Polynomial Functions Objectives: Identify Polynomials and their Degree Graph Polynomial Functions using Transformations Identify the Zeros of a Polynomial and their Multiplicity Analyze the Graph of Polynomial Function

Polynomial Function The function defined by is called a polynomial function of degree n. The number the coefficient of the variable to the highest power is called the leading coefficient. EX 1: Determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tell why not a) b)

Properties of Power Functions Even Power 1. The graph is symmetric with respect to the y-axis 2. Domain: Range: 3. The graph always contains the points 4. As n increases, the graph tends to flatten out near the origin and to increase very rapidly when x is far from 0 Odd Power 1. The graph is symmetric with respect to the origin 2. Domain: Range: 3. The graph always contains the points 4. As n increases, the graph tends to flatten out near the origin and to increase very rapidly when x is far from 0

EX 2: Use transformations to graph each function b)

Steps for Analyzing the Graph of a Polynomial 1. INTERCEPTS: Find the x-intercepts, if any, by solving the equation . Find the y-intercept by finding the value of 2. MULTIPLICITY: Determine whether the graph of crosses or touches the x-axis at each x-intercept 3. END BEHAVIOR: Find the power function that the graph of resembles for large values of x. 4. TURNING POINTS: Determine the maximum number of turning points on the graph of 5. Use the x-intercept(s) to find the intervals on which the graph of is above the x-axis and the intervals on which the graph is below the x-axis 6. Plot the points obtained in steps 1 and 5, and use the remaining information to connect them with a smooth, continuous curve

EX: Graph the polynomial function 3.

EX: Graph the polynomial function 4.