Chapter 3 – Polynomial and Rational Functions Polynomial Functions and Their Graphs

Definition Polynomial Functions and Their Graphs

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

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) Polynomial Functions and Their Graphs

Example Determine which of the graphs below are continuous Polynomial Functions and Their Graphs

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” Polynomial Functions and Their Graphs

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

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

Guidelines for Graphing Polynomial Functions and Their Graphs

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

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 Polynomial Functions and Their Graphs

Shape of a Graph near Zero Polynomial Functions and Their Graphs

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 Polynomial Functions and Their Graphs

Local Extrema Polynomial Functions and Their Graphs

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 Polynomial Functions and Their Graphs

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