The equation

$y = x^2$

represents a parabola in the set of real numbers $\mathbb{R}^2$.

Key Characteristics of the Curve:

1. Shape: The graph of $y = x^2$ is a parabola that opens upwards. The curve is symmetric about the y-axis.

2. Vertex: The vertex of the parabola is at the origin $(0, 0)$. This is the lowest point on the curve because the coefficient of $x^2$ is positive, indicating the parabola opens upwards.

3. Axis of Symmetry: The parabola is symmetric around the y-axis. Mathematically, for every point $(x, y)$ on the curve, the point $(-x, y)$ is also on the curve.

4. Domain: The domain of the function is all real numbers, $\mathbb{R}$, because you can plug in any real value for $x$.

5. Range: The range of the function is all non-negative real numbers, $[0, \infty)$, because the square of any real number $x$ is always non-negative.

Visual Representation:

The parabola starts from the vertex at $(0, 0)$, and as $x$ increases or decreases, the value of $y = x^2$ increases, forming a "U"-shaped curve.

Would you like more details, or do you have any specific questions about this curve?

Related Questions:

1. How does the equation change if the parabola opens downward?
2. What happens to the graph of $y = x^2$ if we add a constant term?
3. How does the graph of $y = ax^2$ differ from $y = x^2$ for different values of $a$?
4. How would you find the focus and directrix of the parabola represented by $y = x^2$?
5. What transformations would move the vertex of this parabola to another point?

Tip: When analyzing quadratic equations like $y = x^2$, always consider the effect of coefficients and constants on the shape and position of the graph.