Disclaimer: This only intended to give you some guidance,
and thus should not be taken as an exhaustive
listing of possible test question topics. Do not expect this every time. No
calculators , notes, or any other
aids will be al lowed on this test .
The test will cover § § 3.1, 3.2, 3.3, 3.5, 3.6, 4.1, 4.2.
§3.1
Know what a quadratic function is: f(x) = ax2 + bx + c.
Know the quadratic equation and the formulae for:
Axis of symmetry:
You should be able to graph a quadratic function using the
roots/zeros, vertex , y- intercept , and axis of
symmetry (see problems 3.1.35-52).
Know how to do the types of application problems that we did in class and on
WebAssign 2.8 and 3.1
(demand and fencing type problems, see problems 3.1.73-76 & 3.1.77-80).
§3.2
Be able to identify a polynomial function .
Know how to find the degree and leading coefficient (an) of a polynomial if it
is given already factored.
Know what the multiplicity of a factor tells you about the behavior of the graph
at that root.
Know how the end behavior relates to the degree and whether an is positive or
negative.
The two main types of problems I expect you to know are:
1. Sketching the graph of a given polynomial. For this type of problem, the
polynomial will already be
completely factored for you; see problems 3.2.57-80 (a couple of these aren’t
completely factored).
2. Given certain conditions and constraints, be able to construct a polynomial
that satisfies them (sort of
like problems 3.2.37-44, although I may also specify what the end behavior
and/or leading coefficient
needs to be).
§3.3
Know what a rational function looks like:
where
and
are polynomials, and q(x) ≠ 0.
Be able to find the domain of a rational function: Domain = {all real numbers
except zeros of p(x)}
Know how to find vertical asymptotes: Get R(x) in lowest terms (cancel as many
factors as you can), then
the vertical asymptotes occur at the zeros of the denominator.
Know how to find horizontal asymptotes:
• If deg(p)>deg(q), then R(x) has no horizontal asymptote.
• If deg(p)=deg(q), then R(x) has a horizontal asymptote at
.
• If deg(p)<deg(q), then R(x) has a horizontal asymptote at y = 0.
Problems from this section would be a combination of 3.3.11-23,41-52 (would
probably ask about domain
and asymptotes for the same function).
§3.5
Know how to solve polynomial and rational inequalities . The functions will be
completely factored, and no
additional algebra should be required to solve the problem. I am testing your
ability to find and plot the
zeros , check to see if the function is positive or negative between those
points, and then write down the
intervals that satisfy the inequalities.
For rational inequalities, remember that the zeros of the denominator will never
be included in the intervals
for the answer; however, the zeros of the numerator may be included depending on
the particular
problem.
Most of the problems from 3.5.3-56 would be fair game for a test (with the
exception of 33 & 34). Difficulty
will be dictated by the length of the test and the difficulty of the other
problems.
§3.6
Be able to use the following theorems (you will not be asked to state them on
the test):
Remainder Theorem: Let f be a polynomial function. If f(x) is divided by x − c,
then the remainder
is f(c).
Factor Theorem: Let f be a polynomial function. Then x − c is a factor of f(x)
if and only if f(c) = 0.
Rational Roots Theorem : Let
be a
polynomial, where
are integers and
. Then all possible
rational roots of f are of the
form
, where p is a factor of
and q is a factor of
.
In termediate Value Theorem: Let f be a continuous function, and let a < b. If f(a) and f(b) have
opposite signs, then f has at least one real zero in the interval (a, b).
The idea of the IVT is that if you have a continuous
function (one without any jumps or gaps, i.e., you
could draw it without lifting your pencil from the paper), then if the function
at one x-value is negative and
then it changes to positive at another x-value (or visa versa), then the
function had to 0 somewhere between
those two x-values. In other words, a continuous function can’t go from being
negative to being positive (or
from being positive to being negative) unless it’s zero somewhere in between
(see figure 79 on page 377).
Problems from this section will be similar to the WebAssign problems and
3.6.11-32.
§4.1
Know about composition of functions and how to compute them:
(f o g)(x) = f(g(x))
See problems 4.1.11-28 & 53-58 (4.1.45-52 would relate to §4.2).
§4.2
Know the definition of what it means for a function to be 1-1. You may have to
show that a function is 1-1:
• For a function defined by a set of points, check that none of the
y-coordinates are repeated.
• For a function defined by an equation, you could either use the formal
definition:
If
,
then
or
If
,
then
or graph the function and use the Horizontal line test :
f is 1-1
Every horizontal line touches/crosses the graph at at most one point.
Given a function f, be able to find it’s inverse:
1. Check that the function is 1-1 (in some cases you’ll be told that the
function is 1-1, but otherwise you
need to check):
• If the function is defined by a set of points, then be sure that none of the
y-values are repeated.
• If the function is defined by an equation, then either use one of the formal
definitions or the
horizontal line test to check that the function is 1-1.
2. Interchange x’s and y’s (this gives an implicit form of f -1).
3. If possible, solve for y (get an explicit form of f -1).
4. Verify your solution by checking that (f o f-1)(x) = x and (f -1 o f)(x) = x.
Know the relation between the domain and range of f and f -1:
Domain of f = Range of f -1
Domain of f -1 = Range of f
Remember that the graph of f and the graph of f -1 are symmetric about the line y
= x. Using this can
make graphing f and f -1 easier.
