To determine which function has an axis of symmetry of [tex]\( x = -2 \)[/tex], let's break down each given function and find the axis of symmetry.
The general form of a quadratic function is [tex]\( f(x) = a(x - h)^2 + k \)[/tex], where [tex]\( (h, k) \)[/tex] is the vertex of the parabola, and [tex]\( x = h \)[/tex] is the axis of symmetry.
Given functions:
1. [tex]\( f(x) = (x - 1)^2 + 2 \)[/tex]
- Here, the function is written in the form [tex]\( (x - h)^2 + k \)[/tex].
- Comparing, we see [tex]\( h = 1 \)[/tex].
- Therefore, the axis of symmetry is [tex]\( x = 1 \)[/tex].
2. [tex]\( f(x) = (x + 1)^2 - 2 \)[/tex]
- This function can be rewritten as [tex]\( (x - (-1))^2 - 2 \)[/tex].
- This gives [tex]\( h = -1 \)[/tex].
- Therefore, the axis of symmetry is [tex]\( x = -1 \)[/tex].
3. [tex]\( f(x) = (x - 2)^2 - 1 \)[/tex]
- Here, it matches the form [tex]\( (x - h)^2 + k \)[/tex] with [tex]\( h = 2 \)[/tex].
- Therefore, the axis of symmetry is [tex]\( x = 2 \)[/tex].
4. [tex]\( f(x) = (x + 2)^2 - 1 \)[/tex]
- This function can be rewritten as [tex]\( (x - (-2))^2 - 1 \)[/tex].
- This gives [tex]\( h = -2 \)[/tex].
- Therefore, the axis of symmetry is [tex]\( x = -2 \)[/tex].
From the detailed analysis above, the function with an axis of symmetry of [tex]\( x = -2 \)[/tex] is:
[tex]\[ f(x) = (x + 2)^2 - 1 \][/tex].
So, the function [tex]\(\boxed{f(x) = (x + 2)^2 - 1}\)[/tex] has an axis of symmetry of [tex]\( x = -2 \)[/tex].
