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7.2 Graphs of Polynomial Functions y=x2 y=x4 y=x3 y=x5 1 • If a function is ___________ and the leading coefficient is _____________, then the graph rises on both sides. • If a function is ___________ and the leading coefficient is _____________, then the graph falls on both sides. • If a function is ___________ and the leading coefficient is _____________, then the graph falls on the left and rises on the right. • If a function is ___________ and the leading coefficient is _____________, then the graph falls on the right and rises on the left 2 Relative Extrema A polynomial of n degree may have as many as ___________ relative extrema. y = x5 y = x5 5x3 + 4x y = x5 + 8x4 + 18x3 27x 3 A polynomial will have a zero that is a real number at each point at which is crosses the x axis. This means that we can graph by finding the ______________ of the function. We can do this by... 1) Factoring (multiple ways) 2) Quadratic Formula 3) Synthetic Division 4 Example 1: Graph f(x) = x5 5x3 + 4x 5 Multiplicity of Roots If (xr) is a factor, the graph goes "smoothly" through the root. If n is odd and greater than 1, the graph will intersect the xaxis at x = r and flatten out as it passes through that point. If n is even, the graph will be tangent to the xaxis at x = r and will not cross the xaxis at that point. 6 Example 2 Graph f(x) = x5 8x4 + 18x3 27x 7 Example 3: Graph f(x) = (x3)(x+2)(x1). yintercept = 6 Example 4: Graph f(x) = x4 x2 8 Example 5: Determine an equation for the polynomial for the graph shown. 9 When using a graphing calculator, it is important to analyze a function. 1. You need to watch for keying mistakes. If you know the basic shape of a graph, it will help you to notice if the graph is incorrect. 2. The standard viewing rectangle may not display the behavior of the function clearly and may need to be changed. You are looking at 3 things: 1. The overall shape of the graph 2. The zeros of the function 3. The relative extrema of the function 10 Example 6: Using your graphing calculator, graph f(x) = x4 + 3x3 4x. Determine each real zero and the number of relative extrema. Then express the function as a product of linear factors. 11