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Chapter 3
Polynomial and Rational Functions
Section 3.1 deals with
polynomial functions
in general.
- Note that linear and quadratic
functions
are special cases of polynomial functions.
- You will need to know how to
recognize a
polynomial function when you see one.
- You will need to be able to determine the
degree of a polynomial.
- Can you make some drawings on paper to
show the difference between concave up and concave
down?
- The graph of a polynomial function is
smooth and connected. There are a number of other features of the
graph of a polynomial that are predicted based on the leading
coefficient and degree of the
function. Summarize these in the tables below.
| |
The graph of a degree
"n" polynomial function has |
| how many real
roots? |
|
| how many changes
of direction? |
|
| how many concavity
changes? |
|
| When the degree
of the function is odd |
When the degree
of the function is even |
If the leading
coefficient is negative, what is the long-range behavior of the
function?
|
If the leading
coefficient is negative, what is the long-range behavior of the
function?
|
If the leading
coefficient is positive, what is the long-range behavior of the
function?
|
If the leading
coefficient is positive, what is the long-range behavior of the
function?
|
- How can you use your calculator to approximate
a local maximum or local minimum?
-
How can you use your calculator to
approximate a zero or root?
-
How can you use the Remainder Theorem,
Factor Theorem, and synthetic division to calculate a
zero or root?
Section 3.2 introduces additional techniques
for locating a real zero or root.
- How does the
Location Theorem allows us to find roots?
- How can you use the Location Theorem to
solve a polynomial inequality?
Section 3.3 introduces techniques for
locating complex zeros or roots.
- How does the multiplicity of a root affect
the formula and the graph of the function?
- How can you use the Rational Zero
Theorem to locate rational zeros of a function?
- How can you combine various methods to
factor a function into the product of linear factors?
How do you know when you have located all of the linear factors?
Section 3.4 introduces
rational functions and inequalities.
- How is a rational function
related
to other types of functions you already know about? (Think
about how rational functions are similar to and different from polynomials.)
- Can the graph of a rational function have
asymptotes?
- You will need to know the difference
between horizontal and vertical
asymptotes.
- How is a horizontal asymptote related to
long-range behavior of a function?
- How can you determine the
equations
of the asymptotes for a rational function?
- How can you determine the
domain
of a rational function?
- How can you determine the
range
of a rational function?
- How can the graph of a rational function
have a hole instead of an asymptote?
- You will need to know how to use
polynomial
long division to find the equation of a slant
(oblique) asymptote.
- How can you find the
x and y intercepts for the graph of a rational
function?
- Can the graph of a function
cross its horizontal asymptote?
- How can you use algebra (and how can
you use a graph) to solve a rational inequality?