So there is a million (essentially equivalent) ways to make it obvious algebraically that the axis of symmetry of the parabola $y=x^2+bx+c$, must be at $x=-\frac{b}{2}$, while its real roots, if they exist, are $\pm\sqrt{\left(\frac{b}{2}\right)^2-c}$ away from that axis.
It feels like there must be a way to make this graphically obvious as well: The line $y=bx+c$ is tangent to the parabola at $x=0$, so the graphical interpretation of $b$ and $c$ are as the slope or $y$-intercept, respectively, of the parabola at $x=0$.
The question: Given these (or some other less obvious) graphical interpretations of $b$ & $c$, and without doing the standard algebra (completing the square etc.), is there a way to "read off" from the parabola graph that $\sqrt{\left(\frac{b}{2}\right)^2-c}$ must be the distance between a root and the symmetry axis? (Intuitely, a Pythagoras-like argument, based on some geometric definition of parabola?)
What I am looking for is an argument like the following, which makes it visually obvious that $(a+b)^2=a^2+2ab+b^2$:
