Unit 3
Graphing Rational
Functions
Objectives:
Part I
• Find the Vertical Asymptotes
• Find the Horizontal Asymptotes
Rational Functions
An asymptote is a vertical or horizontal
line where the function is undefined.
(Dashed line)
A rational function can have more than one
asymptote:
1.Vertical asymptote (are the “excluded values)
2.Horizontal asymptote (at most one, must apply the
rules for)
Vertical Asymptotes
V.A. is a vertical line where x = a, and “a “ represents the
real zeros of the denominator ( excluded valued).
Example:
Find the vertical asymptote of
f ( x) 
2x
x2 1
2
q
(
x
)

x
 1  ( x  1)(x  1) the zeros
Since
are 1 and -1. Thus the vertical asymptotes
are x = 1 and x = -1.
Recall: x cannot be 1 or -1 if you solved it.
.
Horizontal Asymptotes
The horizontal asymptote is
determined by looking at the
degrees of p(x) and q(x).
**In other words, the degree of the
numerator and degree of the denominator**
Rules for the HA
(horizontal asymptote).
a. If the degree of p(x) is less than the degree of
q(x), then the horizontal asymptote is y = 0.
b. If the degree of p(x) is greater than the degree
of q(x), then there is no horizontal asymptote.
c. If the degree of p(x) is equal to the degree of
q(x), then the horizontal asymptote is a ratio
of…
y
leading coefficient of p( x)
.
leading coefficient of q ( x)
Let’s try 1
Example 1:
Find the horizontal asymptote: f ( x) 
3x
2
x 1
Degree of numerator = 1
Degree of denominator = 2
y = 0.
Since the degree of the numerator is
less than the degree of the denominator,
horizontal asymptote is
.
Example 2:
3x  1
.
Find the horizontal asymptote: f ( x) 
2x 1
Degree of numerator = 1
Degree of denominator = 1
3
y
2
Since the degree of the numerator is
equal to the degree of the denominator,
horizontal asymptote is
.
Example 3
Example:
Find the horizontal asymptote:
3 x2  1
f ( x) 
.
2x  1
Degree of numerator = 2
Degree of denominator = 1
There is no horizontal asymptote. Since the
degree of the numerator is greater than the
degree of the denominator.
Copy …..
Graphing a Rational Function
a
The graph of a y 
 k has the following
xh
characteristics.
Vertical asymptotes: x
Horizontal asymptotes:y
h
k
Then plot 2 points to the
left and right of the
center
Break! 
Video on Asymptotes
http://cms.gavirtualschool.org/Shared/Math/CCGPS_AdvancedAlgebra/RationalandRadicalRelationships/index.html
Tab 4 , Video 1/3
Determine the H.A and V. A for
each
Part II: Graphing
Rational Function
p ( x)
f ( x) 
q ( x)
STEPS!
1. Identify your asymptotes and sketch them.
(Horizontal and Vertical)
2. Create a table of values to graph at least 2 points on
either side of the vertical asymptote.
3. Sketch!
“Hyperbola”- Type of Rational
Function.
• One form: y  ax  b
cx  d
• 2nd Form (Standard form of a Hyperbola: y  a  k
xh
If using 2nd form
• Asymptotes: x=h (vertical) Note: If denominator in equation is (+), equation
will be a neg “h”). y=k (horizontal/take whatever sign with it)
• Find 2 points on either side of the vertical asymptote.
(Evaluate to find (x,y) coordinates.
• Graph the asymptotes.
• Plot the points and sketch graph.
Example 1
Graphing a Rational Function
1.Identify your
asymptotes and sketch
them. (Horizontal and
Vertical)
2.Create a table of
values to graph at least
2 points on either side
of the vertical
asymptote.
3.Sketch!
Graph the function. State the domain and range.
2
1. y 
x
Vertical asymptotes: x
Horizontal asymptotes:y
x
2
1
y
2
2
2 1
0
0
3
y
2
x 1
Example 2
.
Vertical Asymptote: x=1
Horizontal Asymptote: y=2
x
y
-5 1.5
-2
1
2
5
4
3
Left of
vert.
asymp.
Right of
vert.
asymp.
1.Identify your
asymptotes and sketch
them. (Horizontal and
Vertical)
2.Create a table of values
to graph at least 2 points
on either side of the
vertical asymptote.
3.Sketch!
Example 3~ You try!
Graphing a Rational Function
Graph the function.
2
6. y 
4
x 1
center:
Vertical asymptotes:
x 1
Horizontal asymptotes: y   4
x
y
1
0
2
3
5
6
2
3
1,  4 
x2
y
3x  3
Example 4
Vertical asymptote:
3x+3=0 (set denominator =0)
3x=-3
x= -1
x
Horizontal Asymptote:
-3
y
.83
-2 1.33
y 
1
3
0
-.67
2
0
Think Pair Share- 5-10
Handout from yesterday!
Day 3: Domain and Range
Monday..
Domain-The domain of each
graph is all real numbers except
what makes the denominator
zero. (X Values)
Range – All y values
COPY! When describing the Domain and Range
from the Standard equation form OF THE
HYPERBOLA. Consider the following!
3
y
E
2
x 1
Ex. 1
State the domain & range.
The Domain &
Range is….
Domain: (-∞, 1) U (1, ∞)
Range: (-∞, 2) U (2, ∞)
Ex. 2 When describing the domain from the graph itself!
Domain: (- ∞, -2) U (-2, 2) U (2, ∞)
Range: (- ∞, 0] U (3, ∞)
State the domain and range.
x
7. y 
x2
 2
y 1
Vertical asymptotes: x
Horizontal asymptotes:
x
4
3
1
0
y
Domain:
2
3 a ll R x   2
 1 Range:
0 a ll R y  1
You try!
State the domain and range.
x 1
10. y 
2x  3
x  3/2
Vertical asymptotes:
Horizontal asymptotes:
x
y
y  1/ 2
Domain:
0 1/ 3
1 0 a ll R x  3 / 2
Range:
1
2
3 2 / 3 a ll R y  1 / 2
Guided Practice 5-10
Describe the Domain and
Range
Characteristics:
Discontinuity and Holes
• Discontinuity- Where the graph breaks ( The Vertical
Asymptotes)
• Holes – Factor and simplify. They are the factor/s
that will cancel. Solve for the “x”.
• Example 4 – Handout 2
• HW- Describe all characteristics
•
You try
AGAIN
!
Ex
Domain: (-∞,∞)
Range:
 1 1 
 2 , 2 
Let’s practice Describe the Domain
and Range # 5, 6 from Handout