Rational Expression It is the quotient of two polynomials.
A rational function is a function defined by a rational expression.
Examples:
2
5
xy
x
2
3 2
3 2 5
4 5 7
x xy
x x x
Not Rational:
4
2
x
yx
2 5
xy
x
Find the domain:1
( )f xx
Graph it:
Find the domain:1
( )f xx
Graph it:
Find the domain:1
( )2
f xx
Graph it:
Find the domain:1
( )2
f xx
Graph it:
Vertical Asymptote
If x – a is a factor of the denominator of a rational function but not a factor of the numerator, then x = a is a vertical asymptote of the graph of the function.
Find the domain: 2
3( )
12
xf x
x x
Graph it using the graphing calculator. What do you see?
Find the domain: 2
3( )
12
xf x
x x
Graph it using the graphing calculator. What do you see?
Hole (in the graph)
If x – b is a factor of both the numerator and denominator of a rational function, then there is a hole in the graph of the function where x = b, unless x = b is a vertical asymptote.
The exact point of the hole can be found by plugging b into the function after it has been simplified.
Find the domain and identify vertical asymptotes & holes.
2
1( )
2 3
xf x
x x
Find the domain and identify vertical asymptotes & holes.
2
1( )
2 3
xf x
x x
Find the domain and identify vertical asymptotes & holes.
2( )
4
xf x
x
Find the domain and identify vertical asymptotes & holes.
2( )
4
xf x
x
Find the domain and identify vertical asymptotes & holes.
2
5( )
2 3
xf x
x x
Find the domain and identify vertical asymptotes & holes.
2
5( )
2 3
xf x
x x
Find the domain and identify vertical asymptotes & holes.
2
2
3 2( )
2
x xf x
x x
Find the domain and identify vertical asymptotes & holes.
2
2
3 2( )
2
x xf x
x x
Horizontal Asymptotes & Graphing
Horizontal Asymptotes Degree of numerator = Degree of denominator