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To graph the parabola \( y = -x^2 \), we need to:

1. Identify the vertex.
2. Calculate two points to the left and two points to the right of the vertex.
3. Plot these points on the graph.

### Step-by-Step Solution:

1. **Vertex**:
   The equation \( y = -x^2 \) is in the form \( y = ax^2 + bx + c \), where \( a = -1 \), \( b = 0 \), and \( c = 0 \).
   - The vertex of a parabola \( y = ax^2 + bx + c \) occurs at \( x = -\frac{b}{2a} \).
   - Since \( b = 0 \), the vertex is at \( x = 0 \).
   - Substituting \( x = 0 \) into \( y = -x^2 \): \( y = -(0)^2 = 0 \).
   - **Vertex**: \( (0, 0) \).

2. **Points to the Left of the Vertex**:
   - Let \( x = -1 \): \( y = -(-1)^2 = -1 \). So, the point is \( (-1, -1) \).
   - Let \( x = -2 \): \( y = -(-2)^2 = -4 \). So, the point is \( (-2, -4) \).

3. **Points to the Right of the Vertex**:
   - Let \( x = 1 \): \( y = -(1)^2 = -1 \). So, the point is \( (1, -1) \).
   - Let \( x = 2 \): \( y = -(2)^2 = -4 \). So, the point is \( (2, -4) \).

4. **Points for Plotting**:
   - Vertex: \( (0, 0) \)
   - Points to the left: \( (-1, -1) \) and \( (-2, -4) \)
   - Points to the right: \( (1, -1) \) and \( (2, -4) \)

These points can be plotted on a coordinate plane to graph the parabola. The graph will open downwards because the coefficient of \( x^2 \) is negative.

Would you like more details or have any questions?

### Additional Questions:
1. How do you determine the direction of a parabola's opening based on its equation?
2. What effect does the coefficient \( a \) have on the width of a parabola?
3. How can you determine the axis of symmetry for a given parabola?
4. What are some characteristics of a parabola with a vertex at the origin?
5. How would the graph change if the equation were \( y = x^2 \) instead of \( y = -x^2 \)?

#### Tip:
The vertex form of a parabola, \( y = a(x-h)^2 + k \), makes it easier to identify the vertex and direction of opening directly.

Graph the parabola y = -x^2. Plot five points on the parabola: the vertex, two points to the left of the vertex, and two points to the right of the vertex.

Learn how to graph the parabola y = -x^2 by plotting five key points, including the vertex and points to the left and right of the vertex. Understand the quadratic function, its vertex form, and how the graph opens downwards.

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