To determine the vertex of the function [tex]\( f(x) = -\frac{1}{2}|x+8| - 5 \)[/tex], we need to identify the form of the function and the parameters involved in its structure.
The given function is an absolute value function of the form:
[tex]\[ f(x) = a|x-h| + k \][/tex]
Here, [tex]\( a \)[/tex] controls the vertical stretch/compression and the direction of the opening, [tex]\( h \)[/tex] represents the horizontal shift, and [tex]\( k \)[/tex] represents the vertical shift.
For the function [tex]\( f(x) = -\frac{1}{2}|x+8| - 5 \)[/tex]:
1. [tex]\( a = -\frac{1}{2} \)[/tex]: This indicates that the absolute value graph is vertically compressed by a factor of [tex]\(\frac{1}{2}\)[/tex] and reflected across the x-axis.
2. The term [tex]\( |x+8| \)[/tex] indicates a horizontal shift. The value inside the absolute value is in the form [tex]\( x - (-8) \)[/tex], which means the graph is shifted 8 units to the left. Therefore, [tex]\( h = -8 \)[/tex].
3. The constant term [tex]\( -5 \)[/tex] indicates a vertical shift downward by 5 units. Hence, [tex]\( k = -5 \)[/tex].
Therefore, the vertex (h, k) of the function [tex]\( f(x) = -\frac{1}{2}|x+8| - 5 \)[/tex] is:
[tex]\[ (-8, -5) \][/tex]
The correct answer is:
A. [tex]\( (-8, -5) \)[/tex]
