The given equation is \( y = -x^2 - 5 \). 

This is a quadratic equation in the form of \( y = ax^2 + bx + c \), where:
- \( a = -1 \) (coefficient of \( x^2 \))
- \( b = 0 \) (coefficient of \( x \))
- \( c = -5 \) (constant term)

### Key Features:
1. **Shape of the Graph (Parabola)**: Since \( a = -1 \), the parabola opens **downwards**.
2. **Vertex**: The vertex of the parabola is at the point \((h, k)\). For this equation, since \( b = 0 \), the vertex is at \( (0, c) \), which gives us the vertex \( (0, -5) \).
3. **Axis of Symmetry**: The axis of symmetry is the vertical line that passes through the vertex, which is \( x = 0 \).
4. **Y-Intercept**: The y-intercept occurs where \( x = 0 \). Substituting \( x = 0 \) into the equation, \( y = -5 \), so the y-intercept is \( (0, -5) \).
5. **X-Intercepts**: The x-intercepts are found by setting \( y = 0 \) and solving for \( x \):

\[
0 = -x^2 - 5
\]
\[
x^2 = -5
\]

Since \( x^2 = -5 \) has no real solutions (as the square of a real number cannot be negative), the graph does not intersect the x-axis. Therefore, there are **no real x-intercepts**.

### Summary:
- The graph is a downward-opening parabola.
- The vertex is at \( (0, -5) \).
- The y-intercept is at \( (0, -5) \).
- There are no real x-intercepts.

Would you like more details or have any questions?

### Follow-up Questions:
1. What is the effect of changing the coefficient \( a \) on the shape of the parabola?
2. How would the graph change if the constant \( c \) were different?
3. Can you find the vertex of a quadratic equation using a different method?
4. What is the significance of the axis of symmetry in a parabola?
5. How do you determine the direction in which a parabola opens?
6. What are the solutions to \( x^2 = -5 \) in the complex number system?
7. How would the graph change if the equation were \( y = x^2 - 5 \)?
8. What are the characteristics of a parabola with no real x-intercepts?

**Tip:** The vertex of a parabola gives you the highest or lowest point on the graph, depending on the direction the parabola opens.

Explore the quadratic equation y = -x^2 - 5 and understand its properties. Learn about the shape of the graph, vertex, y-intercept, and implications of the axis of symmetry. Suitable for Grades 9-12.

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