Lesson 12
Curve Sketching
One of our objectives in this part of the course is to be able to graph functions.
In this lesson,
we’ll add to some tools we already have to be able to sketch an accurate graph of each function.
From prerequisite material, we can find the domain,
y
intercept and end behavior of the graph of a
function, and from the last two sections, we can learn much about a function by analyzing the first
and second derivatives.
We also know how to find the zeros of some functions.
We’ll expand
that group of function before we continue to curve sketching.
The Rational Zeros of a Polynomial Function
The rational zeros of a function are the zeros of the function that can be written as a fraction, such
as 2 or
.
5
1

We can find all of the possible rational zeros of a given function using the Rational
Zeros Theorem.
Then we can use synthetic division to determine which – if any – of the possible
rational zeros are actual zeros of the function.
Here’s the theorem:
Rational Zeros Theorem:
Suppose
0
1
1
)
(
a
x
a
x
a
x
f
x
n
n
n
+
+
+
=


⋯
, where
0
≠
n
a
and
0
0
≠
a
, and all of the coefficients of
the polynomial are integers.
If
q
p
x
=
is a rational zero of the function, where
p
and
q
have no common factors, then
p
is a
factor of the constant term
0
a
and
q
is a factor of the leading coefficient
n
a
.
Example 1
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 Fall '08
 MARKS
 Math, Derivative, Graph of a function, rational zeros

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