The choices are: a)y = (x+3) (x-5) b)y = (x-3) (x+5) c)y = x(x-2) - 15 d)y = (x-1)^2 - 16 the correct answer is d, i just don’t understand why.

Let's take a look at D:   

D)    y = (x-1)^2 - 16     Compare this to 
        y = (x-h)^2 + k       This is the std. equation of a parabola in vertex form.

You can see, by comparison, that h=1 and k= -16; these are the coordinates of the vertex, clearly shown in the diagram.

Since the coefficient of (x-h)^2 is +1, the graph opens upward (which the given graph confirms), and is neither compressed nor stretched vertically.