Let's start from the parent function [tex] y = |x| [/tex] and see which transformation we have applied. The parent function's graph is open upards, and its vertes lies at [tex] (0,0) [/tex]
The first transformation is [tex] |x| \to |x-1| [/tex]. This is a tranformation of the form [tex] f(x)\to f(x+k) [/tex]. This kind of transformation shift the graph horizontally, k units to the left if k is positive, k units to the right if k is negative. In this case [tex] k = -1 [/tex], so the function is shifted one unit to the right. The new vertex lies at [tex] (1,0) [/tex]
The second transformation is [tex] |x-1| \to -3|x-1| [/tex]. This is a tranformation of the form [tex] f(x)\to kf(x) [/tex]. This kind of transformation stretch the graph vertically, compressing it if k is between 0 and 1, expanding it if k is greater than 1. Moreover, if k is negative, the function is reflected along the x axis. In this case [tex] k = -3 [/tex], so the function is reflected and stretched. This means that now the graph opens downwards, the vertex still lies at [tex] (1,0) [/tex].
Finally, we have [tex] -3|x-1| \to -3|x-1|+12 [/tex]. This is a tranformation of the form [tex] f(x)\to f(x)+k [/tex]. This kind of transformation translates the graph vertically, k units up if k is positive, k units down if k is negative. In this case, [tex] k = 12 [/tex], so the graph is shifted 12 units up. The graph still opens downwards, while the new vertex lies at [tex] (1,12) [/tex]
