Determining the Equation of a Graph based on points

Question

Given that an polynomial has the points $(\pm\sqrt{3},0),(\pm5,11)$, the degree is greater than $2$ and it has an even degree, can the equation of the polynomial be determined. I tried just plugging in values, but that didn't work. I do have a picture, but I want the equation that fills in that gap at the bottom:

EDIT: I have determined that the vertex is $(0,-0.25)$. How do I progress?

Answer 1

If you have number of points equal to the degree, look at Lagrange Interpolation

Answer 2

There are an infinite number of even-order polynomials that fit the data, found by doing a least-squares fit to the polynomial model.

Here are just some of them: