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Algebra II

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I'm having a hard time figuring out a quadratic model for this data.

x y
----------
1 3
2 13
3 31
4 57
5 91
6 133

How do I go about doing this?
Is there a website that could help me?

  • Algebra II -

    First show that this is a quadratic relationship by taking differences and second differences:
    y dy d2y
    3 10 8
    13 18 8
    31 26 8
    57 34 8
    91 42 8
    133

    If the second difference is constant, it is a quadratic relationship.

    Assume the quadratic model as:
    f(x)=ax2 + bx + c
    then
    f(x+1)=a(x+1)^2+b(x+1)+c
    and
    f(x+1)-f(x)=2ax+b+a
    Similarly,
    f(x+2)-f(x+1)=2ax+b+3a
    f(x+3)-f(x+2)=2ax+b+5a
    ....

    Thus, the second difference = 2a
    2a=8
    a=4

    By substituting values of x and a into one of the formulas, b can be found
    for x=1,
    dx=10
    2ax+b+a=10
    b=-2

    Substitute a and b in f(1), we get
    2(1)2-2(1)+c=3
    c=1
    Thus
    f(x)=4x2-2x+1

    check:
    f(1)=3
    f(3)=31
    f(6)=133

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