Does there exists a quadratic function for every parabola?

1. Oct 6, 2008

ritwik06

SOLVED

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

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) ????

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

Can this be solved???

Last edited: Oct 6, 2008
2. Oct 6, 2008

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??

Last edited: Oct 6, 2008
3. Oct 6, 2008

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.

4. Oct 6, 2008

ritwik06

What is the definition of parabola?

5. Oct 6, 2008

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).