So I'm a little unclear about the statement of the question, but I'll assume we are to show how transformations can be applied to f(x) = x^{2 }to obtain g(x), h(x), i(x), and j(x).

g(x) = (x-2)^{2} is f(x) = x^{2} shifted 2 units to the right (because we substitute x with x-2), so its graph is a parabola opening up with a vertex at (2,0).

h(x) = x^{2} - 2 is f(x) = x^{2} shifted 2 units down (because we subtract a 2 from f(x)), so its graph is a parabola opening up with a vertex at (0,-2).

i(x) = (x+3)^{2} is f(x) = x^{2} shifted 3 units to the left (because we substitute x with x+3), so its graph is a parabola opening up with a vertex at (-3,0).

j(x) = (x+1)^{2} + 3 is f(x) = x^{2} shifted 1 unit to the left and 3 units up (because we substitute x with x+1 and then add 3), so its graph is a parabola opening up with a vertex at (-1,3).