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To graph the parabola given by the equation \( y = -x^2 \), let's analyze its characteristics:

1. **Vertex**: The equation \( y = -x^2 \) represents a parabola that opens downward because the coefficient of \( x^2 \) is negative. The vertex of the parabola is at the origin, \( (0, 0) \).

2. **Points to plot**: We need to choose two points to the left and two to the right of the vertex to plot five points in total.

Let's choose the following values for \( x \):

- \( x = -2 \), \( x = -1 \), \( x = 0 \), \( x = 1 \), and \( x = 2 \).

Now, we calculate the corresponding \( y \)-values using the equation \( y = -x^2 \):

- For \( x = -2 \), \( y = -(-2)^2 = -4 \) → Point: \( (-2, -4) \).
- For \( x = -1 \), \( y = -(-1)^2 = -1 \) → Point: \( (-1, -1) \).
- For \( x = 0 \), \( y = -(0)^2 = 0 \) → Point: \( (0, 0) \).
- For \( x = 1 \), \( y = -(1)^2 = -1 \) → Point: \( (1, -1) \).
- For \( x = 2 \), \( y = -(2)^2 = -4 \) → Point: \( (2, -4) \).

These five points can be plotted on a coordinate plane to graph the parabola.

Would you like me to provide a detailed graph of the parabola?

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Here are five related questions:

1. How does changing the sign of the coefficient of \( x^2 \) affect the orientation of the parabola?
2. What happens to the graph if we add a constant to the equation, such as \( y = -x^2 + 3 \)?
3. How can we determine the axis of symmetry for a parabola?
4. What are the properties of the vertex form of a quadratic equation?
5. How does the parabola's width change if the equation is modified to \( y = -0.5x^2 \)?

**Tip**: The general form of a parabola's equation is \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) determine the parabola's shape and position on the coordinate plane.

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.

This exercise demonstrates how to graph the parabola y = -x^2 by plotting five key points, including the vertex and points to the left and right of it. Learn how to identify the vertex and symmetry of a downward-facing parabola.

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