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Polynomial Functions and Models Lesson 4.2

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Review General polynomial formula a 0, a 1, …,a n are constant coefficients n is the degree of the polynomial Standard form is for descending powers of x a n x n is said to be the “leading term”

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Turning Points and Local Extrema Turning point A point (x, y) on the graph Located where graph changes from increasing to decreasing (or vice versa)

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Family of Polynomials Constant polynomial functions f(x) = a Linear polynomial functions f(x) = m x + b Quadratic polynomial functions f(x) = a x 2 + b x + c

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Family of Polynomials Cubic polynomial functions f(x) = a x 3 + b x 2 + c x + d Degree 3 polynomial Quartic polynomial functions f(x) = a x 4 + b x 3 + c x 2 + d x + e Degree 4 polynomial

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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 Decide how these functions are alike or different, based on the view of this graph

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

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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?

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Contrast Short Run Behavior Differences? Real zeros Local extrema Complex zeros Note: The standard form of the polynomials does not give any clues as to this short run behavior of the polynomials:

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Assignment Lesson 4.2A Page 270 Exercises 1 – 38 odd

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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)

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

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

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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

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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

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Piecewise Defined Functions Consider Sketch graph Use calculator to display Consider if f(x) is continuous

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Linear Regression Used in section previous lessons to find equation for a line of best fit Other types of regression are available

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Polynomial Regression Consider the lobster catch (in millions of lbs.) in the last 30 some years Enter into Data Matrix Year 1970197519801985199019952000 t 5101520253035 Lobster 17192220273656

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Viewing the Data Points Specify the plot F2, X's from C1, Y's from C2 View the graph Check Y= screen, use Zoom-Data

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Polynomial Regression Try for 4 th degree polynomial

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Other Technology Tools Excel will also do regression Excel Plot data as (x,y) ordered pairs Right click on data series Choose Add Trend Line

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Other Technology Tools Use dialog box to specify regression Try Others

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Try It Out … An object is lifted rapidly into the air and then released. The table shows the height at t seconds after the start of the experiment Use your calculator to plot the data At what time was the object released? What part of the time interval could be represented by a linear function? Find that function. Find a modeling function for the non linear portion. t01234567 h03672108144128800

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Assignment Lesson 4.2B Page 273 Exercises 41 – 89 EOO

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