Rational Expressions,
Vertical Asymptotes,
and Holes
Section 8.2
Rational Expression

It is the quotient of two polynomials.

A rational function is a function defined by a
rational expression.
Examples:
x2
y
x5
3x 2  2 x  5
y 3
x  4 x2  5x  7
Not Rational:
4x
y
x2
y
x
x2  5
Find the domain:
Graph it:
1
f ( x) 
x
Find the domain:
Graph it:
1
f ( x) 
x2
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:
Graph it using the
graphing calculator:
x 3
f ( x)  2
x  x  12
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.
x 1
f ( x)  2
x  2x  3
Find the domain and identify
vertical asymptotes & holes.
x
f ( x)  2
x 4
Find the domain and identify
vertical asymptotes & holes.
x 5
f ( x)  2
2x  x  3
Find the domain and identify
vertical asymptotes & holes.
3  2x  x2
f ( x)  2
x  x2
Homework

Book, page 495 (11-22)