Why do all parabolas have an axis of symmetry? And if that's just part of the definition of a parabola, I guess my question becomes why is the graph of any quadratic a parabola?
My attempt at explaining:
The way I understand it after some thought is that any quadratic can be written by completing the square as a perfect square + a constant, for ex: 
$f(x) = x^2 + x$ can be written as $f(x) = (x+\frac{1}{2})^2 - \frac{1}{4}$.
So essentially, any quadratic is a displaced version of $f(x) = x^2$, and it's pretty obvious why $f(x) = x^2$ has an axis of symmetry and why it's at the vertex.
Is my reasoning correct? And if you have a different way to think about it and explain it, whether geometric, algebraic or other, I would love to see it.

Suppose, $f(x)=(x+a)^2+b$
Then, we have $$f(-a-x)=f(-a+x)=x^2+b$$ for all $x\in \mathbb R$
