Name: __________________________________________________________
Rational Functions: Part 1: Domain and Vertical Asymptotes (Points of Discontinuity)
A rational function is a function of the form
where
and
The domain of a rational function
includes All Real Numbers EXCEPT
x-values that make the denominator
equal to 0. “Restricted” x-values)
This means the graph does not exist
at these points.
Here is the graph of
.
What would make the denominator of
Vertical Asymptotes If
and
These create what are called points
of discontinuity on the graph. There
are two types: “vertical asymptotes”
and “holes.”
equal 0?
and
have no common factors (i.e. nothing cancels), then
the graph of
has vertical asymptotes at the zeros (solutions) of
or down and run along the asymptote.
Holes: If
are polynomials.
and
and
. The curve will either turn up
do have common factors (i.e. something cancels), then the graph
of
has a hole at that particular zero (solution) of
. The graph will be identical to the graph of the
simplified expression with the exception of this blank spot in the graph. The curve will stop at the hole
and pick up again on the other side of the hole.
Does the point of discontinuity for
above create a hole or an asymptote?
Identify the vertical asymptotes of the following graphs. Draw the vertical asymptotes with a dashed line.
1
2
3
x = ____________
x = ____________
Match the graphs with the rational functions.
#__________
x = ___________ and x = ___________
#__________
#__________
State the domain and find the vertical asymptotes and holes for the following rational functions.
4.
Domain: __________________________________________________________
Vertical Asymptotes: ______________________, Hole:______________
5.
Domain: __________________________________________________________
Vertical Asymptotes: ______________________, Hole:______________
6.
Domain: __________________________________________________________
Vertical Asymptotes: ______________________, Hole:______________
7.
Domain: __________________________________________________________
Vertical Asymptotes: ______________________, Hole:______________
8.
Domain: __________________________________________________________
Vertical Asymptotes: ______________________, Hole:______________
9.
Domain: __________________________________________________________
Vertical Asymptotes: ______________________, Hole:______________
10. Factor:
__________________________________________________