https://www.desmos.com/calculator/z2yi8pjyej
In this graph, there are two parabolas -- one is $f(x) = x^2 + tx + t$, while the other is $g(x) = -x^2 + t$.
As long as the '$x^2$' terms are of opposite sign and have the same coefficient; and all of the '$t$' terms in both equations share the same sign; and the lowest order '$t$' terms in each equation share the same coefficient, the vertex of each parabola will seem to (I do not attempt to prove this) trace the other parabola's graph for all times '$t$'.
So, in general, this seems to remain true so long as the equations take the forms:
$f(x) = \pm(Cx^2+Dtx+Ft+G)$, and $g(x) = \mp(Cx^2-Ft-G)$, where '$C$', '$D$', '$F$', and '$G$', are all arbitrary, non-zero (with the exception of '$G$'), real constants, that have the same value in both equations in which they appear (with the exception of '$D$', which is only found in $f(x)$.)
Is there a simple, nontrivial explanation for why this occurs?
(I tried to find some analogous phenomenon in higher order equations, but I wasn't successful, so while this might generalize to polynomial functions of all orders, I didn't figure out if, nor how, it might.)