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Short Run Behavior of Rational Functions. Lesson 9.5. Zeros of Rational Functions. We know that So we look for the zeros of P(x), the numerator Consider What are the roots of the numerator? Graph the function to double check. Zeros of Rational Functions.

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zeros of rational functions
Zeros of Rational Functions
  • We know that
  • So we look for the zeros of P(x), the numerator
  • Consider
    • What are the roots of the numerator?
    • Graph the function to double check
zeros of rational functions3
Zeros of Rational Functions
  • Note the zeros of thefunction whengraphed
  • r(x) = 0 whenx = ± 3
vertical asymptotes
Vertical Asymptotes
  • A vertical asymptote happens when the function R(x) is not defined
    • This happens when thedenominator is zero
  • Thus we look for the roots of the denominator
  • Where does this happen for r(x)?
vertical asymptotes5
Vertical Asymptotes
  • Finding the roots ofthe denominator
  • View the graphto verify
  • The zeros of r(x) arewhere the numeratorhas zeros
  • The vertical asymptotes of r(x)are where the denominator has zeros
drawing the graph of a rational function
Drawing the Graph of a Rational Function
  • Check the long run behavior
    • Based on leading terms
    • Asymptotic to 0, to a/b, or to y=(a/b)x
  • Determine zeros of the numerator
    • These will be the zeros of the function
  • Determine the zeros of the denominator
    • This gives the vertical asymptotes
  • Consider
given the graph find the function
Given the Graph, Find the Function
  • Consider the graphgiven with tic marks = 1
  • What are the zeros of the function?
  • What vertical asymptotes exist?
  • What horizontal asymptotes exist?
  • Now … what is the rational function?
look for the hole
Look for the Hole
  • What happens when both the numerator and denominator are 0 at the same place?
  • Consider
  • We end up with which is indeterminate
    • Thus the function has a point for which it is not defined … a “hole”
look for the hole10
Look for the Hole
  • Note that when graphed and traced at x = -2, the calculator shows no value
  • Note also, that it does not display a gap in the line
  • Lesson 9.5
  • Page 420
  • Exercises 1 – 41 EOO