Choice A is the correct answer. In vertex form $y = a(x - h)^2 + k$, the value $h$ represents the horizontal shift.

1. The equation is $y = (x - 2)^2$, so $h = 2$.
2. A positive $h$ value indicates a shift to the right.

Think: "What x value makes the term zero?" Here, $x=2$, so the vertex moves to $x=2$.

Choice B is incorrect because a left shift would be $(x + 2)^2$. Choice C is incorrect because a down shift would be $x^2 - 2$. Choice D is incorrect because an up shift would be $x^2 + 2$.