How do I find the coordinates of the vertex of the graph of the function

Y = 4x squared - 3

Either graph and look for the minimum or, algebraically, you pretty much have to use guess and check. In quadratics (x^2 functions) that look like this one (an x^2 term and a constant term, - 3 in this case), the minimum is always located at x=0. Thus, the minimum for this graph is (0,-3)