I am trying to model this shape that I have been given an equation for, but can not understand how to make the equations work in GH or if I am even doing this the best way.

My plan was to pass a range of X points into the equation to give a range of Y points, plot them, and join the dots.

Just one more thought: A parabola is an analytical curve which can be modeled from a uniform degree 2 curve with 3 control points. Any parametric and interpolated curve will just be an approximation. Of course, with 500 control points it may be within your tolerance but I’d prefer the perfect and lightweight curve

It shouldn’t be too complicated to create these control points parametrically in GH.
If you just need that single parabola then this macro should do it (copy and paste to command line):

Thats the dims I got after as well!, not sure where 114.3 comes from. Somone below has modelled it with a simple curve. Ill talk to the customer and see whats going on.

114.3 is the distance of the focal point from the vertex and your diameter is 4 * 114.3 = 457.2

All parabolas are geometrical similar and each parabola does have that particular proportions at the focal point - just in different scales.

For this special case the command macro I’ve posted works fine. For general cases something like Laurent’s GH solution is very handy. Attached is an edited version where I modified the parametric part so that the points for the interpolated curve are more evenly spaced.

However at the start and end such an interpolated curve will have some imperfections. To minimize these you can make the curve longer and split it at the desired parameter.

Many years ago when I got a chance to be mentored by an old school lofter from Northrup he stated that a conic curve with a rho value of either 0.5 or 0.75 ( depending on your software) defined a parabola.