A long time ago in another state of existence
(Texas), I was playing around with parabolas when I noticed something
unexpected. If you take a quadratic function of the form f(x)
= a*x^2 + b*x + c and if you let the value of b vary over an interval,
then the vertices of the resulting parabolas seem to trace out
the path of yet another parabola. With a little algebra, one can
show that the graph of this new parabola is given by the function
y = c - a*x^2. Below is an example of what the graph of the function
f(x) = x^2 + b*x + 2 looks like as b varies from -4 to 4. As the
red parabola travels, you can see its vertex following the path
of the function

y = 2 - x^2.

`> `**with(plots):**

`> `**P:=animate(x^2+t*x+2,
x=-5..5, t=-4..4, view=[-5..5, -5..5], thickness=2, color=red):**

`> `**Q:=plot(2-x^2, x=-5..5,
view=[-5..5, -5..5], thickness=3, color=blue):**

`> `**R:=display(seq(pointplot([2-4*i/15,2-(2-4*i/15)^2],symbol=circle,symbolsize=20,color=red),i=0..15),
insequence=true):**

`> `**display([P,Q,R]);**

`>`