Does there exists a quadratic function for every parabola?

[ritwik06]

SOLVED

Homework Statement

While going through a question, I came across a function f(x)=5^{x}+5^{-x}
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 g(x)=ax^{2}+bx+c 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) ?

The Attempt at a Solution

There is only one thing that I see-
\frac{-\Delta}{4a}=2


Can this be solved?

 

[ritwik06]

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 g(x)=3.2x^{2}+2 But the graph for this does not exactly coincide with f(x). Why??
Somebody Please help me.

 

[statdad]

Th graph of

<br /> 5^x + 5^{-x}<br />

is not exactly a parabola, so your attempt simply gives an approximation of this function and its graph, but will not duplicate it.

 

[ritwik06]

Th graph of

<br /> 5^x + 5^{-x}<br />

is not exactly a parabola, so your attempt simply gives an approximation of this function and its graph, but will not duplicate it.

What is the definition of parabola?

 

[statdad]

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

<br /> f(x) = ax^2 + bx + c<br />

If you graph

<br /> x = ay^2 + by + c<br />

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 (\cosh), or exponentials base e, 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).