Rational Functions and Models

Lesson 4.6

Definition

Consider a function which is the quotient of two polynomials

Example:

( )( )

( )

P xR x

Q x Both polynomials

2500 2( )

xr x

x

Long Run Behavior

Given

The long run (end) behavior is determined by the quotient of the leading terms Leading term dominates for

large values of x for polynomial Leading terms dominate for

the quotient for extreme x

11 1 0

11 1 0

...( )

...

n nn nm m

m m

a x a x a x aR x

b x b x b x b

nnm

m

a x

b x

Example Given

Graph on calculator Set window for -100 < x < 100, -5 < y < 5

2

2

3 8( )

5 2 1

x xr x

x x

Example

Note the value for a large x

How does this relate to the leading terms?

2

2

3

5

x

x

Try This One

Consider

Which terms dominate as x gets large

What happens to as x gets large?

Note: Degree of denominator > degree numerator Previous example they were equal

2

5( )

2 6

xr x

x

2

5

2

x

x

When Numerator Has Larger Degree

Try

As x gets large, r(x) also gets large

But it is asymptotic to the line

22 6( )

5

xr x

x

2

5y x

Summarize

Given a rational function with leading terms

When m = n Horizontal asymptote at

When m > n Horizontal asymptote at 0

When n – m = 1 Diagonal asymptote

nnm

m

a x

b x

a

b

ay x

b

Vertical Asymptotes

A vertical asymptote happens when the function R(x) is not defined This happens when the

denominator is zero Thus we look for the roots of the denominator

Where does this happen for r(x)?

( )( )

( )

P xR x

Q x

2

2

9( )

5 6

xr x

x x

Vertical Asymptotes

Finding the roots ofthe denominator

View the graphto verify

2 5 6 0

( 6)( 1) 0

6 or 1

x x

x x

x x

2

2

9( )

5 6

xr x

x x

Zeros of Rational Functions We know that

So we look for the zeros of P(x), the numerator

Consider

What are the roots of the numerator? Graph the function to double check

( )( ) 0 ( ) 0

( )

P xR x P x

Q x

2

2

9( )

5 6

xr x

x x

Zeros of Rational Functions

Note the zeros of thefunction whengraphed

r(x) = 0 whenx = ± 3

Summary

The zeros of r(x) arewhere the numeratorhas zeros

The vertical asymptotes of r(x)are where the denominator has zeros

2

2

9( )

5 6

xr x

x x

Assignment

Lesson 4.6 Page 319 Exercises 1 – 41 EOO

93, 95, 99