To determine over what interval the graph of the function \( f(x) = -(x+8)^2 - 1 \) is decreasing, we need to analyze the behavior of the function.
First, let's consider the structure of the function:
[tex]\[ f(x) = -(x+8)^2 - 1 \][/tex]
This is a quadratic function in the form of \( f(x) = -a(x - h)^2 + k \), where \( a = 1 \), \( h = -8 \), and \( k = -1 \). This type of equation represents a parabola that opens downwards because of the negative sign in front of the squared term.
To analyze where the function is decreasing, we need to find its vertex and then examine the intervals around it. The vertex can be determined from the equation \( x = h \). In this case:
[tex]\[ x = -8 \][/tex]
Thus, the vertex of \( f(x) \) is at \( x = -8 \). Since the parabola opens downwards, it will be decreasing to the left of this point (i.e., for \( x < -8 \)) and increasing to the right of this point (i.e., for \( x > -8 \)).
Therefore, the interval over which the function \( f(x) \) is decreasing is:
[tex]\[ (-\infty, -8) \][/tex]
Thus, the correct interval is:
[tex]\[ \boxed{(-\infty, -8)} \][/tex]
