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Mathematics LibreTexts

21.7: I1.07: Section 5 Part 1

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    Section 5: Using Models.xls to find a quadratic model

    If we wish to find a model for the data on the right, this quick scatter plot of the data shows that a straight line will not be sufficient. Such parabolic shapes (similar to the path of a thrown ball) is instead represented mathematically by a quadratic formula, in which the term that contains the input variable x is squared.

    graph10

    There are several ways to write a quadratic formula, all of which can make the same curves. For use in fitting data, the best kind of quadratic formula is y = a (x h) 2 + v, where the parameters h and v are the x and y coordinates of the vertex of the parabola (that is, its highest or lowest point), and a is a “shape” parameter that determines how sharply (and in which direction) the parabola bends.

    The quadratic formula pattern that is most convenient for fitting models: y=a\cdot{{(x-h)}^{2}}+v

    a is a “shape” parameter controlling how much (and in which direction) the parabola bends

    h is the x coordinate of the vertex (its horizontal distance from the origin)

    v is the y coordinate of the vertex (its vertical distance from the origin)

    Examples: y=3\cdot{{(x-2)}^{2}}+8   y=2.5\cdot{{(x-7)}^{2}}-33.4   y=-2\cdot{{(x+3.5)}^{2}}+37

     

    time height
    x y
    0 0.0
    1 8.6
    2 16.8
    3 24.4
    4 31.5
    5 37.4
    6 43.8
    7 48.9
    8 54.4
    9 58.9
    10 63.3
    11 66.5
    12 69.7
    13 72.2
    14 74.5
    15 76.2
    16 77.8
    17 78.1
    18 79.0
    19 78.5
    20 78.0
    21 76.9
    22 75.6
    23 73.0
    24 70.3
    25 67.3
    26 63.7
    27 60.0
    28 55.0
    Examples of graphs of various quadratic formulas
    graph1 graph2 graph3 graph4
    y=3\cdot{{(x-2)}^{2}}+8 y=-1.2\cdot{{(x-2.5)}^{2}}-50 y=18\cdot{{(x+3)}^{2}}-158 y=3\cdot{{(x+7)}^{2}}-300

    Example 5: For each of the formulas above, state the location of the vertex of the parabola formed.

    Solution: Since the vertex is at (h,v) when a formula is expressed in the form,      the coordinates for the vertices are: (2, 8) (2.5, −50) (−3, −158) (−7, −300)

    Note that the sign of the x vertex coordinate is the opposite of the sign that the same number has in the formula, since the h value is subtracted when forming the formula.

    Quadratic models are somewhat more complicated than linear ones, as is indicated by the fact that a quadratic model has three parameters instead of two. But there is really very little difference in the fitting process from what is done for straight lines: [1] put the data in the appropriate worksheet, [2] spread the C3:E3 formulas down beside the data, [3] make a graph and adjust the vertex (instead of the intercept) and the shape (instead of the slope) until the model and the data match, and [5] write down the formula or use it to predict any values you have been asked for.

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    • Mathematics for Modeling. Authored by: Mary Parker and Hunter Ellinger. License: CC BY: Attribution