To determine which function has a vertex at [tex]\((2, 6)\)[/tex], we need to understand the vertex form of an absolute value function. The vertex form is given by:
[tex]\[ f(x) = a|x - h| + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex.
Let's examine each of the given functions to identify their vertices:
1. Function: [tex]\( f(x) = 2|x - 2| - 6 \)[/tex]
- Here, the form is [tex]\( 2|x - 2| + (-6) \)[/tex].
- This means [tex]\( h = 2 \)[/tex] and [tex]\( k = -6 \)[/tex].
- The vertex of this function is [tex]\((2, -6)\)[/tex], which does not match [tex]\((2, 6)\)[/tex].
2. Function: [tex]\( f(x) = 2|x - 2| + 6 \)[/tex]
- Here, the form is [tex]\( 2|x - 2| + 6 \)[/tex].
- This means [tex]\( h = 2 \)[/tex] and [tex]\( k = 6 \)[/tex].
- The vertex of this function is [tex]\((2, 6)\)[/tex], which matches the given vertex [tex]\((2, 6)\)[/tex].
3. Function: [tex]\( f(x) = 2|x + 2| + 6 \)[/tex]
- Here, the form can be rewritten as [tex]\( 2|x - (-2)| + 6 \)[/tex].
- This means [tex]\( h = -2 \)[/tex] and [tex]\( k = 6 \)[/tex].
- The vertex of this function is [tex]\((-2, 6)\)[/tex], which does not match [tex]\((2, 6)\)[/tex].
4. Function: [tex]\( f(x) = 2|x + 2| - 6 \)[/tex]
- Here, the form can be rewritten as [tex]\( 2|x - (-2)| + (-6) \)[/tex].
- This means [tex]\( h = -2 \)[/tex] and [tex]\( k = -6 \)[/tex].
- The vertex of this function is [tex]\((-2, -6)\)[/tex], which does not match [tex]\((2, 6)\)[/tex].
After analyzing all four functions, we see that the function with a vertex at [tex]\((2, 6)\)[/tex] is:
[tex]\[ \boxed{f(x) = 2|x - 2| + 6} \][/tex]
So, the correct function is the second one, [tex]\(2|x - 2| + 6\)[/tex].
