Sullivan Algebra and Trigonometry: Section 4.3 Rational Functions I

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Sullivan Algebra and Trigonometry Section
4.3Rational Functions I

• Objectives
• Find the Domain of a Rational Function
• Determine the Vertical Asymptotes of a Rational
  Function
• Determine the Horizontal or Oblique Asymptotes of
  a Rational Function

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A rational function is a function of the form
where p and q are polynomial functions and q is
not the zero polynomial. The domain consists of
all real numbers except those for which the
denominator q is 0.
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Find the domain of the following rational
functions.
All real numbers x except 6 and 2.
All real numbers x except 4 and 4.
All Real Numbers
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Recall that the graph of is
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Graph the function
using transformations
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In the previous example, there was a vertical
asymptote at x 2 and a horizontal asymptote at
y 1.
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Examples of Horizontal Asymptotes
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Examples of Vertical Asymptotes
x c
y
x c
y
x
x
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If an asymptote is neither horizontal nor
vertical it is called oblique.
Theorem Locating Vertical Asymptotes
A rational function R(x) p(x) / q(x), in lowest
terms, will have a vertical asymptote x r, if x
r is a factor of the denominator q.
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Example Find the vertical asymptotes, if any, of
the graph of each rational function.
Vertical asymptotes x 1 and x 1
No vertical asymptotes
Vertical asymptote x 4
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Consider the rational function
in which the degree of the numerator is n and the
degree of the denominator is m.
1. If n lt m, then y 0 is a horizontal
asymptote of the graph of R.
2. If n m, then y an / bm is a horizontal
asymptote of the graph of R.
3. If n m 1, then y ax b is an oblique
asymptote of the graph of R. Found using long
division.
4. If n gt m 1, the graph of R has neither a
horizontal nor oblique asymptote. End behavior
found using long division.
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Example Find the horizontal or oblique
asymptotes, if any, of the graph of
Horizontal asymptote y 0
Horizontal asymptote y 2/3
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Oblique asymptote y x 6