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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 .

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Functions

AII.7 e 2009

objectives find the vertical asymptotes find the horizontal asymptotes
Objectives:

Find the Vertical Asymptotes

Find the Horizontal Asymptotes

rational functions
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
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 asymptotes1
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
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 asymptotes1
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 asymptotes2
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 asymptotes3
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 asymptotes4
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
Vertical & Horizontal Asymptotes

Answer Now

  • 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 asymptotes1
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 asymptotes2
Vertical & Horizontal Asymptotes

Answer Now

  • 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 asymptotes3
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 asymptotes4
Vertical & Horizontal Asymptotes

Answer Now

  • 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 asymptotes5
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