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La ecuación que corresponde a la gráfica es:



−𝑥 2 + 𝑦 2 − 𝑧 2 = 1

𝑥 2 − 9 = 𝑧

36𝑥 2 − 𝑦 2 + 9𝑧 2 = 144

The image you provided appears to depict a three-dimensional surface along with a set of equations. These equations describe the mathematical relationships that characterize the geometry of the surface.

The equations listed are:
1. \(-x^2 + y^2 - z^2 = 1\)
2. \(x^2 - 9 = z\)
3. \(36x^2 - y^2 + 9z^2 = 144\)

To understand this surface better, let's analyze each equation:

1. **\(-x^2 + y^2 - z^2 = 1\)**: This equation represents a hyperboloid of one sheet. It's a quadric surface which is a general form of a hyperboloid. 

2. **\(x^2 - 9 = z\)**: This equation seems unconventional when describing typical surfaces since it's setting \(z\) as a function of \(x\), while \(x^2 - 9\) would generally imply \(x\) is related to \(z\) under a shifted parabolic curve if plotted in 2D.

3. **\(36x^2 - y^2 + 9z^2 = 144\)**: This equation can be rearranged to the form \(\frac{x^2}{4} - \frac{y^2}{144} + \frac{z^2}{16} = 1\), which describes a hyperboloid of two sheets depending on the sign in front of \(y^2\).

It's important to note that these equations might represent different sections or constraints on the same surface or they could be describing the relationships and intersections of multiple surfaces in space. Each equation defines a surface, and their interaction could potentially define complex structures in three-dimensional space.

Would you like to delve into more specifics about these equations or their graphical interpretations? 

Here are some related questions you might consider:

1. How do you graph a hyperboloid of one sheet?
2. What does the equation \(x^2 - 9 = z\) represent in three dimensions?
3. Can we find any intersections between these surfaces analytically?
4. How does the form of the equation affect the geometry of the surface?
5. What are the practical applications of studying such complex surfaces?

Tip: For better visualization and understanding of these equations, plotting them using a 3D graphing software can provide insightful perspectives on their geometrical properties.

La ecuación que corresponde a la gráfica es:

−𝑥^2 + 𝑦^2 − 𝑧^2 = 1
𝑥^2 − 9 = 𝑧
36𝑥^2 − 𝑦^2 + 9𝑧^2 = 144

Explore the graph of hyperboloids and quadric surfaces represented by the equations −x^2 + y^2 − z^2 = 1, x^2 − 9 = z, and 36x^2 − y^2 + 9z^2 = 144. Learn the algebraic structure and geometry of these surfaces. Suitable for Grades 10-12 and College students.

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