet that revenue exercises will NOT be an issue again, i’ll be able to do em’ in my sleep.

5. for anybody not in the class:
usually the main problem with problems
like my number 4 is the first step:
we require that the student know
that “R = xp” —
revenue is number-sold times price-per-item —
so that, for example, one has
R(x) = 100x – .2x^2
in the stated problem.
everything else tends to fall into place.
or anyway, there are all *kinds* of issues
that may arise *after* this step;
the trouble is that *without* this step,
one can’t even really get properly started.
so “not getting started” is unfortunately
the commonest error on problems like this.

it appears to be just the kind of thing
that we teachers are more-or-less *supposed*
to require students to know … so i try to stress
the importance of this step in at least two classroom
presentations before a test …

note that the actual *math* involved here
amounts to little more than “common sense”;
it’s easy to *understand* R = xp once you know
that it’s going to be useful. it’s just far from obvious
that it’s useful in the given context.

6. ellie

dont feel bad brutus! i left letter #4 (d) blank too… I was trying to visualize the notes in my head but something was blocking my view and i couldnt remember how to do it.

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Vlorbik On Math Ed ('07—'09)
(a good place to start!)