Today we are going to look further at the behavior of the graphs of different functions. Now we are going to determine if the graphs have horizontal and/or vertical asymptotes.

Vertical AsymptotesVertical Asymptotes

Horizontal AsymptotesHorizontal Asymptotes

Imaginary vertical lines to which a graph gets closer to but usually never crosses. These are represented by dotted lines on the graph.

Imaginary horizontal lines to which a graph gets closer to but usually never crosses. These are represented by dotted lines on the graph.

A vertical asymptote comes from the denominator. It exists at the value(s) that will make the denominator zero.

Example:

f(x)= 9 x – 2

If there are no variables in the denominator, there are no vertical asymptotes.

Example:

f(x) = x2 – 9 6

To determine horizontal asymptotes, there are 3 rules that you must memorize.

← degree is 0

← degree is 1

← degree is 1

← degree is 2

If the degree of the numerator is < than the degree of the denominator, then the HA is y = 0.

Example:

f(x) = 5 x - 2

HA @ y = 0 Example:

f(x) = 2x x2 - 5

HA @ y = 0

If the degree of the numerator = degree of denominator, then divide the leading coefficient of the numerator by the leading coefficient of the denominator.

← degree is 1

← degree is 1

Example:

f(x) = 6x 5x + 10

HA @ y = 6/5 and VA @ x = -2

← degree is 1

← degree is 1

Example:

f(x) = x – 2 x + 2

HA @ y = 1 and VA @ x = -2

← degree is 2

← degree is 2

Example:

f(x) = 2(x2 – 9) x2 – 4

HA @ y = 2 and VA @ x = ±2

If the degree of the numerator the degree of denominator , then there are no horizontal asymptotes.

In this case there may be asymptotes, but they are oblique or parabolic.

In calculus you will learn another way to find horizontal asymptotes when studying the limits of functions.

                         

                         

                         

                         

                         

                         

                         

                         

                         

                         

                         

Let’s graph some rational graphs:

y = 3 x - 1

VA:

HA:

x – 1 = 0 x = 1

y = 0

x y

-1 -3/2

0 -3

1 V.A.

2 3

3 3/2

Domain: Range: x 1 y 0

Let’s graph some rational graphs:

                         

                         

                         

                         

                         

                         

                         

                         

                         

                         

                         

y = 3x x + 2

VA:

HA:

x + 2= 0 x = -2

y = 3

x y

-4 6

-3 9

-2 VA

-1 -3

0 0

Domain: Range: x -2

y 3

Let’s graph some rational graphs:

                         

                         

                         

                         

                         

                         

                         

                         

                         

                         

                         

y = 2 - x x - 1

VA:

HA:

x - 1= 0 x = 1

y = -1

x y

-1 -3/2

0 -2

1 VA

2 0

3 -1/2

Domain: Range: x 1 y -1

Let’s graph some rational graphs:

                         

                         

                         

                         

                         

                         

                         

                         

                         

                         

                         

y = 1 x2 + 3

VA:

HA:

x2 + 3= 0 x2 = -3 none

y = 0

x y

-2 1/7

-1 1/4

0 1/3

1 1/4

2 1/7

Domain: Range: (-,) y 0

1

1-1

-1