How can one interpret just by seeing the equation of parabola that it opens UP/DOWN and or centered around $x$- or $y$-axis.
I know parabola equation as $ax^2+bx+c$, I know that it opens up when $a>0$ and down when $a<0$, but I am not sure to about the orientation of parabola over $x$- or $y$-axis.
For example - how could I have interpreted that $y =x^2-2$ will open up and is oriented on $y$-axis.
My biggest doubt is how can $-2$ in $y=x^2-2$ can help in revealing that the parabola equation is centered around $y$-axis.
Thanks in advance.
The short answer is: it cannot. The $-2$ has nothing to do with finding the axis of the parabola.
If your parabola has an equation of the form $y = ax^2 + bx + c,$ you find the axis of the parabola by looking at the coefficients $a$ and $b.$ Specifically, the axis will be the line $x = -\frac{b}{2a},$ because $$ ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right). $$
When the particular parabola you have to deal with is $y = x^2 - 2,$ then in terms of the usual $a,b,c$ form of the equation you have $a = 1,$ $b = 0,$ and $c = -2.$ Therefore $-\frac{b}{2a} = 0$ and the axis is the line $x = 0,$ that is, the $y$-axis.
Alternatively, you could simply observe that $y = x^2 - 2$ is just the parabola $y = x^2$ shifted downward $2$ units. Since $y = x^2$ has $x = 0$ as its axis, so does $y = x^2 - 2.$ (It only went downward, not sideways.)