https://file.aiquickdraw.com/mathbot/file/1727664607961ks9f50er.jpg

Let's analyze the graphs and match them with the equations. The given functions describe different 3D surfaces, and we need to pair each equation with the appropriate graph.

### 1. Analyze the equations:

1. \( x^2 + y^2 + \frac{1}{4} z^2 = 1 \)  
   This is an ellipsoid, as the equation resembles the general form of an ellipsoid:  
   \( \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \), where \(a = 1\), \(b = 1\), and \(c = 2\). This suggests that the graph is a closed, oval-like shape.
   
2. \( z = -\sqrt{4 - x^2 - y^2} \)  
   This is a hemisphere (the lower part), since the equation restricts the values of \(z\) to be negative and \(x^2 + y^2 \leq 4\), corresponding to a portion of a sphere.

3. \( y^2 + \frac{1}{4} z^2 = 1 \)  
   This is a cylinder with elliptical cross-sections in the \(yz\)-plane, indicating the shape should resemble an elliptical cylinder extended along the \(x\)-axis.

4. \( \frac{1}{4} x^2 - y^2 - z^2 = 1 \)  
   This represents a hyperboloid of two sheets, as the equation is of the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 \), which describes a hyperboloid shape.

5. \( |z| = \sqrt{x^2 + y^2} \)  
   This is a double cone (an absolute value sign shows two symmetrical surfaces), which suggests the graph should display a cone with vertex at the origin.

6. \( \frac{1}{9} z^2 - \frac{1}{4} y^2 = 1 \)  
   This is a hyperboloid of one sheet, where the equation describes a saddle shape or a surface that curves upwards and downwards in different directions.

### 2. Match the graphs:

- **Graph I**: This graph shows a hemisphere, which matches the equation \( z = -\sqrt{4 - x^2 - y^2} \) (equation 2).
- **Graph II**: This is a cylindrical shape, which corresponds to \( y^2 + \frac{1}{4} z^2 = 1 \) (equation 3).
- **Graph III**: This is a double cone, matching the equation \( |z| = \sqrt{x^2 + y^2} \) (equation 5).
- **Graph IV**: This shows a hyperboloid of one sheet, corresponding to \( \frac{1}{9} z^2 - \frac{1}{4} y^2 = 1 \) (equation 6).
- **Graph V**: This is an ellipsoid, matching \( x^2 + y^2 + \frac{1}{4} z^2 = 1 \) (equation 1).
- **Graph VI**: This graph shows a hyperboloid of two sheets, which matches \( \frac{1}{4} x^2 - y^2 - z^2 = 1 \) (equation 4).

### Final Matching:
- **I** → \( z = -\sqrt{4 - x^2 - y^2} \)
- **II** → \( y^2 + \frac{1}{4} z^2 = 1 \)
- **III** → \( |z| = \sqrt{x^2 + y^2} \)
- **IV** → \( \frac{1}{9} z^2 - \frac{1}{4} y^2 = 1 \)
- **V** → \( x^2 + y^2 + \frac{1}{4} z^2 = 1 \)
- **VI** → \( \frac{1}{4} x^2 - y^2 - z^2 = 1 \)

Would you like more details or explanations on any part of this process?

Here are 5 related questions for you to consider:
1. How does the ellipsoid equation change with different scaling factors for \(x\), \(y\), and \(z\)?
2. What are the differences between a hyperboloid of one sheet and two sheets in terms of their equation and graph?
3. Can a cylinder have circular cross-sections? How would its equation differ from an elliptical cylinder?
4. How do you find the vertices and axes of a hyperboloid from its equation?
5. What transformations are involved in obtaining a cone from the equation \( |z| = \sqrt{x^2 + y^2} \)?

**Tip:** When analyzing 3D surfaces, pay close attention to whether terms are squared or involve square roots, as these often indicate specific surface types like spheres, cones, or hyperboloids.

Match each function with its graph from the given options. Determine which of the provided equations corresponds to each of the 3D graphs displayed in the image.

Learn how to match 3D graphs with equations involving ellipsoids, hyperboloids, and conic sections. Understand key geometric properties and algebraic representations to identify the correct graph for each equation. This problem is suitable for Grades 11-12 and college-level mathematics.

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