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

$$5^x + 5^{-x}$$

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

$$5^x + 5^{-x}$$

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

$$f(x) = ax^2 + bx + c$$

If you graph

$$x = ay^2 + by + c$$

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).