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Does there exists a quadratic function for every parabola?

  1. Oct 6, 2008 #1
    SOLVED

    1. The problem statement, all variables and given/known data

    While going through a question, I came across a function [tex]f(x)=5^{x}+5^{-x}[/tex]
    I saw its graph through a software. It was a parabola with minimum value 2.

    Now a question arises in my mind.
    Every function of the type [tex]g(x)=ax^{2}+bx+c[/tex] is a parabola.
    Can I assume the corollary to be true, that is for every parabola, there exists a quadratic function??
    If yes, how may I find the coefficients a,b,c such that f(x)=g(x) ????



    3. The attempt at a solution
    There is only one thing that I see-
    [tex]\frac{-\Delta}{4a}=2[/tex]


    Can this be solved???
     
    Last edited: Oct 6, 2008
  2. jcsd
  3. Oct 6, 2008 #2
    Re: Graph+Quadratic

    3. The attempt at a solution
    For f(x)
    if x=0, f(x)=2
    if x=1, f(x)=5.2
    if x=-1, f(x)=5.2
    And if I use these equations to solve for the quadratics to solve for a,b,c the coefficients of g(x), I find that a=3.2, b=0, c=2.
    which makes [tex]g(x)=3.2x^{2}+2[/tex] But the graph for this does not exactly coincide with f(x). Why??
    Somebody Please help me.
     
    Last edited: Oct 6, 2008
  4. Oct 6, 2008 #3

    statdad

    User Avatar
    Homework Helper

    Re: Graph+Quadratic

    Th graph of

    [tex]
    5^x + 5^{-x}
    [/tex]

    is not exactly a parabola, so your attempt simply gives an approximation of this function and its graph, but will not duplicate it.
     
  5. Oct 6, 2008 #4
    Re: Graph+Quadratic

    What is the definition of parabola?
     
  6. Oct 6, 2008 #5

    statdad

    User Avatar
    Homework Helper

    Re: Graph+Quadratic

    A parabola is the graph of a function that has the form

    [tex]
    f(x) = ax^2 + bx + c
    [/tex]

    If you graph

    [tex]
    x = ay^2 + by + c
    [/tex]

    you get a parabola shape, but this is not a function.

    The equation you encountered (and its graph) are a form of a catenary . The classical equation for this graph involves the hyperbolic cosine ([tex] \cosh [/tex]), or exponentials base [tex] e [/tex], but the form you give works as well. A catenary can be loosely described as the shape a hanging chain takes (or the graph of power lines between towers).
     
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