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Math Help - How to create a polynomial equation from a given set of data.

  1. #1
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    How to create a polynomial equation from a given set of data.

    I have a problem in which I am given following data

    Year Number of Students
    1980 3401
    1983 7462
    1986 4492
    1989 4288
    1992 6493
    1995 7538
    1998 12699

    I am being asked to calculate when the enrollment was increasing and decreasing, also to find the maximum and the minimum of the curve. I do not have any problems doing all of that as long as I have a polynomial function, but that is where my problem lies... I do not know how to determine the equation of the polynomial that best models the given data.

    On the question it says "Using graphing technology, determine the equation of the polynomial that best models the given data. However, I do not have access to any scientific calculator that can do that for me.

    Any help with this would be appreciated. I just need to know the principle behind determining polynomial equations from table data.

    Thank you!
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  2. #2
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    Re: How to create a polynomial equation from a given set of data.

    Do you have ms-excel?

    I tried the data set with polynomial degree 6 and found an equation.
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  3. #3
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    Re: How to create a polynomial equation from a given set of data.

    There exist a unique polynomial of degree n-1 or less passing through n given points. Here you have 7 points so you can fit a sixth degree polynomial, y= ax^6+ bx^5+ cx^4+ dx^3+ ex^2+ fx+ g. Putting the given values in for x and y gives you 7 linear equations to solve for a, b, c, d, e, f, and g.

    Another way to get the same polymnomial is to use the Lagrange form: Given the n points (x_1, y_1), (x_2, y_2), (x_3, y_3), \cdot\cdot\cdot, (x_n, y_n) form the sum of products

    y_1\frac{x- x_2}{x_1- x_2}\frac{x- x_3}{x_1- x_3}\cdot\cdot\cdot\frac{x- x_n}{x_1-x_n} (notice that x- x_1 is missing from this product)
    + y_2\frac{x- x_1}{x_2- x_1}\frac{x- x_3}{x_2- x_3}\cdot\cdot\cdot\frac{x- x_n}{x_2-x_n} (notice that x- x_2 is missing from this product)
    + \cdot\cdot\cdot
    + y_n\frac{x- x_1}{x_n- x_1}\frac{x- x_2}{x_n-x_2}\cdot\cdot\cdot\frac{x- x_{n-1}}{x_n- x_{n-1}} (notice that x- x_n is missing from this product.
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  4. #4
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    Re: How to create a polynomial equation from a given set of data.

    Quote Originally Posted by pickslides View Post
    Do you have ms-excel?
    Yes I do have Excel 2007. How do you do it?
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  5. #5
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    Re: How to create a polynomial equation from a given set of data.

    Plots the data as a scatterplot.

    After you have the chart, right click on the series and choose "Add Trendline"
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  6. #6
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    Re: How to create a polynomial equation from a given set of data.

    As a matter of practicality, I would do two things when you follow pickslides's excellent advice in Post # 5. 1. Display the equation generated. 2. Display the R^2 value. If your R^2 value is close to 1, you have a good fit for your curve.

    That's important, because if you're trying to find max's and min's, then a higher-order polynomial is going to be much harder to work with. If a third-degree or even a quadratic fits the data very well, then why bother with a higher-order polynomial, unless it needs to be exact?
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