RATIONAL FUNCTIONS IObjectives:

Find the domain of a Rational FunctionDetermine the Vertical Asymptotes of a Rational Function

Determine the Horizontal or Oblique Asymptotes of a Rational Function

Rational Function

A quotient of polynomial functions where are polynomial functions and . The domain is the set of all real numbers except those for which the denominator is 0.

EX: Find the domain of each rational function

1. 2.

3.

p x

R xq x

and p x q x 0q x

q x

2

1

25R x

x

23

3 5

xH x

x x

2

3 1

2 5 3

x xF x

x x

EX; Graph each Rational Function using Transformations

1. 2. 2

12H x

x

2

2

3G x

x

3 types of Asymptotes: Vertical, Horizontal and Oblique

A graph will never intersect a vertical asymptote but may intersect a horizontal or oblique.

Locating Asymptotes: Vertical: factor the numerator and denominator, cancel out the

common factors and find the zeros of the denominator The degree of the numerator is n and the degree of the denominator

is m. 1. If , the x-axis or is the horizontal asymptote

2. If , the line the horizontal asymptote

3. If , the line is the oblique asymptote 4. If , the graph has no horizontal or oblique asymptote

The graph of a rational function either has one horizontal or one oblique asymptote or else has no horizontal and no oblique asymptote

x a

n m 0y

n m n

n

ay

b

1n m y ax b 1n m

EX: Find the vertical, horizontal and oblique asymptotes, if any, of each rational function

1. 2.

3. 4.

2

1

1

xF x

x

3

4

xG x

x

3

2

1xH x

x x

4 2

2

2 1

1

x xP x

x x