Rational Functions

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Rational Functions

• Sec. 2.7a
• Homework p. 246 1-29 odd

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Definition Rational Functions
Let f and g be polynomial functions with g (x )
0.
Then the function given by
is a rational function.

• The domain of a rational function is all reals
  except the zeros
• of its denominator.

• Every rational function is continuous on its
  domain.

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Finding the Domain of aRational Function
Find the domain of the given function and use
limits to describe its behavior at value(s) of x
not in its domain.
Now, sketch the graph
Domain?
What does the function approach as x approaches 2?
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A Reminder about Asymptotes
The line y b is a horizontal asymptote if
or
The line x a is a vertical asymptote if
or
Does this make sense with our previous example???
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Now, Lets Analyze the Reciprocal Function
Domain
Range
Continuous on D
Continuity
Inc/Dec
Dec. on
H.A.
Origin (odd function)
Symmetry
V.A.
Boundedness
Unbounded
End Behavior
Local Extrema
None
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Transforming theReciprocal Function
Describe how the graph of the given function can
be obtained by transforming the graph of the
reciprocal function. Identify the horizontal and
vertical asymptotes and use limits to describe
the corresponding behavior. Sketch the graph of
the function.
Translate f (x) left 3 units, then vertically
stretch by 2
H.A
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Transforming theReciprocal Function
Describe how the graph of the given function can
be obtained by transforming the graph of the
reciprocal function. Identify the horizontal and
vertical asymptotes and use limits to describe
the corresponding behavior. Sketch the graph of
the function.
Translate f (x) left 3 units, then vertically
stretch by 2
V.A
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Transforming theReciprocal Function
Lets do the same thing with a new function
Begin with polynomial division
Translate f (x) right 2, reflect across x-axis,
translate up 3
H.A
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Transforming theReciprocal Function
Lets do the same thing with a new function
Begin with polynomial division
Translate f (x) right 2, reflect across x-axis,
translate up 3
V.A
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NowLimits and Asymptotes of Rational Functions
Find the horizontal and vertical asymptotes of
the given function. Use limits to describe the
corresponding behavior of the function.
First, lets solve this algebraically
Whats the Domain?
? So there are no vertical asymptotes!!! Why
not???
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Find the horizontal and vertical asymptotes of
the given function. Use limits to describe the
corresponding behavior of the function.
First, lets solve this algebraically
Verify graphically?
To find horizontal asymptotes, first use
polynomial division
As x becomes very large or very small, this last
term approaches zero Why?
So, the horizontal asymptote is the line y 1
Using limit notation
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Graphs of Rational Functions
The graphs of
have the following characteristics
1. End Behavior Asymptote
If n lt m, the end behavior asymptote is the
horizontal asymptote of y 0.
If n m, the end behavior asymptote is the
horizontal asymptote .
If n gt m, the end behavior asymptote is the
quotient polynomial function y q(x), where f
(x) g(x)q(x) r(x). There is no horizontal
asymptote.
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Graphs of Rational Functions
The graphs of
have the following characteristics
2. Vertical Asymptotes
These occur at the zeros of the denominator,
provided that the zeros are not also zeros of the
numerator of equal or greater multiplicity.
3. x-intercepts
These occur at the zeros of the numerator, which
are not also zeros of the denominator.
4. y-intercept
This is the value of f (0), if defined.
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Find the asymptotes and intercepts of the given
function, and then graph the function.
Degree of Numerator gt Degree of Denominator ?
long division!
The quotient q(x) x is our slant asymptote
Factor the denominator
Vertical Asymptotes are at x 3 and x 3
x-intercept 0, y-intercept f (0) 0
Verify all of this graphically???
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Guided Practice
Find the horizontal and vertical asymptotes of
the given function. Use limits to describe the
corresponding behavior.
No vertical asymptotes
H.A. y 3
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Guided Practice
Find the horizontal and vertical asymptotes of
the given function. Use limits to describe the
corresponding behavior.
H.A. y 0
V.A. x 0, x 3
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Whiteboard Practice
Find the asymptotes and intercepts of the given
function, then graph the function.
Intercepts (0, 2/3), (2, 0)
Asymptotes x 3, x 1, y 0
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Whiteboard Practice
Find the asymptotes and intercepts of the given
function, then graph the function.
No Intercepts
Asymptotes x 2, x 0, x 2, y 0
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Whiteboard Practice
Find the asymptotes and intercepts of the given
function, then graph the function.
Intercepts (0, 3), (1.840, 0), (2.174, 0)
Asymptotes x 2, x 2, y 3
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Whiteboard Practice
Find the asymptotes and intercepts of the given
function, then graph the function.
Intercepts (0, 7/3), (1.541, 0), (4.541, 0)
Asymptotes x 3, y x 6