Rationals

1
Rationals

• Domain and Vertical and Horizontal Asymptotes

2
Learning Goal

• I will be able to determine vertical and
  horizontal asymptotes of rational functions.

3
Rational Functions
Definition A rational function is one that can
be written in the form
where p(x) and q(x) are polynomials and q(x) is
not the zero polynomial. In general, the domain
of a rational function includes all real numbers
except x-values which make the denominator zero.
4
Domain of Rationals
Find the domain of
Denominator cant equal 0
5
Domain of Rationals
Find the domain of
Denominator cant equal 0
6
Domain of Rationals
Find the domain of
Denominator cant equal 0
7
Vertical Asymptotes
At the value(s) for which the domain is
undefined, there may be a vertical asymptote(s).
List the vertical asymptotes for the previous
examples.
none
8
Vertical Asymptotes
The figure below shows the graph of
The equation of the vertical asymptote is
9
Vertical Asymptotes
Definition The line x a is a vertical
asymptote of the graph of f(x) if
as x approaches a either from the left or from
the right.
or
Look at the table of values for
10
Vertical Asymptotes
As x approaches____ from the _______, f(x)
approaches _______.
As x approaches____ from the _______, f(x)
approaches _______.
-2
-2
right
left
Therefore, by definition, there is a vertical
asymptote at
11
Vertical Asymptotes
Describe what is happening to x and determine if
a vertical asymptote exists, given the following
information
Therefore, a vertical asymptote occurs at x -3.
As x approaches____ from the _______, f(x)
approaches _______.
As x approaches____ from the _______, f(x)
approaches _______.
-3
-3
left
right
12
Vertical Asymptotes
Describe what is happening to x and determine if
a vertical asymptote exists, given the following
information
Therefore, a vertical asymptote does not occur at
x 0.
As x approaches____ from the _______, f(x)
approaches _______.
As x approaches____ from the _______, f(x)
approaches _______.
0
0
left
right
0.5
0.5
13
Horizontal Asymptotes
DefinitionThe line y b is a horizontal
asymptote if
as
or
Look at the table of values for
14
Horizontal Asymptotes
0
0
y?_____ as x?________
y?____ as x?________
Therefore, by definition, there is a horizontal
asymptote at y 0.
15
Horizontal Asymptotes
Looking at end behavior of f(x) The Question
What happens to f(x) as x gets sufficiently
large or small? The Answer Reduce f(x) to only
the terms of highest degree in the numerator and
denominator.
WHY??
16
End Behavior
Rewrite each f(x) and determine the end behavior
of the function.
What similarities do you see between these
problems?
Denominator degree is larger
17
End Behavior
Rewrite each f(x) and determine the end behavior
of the function.
What similarities do you see between these
problems?
Denom. Numerator degrees are same
18
End Behavior
Rewrite each f(x) and determine the end behavior
of the function.
What similarities do you see between these
problems?
Numerator degree is larger
19
Asymptotes
Summing up Asymptotes of a Rational Function  Let
f be the rational function given by
where p(x) and q(x) have no common factors.
20
Asymptotes
1. The graph of f has vertical asymptotes at the
_________ of q(x).  2. The graph of f has at
most one horizontal asymptote, as follows.  a)  
If n lt m, then the ____________ is a horizontal
asymptote. b)    If n m, then the line
____________ is a horizontal asymptote. c)   If
n gt m, then the graph of f has ______ horizontal
asymptote.
zeros
line y 0
y an/bm
no
21
Asymptotes
Find all asymptotes of the following function
Vertical Asymptote x -1
Horizontal Asymptote y 2
22
Asymptotes
Find all asymptotes of the following function
Vertical Asymptote none
Horizontal Asymptote y 0
23
Asymptotes
Find all asymptotes of the following function
Vertical Asymptote x 2
Horizontal Asymptote none
24
Learning Goal

• I will be able to determine vertical and
  horizontal asymptotes of rational functions.

25
End of notes.

• Homework p. 148 All Vocab Check and Exercises
  1, 3, 7 12, 15, 19, 21, 23, 33, 41, 43, 45