1. Chapter 3 – Polynomial and Rational Functions 3.2 - Polynomial Functions and Their Graphs

2. Definition 3.2 - Polynomial Functions and Their Graphs

3. Example Find the degree, leading coefficient, leading term, coefficients, and constant terms of the below polynomial function. 3.2 - Polynomial Functions and Their Graphs

4. Definition The graph of a polynomial function is continuous meaning the graph has no breaks or holes. The graph is also smooth and has no corners or cusps (sharp points). 3.2 - Polynomial Functions and Their Graphs

5. Example Determine which of the graphs below are continuous. 3.2 - Polynomial Functions and Their Graphs

6. Definition The end behavior of a polynomial is a description of what happens as x becomes large in the positive or negative direction. We use the following notation: x →∞ means “x becomes large in the positive direction” x →-∞ means “x becomes large in the negitive direction” 3.2 - Polynomial Functions and Their Graphs

7. End Behavior The end behavior of a polynomial is determined by the degree n and the sign of the leading coefficient a n. 3.2 - Polynomial Functions and Their Graphs

8. End Behavior The end behavior of a polynomial is determined by the degree n and the sign of the leading coefficient a n. 3.2 - Polynomial Functions and Their Graphs

9. Guidelines for Graphing 3.2 - Polynomial Functions and Their Graphs

10. Example – pg. 244 #19 Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits proper end behavior. 3.2 - Polynomial Functions and Their Graphs

11. Definition If c is a zero of a polynomial P, and the corresponding factor x – c occurs exactly m times in the factorization of P, then we say that c is a zero of multiplicity m. Multiplicity helps determine the shape of a graph. 3.2 - Polynomial Functions and Their Graphs

12. Shape of a Graph near Zero 3.2 - Polynomial Functions and Their Graphs

13. Definitions If the point (a, f (a)) is the highest point on the graph of f within some viewing rectangle, then f (a) is a local maximum. If the point (a, f (a)) is the lowest point on the graph of f within some viewing rectangle, then f (a) is a local minimum. The local minimum and maximum on a graph of a function are called its local extrema. 3.2 - Polynomial Functions and Their Graphs

14. Local Extrema 3.2 - Polynomial Functions and Their Graphs

15. Local Extrema of Polynomials If P(x) = a n x n + a n-1 x n-1 + … + a 1 x + a 0 is a polynomial of degree n, then the graph of P has at most n – 1 local extrema. 3.2 - Polynomial Functions and Their Graphs

16. Example Graph the polynomial below and identify all extrema, solutions, multiplicity, and end behavior. 3.2 - Polynomial Functions and Their Graphs