Question

How can you use the vertex and the sign of "a" to determine the range of the absolute value function?

abs() = absolute value sign

y = -2abs(x-1)+4

Answer 1

as i wrote in the last post,
\[y=-2|x-1|+4\leq 4\] that is a synonym for saying that the range is
\[y\leq 4\]

Answer 2

you know it is less than or equal to (not greater than) also for the reasons stated in the last post. you know

\[|x|\geq 0\] for all x and therefor, because -2 is negative, you know
\[-2|x-1|\leq 0\] for all x