Graphing Rational Functions
1) Find the y-intercept: Substitute 0 in for x2) Find the x-intercepts: Set the numerator equal to 0 and
solve.3) Find the Vertical Asymptote: Set the denominator
equal to zero and solve.4) Find the Horizontal or Slant Asymptote: Use the
numerator’s degree (m) and the denominator’s degree (n) to determine which method to use.
5) Find Extra Points: Find points to the left and right of each vertical asymptote.
6) Graph: Graph steps 1-6.
( )( )
( )
m
n
p x ax bf x
q x cx d
Degree Comparison Alternate Asymptote
nm
nm
dcx
bax
xq
xpxf
n
m
)(
)()(
0y
Slant Asymptote:Long Division
Horizontal Asymptote:
( )
( )
p xy
q x
nm ay
c
Horizontal Asymptote:
1int . :
4y
Graphing Rational Functions (m=n)
2
21.
1
2
4
x xf x
x
2
2
2 0 0 1 - intercept :) 0
0 4fb y
1int . : ,1
2x
2
2
2 1 - intercepts : 0
4)
x x
xc x
0 2 1 1x x
2 1 1
2 2
x x
x x
a
b
2y
Graphing Rational Functions (m=n)
2 Vertical Asymptote : 4 0)d x 2,2x
2 Horizontal Asymptote :
1)e y
2 2 0x x
2
21.
1
2
4
x xf x
x
2 1 1
2 2
x x
x x
c
Graphing Rational Functions (m=n)
x y3 43 1
)f
2
2
2 3 3 13
3 4f
18 3 13
9 4f
203 4
5f
2
2
2 3 3 13
3 4f
18 3 13
9 4f
143 2.8
5f
2
21.
1
2
4
x xf x
x
2 1 1
2 2
x x
x x
d
1int . :
2y
Graphing Rational Functions (m<n)
2
3
5 62.
xf x
x x
2
0 3 - intercept : 0
0)
5 0 6y fb
int . : 3x
2
3 - intercepts) : 0
5 6
xx
xc
x
0 3x
3
3 2
x
x x
a
b
Graphing Rational Functions (m<n)
2 Vertical Asym) ptote : 5 6 0d x x 2,3x
Horizontal Asymptote) : 0e y
3 2 0x x
2
3
5 62.
xf x
x x
3
3 2
x
x x
d
c
1int . :
3y
Graphing Rational Functions (m>n)
2
3 2
.1
3
x xf x
x
20 2 0 1
- intercept :) 00 3
y fb
int . :1x
2 2 1 - intercepts : 0
3)
x xxc
x
20 1x
21
3
x
x
b
a
Graphing Rational Functions (m>n)
Vertical Asymptot 3) e : 0xd 3x
Slant Asymp :) totee
2
3 2
.1
3
x xf x
x
2
1
3
x
x
23 2 1x x x x
2 3x x1x
1
3x
1y x d
c
int . : 0y
Special Rational Functions
2
2
44.
2 1
xf x
x
2
2
4 0 - intercept : 0
2 1)
0y fb
int . : 0x
2
2
4 - intercept : 0
2 1) s
xx
xc
20 4x
b
a
Special Rational Functions
2 Vertical Asymptote : 2 1 0)d x
none
4 Horizontal Asymptote :
2)e y
2
2
44.
2 1
xf x
x
2y
1
2x i
d
c
Special Rational Functions
x y
1 0.8
41
5f
2
2
4 11
4 1 1f
2
2
44.
2 1
xf x
x
41
5f
2
2
4 11
4 1 1f
1 0.8
e
int . : 3y
Special Rational Functions
20 0 6
- intercept) : 00 2
y fb
int . : 3x - intercepts : 0 3) xc x
2
56
. 2
x xf x
x
3 2
2
x x
x
3x
b
a
Special Rational Functions
Vertical Asymptot 2) e : 0xd 2x
none
2
56
. 2
x xf x
x
3 2
2
x x
x
3x
d
c
Horizontal Asymptote:
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