The given function is:

$f(x, y, z) = x^2 + y^2 - 3z$

To classify this surface, we should rearrange the equation:

$x^2 + y^2 - 3z = 0$

This can be rewritten as:

$z = \frac{1}{3}(x^2 + y^2)$

Surface Type:

This equation represents a paraboloid because it has a quadratic form in $x$ and $y$ and is linear in $z$.

To be more precise, it is an elliptic paraboloid, which is a surface generated by the equation:

$z = ax^2 + by^2$

where $a = b = \frac{1}{3}$ in this case, making the surface symmetric about the $z$-axis.

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Do you want more details, or do you have any further questions?
Here are some related questions:

1. How do you distinguish between elliptic and hyperbolic paraboloids?
2. What is the general form of a paraboloid?
3. Can you give an example of a real-world object modeled by a paraboloid?
4. How does the coefficient of $z$ influence the shape of the paraboloid?
5. How would the surface change if one of the terms in $x^2$ or $y^2$ was negative?

Tip: Visualizing 3D surfaces can help in understanding their geometric properties better. Graphing calculators or 3D plotting tools can assist in this.