The function given is:

\[
f(x) = -x^2 - 1
\]

### Analysis:
1. **Type of Function**: This is a **quadratic function** in the form \( f(x) = ax^2 + bx + c \), where \( a = -1 \), \( b = 0 \), and \( c = -1 \).
   
2. **Shape of the Graph**: The graph will be a **parabola**. Since the coefficient \( a = -1 \) is negative, the parabola opens **downwards**.

3. **Vertex**: The vertex of the parabola can be calculated using the formula for the \( x \)-coordinate of the vertex:  
   \[
   x = -\frac{b}{2a}
   \]
   Substituting \( b = 0 \) and \( a = -1 \):
   \[
   x = -\frac{0}{2(-1)} = 0
   \]
   Now, substituting \( x = 0 \) into \( f(x) \):
   \[
   f(0) = -(0)^2 - 1 = -1
   \]
   So, the **vertex** is at \( (0, -1) \).

4. **Axis of Symmetry**: The axis of symmetry is the vertical line through the vertex:
   \[
   x = 0
   \]

5. **Y-intercept**: The y-intercept occurs when \( x = 0 \):
   \[
   f(0) = -1
   \]
   So, the **y-intercept** is \( (0, -1) \).

6. **X-intercepts**: To find the x-intercepts, set \( f(x) = 0 \):
   \[
   -x^2 - 1 = 0 \Rightarrow -x^2 = 1 \Rightarrow x^2 = -1
   \]
   Since \( x^2 = -1 \) has no real solutions, there are **no x-intercepts**.

### Summary:
- The vertex is at \( (0, -1) \).
- The parabola opens downwards.
- The axis of symmetry is \( x = 0 \).
- There is a y-intercept at \( (0, -1) \).
- No x-intercepts.

Would you like to see a graph or have more details on this?

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### Related Questions:
1. What would be the vertex if the function was \( f(x) = -x^2 + 2x - 1 \)?
2. How does the sign of the coefficient \( a \) affect the direction in which the parabola opens?
3. What transformations would shift the vertex of this function to \( (2, -1) \)?
4. How do you find the maximum or minimum value of a quadratic function?
5. How would you solve for \( x \) if the function were \( f(x) = -x^2 + 3x - 4 \)?

### Tip:
For any quadratic function of the form \( ax^2 + bx + c \), the vertex is always given by \( x = -\frac{b}{2a} \). This helps quickly locate the highest or lowest point on the parabola.

Explore the quadratic function f(x) = -x^2 - 1 with a detailed step-by-step analysis. Understand the key concepts of the vertex, axis of symmetry, intercepts, and graphing the parabola. Learn how the negative coefficient affects the shape of the graph. Suitable for students in Grades 9-10.

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