9.3 Rational Functions and Their Graphs

Rational Function – A function that is written as

, where P(x) and Q(x) are polynomial functions. The domain of f(x) is all real numbers except for those which Q(x) = 0.

To find points of discontinuity, find the values of x that makes the denominator 0.

)(

)()(

xQ

xPxf

Find the points of discontinuity for each rational function.

Ex 1:

Ex 2:

Ex 3:

96

12

xx

y

3

12

2

x

xy

Vertical Asymptotes and Holes

- Both are types of discontinuity

- Holes occur if there are common factors in the numerator and denominator

- Vertical asymptotes occur when a factor in the denominator does not have a common factor in the numerator.

Finding Vertical Asymptotes and Holes

Ex 4:

Ex 5:

65

12

xx

xy

1

12

x

xy

Finding Horizontal AsymptotesThree rules:  1.) If the degree of the denominator is larger than

the degree of the numerator, then the horizontal asymptote is zero.

2.) If the degree of the denominator is smaller than the degree of the numerator, then there is no horizontal asymptote.

 3.) If the degree of the denominator is equal to the

degree of the numerator, then the horizontal asymptote is the quotient of the coefficients of the highest degree terms in the rational expression.

Note: k values can change the value of an H.A in addition with these rules.

Practice ProblemsFind the vertical asymptotes, holes, and

horizontal asymptotes of each function if they exist.

53

1

x

y56

46

x

xy

124

52

xx

y3

22

2

x

xy

1

23

x

xxy

)2)(7(

5

xx

xy

1.) 2.)

3.) 4.)

5.) 6.)