3.5C Rational Functions and Asymptotes

3.5C Rational Functions and Asymptotes

A. Definition of a Rational Function

is said to be a rational function if ´Üµ ,´Üµ where and are polynomial functions.

´Üµ

That is, rational functions are fractions with polynomials in the numerator and denominator.

B. Asymptotes/Holes

Holes are what they sound like:

is a hole
Rational functions may have holes or asymptotes (or both!).
Asymptote Types:

1. vertical
2. horizontal
3. oblique (“slanted-line”)
4. curvilinear (asymptote is a curve!)
We will now discuss how to find all of these things.

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C. Finding Vertical Asymptotes and Holes

Factors in the denominator cause vertical asymptotes and/or holes.
To find them:

1. Factor the denominator (and numerator, if possible).
2. Cancel common factors.
3. Denominator factors that cancel completely give rise to holes. Those that don’t
give rise to vertical asymptotes.

D. Examples

Example 1: Find the vertical asymptotes/holes for where ´Üµ  ´¿Ü·½µ´Ü µ´Ü· µ
  .´Ü µ¾´Ü· µ

Solution

 Canceling common factors: ´Üµ
 ¿Ü·½

ÜÜ

µ  Ü · factor cancels completely hole at Ü

  µÜ factor not completely canceled vertical asymptote with equation Ü

Example 2: Find the vertical asymptotes/holes for where ´Üµ ¾Ü¾     Ü ½¾ .
ܾ
Ü·

Solution

Factor: ´Üµ  ´Ü µ´¾Ü·¿µ
   ´Ü µ´Ü ½µ

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Cancel: ´Üµ  ¾Ü·¿

ܽ Ü

Ans Hole at Ü
Vertical Asymptote with equation Ü ½

E. Finding Horizontal, Oblique, Curvilinear Asymptotes

Suppose ´Üµ ¡ ¡ ¡ÒÜÒ · · ½Ü · ¼
¡ ¡ ¡ÑÜÑ ·
· ½Ü · ¼

If

1. degree top degree bottom: horizontal asymptote with equation Ý ¼

2. degree top degree bottom: horizontal asymptote with equation Ý ÑÒ

3. degree top degree bottom: oblique or curvilinear asymptotes

To find them: Long divide and throw away remainder

F. Examples

Example 1: Find the horizontal, oblique, or curvilinear asymptote for where ´Üµ Ü  Ü·¾ .
Ü  ·¾Ü ½

Solution

degree top degree bottom . Since , we have

Ans horizontal asymptote with equation Ý ¼

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Example 2: Find the horizontal, oblique, or curvilinear asymptote for where ´Üµ   .Ü¿ ¾Ü¾·½

¾Ü¿ ·

Solution

degree top ¿ degree bottom ¿.

Since ¿ ¿, we have a horizontal asymptote of Ý ¾ ¿. Thus

Ans horizontal asymptote with equation Ý ¿

Example 3: Find the horizontal, oblique, or curvilinear asymptote for where ´Üµ  ¾Ü¿ ¿ .
 Ü¾ ½

Solution

degree top ¿ degree bottom ¾.

Since ¿ ¾, we have an oblique or curvilinear asymptote. Now long divide:

¾Ü

   Ü¾ · ¼Ü ½ ¾Ü¿ · ¼Ü¾ · ¼Ü ¿
   ´¾Ü¿ · ¼Ü¾ ¾Üµ
 ¾Ü ¿

 ¾Ü¿ ¿  ¾Ü ¿ , we have that
 Since  ¾Ü ·
ܾßÞ ½
ܾ ½

Throw away

Ans Ý ¾Ü defines a line, and is the equation for the oblique asymptote

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Example 4: Find the horizontal, oblique, or curvilinear asymptote for where

 ¿Ü Ü · ¾Ü¾ · Ü · ½

´Üµ
ܾ · ½

Solution

degree top degree bottom ¾.

Since ¾, we have an oblique or curvilinear asymptote. Now long divide:

   ¿Ü¿ ܾ ¿Ü · ¿

 Ü¾ · ¼Ü · ½ ¿Ü Ü · ¼Ü¿ · ¾Ü¾ · Ü · ½

 ´¿Ü · ¼Ü · ¿Ü¿µ

   Ü ¿Ü¿ · ¾Ü¾
     ´ Ü · ¼Ü¿ ܾµ
 ¿Ü¿ · ¿Ü¾ · Ü
     ´ ¿Ü¿ · ¼Ü¾ ¿Üµ

¿Ü¾ · Ü · ½

 ´¿Ü¾ · ¼Ü · ¿µ
 Ü ¾

 Since ¿Ü Ü · ¾Ü¾ · Ü · ½      ¿Ü¿ ¾
¾ Ü
ܾ · ½ Ü , we have that
¿Ü · ¿ ·
Ü¾ß·Þ ½

Throw away

   Ans Ý ¿Ü¿ ܾ ¿Ü · ¿ defines a curvilinear asymptote

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G. Asymptote Discussion for Functions

¦½1. As the graph of a function approaches a vertical asymptote, it shoots up or down toward .

approach

approach

2. Graphs approach horizontal, oblique, and curvilinear asymptotes as Ü  ½ or

½Ü .

approach approach

approach

approach

3. Graphs of functions never cross vertical asymptotes, but may cross other asymptote
types.

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