What does the equation 
y = x2
 represent as a curve in the set of real numbers2?

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.

What does the equation y = x^2 represent as a curve in the set of real numbers?

Explore the graph of the quadratic function y = x^2, which represents a parabola in the set of real numbers. Learn key characteristics such as the vertex, axis of symmetry, domain, and range. Suitable for students in Grades 9-10.

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