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Modeling Polynomial Functions

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Introduction

Colin Wick                12/2/08

Period 7                Precalc

Modeling Polynomial Functions

        Polynomial functions are power functions, or sums of two or more power functions.  Furthermore, a polynomial function must be made up entirely of nonnegative integer powers.  These polynomial functions are commonly used to graph changes in a population or amount over a specified period of time.  Derivative functions are functions that represent the slope of an exponential or polynomial function in relation to time.  Therefore, they can assist in the calculation of a rate of change at a specific moment in time.

        To visually see how the derivative of a function relates to the original power function you must graph them both on the same set of axis.  To graph the original power function either input the equation into a graphing calculator or calculate the zeros by hand and approximate.  In this case, the equation for the polynomial function is: x3+6x2+9x. Graphing the derivative function without using power rule is more complicated in that you must use the equation f ‘(x)=f(x+0.001)-f(x)/0.001 and plug in the original power function.  This equation finds a little bit above and below the point and calculates a close to instantaneous rate of change.

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Middle

-6

189

-5

144

-4

105

-3

72

-2

45

-1

24

0

9

1

0

2

-3

3

0

4

9

5

24

6

45

7

72

8

105

By graphing both the derivative and the original polynomial on the same set of axis, the two graphs and juxtaposed and it is now visible how the derivative is based off of the slope of the graph (the derivative being the parabola).  To conclude, graphing the polynomial function as well as an estimate of the derivative function is a viable way to see the relationship between the two graphs.

        The zeros of a derivative function correlate directly with the local minima and maxima of the corresponding polynomial because it is at the minima and maxima where the slope, or rate of change, is zero.  For example, the zeros to the derivative function, f’(x), are (1,0) and (3,0).  These zeros can be found by analyzing the table created for the approximate graph of the derivative function by using the equation f ‘(x)=f(x+0.001)-f(x)/0.001.  Furthermore, the local minima and maxima of the polynomial occur at the same points as the zeroes of the derivative.  At these local minima and maxima, the graph is no longer increasing or decreasing therefore making its slope zero.  Because the graph

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Conclusion

        To conclude, a power function’s zeros are based directly off of the degree of the equation, and the total number of local maxima and minima of the equation is one less than the degree of the equation because for every two zeros there is only one maximum or minimum.  Furthermore, because a derivative function is based off of the slope of the initial polynomial, the polynomial’s minima and maxima and the derivative function’s zeros.  Therefore, the derivative function is to a degree of no more than one less the degree of the original polynomial.  A general equation can therefore be created to find the derivative function for any polynomial function to a degree of n: if a polynomial function is represented by axn+an-1xn-1+…+a2x2+ax+a, then the derivative of that function is represented as: nanxn-1+(n-1)an-1xn-2+…+2a2x+a1.  This equation could therefore be used to find the instantaneous rate of change for any polynomial function at any given point and could be used to track population changes or other data based off of polynomial functions.

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