9.2 Vertical Asymptotes and Holes
November 19, 2015
Unit 5 - Graphing Rational Functions
In this Unit we will:
1) Identify Rational Functions
2) Find the domain of Rational Functions
3) Identify types of discontinuity
4) Find discontinuity in rational functions
5) Sketch the graphs of rational functions
*Discontinuities is a skill that is highly
necessary for success in Pre-Calculus and
Calculus.
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9.2 Vertical Asymptotes and Holes
November 19, 2015
Recall....
Rational Numbers are numbers that can be
written in the form of a fraction.
Non-Examples
Examples
A Rational Function can be written in the form,
R(x) =
N(x)
D(x)
Where N(x) and D(x) are both polynomials.
* D(x) ≠ 0 *
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9.2 Vertical Asymptotes and Holes
November 19, 2015
We have already worked with many rational functions!!
Domain Connection
*We know how to find
the domain of rational
functions!
Which of our parent
graphs is a rational
function?
f(x)=1/x
g(x) = x+4
x2+7x+12
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9.2 Vertical Asymptotes and Holes
November 19, 2015
Domain Connection
g(x) = x+4
x2+7x+12
Domain:
(-∞, -4) U (-4,-3) U (-3,∞)
Because the denominator of the rational
f(x)=1/x
function can never equal zero, any x-values
that cause the denominator to be zero
cannot be included on the graph
This should make sense using what we found for
the domain (think about the restrictions on the
domain!) At these places the function is
considered discontinuous.
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9.2 Vertical Asymptotes and Holes
November 19, 2015
Some Rational Functions are discontinuous functions.
Examples of discontinuous function:
Example 1
1
g(x) = x-2
Example 2
1
f(x) = x2-4
NonExample
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h(x) =
4
x2+1
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9.2 Vertical Asymptotes and Holes
November 19, 2015
There are two types of discontinuity that
we will look for in Rational Functions
1) Line Discontinuity
• Vertical Asymptotes (VA)
HA at y=3
VA at x=2
• Horizontal Asymptotes (HA)
• Slant Asymptotes (SA)
SA at y=x-2
2) Point Discontinuity
• Holes
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9.2 Vertical Asymptotes and Holes
November 19, 2015
Vertical Asymptotes
The graph of a rational function will have a
vertical asymptote in the location of any
value of x which causes the denominator
to equal zero and it is NOT in a factor
that can be cancelled.
Examples
f(x) =
f(x) =
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Vertical Asymptote
3
x-5
7
x2-x-6
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9.2 Vertical Asymptotes and Holes
November 19, 2015
Holes
The graph of a rational function will have a hole in the
location of any x value that causes the denominator to
equal zero AND it is a factor which can be cancelled.
To find the hole in the graph (x,y):
1) Set the canceled factor equal to zero
2) Solve for x
3) Plug the answer into the SIMPLIFIED EQUATION
to get y
4) (x,y) is the Hole
5) Use the SIMPLIFIED EQUATION for the rest of
the problem.
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9.2 Vertical Asymptotes and Holes
November 19, 2015
Holes
To find the hole in the graph
(x,y):
1) Set the canceled factor equal
to zero
2) Solve for x
3) Plug the answer into the
SIMPLIFIED EQUATION to
get y
4) (x,y) is the Hole
5) Use the SIMPLIFIED
EQUATION for the rest of the
problem.
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9.2 Vertical Asymptotes and Holes
November 19, 2015
Finding Holes, Vertical Asymptotes and Domain
Example 1:
Hole(s):________________________________
V.A.s:__________________________________
Domain:________________________________
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9.2 Vertical Asymptotes and Holes
November 19, 2015
Finding Holes, Vertical Asymptotes and Domain
Example 2:
Hole(s):________________________________
V.A.s:__________________________________
Domain:________________________________
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9.2 Vertical Asymptotes and Holes
November 19, 2015
Finding Holes, Vertical Asymptotes and Domain
Example 3:
Hole(s):________________________________
V.A.s:__________________________________
Domain:________________________________
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9.2 Vertical Asymptotes and Holes
November 19, 2015
Finding Holes, Vertical Asymptotes and Domain
Example 4:
Hole(s):________________________________
V.A.s:__________________________________
Domain:________________________________
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9.2 Vertical Asymptotes and Holes
November 19, 2015
Finding Holes, Vertical Asymptotes and Domain
Example 5:
Hole(s):________________________________
V.A.s:__________________________________
Domain:________________________________
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9.2 Vertical Asymptotes and Holes
November 19, 2015
Finding Holes, Vertical Asymptotes and Domain
Example 6:
Hole(s):________________________________
V.A.s:__________________________________
Domain:________________________________
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