https://file.aiquickdraw.com/mathbot/file/1730491570306l8qqx6az.jpg

The image shows the function \( y = x^2 \), which represents a basic quadratic function. Here’s a breakdown of key properties for \( y = x^2 \):

1. **Shape**: The graph of \( y = x^2 \) is a parabola that opens upwards.
2. **Vertex**: The vertex of this parabola is at the origin, \( (0, 0) \).
3. **Axis of Symmetry**: The axis of symmetry is the vertical line \( x = 0 \).
4. **Domain**: The domain of \( y = x^2 \) is all real numbers \( (-\infty, \infty) \).
5. **Range**: Since \( y = x^2 \) only produces non-negative values, its range is \( [0, \infty) \).

Would you like to see a graph of this function, or perhaps explore transformations of this parabola?

Here are some related questions for further exploration:

1. What happens to the graph of \( y = x^2 \) if you add or subtract a constant?
2. How does the graph change if we replace \( x^2 \) with \( (x - h)^2 \) for some constant \( h \)?
3. What if we introduce a coefficient, such as \( y = ax^2 \)? How does the value of \( a \) affect the graph?
4. How do we find the roots of a quadratic function like \( y = x^2 - c \)?
5. What is the relationship between the derivative of \( y = x^2 \) and the slope of the tangent to the curve at any given point?

**Tip**: For quadratic functions, remember that the vertex form \( y = a(x - h)^2 + k \) can be particularly useful for identifying the vertex and transformations of the basic parabola.

Identify and explain the properties of the function y = x^2.

Explore the properties of the quadratic function y = x^2, including its parabolic shape, vertex, axis of symmetry, domain, and range. This explanation provides a breakdown of key features for students in Grades 8-10.

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