The graph of the parent quadratic function, f(x)=x^2, is shifted right 6 units and down 2 units. what is the function written in standard form for the translated parabola?

In general the standard form of a parabola is:

                                      (y-k) =a(x-h)²
Where k indicates the vertical shift & h, the horizontal one.
In your original function f(x) = x², k=h=0. That means the parabola passes by the origine & since a=1>0, it opens up,
Now if you want to shifted it 6 units at the right that means h=6
& 2 units at 2 units down that means k= -2
Plug these value into the standard form & you will get:
y-(-2)= (x-6)² or ==> y+2 = (x-6)² or y= (x-6)² -2