vertical asymptotes and holes

1

• Vertical , Horizontal Asymptotes
• and
• Holes
• of
• Rational Functions

2
Asymptotes

• 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

3
The 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
4
Vertical 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.

5
Find the vertical asymptote
Graph Y 6/(x4)
The graph approaches but never touches the line x
4
6
Find the vertical asymptote
Graph Y x-5)/(x2)
The graph approaches but never touches the line x
2
7
How 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.

8
Find the Holes and Vertical Asymptotes
Graph Y ( x5)/((x1)(x2))
holes
VA
The graph approaches but never touches the line x
1
9
Find 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
10
Find 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
11
Find the Holes and Vertical Asymptotes
holes
VA