Final answer:
The function with a vertex at (2, 6) is f(x)=2|x-2|+6, which is Option B. This is because it translates the absolute value function right 2 units and up 6 units.
Step-by-step explanation:
The student is asking for the function that has a vertex at the point (2, 6). A vertex is a point that describes the highest or lowest point on the graph of a function. In the context of the provided options which involve absolute value functions, the vertex is the point at which the graph changes direction.
Option A, f(x)=2|x-2|-6, translates the parent function |x| right 2 units and down 6 units. However, the vertex for this function would be at (2, -6), not (2, 6).
Option B, f(x)=2|x-2|+6, also translates the parent function |x| right 2 units, but this time it is translated up 6 units, which gives us a vertex at (2, 6). This is the correct answer.
Options C and D translate the function left and either up or down, therefore their vertices would not have x-coordinate 2.