2.6 Rational Functions and Asymptotes

• Students will find the domains of rational functions.

• Students will find vertical and horizontal asymptotes of rational functions.

• Students will use rational functions to model and solve real-life problems.

Example 1: The domain of a rational function

Find the domain of f(x) = 1/x and discuss the behavior of f near any excluded x-values.

Asymptotes of a Rational FunctionLet f be the rational function

Where N(x) and D(x) have no common factors.

Vertical Asymptotes occur at the zeros of D(x)

V.A. : Set D(x) = 0

Horizontal Asymptotes

1. If the graph approaches the x-axis. H.A. : y=0

2. If n = m the graph has a horizontal asymptote of y = the fraction of the leading coefficients.

3. If the graph has no horizontal asymptote.

Num era tor Po lynom ia l

D eno a tor Po lynom ia l

m inf x

N x

D x

ree n

ree d( )

( )

( )

d eg ( )

d eg ( )

n d

n d

Example 2

Find all horizontal and vertical asymptotes of the graph of each rational function.

a. b.f xx

x( )

2

3 12f x

x

x( )

2

1

2

2

Example 3:

Find all horizontal and vertical asymptotes of the graph of

f xx x

x x( )

2

2

2

6

Example 4

For the function f, find (a) the domain of f, (b) the vertical asymptotes of f, (c) the horizontal asymptotes of f.

f xx x

x( )

3 7 2

4 5

3 2

3

Example 5 A graph with 2 horizontal asymptotes

The function f xx

x( )

10

2

Example 6

A utility company burns coal to generate electricity. The cost C (in dollars) of removing p% of the smokestack pollutants is given by C = 80,000p/(100-p) for . Use a graphing utility to graph the function. You are a member of a state legislature that is considering a law that is considering a law that would require utility companies to remove 90% of the pollutants from their smokestack emissions. The current law requires 85% removal. How much additional cost would there be to the utility company because of the new law?

0 100 p

C = 80,000p/(100-p)How much additional cost would there be to

the utility company because of the new law?

Homework

p.148-149 #6-33x3,

7-12 all (give V.A. and H.A.)