To determine the vertex of the function [tex]\( f(x) = -\frac{1}{2}|x+8| - 5 \)[/tex], we need to express the function in its vertex form. The vertex form of an absolute value function is given by:
[tex]\[ f(x) = a|x - h| + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the function, and [tex]\(a\)[/tex] determines the vertical stretch/compression and the direction of the opening.
The given function is:
[tex]\[ f(x) = -\frac{1}{2} |x + 8| - 5 \][/tex]
We can rewrite this function to match the vertex form:
[tex]\[ f(x) = -\frac{1}{2} |x - (-8)| + (-5) \][/tex]
From this form, we identify the parameters [tex]\(h\)[/tex] and [tex]\(k\)[/tex]:
[tex]\[ h = -8 \][/tex]
[tex]\[ k = -5 \][/tex]
Thus, the vertex of the function is [tex]\((-8, -5)\)[/tex].
So, the correct answer is:
A. [tex]\((-8, -5)\)[/tex]
