2.6

Rational Functions

Steps for Graphing guidelines.

y-int. ( , )

x-int. ( , )

Domain:

Asymptote(s)

2

3)(

−=x

xg

let x = 0 to find y-int.

0 2

3−

let y = 0 to find x-int.(s)

none

where is g(x) undefined2, ≠ℜ x

if x is undefined at a number,there is a vertical asymptote at that number.

V.A. @ x = 2

Compare the exponents. Do wehave a horizontal at y = 0, a horz.at y = a/b, or a slant asymptote?

Deg. of N < Deg. of D

0=∴ yis horz. asymptote

x = 2

y = 0

y-int. ( , )

x-int. ( , )

Domain:

Asymptote(s)

x

xxf

12)(

−=

none

02

1

0, ≠ℜ x

V.A. @ x = 0

H.A.

21

2@ ==y

x = 0

y = 2

y-int. ( , )

x-int. ( , )

Domain:

Asymptote(s)

2)(

2 −−=

xxx

xf

(x-2)(x+1)

0 0

0 0

2,1, −≠ℜ x

V.A. @ x = -1 x = 2

H.A. y = 0b/c N < D

x = -1 x = 2

y-int. ( , )

x-int. ( , ) ( , )

Domain:

Asymptote(s)

Slant asymptotes

1

2)(

2

−−−

=xxx

xg

(x-2)(x+1)0 2

2 0-1 0

1, ≠ℜ x

V.A. x = 1

Slant asym.

y = x

x = 1

y = x

y-int. ( , )

x-int. ( , ) ( , )

Domain:

Asymptote(s)

4

9)(

2

2

−−

=xx

xf(x-3)(x+3)

(x-2)(x+2)

4

90

2,2, −≠ℜ x

V.A. x = -2 x = 2

H.A. 11

1==y

3 0-3 0

y-int. ( , )

x-int. ( , )

Domain:

Asymptote(s)

( )22

1

)1()(

+

−−=x

xxf

0 -1

1 0

1, −≠ℜ x

V.A. x = -1

H.A.

11

1−=−=y

y-int. ( , )

x-int. ( , ) ( , )

Domain:

Asymptote(s)

2

1.1

++

=xx

y

y-int. ( , )

x-int. ( , ) ( , )

Domain:

Asymptote(s)

32.2

2 −−=

xxx

y

y-int. ( , )

x-int. ( , ) ( , )

Domain:

Asymptote(s)

x

xy

2)1(.3

−=

y-int. ( , )

x-int. ( , ) ( , )

Domain:

Asymptote(s)

1

1.4

2

−++

=xxx

y

y-int. ( , )

x-int. ( , ) ( , )

Domain:

Asymptote(s)

( )21.5

−=x

xy

y-int. ( , )

x-int. ( , ) ( , )

Domain:

Asymptote(s)

( )( )( )( )31

232.6

−+

−+=

xx

xxy

y-int. ( , )

x-int. ( , ) ( , )

Domain:

Asymptote(s)

1

1.7

−=x

y

y-int. ( , )

x-int. ( , ) ( , )

Domain:

Asymptote(s)

( )( )( ) ( )32

21.8 2

2

−+

−+=

xx

xxy

y-int. ( , )

x-int. ( , ) ( , )

Domain:

Asymptote(s)

y-int. ( , )

x-int. ( , ) ( , )

Domain:

Asymptote(s)

y-int. ( , )

x-int. ( , ) ( , )

Domain:

Asymptote(s)