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Rational Functions and Models. Lesson 4.6. Both polynomials. Definition. Consider a function which is the quotient of two polynomials Example: . Long Run Behavior. Given The long run (end) behavior is determined by the quotient of the leading terms

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definition

Both polynomials

Definition
  • Consider a function which is the quotient of two polynomials
  • Example:
long run behavior
Long Run Behavior
  • Given
  • The long run (end) behavior is determined by the quotient of the leading terms
    • Leading term dominates forlarge values of x for polynomial
    • Leading terms dominate forthe quotient for extreme x
example
Example
  • Given
  • Graph on calculator
    • Set window for -100 < x < 100, -5 < y < 5
example1
Example
  • Note the value for a large x
  • How does this relate to the leading terms?
try this one
Try This One
  • Consider
  • Which terms dominate as x gets large
  • What happens to as x gets large?
  • Note:
    • Degree of denominator > degree numerator
    • Previous example they were equal
when numerator has larger degree
When Numerator Has Larger Degree
  • Try
  • As x gets large, r(x) also gets large
  • But it is asymptotic to the line
summarize
Summarize

Given a rational function with leading terms

  • When m = n
    • Horizontal asymptote at
  • When m > n
    • Horizontal asymptote at 0
  • When n – m = 1
    • Diagonal asymptote
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 asymptotes1
Vertical Asymptotes
  • Finding the roots ofthe denominator
  • View the graphto verify
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 functions1
Zeros of Rational Functions
  • Note the zeros of thefunction whengraphed
  • r(x) = 0 whenx = ± 3
summary
Summary
  • The zeros of r(x) arewhere the numeratorhas zeros
  • The vertical asymptotes of r(x)are where the denominator has zeros
assignment
Assignment
  • Lesson 4.6
  • Page 319
  • Exercises 1 – 41 EOO 93, 95, 99