To find the center of a circle, it's enough to choose three points on the circle and find the circumcenter of a triangle with those three points.
When a parabola is drawn and the formula is not given, how can I find its focus?
I add an example.
Can you find the center, vertex, axis, or directrix of the parabola?
If I well understand the problem, I think that here there is a solution. Note that the key problem is to construct the axis, than you can use the answer in the related question cited in the comment of @Blue