To determine the interval over which the function [tex]\( f(x) = -(x + 8)^2 - 1 \)[/tex] is decreasing, we should examine certain characteristics of the function:
1. Function Form: The given function is a quadratic function of the form [tex]\( f(x) = -(x + 8)^2 - 1 \)[/tex]. Quadratic functions form parabolas and can be written in the vertex form [tex]\( a(x - h)^2 + k \)[/tex].
2. Vertex Identification: In the given function, it is clear that:
- The coefficient of [tex]\((x + 8)^2\)[/tex] is negative ([tex]\(-1\)[/tex]), meaning the parabola opens downwards.
- The term [tex]\((x + 8)\)[/tex] indicates a horizontal shift 8 units to the left from the origin.
- [tex]\(-1\)[/tex] is the vertical shift downward.
Therefore, the vertex (the maximum point of the parabola) is located at [tex]\( x = -8 \)[/tex]. Specifically, the vertex is at the point [tex]\((-8, -1)\)[/tex].
3. Function Behavior:
- Since the parabola opens downward (as indicated by the negative leading coefficient), the function reaches a maximum at the vertex and then decreases on both sides of this maximum point.
For a parabola that opens downwards, the function behavior can be summarized as:
- Increasing on the interval [tex]\((- \infty, -8)\)[/tex]
- Decreasing on the interval [tex]\((-8, \infty)\)[/tex]
4. Identifying the Correct Interval: From the previous points, the function [tex]\( f(x) \)[/tex] is decreasing for all [tex]\( x \)[/tex] values greater than the x-coordinate of the vertex.
Thus, the interval over which [tex]\( f(x) = -(x + 8)^2 - 1 \)[/tex] is decreasing is:
[tex]\[ (-8, \infty) \][/tex]
So, the correct answer is [tex]\( (-8, \infty) \)[/tex].
