To determine which functions have an axis of symmetry of [tex]\( x = -2 \)[/tex], we will use the formula for the axis of symmetry for a quadratic function in the form [tex]\( ax^2 + bx + c \)[/tex], which is given by:
[tex]\[ x = -\frac{b}{2a} \][/tex]
We'll analyze each provided function to find their axis of symmetry.
1. For the function [tex]\( f(x) = x^2 + 4x + 3 \)[/tex]:
- Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = 4 \)[/tex].
- The axis of symmetry is calculated as [tex]\( x = -\frac{b}{2a} = -\frac{4}{2 \cdot 1} = -\frac{4}{2} = -2 \)[/tex].
This function has an axis of symmetry at [tex]\( x = -2 \)[/tex].
2. For the function [tex]\( f(x) = x^2 - 4x - 5 \)[/tex]:
- Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = -4 \)[/tex].
- The axis of symmetry is calculated as [tex]\( x = -\frac{b}{2a} = -\frac{-4}{2 \cdot 1} = \frac{4}{2} = 2 \)[/tex].
This function does not have an axis of symmetry at [tex]\( x = -2 \)[/tex].
3. For the function [tex]\( f(x) = x^2 + 6x + 2 \)[/tex]:
- Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = 6 \)[/tex].
- The axis of symmetry is calculated as [tex]\( x = -\frac{b}{2a} = -\frac{6}{2 \cdot 1} = -\frac{6}{2} = -3 \)[/tex].
This function does not have an axis of symmetry at [tex]\( x = -2 \)[/tex].
4. For the function [tex]\( f(x) = -2x^2 - 8x + 1 \)[/tex]:
- Here, [tex]\( a = -2 \)[/tex] and [tex]\( b = -8 \)[/tex].
- The axis of symmetry is calculated as [tex]\( x = -\frac{b}{2a} = -\frac{-8}{2 \cdot (-2)} = \frac{8}{-4} = -2 \)[/tex].
This function has an axis of symmetry at [tex]\( x = -2 \)[/tex].
5. For the function [tex]\( f(x) = -2x^2 + 8x - 2 \)[/tex]:
- Here, [tex]\( a = -2 \)[/tex] and [tex]\( b = 8 \)[/tex].
- The axis of symmetry is calculated as [tex]\( x = -\frac{b}{2a} = -\frac{8}{2 \cdot (-2)} = -\frac{8}{-4} = 2 \)[/tex].
This function does not have an axis of symmetry at [tex]\( x = -2 \)[/tex].
So, the functions that have an axis of symmetry at [tex]\( x = -2 \)[/tex] are:
- [tex]\( f(x) = x^2 + 4x + 3 \)[/tex]
- [tex]\( f(x) = -2x^2 - 8x + 1 \)[/tex]
