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].
