Getting rough graphs of polynomials is really no more difficult than
analyzing inequalities involving
polynomials.
Suppose we’d like to sketch the graph of
,
some polynomial. Here’s a
step-by- step process we can fol low to help us come up with a graph:
1. One way or another, find the roots of the polynomial (this can be done
with the help of techniques
like the ________ Theorem, the ___________ Formula , and long division of
polynomials).
2. The graph we seek passes through the ______ at the roots you’ve found in
(1). The graph we
seek is either above the ________ or below the ________ on each of the intervals
induced by the
roots.
3. Interval by interval, test the inequality p(x) > 0 for some test point x
on each of the intervals
induced by the roots. The intervals where this inequality holds are intervals on
which the function
is positive.
4. Use the information from (1) through (3) to sketch a graph! (If desired,
plot a few points to get a
“finer” picture of the graph of p(x).)
Let’s try it out on a few basic
Examples. Sketch rough graphs of each of the following polynomials.
(No need to factor here !)
The method we ’ve highlighted above gives us a means of understanding the
“fine” structure of a polynomial’s
graph. To understand “coarse” or “long-term” structure, it might help to keep
the following in
mind:
Fact. If
is a polynomial, then for large values of |x|, p(x)
behaves essentially like
.
Roughly speaking, a polynomial behaves like its leading term when x is either
a very large positive
number or a very large negative number .
This leads to 4 fundamentally different graphs , depending on whether n is odd
or even, and whether an
is positive or negative . In the space below you should sketch the rough shapes
of these 4 possible graphs:
You should now be ready to tackle Homework 4.5 on- line ;
this homework is due by Friday, November
14th.
