HA2 5.1 Polynomial Basics Objectives: To classify polynomials To graph polynomial functions To describe the end behavior Polynomial functions have distinguishing “behaviors.” Looking at its algebraic form tells a lot about its graph and vise-versa. Using algebraic and graphical analysis, one can determine the degree and basic shape. ESSENTIAL UNDERSTANDING: KEY TERMS_: __ _____ Monomial Polynomial Degree Polynomial Function Turning Point End Behavior You can classify a polynomial by its degree or by its number of terms. These vocabulary words must be used correctly to describe a polynomial expression. Name Using Degree Polynomial Example Number of Terms 0 5 1 1 x4 2 2 4x 2 1 3 4x 3 3x 2 x 3 4 2x 4 5x 2 2 5 x 5 4x 2 2x 1 4 Degree Example Name Using Number of Terms polynomial of 4 terms Classifying Polynomials Write –3x + 4x + 7x – 3 in standard form. What is the classification of this polynomial by its degree? by its number of terms? 3 Got It? Write each polynomial in standard form. What is the classification of each by degree? by number of terms? a. 3x 3 x 5x 4 b. 3 4x 5 2x 2 10 Exploration Activity of Polynomial End Behaviors Example Describing End Behavior of Polynomial Functions Consider the leading term of y = 3x4 – 2x3 + x – 1. What is the end behavior of the graph? Got It? Consider the leading term of y 4x 3 2x 2 7 . What is the end behavior of the graph? . You can determine the end behavior of a polynomial function of degree n from the leading term axn of the standard form. Look at the polynomials above. What do you think is meant by the phrases “up and up,” “down and down,” “down and up,” and “up and down?” How can you use the leading term of the polynomial to determine the end behavior? Why do you think that the end behavior of a polynomial function does not rely on any other term than the leading term? You can tell, by the degree of the leading coefficient, the maximum number of turning points of a polynomial. Look at the shapes of the following polynomials. Then use a graphing calculator to look at polynomials of different degrees. Describe a pattern you notice about the number of turning points and the degree of the polynomial. If the graph of a polynomial function has “t” turning points, how can you describe its degree? Example Graphing Cubic Functions What is the graph of y = 3 – 2x3 + x? Describe the graph. x -2 -1 0 1 2 y Got It? What is the graph of each cubic function? Describe the graph. a. y x 3 2x 2 x 2 b. y x 3 1 Before you start these problems, think: what should the end behaviors be? x -2 -1 0 1 2 y x -2 -1 0 1 2 y Suppose you are given a set of polynomial function ordered pairs. When the inputs (x-values) of the polynomial differ by a constant, it is possible to analyze the differences of consecutive outputs (y-values) to determine the leastdegree polynomial function that could generate the data. Example Using Finite Differences to Determine Degree What is the degree of the polynomial function that generates the data shown in the table? Got It? What is the degree of the polynomial function that generates the data shown at the right? Reasoning What is an example of a polynomial function whose fifth differences are constant but whose fourth differences are not constant? Homework: Pg 285: #1-7, 21-39 odd, 40-56 Exit Slip Questions Answer these questions on this slip of paper, then remove from your packet and give to your teacher before you leave the room. Name: _______________________________ Per: 4 December 5, 2012 1. Wrtie 2x 3 3x 2 x 5 in standard form. What is the classification of this polynomial by its degree? By its number of terms? 2. Consider the leading term of y 2x 2 3x 3 . What is the end behavior of the graph? Include a sketch in your explanation. 3. Describe the end behavior and number of turning points in the graph of y x 3 x 3 . 4. Do you Understand? What is the degree of the polynomial function that generates the data shown in the table? Explain how you determined your answer.