Polynomial and Rational Functions

1. Polynomial and Rational Functions Lesson 2.3

2. Animated Cartoons • Note howmathematicsare referencedin the creation of cartoons

3. This lesson studiespolynomials Animated Cartoons • We need a wayto take a numberof pointsand makea smoothcurve

4. Polynomials • General polynomial formula • a0, a1, … ,an are constant coefficients • n is the degree of the polynomial • Standard form is for descending powers of x • anxn is said to be the “leading term” • Note that each term is a power function

5. Family of Polynomials • Constant polynomial functions • f(x) = a • Linear polynomial functions • f(x) = m x + b • Quadratic polynomial functions • f(x) = a x2 + b x + c

6. Family of Polynomials • Cubic polynomial functions • f(x) = a x3 + b x2 + c x + d • Degree 3 polynomial • Quartic polynomial functions • f(x) = a x4 + b x3 + c x2+ d x + e • Degree 4 polynomial

7. • • • Properties of Polynomial Functions • If the degree is n then it will have at most n – 1 turning points • End behavior • Even degree • Odd degree or or

8. Properties of Polynomial Functions • Even degree • Leading coefficient positive • Leading coefficient negative • Odd degree • Leading coefficient positive • Leading coefficient negative

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

10. 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

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

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

13. 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

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

15. 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

16. 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)?

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

18. Assignment • Lesson 2.3 • Page 91 • Exercises 3 – 59 EOO