To identify the vertex of the function [tex]\( f(x) = -\frac{1}{2} |x + 8| - 5 \)[/tex], we need to understand the transformations applied to the basic absolute value function [tex]\( f(x) = |x| \)[/tex].
1. Horizontal Shift: The expression inside the absolute value, [tex]\( x + 8 \)[/tex], indicates a horizontal shift. The general form [tex]\( |x - h| \)[/tex] shifts the graph [tex]\( h \)[/tex] units to the right if [tex]\( h \)[/tex] is positive, and [tex]\( h \)[/tex] units to the left if [tex]\( h \)[/tex] is negative. In this case, [tex]\( x + 8 \)[/tex] can be interpreted as [tex]\( x - (-8) \)[/tex], indicating a shift 8 units to the left. Thus, the x-coordinate of the vertex is [tex]\( -8 \)[/tex].
2. Vertical Shift: The constant term outside the absolute value function, [tex]\( -5 \)[/tex], shifts the graph vertically. A positive value shifts the graph up, and a negative value shifts it down. Here, [tex]\( -5 \)[/tex] indicates a downward shift of 5 units. Thus, the y-coordinate of the vertex is [tex]\( -5 \)[/tex].
3. Combining these shifts, the vertex of the function is at the coordinate [tex]\( (-8, -5) \)[/tex].
Therefore, the correct answer is:
A. [tex]\((-8, -5)\)[/tex]
