Quadratic Function
Let $f(x)=ax^2+bx+c$
It's obvious that when we change absolute term the vertex of the parabola graphs a vertical line with equation $x=-b/2a$
When we change $b$, the vertex traces a parabola with equation $y=-ax^2+c$ as shown here:
Define $b$ as the parameter $t$ so the parametric equation traces the vertex is $$\left({-t\over2a},f\left({-t\over2a}\right)\right)$$ $$\left({-t\over2a},{-t^2+4ac\over4a}\right)$$ Now, eliminate the parameter: $$x={-t\over2a}\iff t=-2ax\\y={-t^2+4ac\over4a}={-4a^2x^2+4ac\over 4a}=-ax^2+c\space\square$$
By the same method, when we change $a$ it traces: $$y={b\over 2}x+c$$
My question is "Does other polynomials have this pattern? Does this tracing have a name? Is there a generalization?"
NB. for cubic we can trace the inflection point. Or the mid-point between the two extrema.