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Short Run Behavior of Rational Functions

Learn about finding zeros, vertical asymptotes, graphing, and long-run behavior of rational functions. Study roots of numerators and denominators, identify vertical asymptotes, and locate holes in functions. Complete exercises for practice.

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Short Run Behavior of Rational Functions

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  1. Short Run Behavior of Rational Functions Lesson 9.5

  2. 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

  3. Zeros of Rational Functions • Note the zeros of thefunction whengraphed • r(x) = 0 whenx = ± 3

  4. 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)?

  5. Vertical Asymptotes • Finding the roots ofthe denominator • View the graphto verify

  6. Summary • The zeros of r(x) arewhere the numeratorhas zeros • The vertical asymptotes of r(x)are where the denominator has zeros

  7. 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

  8. 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?

  9. 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”

  10. 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

  11. Assignment • Lesson 9.5 • Page 420 • Exercises 1 – 41 EOO

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