Here is the graph of the function $f(x) = -x^2 - 1$. It represents a downward-opening parabola with its vertex at $(0, -1)$.

Would you like more details or have any questions about this graph?

Here are five related questions for further exploration:

1. What is the vertex of the parabola $f(x) = -x^2 - 1$?
2. How does the graph of $f(x) = -x^2 - 1$ compare to $f(x) = x^2 + 1$?
3. How do you find the x-intercepts and y-intercepts of the graph?
4. What is the axis of symmetry of the parabola?
5. How does the value of the coefficient of $x^2$ affect the shape of the parabola?

Tip: The sign of the coefficient in front of $x^2$ determines whether the parabola opens upwards or downwards.