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