0-3: Rational Functions & AsymptotesObjectives:

•Determine horizontal, vertical & slant asymptotes•Graph rational functions

©2002 Roy L. Gover (roygover@att.net)Modified by Mike Efram 2004

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DefinitionsRational Function: the ratio of two functions of the form: ( )( ) , ( ) 0

( )g xf x h xh x

Example

2

3

2 3( )2

xf xx x

g(x)

h(x)

More Examples

2

1( )2 5

f xx x

3 2

2

2( ) x xp xx

Definitions•Degree of a Function:The largest exponent in the function.•Leading Coefficient: the coefficient of the term with the largest exponent.

Example1. What is the degree of f(x)=x+2x4-3?

2. What is the leading coefficient of f(x)?

Definition

The line x=a is a vertical asymptote for f(x) if f(x) as x a from the left or from the right

( )2

xf xx

What is true about f(x) when x = -2 ?

Example

Try ThisFind all vertical asymptotes for:

( )( 2)( 3)

xf xx x

x=2,-3

ExampleFind all vertical asymptotes of:

2

1( )2 3

p xx x

Hint: Factor to find where the denominator = 0!!

Definition

The line y=b is a horizontal asymptote of f(x) if f(x) b as x

Example

2

1( ) 3f xx

y=-3 is a horizontal asymptote

Rules for Finding Horizontal Asymptotes1. If degree of numerator < degree of denominator, horizontal asymptote is the line y=0 (x axis)

Rules for Finding Horizontal Asymptotes

(cont.)2. If degree of numerator =degree of denominator, horizontal asymptote is the line y=ratio of leading coefficients.

Rules for Finding Horizontal Asymptotes

(cont.)3. If degree of numerator >degree of denominator, there is no horizontal asymptote.

ExampleFind the horizontal asymptotes, if any, of:

4

3

3( )1

xq xx

Graph and confirm

ExampleFind the horizontal asymptotes, if any, of:

2

3

34 5

xyx

Graph and confirm

Try ThisFind the horizontal asymptotes, if any, of:

3

3

84 5

xyx

Graph and confirm

DefinitionThe slant line y=mx+b is a slant asymptote of f(x) if the degree of the numerator is exactly one greater than the degree of the denominator.

Guidelines for Finding Slant Asymptotes

1. Divide denominator into numerator using long division. Ignore any remainder.2. Slant asymptote is y=the result of the above division.

Long Division of Polynomials(Needed to find slant asymptotes)

Examples2 7 21.

2x x

x

Try This Divide using long division: 22 1

4x x

x

272 74

xx

Remainder

Example

2 5 8( )3

x xq xx

Find the slant asymptotes, if any for:

Graph and confirm

Try This

2

( )2

x xq xx

Find the slant asymptotes, if any for:

Graph and confirm

y=x+3

Example

2 2( )1

x xf xx

Consider the rational function

1. Find all asymptotes.2. What’s happening at f(-1)?

Try This2

( ) x xf xx

Consider the rational function1. Find all asymptotes.2. What’s happening at f(0)?3. Graph (don’t use a calculator)

Example

2 3 2( )2

x xp xx

Consider the rational function:

What is different? Hint: factor numerator & simplify.

Some rational functions are discontinuous... asymptote rules don’t apply. How do you know?

Important Idea