My equation is located on the following url: https://www.desmos.com/calculator/7iewlnmfwv
Basically trying to make my ellipse move on my polynomial equation. Almost, like a roller coaster.
I'm aiming for the vertex of my ellipse to be on the line of my polynomial equation. As well as move along that line when the value is changed.
You have a long way to go on a long roller coaster ride ...First you have not defined the shape of your roller coaster path!
Second.. to describe rolling on a straight path without slipping, go with an exercise on cycloid motion.
Thirdly, learn up what are natural or intrinsic equations of curves.
Fourthly, find out connection between locus of ellipse focus and constant mean curvature curves or DeLaunay Unduloids.
Good Luck!