1)

Write an absolute value equation that is reflected, vertically shifted 2/3 units down and 1/2 units to the right.

2)

Write an absolute value equation with a horizontal shift 4 units to the right.

I assuming it can be an equation of an arbitrary choice. And I am further assuming that by "equation" they mean a function.

so I choose a simple f (x) = |x|, then

(1)

reflected (around the x axis here)

f (x) = - |x|

vertically shifted 2/3

f (x) = - |x|+2/3

and 1/2 units to the right

f (x) = - |x-1/2|+2/3 (final answer)

(2)

f (x) = |x-4| (final answer)