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?

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.