Quadratic Function (x-4)²: Shifting 2 Right and 3 Up

Choose which equation represents the function

$y=(x-4)^2$

moved 2 spaces to the right

and 3 spaces upwards upwards.

To solve this problem, we need to perform two transformations on the original function $y = (x-4)^2$: a shift 2 units to the right and a shift 3 units upwards.

Step 1: Horizontal Shift (2 units to the right)
When a function is shifted to the right by $a$, we replace $x$ with $x - a$. In this case, $a = 2$. Thus, replacing $x$ with $x-2$ in the original function $y = (x-4)^2$ results in $y = ((x-2)-4)^2 = (x-6)^2$.

Step 2: Vertical Shift (3 units upwards)
To shift a function upwards by $k$, add $k$ to the entire function. Here, $k = 3$, so the transformed equation becomes $y = (x-6)^2 + 3$.

Thus, the equation of the function after these transformations is $y = (x-6)^2 + 3$.

The correct answer, as given in the problem, is indeed: $y = (x-6)^2 + 3$. This corresponds to choice 4.