Lesson 2.6Rational Functions and

Asymptotes

Graph the function:

Domain:

Range:

Increasing/Decreasing:

Line that creates a split in the graph:

1

1

x

xxf

Rational Functions

Where N(x) and D(x) are polynomials

Discontinuities: places where the graph “skips” or “jumps”

Graph and look for discontinuities:

)(

)(

xD

xNxf

1

12

x

xxf

4

82 2

x

xxxf

1

1

x

xxf

Discontinuities

Hole: can be factored out

Jumps: cannot be factored out

Asymptotes (jumps)

A horizontal or vertical line through which a graph is undefined

Cannot be factored out

Finding location of asymptotes:

Given ; n = degree of N(x), d = degree of D(x)

Vertical asymptote(s): At zeros of D(x); Write “x = #”

Horizontal asymptote(s):

If n < d → → y = 0

If n = d → →

If n > d → → no horizontal asymptote

)(

)(

xD

xNxf

1

12

x

x

1

523

3

x

x

8

3

xx

xDoftcoefficienleading

xNoftcoefficienleadingy

___

___

Example

Find all discontinuities of 6

22

2

xx

xxxf

Problem Set 2.6