we will proceed to verify each of the cases to determine the solution of the problem
case A
\(f\left(x\right)=2\left|x-2\right|-6\)
using a graphing tool
see the attached figure N \(1\)
The vertex is the point \((2,-6)\)
therefore
the function \(f\left(x\right)=2\left|x-6\right|-6\) is not the solution of the problem
case B
\(f\left(x\right)=2\left|x-2\right|+6\)
using a graphing tool
see the attached figure N \(2\)
The vertex is the point \((2,6)\)
therefore
the function \(f\left(x\right)=2\left|x-2\right|+6\) is the solution of the problem
case C
\(f\left(x\right)=2\left|x+2\right|+6\)
using a graphing tool
see the attached figure N \(3\)
The vertex is the point \((-2,6)\)
therefore
the function\(f\left(x\right)=2\left|x+2\right|+6\) is not the solution of the problem
case D
\(f\left(x\right)=2\left|x+2\right|-6\)
using a graphing tool
see the attached figure N \(4\)
The vertex is the point \((-2,-6)\)
therefore
the function\(f\left(x\right)=2\left|x+2\right|-6\) is not the solution of the problem
the answer is the function
\(f\left(x\right)=2\left|x-2\right|+6\)
