Rational Expressions, Vertical Asymptotes, and Holes

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Rational Expressions, Vertical Asymptotes, and
Holes
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Rational Expression

• It is the quotient of two polynomials.
• A rational function is a function defined by a
  rational expression.

Examples
Not Rational
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Find the domain
Graph it
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Find the domain
Graph it
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Find the domain
Graph it
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Find the domain
Graph it
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Vertical Asymptote

• If x a is a factor of the denominator of a
  rational function but not a factor of the
  numerator, then x a is a vertical asymptote of
  the graph of the function.

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Find the domain
Graph it using the graphing calculator. What do
you see?
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Find the domain
Graph it using the graphing calculator. What do
you see?
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Hole (in the graph)

• If x b is a factor of both the numerator and
  denominator of a rational function, then there is
  a hole in the graph of the function where x
  b, unless x b is a vertical asymptote.
• The exact point of the hole can be found by
  plugging b into the function after it has been
  simplified.

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Find the domain and identify vertical asymptotes
holes.
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Find the domain and identify vertical asymptotes
holes.
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Find the domain and identify vertical asymptotes
holes.
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Find the domain and identify vertical asymptotes
holes.
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Find the domain and identify vertical asymptotes
holes.
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Find the domain and identify vertical asymptotes
holes.
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Find the domain and identify vertical asymptotes
holes.
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Find the domain and identify vertical asymptotes
holes.
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Horizontal Asymptotes Graphing
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Horizontal Asymptotes

• Degree of numerator Degree of denominator
• Degree of numerator lt Degree of denominator
• Degree of numerator gt Degree of denominator

Horizontal Asymptote
Horizontal Asymptote
Horizontal Asymptote
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Find all asymptotes holes then graph
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Find all asymptotes holes then graph
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Find all asymptotes holes then graph
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Find all asymptotes holes then graph
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Find all asymptotes holes then graph
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Find all asymptotes holes then graph
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Find all asymptotes holes then graph
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Find all asymptotes holes then graph
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Find all asymptotes holes then graph
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Find all asymptotes holes then graph
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Find all asymptotes holes then graph
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Find all asymptotes holes then graph
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Find all asymptotes holes then graph
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Find all asymptotes holes then graph