Given the quadratic function g(x) defined as:

We can go from the function f(x) = x² to g(x) making the transformations:
1) A reflection about the x-axis:

2) A horizontal dilation by a factor of 1/2:

3) A shift of 2 units down:

4) A shift of 2 units right:

Combining all these transformations:

Then, the graphs of f(x) (red) and g(x) (green) are: