Short Run Behavior of PolynomialsCompare Long Run BehaviorSlide 3Contrast Short Run BehaviorSlide 5Factored FormSlide 7Local Max and MinMultiple ZerosSlide 10From Graph to FormulaSlide 12AssignmentShort Run Behavior of PolynomialsLesson 11.3Compare Long Run BehaviorConsider the following graphs:• f(x) = x4 - 4x3 + 16x - 16 •g(x) = x4 - 4x3 - 4x2 +16x •h(x) = x4 + x3 - 8x2 - 12x •Graph these on the window -8 < x < 8 and 0 < y < 4000•Decide how these functions are alike or different, based on the view of this graphCompare Long Run Behavior•From this view, they appear very similarContrast Short Run Behavior•Now Change the window to be-5 < x < 5 and -35 < y < 15•How do the functions appear to be different from this view?Contrast Short Run BehaviorDifferences? •Real zeros •Local extrema •Complex zeros•Note: The standard form of the polynomials do not give any clues as to this short run behavior of the polynomials:Factored Form•Consider the following polynomial:p(x) = (x - 2)(2x + 3)(x + 5)•What will the zeros be for this polynomial? x = 2 x = -3/2 x = -5 •How do you know? We see the product of two values a * b = 0 We know that either a = 0 or b = 0 (or both)Factored Form•Try factoring the original functions f(x), g(x), and h(x) (enter factor(y1(x)) what results do you get?Local Max and Min•For now the only tools we have to find these values is by using the technology of our calculators:Multiple Zeros•Given•We say the degree = n •With degree = n, the function can have up to n different real zeros •Sometimes the zeros are repeated, as seen in y1(x) and y3(x) below11 1 0( ) ...n nn np x a x a x a x a--= + + + +Multiple Zeros•Look at your graphs of these functions, what happens at these zeros? Odd power, odd number of duplicate roots => inflection point at root Even power, even number of duplicate roots => tangent point at rootFrom Graph to Formula•If you are given the graph of a polynomial, can the formula be determined?•Given the graph below:•What are the zeros? •What is a possible set of factors? Note the double zeroNote the double zeroFrom Graph to Formula•Try graphing the results ... does this give the graph seen above (if y tic-marks are in units of 5 and the window is -30 < y < 30)•The graph of f(x) = (x - 3)2(x+ 5) will not go through the point (-3,-7.2) •We must determine the coefficient that is the vertical stretch/compression factor...f(x) = k * (x - 3)2(x + 5) ... How?? Use the known point (-3, -7.2)-7.2 = f(-3) Solve for kAssignment•Lesson 11.3•Page 452•Exercises 1 – 45
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