2.7 Rational Functions and

Their Graphs

Graphing Rational Functions

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

Graphing Rational Functions (m<n)

x y

41

42

2.5 22

)f

2

3

5 62.

xf x

x x

3

3 2

x

x x

4 3.5

e

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

Graphing Rational Functions (m>n)

x y

4 9

)f

94 9

1f

24 1

44 3

f

2

3 2

.1

3

x xf x

x

2

1

3

x

x

e

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:

Special Rational Functions

)f

2

56

. 2

x xf x

x

3 2

2

x x

x

3x

e