Rational Functions and Models

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# Rational Functions and Models - PowerPoint PPT Presentation

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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### 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
• Leading term dominates forlarge values of x for polynomial
• Leading terms dominate forthe quotient for extreme x
Example
• Given
• Graph on calculator
• Set window for -100 < x < 100, -5 < y < 5
Example
• Note the value for a large x
• How does this relate to the leading terms?
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
• Try
• As x gets large, r(x) also gets large
• But it is asymptotic to the line
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
• 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 Asymptotes
• Finding the roots ofthe denominator
• View the graphto verify
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
• Note the zeros of thefunction whengraphed
• r(x) = 0 whenx = ± 3
Summary
• The zeros of r(x) arewhere the numeratorhas zeros
• The vertical asymptotes of r(x)are where the denominator has zeros
Assignment
• Lesson 4.6
• Page 319
• Exercises 1 – 41 EOO 93, 95, 99