Rational Functions and Models

1. Rational Functions and Models Lesson 4.6

2. Both polynomials Definition • Consider a function which is the quotient of two polynomials • Example:

3. 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

4. Example • Given • Graph on calculator • Set window for -100 < x < 100, -5 < y < 5

5. Example • Note the value for a large x • How does this relate to the leading terms?

6. 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

7. When Numerator Has Larger Degree • Try • As x gets large, r(x) also gets large • But it is asymptotic to the line

8. 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

9. 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)?

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

11. 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

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

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

14. Assignment • Lesson 4.6 • Page 319 • Exercises 1 – 41 EOO 93, 95, 99