Section 4'1Rational Functions and Asymptotes

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• Section 4.1 Rational Functions and
  Asymptotes

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(No Transcript)
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Reciprocal function y1/x
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Know the following basic facts about rational
functions.

• Definition of rational function
• Domain is set of reals numbers which DO NOT make
  the denominator zero
• Find vertical asymptotes
• Find horizontal asymptotes

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Definition of Rational Function

• A function of the form
• D(x) ? 0, where N(x) and D(x) are polynomials,
  is called a rational function.

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Domain

• The domain of a rational function is the set of
  all real numbers except those which make the
  denominator zero.

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• X a is vertical asymptote if a is such a value
  the D(x) 0
• The line y b is a horizontal asymptote of the
  graph of f if

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Where N(x) and D(x) have no common factors
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Horizontal asymptotes

• x 0
• x
• No horizontal asymptotes

• 1. If n lt m, then the x-axis is a horizontal
  asymptote.
• 2. If n m,
• 3. If n gt m,

• y 0
• y leading coefficient of n(x)
• Leading coefficient of d(x)

• 1. If n lt m, then the x-axis is a horizontal
  asymptote.
• 2. If n m,
• 3. If n gt m,

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• Domain
• All reals except 1
• Horizontal asymptote
• y 0 since degree of numerator is smaller than
  degree of denominator

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• Domain
• Zero denominator are 1 and -1
• Vertical asymptotes are 1 and -1
• Horizontal asymptotes is y 3

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• Domain
• Zero denominator is 1
• Vertical asymptotes are 1
• Horizontal asymptotes none because degree of
  numerator is larger than degree of denominator

X ? 1
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Find a function
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Find a function
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4.2 Graphs of Rational functions
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Guidelines for analyzing graphs of rational
functions

• Find and plot y-intercepts (if any) by evaluating
  f(0)
• Find the zeros of numerator (if any) by solving
  N(x) 0. These are the x-intercepts
• Find the zeros of the denominator (if any) by
  solving D(x) 0. Then sketch the vertical
  asymptotes

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• 4. Find and sketch the horizontal asymptotes (if
  any) using the rule for horizontal asymptotes

• 1. If n lt m, then the x-axis is a horizontal
  asymptote.
• 2. If n m,
• 3. If n gt m,

• y 0
• y leading coefficient of n(x)
• Leading coefficient of d(x)
• No horizontal asymptotes

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5. Test for symmetry

• Remember the graph of f(x) 1/x is symmetric
  with respect to the origin.

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Slant asymptotes

• A rational function whose denominator is of
  degree 1 or greater and if the degree of
  numerator is exactly one more than degree of the
  denominator. The graph of the function has a
  slant asymptotes

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Find slant asymptotes

• To find the equation of slant asymptote, use long
  division to divide numerator by the denominator.
  Disregard and remainder.

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Find slant asymptote
Slant asymptote y x - 2
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• Domain
• Zero denominator is 1
• Vertical asymptotes are 1
• Horizontal asymptotes none because degree of
  numerator is larger than degree of denominator

X ? 1
Slant asymptote ?
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Review process

• y-intercepts
• X-intercepts
• Vertical asymptotes
• Horizontal asymptote or Slant asymptote
• Additional points

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Examples