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Polynomials and Graphs I. Midterm II II. Left and Right Behavior. We will investigate the outer shape of several polynomials and explore the following rules:
Where Even and Odd refers to the degree of the polynomial, Pos and Neg refers to the leading coefficient, And a U or a D refers to the left and right behavior of the curve. Example -3x7 + 4x4 - 1 has degree 7 which is odd and has leading coefficient -3 which is negative. Hence the left and right behavior is UD, i.e. the curve goes up on the left and down on the right. In class we will do several of these. III. Max and Min Theorem: If f(x) is a polynomial of degree n then f(x) has at most n -1 relative extrema. Where relative extrema are lumps of the graph. Example: 4x5 + 2x3 - x2 + 7x + 12 Has at most 4 relative extrema. IV. Three Step Procedure For Graphing Polynomials Step 1: Factor the polynomial into linear factors of the form ax + b. Step 2 : Determine the left and right behavior of the graph and the shape of the graph near each x intercept. Step 3: Connect the dots Example: Graph y = x4 - 10x2 + 9 1) We have y = (x2 -9)(x2 - 1) = (x - 3)(x + 3)(x - 1)(x + 1) 2) The left behavior is up and the right behavior is up. Near x = -3 the graph is positive on the left and negative on the right. Near x = -1 the graph is negative on the left and positive on the right Near x = 1 the graph is positive on the left and negative on the right. Near x = 3 the graph is negative on the left and positive on the right 3) Graph it! We will do problems 48, 53, and 59 in class in groups. |