Rational Functions & Their Graphs
1
2
3
Simplifying
Discontinuity, Intercepts &
Asymptotes
Practice Problems
Rational Functions
2

Rational Functions
 can

be written as
f ( x) 
P( x)
Q( x)
Continuous Functions
 can
be graphed w/o lifting the pencil
 are not undefined at any value

Discontinuous Functions
 Can
not be graphed w/o lifting the pencil
 Are undefined at one or more values
Simplifying
3
Steps
 COMPLETELY Factor the Numerator
 COMPLETELY Factor the Denominator
 Cancel Matching Factors/Terms
Simplifying Examples
4
Simplify
x 9
x 2  7 x  18
x 9
( x  9)( x  2)
1
( x  2)
x 4
3x  6
2
( x  2)( x  2)
3( x  2)
( x  2)
3
Discontinuity
5

Point of Discontinuity is any value that makes the
function undefined (divide by zero)
 Removable
 Can
Discontinuity
be removed by Simplifying
 Non-Removable
 Can
Discontinuity
not be removed by simplifying
Discontinuity Example
6
What are the domain points of discontinuity? Are they
removable or non-removable?
x3
y 2
x  4x  3
x3
y
( x  3)( x  1)
Discontinuous at x=3 and x=1
Non-removable
Horizontal Asymptotes
7
m
P( x)
(
x

a
)
f ( x) 

Q( x)
( x  a) n
mn
has a horizontal asymptote at y  0
mn
no horizontal asymptote
mn
Horizontal asymptote at y=a/b where “a” is
the coefficient of the term of the greatest
power in the numerator and “b” is the
coefficient of the term of the greatest power
in the denominator.
Vertical Asymptotes
8
m
P( x)
(
x

a
)
f ( x) 

Q( x)
( x  a) n
mn
has a vertical asymptote at x  a
Non-removable points of discontinuity
are vertical asymptotes !
Asymptote Example
9

What are the vertical asymptotes of y 
x 1
y
( x  2)( x  3)
x2
x3
x 1
x 2  5x  6
Another Asymptote Example
10

What are the horizontal asymptotes of
2x
y
x 3
D1
D1
2
y
1
y2
x2
y 2
x  2x  3
D1
D2
y0
x2
y
2x  5
D2
D1
No Horizontal
Asymptote
Intercepts
11

Y-Intercept
 Set
x=0 and solve
 (0, ?)

X-Intercept
 Set
y=0 and solve
 (?,0)
Intercept Example
12
Find the intercepts
Y-intercept
03
y
(0  3)(0  1)
3
y
(3)(1)
3
y
3
(0,1)
x3
y 2
x  4x  3
X-intercept
x3
0 2
x  4x  3
0  x3
x  3
(3,0)