To graph the quadratic equation \( y = -x^2 - 5 \), follow these steps:

### Step 1: Identify Key Components
1. **Equation**: \( y = -x^2 - 5 \)
2. **Vertex**: The vertex is at \( (0, -5) \).
3. **Direction**: Since the coefficient of \( x^2 \) is negative, the parabola opens downwards.
4. **Axis of Symmetry**: The axis of symmetry is the vertical line \( x = 0 \).
5. **Y-Intercept**: The y-intercept is \( (0, -5) \).
6. **X-Intercepts**: There are no real x-intercepts, as \( x^2 = -5 \) has no real solutions.

### Step 2: Plot the Vertex
- Start by plotting the vertex at \( (0, -5) \) on the coordinate plane.

### Step 3: Plot Additional Points
- Choose values of \( x \) on either side of the vertex to plot additional points:
  - For \( x = 1 \):
    \[
    y = -(1)^2 - 5 = -1 - 5 = -6
    \]
    Plot the point \( (1, -6) \).
  
  - For \( x = -1 \):
    \[
    y = -(-1)^2 - 5 = -1 - 5 = -6
    \]
    Plot the point \( (-1, -6) \).

  - For \( x = 2 \):
    \[
    y = -(2)^2 - 5 = -4 - 5 = -9
    \]
    Plot the point \( (2, -9) \).

  - For \( x = -2 \):
    \[
    y = -(-2)^2 - 5 = -4 - 5 = -9
    \]
    Plot the point \( (-2, -9) \).

### Step 4: Draw the Parabola
- Connect the points smoothly, forming a U-shaped curve that opens downward. The graph should be symmetrical about the axis of symmetry \( x = 0 \).

### Step 5: Label Key Points
- Label the vertex at \( (0, -5) \).
- Label the y-intercept at \( (0, -5) \).
- Optionally, label the points \( (1, -6) \), \( (-1, -6) \), \( (2, -9) \), and \( (-2, -9) \).

### Example Graph
Below is an example of how the graph might look:

- The parabola opens downward with its highest point at \( (0, -5) \).
- The graph is symmetrical around the y-axis.

Would you like any additional details on this process?

### Follow-up Questions:
1. How do you determine the y-intercept of a quadratic equation?
2. Why is it important to choose points on both sides of the vertex?
3. How would the graph look if the coefficient of \( x^2 \) was positive?
4. What are the implications of having no real x-intercepts?
5. How does the vertex form help in graphing quadratic equations?
6. How can you check if your graph is correct?
7. What would happen if you changed the constant \( -5 \) in the equation?
8. How can you find the minimum or maximum value of a quadratic equation?

**Tip:** Always start graphing a parabola by plotting the vertex and using symmetry to plot additional points. This ensures an accurate graph.

Learn how to graph a quadratic equation y = -x^2 - 5 with detailed steps and example graphs. Understand key concepts such as the vertex, axis of symmetry, and plotting points for accurate graphing. Suitable for high school students studying quadratic equations.

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