To find the vertex of the function [tex]\( f(x) = -|x + 7| - 4 \)[/tex], we will analyze its structure, specifically focusing on the form and parameters of the absolute value function.
1. Identify the standard form: The general form of an absolute value function can be written as [tex]\( f(x) = a|x - h| + k \)[/tex], where [tex]\( (h, k) \)[/tex] represents the vertex of the function.
2. Recognize the given function parameters:
- Note that the given function is [tex]\( f(x) = -|x + 7| - 4 \)[/tex].
- We need to rewrite [tex]\( x + 7 \)[/tex] in a form that matches [tex]\( x - h \)[/tex]. This can be done by observing that [tex]\( x + 7 \)[/tex] is equivalent to [tex]\( x - (-7) \)[/tex]. Hence, [tex]\( h = -7 \)[/tex].
3. Identify the vertical shift:
- The constant term outside the absolute value function, [tex]\(-4\)[/tex], represents the vertical shift [tex]\( k \)[/tex].
- Hence, [tex]\( k = -4 \)[/tex].
4. Determine the vertex: Using the parameters identified for [tex]\( h \)[/tex] and [tex]\( k \)[/tex]:
- The vertex [tex]\( (h, k) = (-7, -4) \)[/tex].
5. Conclusion: The vertex of the function [tex]\( f(x) = -|x + 7| - 4 \)[/tex] is [tex]\( (-7, -4) \)[/tex].
Therefore, the correct choice is:
A. [tex]\( (-7, -4) \)[/tex].
