To identify the location of the axis of symmetry for the function [tex]\( f(x) = 3(x + 5) - 4 \)[/tex], we will carefully analyze the components of the function.
First, let's rewrite the function in a more familiar quadratic form. We notice that the function given is:
[tex]\[ f(x) = 3(x + 5) - 4. \][/tex]
However, it seems there might be a slight notation issue since [tex]\( f(x) = 3(x + 5) \)[/tex] might need further clarification. Let's correct it and assume it should reflect a standard quadratic form [tex]\( f(x) = a(x - h)^2 + k \)[/tex].
Considering the standard form [tex]\( f(x) = 3(x + 5)^2 - 4 \)[/tex], we proceed with identifying the vertex form:
1. The term inside the parenthesis [tex]\( (x + 5) \)[/tex] indicates the horizontal shift. In vertex form [tex]\( f(x) = a(x - h)^2 + k \)[/tex], this would normally look like [tex]\( (x - (-5)) \)[/tex].
2. This horizontal shift indicates a shift to the left by 5 units.
3. The vertex form of a quadratic function is given by [tex]\( (h, k) \)[/tex], where [tex]\( x = h \)[/tex].
4. Here, [tex]\( h \)[/tex] is [tex]\(-5\)[/tex] and [tex]\( k \)[/tex] is [tex]\(-4\)[/tex] based on the vertex form.
The axis of symmetry for a quadratic function in the form [tex]\( f(x) = a(x - h)^2 + k \)[/tex] is given by the line [tex]\( x = h \)[/tex].
Therefore, for this function, [tex]\( h = -5 \)[/tex].
Thus, the axis of symmetry is:
[tex]\[ x = -5. \][/tex]
So, the correct answer is [tex]\( x = -5 \)[/tex].
