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Short Run Behavior of Polynomials

Short Run Behavior of Polynomials. Lesson 9.3. Compare Long Run Behavior. Consider the following graphs: f(x) = x 4 - 4x 3 + 16x - 16 g(x) = x 4 - 4x 3 - 4x 2 +16x h(x) = x 4 + x 3 - 8x 2 - 12x Graph these on the window -8 < x < 8       and      0 < y < 4000

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Short Run Behavior of Polynomials

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  1. Short Run Behavior of Polynomials Lesson 9.3

  2. Compare Long Run Behavior Consider 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 graph

  3. Compare Long Run Behavior • From this view, they appear very similar

  4. Contrast 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?

  5. Contrast Short Run Behavior Differences? • 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:

  6. 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)

  7. Factored Form • Try factoring the original functions f(x), g(x), and h(x)  • (enter    factor(y1(x))  • what results do you get?

  8. Local Max and Min • For now the only tools we have to find these values is by using the technology of our calculators:

  9. 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) below

  10. 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 root

  11. From 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 zero

  12. From 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 k

  13. Assignment • Lesson 9.3 • Page 406 • Exercises 1 – 45 EOO

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