Precalculus Section 2.72015

Objective:To sketch graphs of rational

functions

Refer to “Quick Guide to Rational Functions.”

y-int. ( , )

x-int. ( , )

Domain:

Asymptote(s)

xxxf 12)(

none

021

0, x

V.A. @ x = 0H.A. y=2@

x = 0

y = 2

y-int. ( , )x-int. ( , )

Domain:

Asymptote(s)

2

2

2( )6

x xf xx x

You try:

Slide 4.6 - 8 Copyright © 2010 Pearson Education, Inc.

Slant, or Oblique, Asymptotes

A third type of asymptote, which is neither vertical nor horizontal, occurs when the numerator of a rational function has degree one more than the degree of the denominator. To find a slant asymptote, divide the rational function and ignore the remainder.

Ex: Find all asymptotes on the graph of the function:

2 2( )1

xf xx

Slide 4.6 - 10 Copyright © 2010 Pearson Education, Inc.

Slant, or Oblique, Asymptotes

The line y = x + 1 is a slant asymptote, or oblique asymptote of the graph of f.

f (x)

x2 2x 1

x 13

x 1

y-int. ( , )x-int. ( , )

( , )

Domain:

Asymptote(s)

12)(

2

x

xxxg

(x-2)(x+1) 0 22 0-1 0

1, x

V.A. x = 1Slant asym.

y = x

x = 1

y = x

(You Try)

Try This:

Write the equation of a rational function that has a vertical asymptote at x = 1, a horizontal asymptote at y = -2, a hole at x = -3, and an x-intercept at x = 0.

2

2

2 62 3

x xx x

Homework:

Graphing Rational Functions WS

Additional Examples

y-int. ( , )

x-int. ( ) ( )

Domain:

Asymptote(s)

(You Try)

3 2

2

6( )2

x x xf xx x

(x)(x-2)(x+3) 0 0

0 , 0-3, 0

, 2, 1x

V.A. x = -1

Slant asym.y = x+2y = x+2

(x-2)(x+1)

Hole @ x=2

y-int. ( , )

x-int. ( , ) ( , )

Domain:

Asymptote(s)

49)( 2

2

xxxf

(x-3)(x+3)

(x-2)(x+2)

490

2,2, x

V.A. x = -2 x = 2

H.A. 111y

3 0-3 0

(You Try)

y-int. ( , )x-int. ( , )

Domain:

Asymptote(s)

2

2

1)1()(

xxxf

0 -11 0

1, x

V.A. x = -1

H.A.

111

y

(If Time Permits)