Rational Expressions

GRAPHING

Objectives Graph a rational function, find its

domain, write equations for its asymptotes, and identify any holes (point discontinuity) in its graph.

Glossary Termsasymptote

horizontal asymptote

point discontinuity

rational function

vertical asymptote

y-intercept

Rational Function

An equation in the form

Where p(x) and q(x) are polynomial functions and q(x)0.

p x

f xq x

When graphing rational functions

State domainFind Vertical Asymptote(s)Find Point of Discontinuity in the graph

(HOLES) Find Horizontal Asymptote Find the y-intercepts & x-intercepts Sketch

3f x

x 1 x 5

State DomainFind Vertical Asymptote(s)Find Point of Discontinuity in the graph

Domain: (-∞,-5) U (-5,1) U (1,∞)Vertical Asymptote(s) x=1 & x=-5Find Point of Discontinuity in the graph - none

3f x

x 1 x 5

Find Horizontal Asymptote

Compare the degree of the numerator to the degree of the denominator

degree of the numerator < degree of the denominator

H.A. y = 0

3f x

x 1 x 5

Find Horizontal Asymptote

Compare the degree of the numerator to the degree of the denominator

degree of the numerator < degree of the denominator

H.A. y = 0

degree of the numerator = degree of the denominator

H.A. y = the ratio of the lead coefficients.

degree of the numerator > degree of the denominator

none

3f x

x 1 x 5

Find y-intercept

Substitute zero for x and find the value of the function

3

f 00 1 0 5

3f 0

5

Sketch the graphVertical Asymptotes

Horizontal Asymptote

y-intercept

3f x

x 1 x 5

Plot a few points

x y

-10 .05

-5.5 .92

-4.5 -1.1

.5 -1.1

1.5 .92

10 .02

2

2

x 36f x

x 7x 6

State the domainFind Vertical Asymptote(s)Find Point of Discontinuity in the graph

Domain: (-∞,-6) U (-6,-1) U (-1,∞)Vertical Asymptote(s) x=-1Find Point of Discontinuity at -6

4

x 6 x 6

f xx 6 x 1

Find Horizontal Asymptote

Compare the degree of the numerator to the degree of the denominator

2

2

x 36f x

x 7x 6

degree of the numerator = degree of the denominator

H.A. y = the ratio of the lead coefficients.

y = 1

Find y-intercept Substitute zero for x and find the value of

the function

2

0 36f 0

0 7 0 6

36f 0

6

2

2

x 36f x

x 7x 6

f 0 6

Sketch the graphVertical Asymptote

Point of Discontinuity

Horizontal Asymptote

y-intercept

Plot a few points

x y

-10 1.8

-2 8

-.5 -13

1 -2.5

10 .4

2

2

x 36f x

x 7x 6

2x 16

f xx 4

State the Domain

Find Vertical Asymptote(s)Find Point of Discontinuity in the graph

Domain: (-∞,-4) U (-4,∞)Vertical Asymptote(s) NONEFind Point of Discontinuity at -4

6

x 4 x 4f x

x 4

Find Horizontal Asymptote

Compare the degree of the numerator to the degree of the denominator

degree of the numerator > degree of the denominator

H.A. there is no horizontal asymptote

2x 16

f xx 4

Find y-intercept Substitute zero for x and find the value of

the function

0 16f 0

0 4

16f 0

4

2x 16

f xx 4

f 0 4

Sketch the graphPoint of Discontinuity

y-intercept

Plot a few points

x y

-5 -9

-3 -7

4 0

2x 16

f xx 4

Homework – day 1

Rational Functions Worksheet 1, part A & B

Graphing Rational Function Worksheet #1-3, 5

Graphing Rational ExpressionsOblique Asymptotes

To find Oblique (Slant) Asymptotes you will need to divide.

2 9 20

3x x

f xx

3 2

2

3 2 3 41

x x xg x

x

Homework – day 2

Page 405 21-43 odd