Lesson 2.6
Rational Functions and
Asymptotes
Graph the function:
f x  
x 1
x 1
Domain:
Range:
Increasing/Decreasing:
Line that creates a split in the graph:
Rational Functions
f x  
N ( x)
D( x )
Where N(x) and D(x) are polynomials
Discontinuities: places where the graph “skips” or “jumps”
Graph and look for discontinuities:
2 x 2  8x
f x  
x4
f x  
x 1
x2 1
f x  
x 1
x 1
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 ( x)
f x  
D( x )
; 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 →
x 1
x2 1
If n = d →
2 x3  5
x3  1
If n > d →
x3
x 8
→ y=0
leading_ coefficient _ of _ N x 
→ y
leading_ coefficient _ of _ Dx 
→ no horizontal asymptote
Example
Find all discontinuities of
x2  x  2
f x   2
x  x6
Problem Set 2.6