Polynomial Functions Objectives Identify Polynomials and their Degree
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 � a) � 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.
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