Honors Pre-Calculus A, 3rd Hour, Winter 2016: 2.6 Rational Functions and Asymptotes

Rational Functions:
A rational function is a function that can be written as:

Where

and

are polynomials and

and

have no common factors

X Intercepts
To find X Intercepts you must find when

-You can do this through the factorization of the numerator

For Example:

after factoring the numerator into

and

You find there is an X intercept at (-3, 0) and (3, 0)

Y Intercepts
To find Y Intercepts you must find what

is equal to
-You do this by plugging in 0 for every instance of X in a function
(If the denominator comes out as 0 then there is no Y Intercept, it is a Vertical Asymptote)
For Example:

after plugging in 0 for X you find that

making there a Y Intercept at (0,-5)

Horizontal and Vertical Asymptotes:
An asymptote is defined as a line that continually approaches a given curve but never fully meets at any finite distance.

Vertical Asymptote:

To find a vertical asymptote you must find when

-You can do this through the factorization of the denominator

(Because where

it makes

undefined)

For example in the function

you can find the Vertical Asymptotes by factoring the denominator into

making two Vertical Asymptotes at

and

As you can see no line ever crosses the Asymptotes

and

they only continue to approach to as X approaches

Horizontal Asymptote:

When finding the horizontal asymptote you must find

-You can do this by finding the end behavior of the given function

There are 3 things to check for when determining if there is a horizontal asymptote and if one exists what it is

1.) When the degree of the numerator is less than that of the denominator the Horizontal Asymptote is Y=0

Ex.)

2.) When the degree of the numerator is equal to that of the denominator the Horizontal Asymptote is the ratio between their leading coefficients

Ex.)