Title: vertical asymptotes and holes
1- Vertical , Horizontal Asymptotes
- and
- Holes
- of
- Rational Functions
2Asymptotes
- Rational function has a variable in the
denominator - vertical asymptotes the vertical line x
____ that the graph of a function
approaches but does not cross - any values for x, that make the denominator
zero are undefined, so - find the zeros of the denominator and you
find the vertical asymptotes
3The graph of looks
like this
If you choose x values close to 0, the graph gets
close to the asymptote, but never touches it.
Since x ? 0, the graph approaches 0 but never
crosses or touches 0. A vertical line drawn at x
0 is called a vertical asymptote. It is a
sketching aid to figure out the graph of a
rational function. There will be a vertical
asymptote at x values that make the denominator
0
4Vertical Asympotes
- A vertical asymptote comes from the denominator.
It exists at the value(s) that will make the
denominator zero. - For example
- f(x) 9
- x 2
- For example
- f(x) x2 9
- 6
- Because there are no variables in the
denominator, there are no vertical asymptotes.
5Find the vertical asymptote
Graph Y 6/(x4)
The graph approaches but never touches the line x
4
6Find the vertical asymptote
Graph Y x-5)/(x2)
The graph approaches but never touches the line x
2
7How to find the points of discontinuity (holes)
- When simplifying the function, if you cancel a
polynomial from the numerator and denominator,
then you have a hole! - a single point at which the function has no
value. - Set the cancelled factor equal to zero and solve.
8Find the Holes and Vertical Asymptotes
Graph Y ( x5)/((x1)(x2))
holes
VA
The graph approaches but never touches the line x
1
9Find the Holes and Vertical Asymptotes
Graph Y ( x5)/((x1)(x2))
holes
VA
You cant tell from the graph but there is also a
hole in the graph where x 2, check the table
10Find the Holes and Vertical Asymptotes
Graph Y ( x5)/((x1)(x2))
holes
VA
You cant tell from the graph but there is also a
hole in the graph where x 2, check the table
11Find the Holes and Vertical Asymptotes
holes
VA