Section 8.2 – Rational Functions and their GraphsObjectiveso Identify and evaluate rational functions.

o Graph a rational function, find its domain, write equations for its asymptotes, and identify any holes in the graph.

Warm – Up

What is a rational expression?

o A rational expression is the quotient of two polynomials.

o A rational function is a function defined by a rational expression.

o An example is 2 5 7

8 2

x xy

x

How do you find the domain of a rational function?o The domain is all real numbers except for where

the denominator is zero.o Example…

2

2

7 12( )

9 20

x xg x

x x

Example…

o What is the domain of 2

2

3 2( )

2 3

x xj x

x x

How do I find vertical asymptotes and holes of a rational function?o If a factor is in the denominator of a rational

function but not in the numerator of a rational function, then it is a vertical asymptote of the graph of the function. (Non-removable discontinuity)

o If a factor is in both the numerator and the denominator of a rational function, then there is a hole in the graph of the function. (Removable discontinuity)

Example of vertical asymptotes…

o Find all the vertical asymptotes of

o Set denominator equal to zero.o Solve for x.o Write equations of vertical asymptotes.

2

2( )

1

xr x

x

Example with vertical asymptotes and holes.o Identify all vertical asymptotes and holes in the

graph.

o REMEMBER: If a factor is in both the numerator and denominator, then it’s a hole. If it is only in the denominator, then it’s a vertical asymptote.

2

2

2 3

2

x xy

x x

Example…

o Identify all vertical asymptotes and holes in the graph.

3 2

2

3

2 3

x xy

x x

What are horizontal asymptotes?

o Let , where P and Q are polynomials.

o If the degree of P is less than the degree of Q, then y=0 is the equation of the horizontal asymptote.

o If the degree of P equals the degree of Q and a and b are the leading coefficients of P and Q respectively, then is the equation of the horizontal asymptote.

o If the degree of P is greater than the degree of Q, then the graph has no horizontal asymptotes.

( )P

R xQ

ay

b

Example of Horizontal Asymptotes…

o Determine horizontal asymptotes (if any).

2( )

2 3

xR x

x x

2

2

2 2 1( )

12

x xR x

x x

Exit Slip and Homework

o Exit Slip is…o p. 495 #11-13, 17-19

o Homework is…o p. 495 #14-16, 20-22