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

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

anonymous · Answer

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

anonymous · Answer

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