To determine the vertex of the given absolute value function [tex]\( f(x) = -\frac{2}{3} |x + 4| \)[/tex], we need to understand the structure and properties of absolute value functions.
An absolute value function of the form [tex]\( f(x) = a |x - h| + k \)[/tex] has its vertex at the point [tex]\((h, k)\)[/tex]. Here, [tex]\(a\)[/tex] affects the steepness and direction of the function, [tex]\(h\)[/tex] is the horizontal shift, and [tex]\(k\)[/tex] is the vertical shift.
For the given function [tex]\( f(x) = -\frac{2}{3} |x + 4| \)[/tex]:
- The coefficient [tex]\( a = -\frac{2}{3} \)[/tex], which indicates the absolute value function is vertically compressed by a factor of [tex]\(\frac{2}{3}\)[/tex] and oriented downwards due to the negative sign.
- The term inside the absolute value is [tex]\(x + 4\)[/tex], which can be rewritten as [tex]\( x - (-4) \)[/tex]. This shows a horizontal shift of the function 4 units to the left ([tex]\(h = -4\)[/tex]).
- There is no [tex]\(k\)[/tex] term present, so [tex]\(k = 0\)[/tex], indicating no vertical shift.
Putting these together, the vertex of the function [tex]\( f(x) = -\frac{2}{3} |x + 4| \)[/tex] is at the point [tex]\((-4, 0)\)[/tex].
Thus, the vertex of the function is at [tex]\(\boxed{(-4, 0)}\)[/tex].
