Short Run Behavior of Polynomials Lesson 11.3

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

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

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?

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:

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) below

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  

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

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

Assignment Lesson 11.3 Page 452 Exercises 1 – 45 EOO