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.