Answer :

To determine which function has a vertex at [tex]\((2,6)\)[/tex], we first need to understand the general form of an absolute value function's vertex form, which is given by:

[tex]\[ f(x) = a| x - h | + k \][/tex]

Here, [tex]\((h, k)\)[/tex] represents the vertex of the function.

Now, let’s analyze each given function and identify their vertices step-by-step:

### 1. Function: [tex]\( f(x) = 2|x-2|-6 \)[/tex]
- The expression inside the absolute value, [tex]\(|x-2|\)[/tex], suggests [tex]\( h = 2 \)[/tex].
- After applying the absolute value, we multiply by 2 and then subtract 6.
- Thus, the vertex of this function is [tex]\((2, -6)\)[/tex].

### 2. Function: [tex]\( f(x) = 2|x-2|+6 \)[/tex]
- The expression inside the absolute value, [tex]\(|x-2|\)[/tex], suggests [tex]\( h = 2 \)[/tex].
- After applying the absolute value, we multiply by 2 and then add 6.
- Thus, the vertex of this function is [tex]\((2, 6)\)[/tex], which matches the given vertex.

### 3. Function: [tex]\( f(x) = 2|x+2|+6 \)[/tex]
- The expression inside the absolute value, [tex]\(|x+2|\)[/tex], rewrites as [tex]\(|x - (-2)|\)[/tex], suggesting [tex]\( h = -2 \)[/tex].
- After applying the absolute value, we multiply by 2 and then add 6.
- Thus, the vertex of this function is [tex]\((-2, 6)\)[/tex].

### 4. Function: [tex]\( f(x) = 2|x+2|-6 \)[/tex]
- The expression inside the absolute value, [tex]\(|x+2|\)[/tex], rewrites as [tex]\(|x - (-2)|\)[/tex], suggesting [tex]\( h = -2 \)[/tex].
- After applying the absolute value, we multiply by 2 and then subtract 6.
- Thus, the vertex of this function is [tex]\((-2, -6)\)[/tex].

Upon reviewing each function, we can see that the function with a vertex at [tex]\((2,6)\)[/tex] is:

[tex]\[ f(x) = 2|x-2|+6 \][/tex]

Therefore, the correct function is the second one: [tex]\( f(x) = 2|x-2|+6 \)[/tex].