2.6 Rational Functions2.6 Rational Functions

Asymptotes;Asymptotes;

Can’t touch this stuffCan’t touch this stuff

Asymptote

as·ymp·tote /ˈasəm(p)ˌtōt/

Noun: A line that continually approaches a given curve but does not meet it at any finite distance.

Math wants to know the behavior of the function coming at it from the Right or Left.

x is all real numbers but what value in the function 1

1)(

x

xxf

x is all real numbers except what value in the function

x ≠ 1 The asymptotes would be x = 11

1)(

x

xxf

x is all real numbers except what value in the function

x ≠ 1 The asymptotes would be x = 11

1)(

x

xxf

x is all real numbers except what value in the function

x ≠ 1 The asymptotes would be x = 1

Vertical asymptotes

is 1

Since the you

can not have zero

in the denominator.

1

1)(

x

xxf

x is all real numbers but what value in the function

x ≠ 1 The asymptotes would be x = 1

Horizontal

asymptotes

Is also 1

1

1)(

x

xxf

To find the Horizontal Asymptote

Look the equation

If n = m, then the horizontal asymptote is

If n < m, then the horizontal asymptote is 0.

If n > m, then No horizontal asymptote

If n = m + 1, then we have a slant asymptote.

Slant asymptote is y =

dcx

baxm

n

c

ay

dcxbax mn )(

Behavior as the function approaches the asymptote

Behavior as the function approaches the asymptote

From the Right

From the Left

Also,

1,)( xxf

1,)( xxf

xxf

xxf

,1)(

,1)(

Lets graph

2

222

x

xx

Homework

Page 174 – 177

# 5, 9, 17, 27,

45, 53, 69, 73,

91

HomeworkHomework

Page 174 – 177Page 174 – 177

## 7, 11, 19, 37, 7, 11, 19, 37,

49, 57, 71, 8749, 57, 71, 87