getting the coordinates

With the given function from a previous problem I was solving link, I want to create/collect the coordinate points of Min. value for 21 values of parameter a (from -7 to 7), and find a,b, and c such that the points are on the curve of the quadratic equation $ax^2+bx+c=0$. I also want to use the plot command to show this coordinate system with the parabola obtained from it.