The equation you've provided is:

\[
y = x^2 + 3
\]

This represents a quadratic function. Let's break it down:

1. **Parabola**: The graph of this equation is a parabola, opening upwards, since the coefficient of \(x^2\) is positive.
2. **Vertex**: The vertex of the parabola is at \((0, 3)\), because the constant term \(+3\) shifts the graph upwards by 3 units.
3. **Axis of symmetry**: The axis of symmetry is the vertical line \(x = 0\), which passes through the vertex.
4. **Y-intercept**: The y-intercept is at \((0, 3)\), because when \(x = 0\), \(y = 3\).
5. **X-intercepts**: There are no real x-intercepts, as the equation \(x^2 + 3 = 0\) does not have real solutions (it only has complex solutions).

Would you like more details or have any questions?

Here are five questions related to this:

1. How would the graph change if we modified the equation to \(y = x^2 - 3\)?
2. What are the complex solutions to the equation \(x^2 + 3 = 0\)?
3. How does the value of the constant term affect the graph of a quadratic equation?
4. Can you describe how to find the axis of symmetry for any quadratic function?
5. What is the general form of a quadratic equation?

**Tip**: For quadratic equations, the vertex form is useful for identifying the vertex and transformations of the graph easily!

Explore the properties of the quadratic function y = x^2 + 3, including its parabolic shape, vertex at (0, 3), and lack of real x-intercepts. Learn how changes in the constant term affect the graph and key concepts such as the axis of symmetry and vertex. Suitable for students in Grades 9-10.

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