f(x, y, z) = x^2 + y^2− 3z   is it a....
ellipsoids
paraboloids    
hyperbolic paraboloids
planes
generalized cylinders
cones
Incorrect: Your answer is incorrect.

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.

f(x, y, z) = x^2 + y^2− 3z. Is it a(n): ellipsoid, paraboloid, hyperbolic paraboloid, plane, generalized cylinder, or cone?

Explore how to classify the surface represented by f(x, y, z) = x^2 + y^2 - 3z as an elliptic paraboloid. Learn about surface geometry, quadratic forms, and how this equation fits into the classification of geometric shapes. Suitable for students in Grades 10-12.

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