Rational Functions

xqxp

xf

5 values to consider

1) Domain

2) Horizontal Asymptotes

3) Vertical Asymptotes

4) Holes

5) Zeros

Domain

• In general, the domain of a rational function includes all real numbers except those x-values that make the denominator zero

Examples• Find the domain of each:

2

4

x

xf 209

1272

2

xx

xxxP

Horizontal Asymptote

• Describes the end behavior of the graph as x approaches

• If the degree of p(x) < the degree of q(x), there is a horizontal asymptote at y = 0

• If the degree of p(x) > the degree of q(x), there is NO horizontal asymptote

• If the degree of p(x) = the degree of q(x), there is a horizontal asymptote at

-or

q(x)t coefficien leading

p(x)t coefficien leadingy

Holes in the graph

• Do not occur unless there are factors in p(x) that are the same as factors in q(x)

• Occur at the places where the numerator and denominator have the same solution (Cancels out of top and bottom)

Vertical Asymptote

• Shows excluded values for which the function, f(x) is not defined for x

• The graph of f has vertical asymptotes at the solutions to the denominator

Zeros

• The x-intercepts• Occur when the numerator is equal to

zero.

Ex1) Give the domain, asymptotes, holes and zeros. Then graph the function.

2

3

x

xf

Ex2) Find the domain, asymptotes, holes and zeros. Then graph the function.

2

322

2

xx

xxxf

Ex3) Find the domain, asymptotes, holes and zeros. Then graph the function.

52

2

xx

xy

Ex4) Find the domain, asymptotes, holes and zeros. Then graph the function.

45

162

2

xx

xy

Practice• Complete WS