Let's analyze the given function [tex]\( f(x) = -3|x-1| + 12 \)[/tex] using our understanding of absolute value functions in the form [tex]\( a|x-h| + k \)[/tex].
The standard form for an absolute value function is:
[tex]\[ f(x) = a|x-h| + k \][/tex]
1. Identifying Parameters:
- [tex]\( a = -3 \)[/tex]
- [tex]\( h = 1 \)[/tex]
- [tex]\( k = 12 \)[/tex]
2. Vertex of the Graph:
- The vertex of the absolute value function is given by the point [tex]\( (h, k) \)[/tex].
- Therefore, in this case, the vertex is at [tex]\( (1, 12) \)[/tex].
3. Direction of the Graph:
- The direction in which the graph opens is determined by the coefficient [tex]\( a \)[/tex].
- If [tex]\( a \)[/tex] is positive, the graph opens upward.
- If [tex]\( a \)[/tex] is negative, the graph opens downward.
- Since [tex]\( a = -3 \)[/tex] is negative in this case, the graph opens downward.
Thus, putting these pieces of information together, the correct statement describing the graph of the function [tex]\( f(x) = -3|x-1| + 12 \)[/tex] is:
B. The graph opens downward, and its vertex lies at [tex]\((1, 12)\)[/tex].
