So to solve problems like these you can work out each equation by substituting the x and y with the coords from the vertex (2,-4) and which ever one is true is the corresponding equation.
Lets try the first one
A) y = 2( x-2)^2-4 this would become
-4 = 2(2-2)^2-4 we solve this and get
-4 = 2(0)-4
-4 = -4
so it seems like A is the correct answer, of course we'd wanna check out the other answers just to be sure.
Lets try one and do C
C) -2 = 2(-4-2)^2+2
-2 = 2(-8)^2+2
-2 = 130
These aren't equal so this can't be our equation
So you can also do one more just to be super sure you have the right answer but I think A is the correct one :)