Rational Functions

Intro - Chapter 4.4

Let x = ___ to find y – intercepts

A rational function is the _______ of two polynomials

RATIO Graphs of Rational FunctionsStep 1: Write in_____________ form.FACTORED

Step 2: Find the __________.DOMAIN

It is the set of all real numbers that are not _________ of the denominator ZEROS

Step 3: Find the _______________.INTERCEPTS Let y = ___ to find x – intercepts (when the REDUCED _____________ is zero)

0 NUMERATOR

0

Example 24:

Step 4: Find _______________.ASYMPTOTES Vertical Asymptotes: Find where the ________________ zero.

Then find _____________. If _________________, then ________ is a vertical asymptote.

DENOMINATOR

Horizontal Asymptotes:

Find ____________ limx

f x

Example 24:

x c lim

x cf x

lim

x cf x

x c

If ___________ then _____ is a horizontal asymptote

limx

f x a

y a

If ___________ then _____ is a horizontal asymptote

limx

f x b

y b

If ____________ then there is an _________asymptote

limx

f x

SLANTED

Use ________________ to write the rational function , , into the form ________. ____________ will be the ___________asymptote and ____________________ is used to find the ___________ asymptote.

Step 5: Find the points (x, y) that are _______HOLES The value of x that makes the _____________ and ________________ zero.(same number of factors)

NUMERATORDENOMINATOR

Plug the x into the ___________form of f(x) the value of y .

REDUCED

Example 24:Other Asymptotes: LONG DIVISION

P x /Q R D , the quotientQ

SLANTED , the remainder/divisorR

VERTICAL

Step 6: __________ the behavior at the _____________

ANALYZE ASYMPTOTES

Step 7: ________ any “pretty” pointsPLOT Example 24:

Example 24: Find the sketch of the rational function.

22

128423

2

xxx

xxxf

4 1 3

2 1 1

x x

x x x

Domain: x

2 1 1

x - intercepts:

4 1 3x x

03x

y - intercept:

4 0 3 12

0 2 0 1 2

6

Vertical Asymptotes:

4 3

2 1

x

x x

2 1x x

02, 1x

Holes:

1x 1x

1x

4 3

2 1

x

x x

4 1 38

1 2 1 1

1, 8

1x 1x

Horizontal Asymptotes:

0y

Horizontal Asymptotes:

2

3 2

4 8 12lim

2 2x

x x

x x x

2 3

2 3

4 8 12

lim2 1 2

1x

x x x

x x x

Pretty Points:2x

2f 12

112

3x 3f

486

8

00

1

2

3 2

4 8 12lim

2 2x

x x

x x x

2 3

2 3

4 8 12

lim2 1 2

1x

x x x

x x x

00

1

Vertical Asymptotes:

2

4 3lim

2 1x

x

x x

2

4 3lim

2 1x

x

x x

1

4 3lim

2 1x

x

x x

1

4 3lim

2 1x

x

x x

Asymptote Analysis

Example 25: Find the sketch of the rational function.

2 2 3

2

x xf x

x

1 3

2

x x

x

Domain: 2x x - intercepts:

1 3 0x x 3,1x

y - intercept:

0 1 0 3 3

0 2 2

Vertical Asymptotes:

2 0x 2x

Other Asymptotes:

4y x

Holes: NONE

2 1 2 3

2 8

1 4 5

Pretty Points:

3

3 12

x

f

7

7 12

x

f

Horizontal Asymptotes:

2

2

2 31

lim1 2x

x x

x x