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Rational Functions

A rational function can have more than one vertical asymptote, but it can have at most one horizontal asymptote.

A rational function f(x) is a function that can be written as

)(

)()(

xq

xpxf

where p(x) and q(x) are polynomial functions and q(x) 0 .

Vertical Asymptotes

If p(x) and q(x) have no common factors, then f(x) has vertical asymptote(s) when q(x) = 0. Thus the graph has vertical asymptotes at the zeros of the denominator.

Horizontal Asymptotes

The horizontal asymptote is determined by looking at the degrees of p(x) and q(x).

A rational function f(x) is a function that can be written as

)(

)()(

xq

xpxf

where p(x) and q(x) are polynomial functions and q(x) 0 .

Horizontal Asymptotes

a. If the degree of p(x) is less than the degree of q(x), then the horizontal asymptote is y = 0.

b. If the degree of p(x) is equal to the degree of q(x), then the horizontal asymptote is

.)( oft coefficien leading

)( oft coefficien leading

xq

xpy

c. If the degree of p(x) is greater than the degree of q(x), then there is no horizontal asymptote.

)(

)()(

xq

xpxf

Practice:Practice: Find the vertical and horizontal asymptotes:

12

13)(

2

x

xxf

1

2)(

2 x

xxf

Quick Check:

• Vertical and Horizontal Asymptotes Worksheet.

Holes and Vertical Asymptotes

• Values for which a ration function ins undefined results in a vertical asymptote or a hole in the graph.

• Our vertical function will have a hole if both numerator and denominator have the same factor (x – b). The hole will be at x = b.

• One exception. If x = b is a vertical asymptote then there is no hole.

Example:

• Find all the horizontal and vertical asymptotes and holes in the graph of:

Example:

• Find all the horizontal and vertical asymptotes and holes in the graph of:

Student Check:

• Find all horizontal and vertical asymptotes and holes in the graph of:

Practice:

• Rational functions with holes worksheet.