The function provided is:

$f(x) = -x^2 - 1$

Detailed Analysis:

1. Type of Function: This is a quadratic function in the general form $f(x) = ax^2 + bx + c$, where:

   • $a = -1$,
   • $b = 0$, and
   • $c = -1$.

2. Shape and Direction: Since $a = -1$, the parabola opens downwards. A negative coefficient for $x^2$ means the parabola is "upside down."

3. Vertex of the Parabola: The vertex can be found using the vertex formula:

   [ x = -\frac{b}{2a}