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 Terms

asymptote 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.

When graphing rational functions

 State domain  Find Vertical Asymptote(s)  Find Point of Discontinuity in the graph (HOLES)  Find Horizontal Asymptote  Find the y-intercepts & x-intercepts  Sketch

 

3

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

 

3

    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

 

3

    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

 

3

    Find y-intercept  Substitute zero for x and find the value of the function

3

    

3 5

Sketch the graph

Vertical Asymptotes Horizontal Asymptote y-intercept Plot a few points x -10 -5.5

-4.5

.5

1.5

10 .05

y .92

-1.1

-1.1

.92

.02

 

3

  

4 

x 2 x 2

 

36

 State the domain  Find Vertical Asymptote(s)  Find Point of Discontinuity in the graph          Domain: ( ∞,-6) U (-6,-1) U (-1,∞) Vertical Asymptote(s) x=-1 Find Point of Discontinuity at -6



x 2 x 2

 

36

 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 = the ratio of the lead coefficients.

y = 1

 Find y-intercept 

x 2 x 2

 

36

 Substitute zero for x and find the value of the function 

0 2

 

6

 

36 6

 

6

Sketch the graph

Vertical Asymptote Point of Discontinuity Horizontal Asymptote y-intercept Plot a few points x -10 -2 -.5

1 10 1.8

y 8 -13 -2.5

.4



x 2 x 2

 

36

6 

x 2



16

 State the Domain  Find Vertical Asymptote(s)  Find Point of Discontinuity in the graph        Domain: ( ∞,-4) U (-4,∞) Vertical Asymptote(s) NONE Find Point of Discontinuity at -4



x 2



16

 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



x 2



16

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

0



4

 

16 4

 

4

Sketch the graph

Point of Discontinuity y-intercept Plot a few points -5 x -3 4 -9 -7 0 y 

x 2



16

Homework – day 1

 Rational Functions Worksheet 1, part A & B  Graphing Rational Function Worksheet #1-3, 5

Graphing Rational Expressions Oblique Asymptotes

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



x

2 

x

9

x

 3  20  3

x

3  2

x x

2 2   1 3

x

 4

Homework – day 2

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