Functions

1 / 16

# Functions - PowerPoint PPT Presentation

Functions. AII.7 e 2009. Objectives: Find the Vertical Asymptotes Find the Horizontal Asymptotes. Rational Functions . A rational function f ( x ) is a function that can be written as . where p ( x ) and q ( x ) are polynomial functions and q ( x ) 0 .

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## Functions

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

Functions

AII.7 e 2009

Objectives:

Find the Vertical Asymptotes

Find the Horizontal Asymptotes

Rational Functions

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

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

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

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.

Vertical Asymptotes

V.A. is x = a, where a represents real zeros of q(x).

Example:

Find the vertical asymptote of

Since the zeros are 1 and -1. Thus the vertical asymptotes are x = 1 and x = -1.

Horizontal Asymptotes

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

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

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

Horizontal Asymptotes

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

b. If the degree of p(x) is equal to the degree of

q(x), then the horizontal asymptote is

c. If the degree of p(x) is greater than the degree

of q(x), then there is no horizontal asymptote.

Horizontal Asymptotes
• deg of p(x) < deg of q(x), then H.A. isy = 0
• deg of p(x) = deg of q(x), then H.A. is
• deg of p(x) > deg of q(x), then no H.A.

Example:

Find the horizontal asymptote:

Degree of numerator = 1

Degree of denominator = 2

Since the degree of the numerator is less than the degree of the denominator, horizontal asymptote isy = 0.

Horizontal Asymptotes
• deg of p(x) < deg of q(x), then H.A. isy = 0
• deg of p(x) = deg of q(x), then H.A. is
• deg of p(x) > deg of q(x), then no H.A.

Example:

Find the horizontal asymptote:

Degree of numerator = 1

Degree of denominator = 1

Since the degree of the numerator is equal to the degree of the denominator, horizontal asymptote is.

Horizontal Asymptotes
• deg of p(x) < deg of q(x), then H.A. isy = 0
• deg of p(x) = deg of q(x), then H.A. is
• deg of p(x) > deg of q(x), then no H.A.

Example:

Find the horizontal asymptote:

Degree of numerator = 2

Degree of denominator = 1

Since the degree of the numerator is greater than the degree of the denominator, there is nohorizontal asymptote.

Vertical & Horizontal Asymptotes

• deg of p(x) < deg of q(x), then H.A. isy = 0

V.A. :

x = a, where a

represents real

zeros of q(x).

H.A. :

• deg of p(x) = deg of q(x), then H.A. is
• deg of p(x) > deg of q(x), then no H.A.

Practice:

Find the vertical and horizontal asymptotes:

Vertical & Horizontal Asymptotes

H.A. :

V.A. :

x = a, where a

represents real

zeros of q(x).

• deg of p(x) < deg of q(x), then H.A. isy = 0
• deg of p(x) = deg of q(x), then H.A. is
• deg of p(x) > deg of q(x), then no H.A.

Practice:

Find the vertical and horizontal asymptotes:

V.A. : x =

H.A.: none

Vertical & Horizontal Asymptotes

• deg of p(x) < deg of q(x), then H.A. isy = 0

V.A. :

x = a, where a

represents real

zeros of q(x).

H.A. :

• deg of p(x) = deg of q(x), then H.A. is
• deg of p(x) > deg of q(x), then no H.A.

Practice:

Find the vertical and horizontal asymptotes:

Vertical & Horizontal Asymptotes
• deg of p(x) < deg of q(x), then H.A. isy = 0

V.A. :

x = a, where a

represents real

zeros of q(x).

H.A. :

• deg of p(x) = deg of q(x), then H.A. is
• deg of p(x) > deg of q(x), then no H.A.

Practice:

Find the vertical and horizontal asymptotes:

V.A. : none

H.A.: y = 0

is not factorable and thus has no real roots.

Vertical & Horizontal Asymptotes

• deg of p(x) < deg of q(x), then H.A. isy = 0

V.A. :

x = a, where a

represents real

zeros of q(x).

H.A. :

• deg of p(x) = deg of q(x), then H.A. is
• deg of p(x) > deg of q(x), then no H.A.

Practice:

Find the vertical and horizontal asymptotes:

Vertical & Horizontal Asymptotes
• deg of p(x) < deg of q(x), then H.A. isy = 0

V.A. :

x = a, where a

represents real

zeros of q(x).

H.A. :

• deg of p(x) = deg of q(x), then H.A. is
• deg of p(x) > deg of q(x), then no H.A.

Practice:

Find the vertical and horizontal asymptotes:

V.A. : x = -1

H.A.: y = 2