To determine the axis of symmetry for the quadratic function [tex]\( f(x) = 3(x + 4)^2 - 6 \)[/tex], we begin by identifying the standard form of a quadratic function.
The standard 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 the line [tex]\( x = h \)[/tex] is the axis of symmetry.
First, let's compare the given function with the standard form:
1. The given function is:
[tex]\[ f(x) = 3(x + 4)^2 - 6 \][/tex]
2. Express the function in the form [tex]\( (x - h) \)[/tex]:
[tex]\[ f(x) = 3(x - (-4))^2 - 6 \][/tex]
In this form, it is clear that [tex]\( h = -4 \)[/tex].
Therefore, the axis of symmetry for the function [tex]\( f(x) = 3(x + 4)^2 - 6 \)[/tex] is given by the line [tex]\( x = h \)[/tex].
So, the axis of symmetry is:
[tex]\[ x = -4 \][/tex]
Among the provided options:
- [tex]\( x = -4 \)[/tex]
- [tex]\( x = 4 \)[/tex]
- [tex]\( x = -6 \)[/tex]
- [tex]\( x = 6 \)[/tex]
The correct answer is:
[tex]\[ x = -4 \][/tex]
