Rational Expressions, Vertical Asymptotes,

and Holes

Rational Expression It is the quotient of two polynomials.

A rational function is a function defined by a rational expression.

Examples:

2

5

xy

x

2

3 2

3 2 5

4 5 7

x xy

x x x

Not Rational:

4

2

x

yx

2 5

xy

x

Find the domain:1

( )f xx

Graph it:

Find the domain:1

( )f xx

Graph it:

Find the domain:1

( )2

f xx

Graph it:

Find the domain:1

( )2

f xx

Graph it:

Vertical Asymptote

If x – a is a factor of the denominator of a rational function but not a factor of the numerator, then x = a is a vertical asymptote of the graph of the function.

Find the domain: 2

3( )

12

xf x

x x

Graph it using the graphing calculator. What do you see?

Find the domain: 2

3( )

12

xf x

x x

Graph it using the graphing calculator. What do you see?

Hole (in the graph)

If x – b is a factor of both the numerator and denominator of a rational function, then there is a hole in the graph of the function where x = b, unless x = b is a vertical asymptote.

The exact point of the hole can be found by plugging b into the function after it has been simplified.

Find the domain and identify vertical asymptotes & holes.

2

1( )

2 3

xf x

x x

Find the domain and identify vertical asymptotes & holes.

2

1( )

2 3

xf x

x x

Find the domain and identify vertical asymptotes & holes.

2( )

4

xf x

x

Find the domain and identify vertical asymptotes & holes.

2( )

4

xf x

x

Find the domain and identify vertical asymptotes & holes.

2

5( )

2 3

xf x

x x

Find the domain and identify vertical asymptotes & holes.

2

5( )

2 3

xf x

x x

Find the domain and identify vertical asymptotes & holes.

2

2

3 2( )

2

x xf x

x x

Find the domain and identify vertical asymptotes & holes.

2

2

3 2( )

2

x xf x

x x

Horizontal Asymptotes & Graphing

Horizontal Asymptotes Degree of numerator = Degree of denominator

Degree of numerator < Degree of denominator

Degree of numerator > Degree of denominator

0y Horizontal Asymptote:

Horizontal Asymptote:

Horizontal Asymptote:

coefficient of numerator

coefficient of denominatory

none

Find all asymptotes & holes & then graph:

2( )

2 3

xf x

x

Find all asymptotes & holes & then graph:

2( )

2 3

xf x

x

Find all asymptotes & holes & then graph:

2

1( )

2 3

xf x

x x

Find all asymptotes & holes & then graph:

2

1( )

2 3

xf x

x x

Find all asymptotes & holes & then graph:2 3 4

( )4

x xf x

x

Find all asymptotes & holes & then graph:2 3 4

( )4

x xf x

x

Find all asymptotes & holes & then graph:

2

2

3 7 6( )

9

x xf x

x

Find all asymptotes & holes & then graph:

2

2

3 7 6( )

9

x xf x

x

Find all asymptotes & holes & then graph:

23

( )2

xf x

x

Find all asymptotes & holes & then graph:

23

( )2

xf x

x

Find all asymptotes & holes & then graph:

2

2

3( )

2 5

xf x

x

Find all asymptotes & holes & then graph:

2

2

3( )

2 5

xf x

x

Find all asymptotes & holes & then graph:

2( )

2 3

xf x

x x

Find all asymptotes & holes & then graph:

2( )

2 3

xf x

x x