The equation provided in the image is:

$\frac{x^2}{4} + \frac{y^2}{9} - \frac{z^2}{16} = 1$

This is the equation of a hyperboloid of one sheet. The general form of a hyperboloid of one sheet is:

$\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$

For this equation:

• $a^2 = 4$ implies $a = 2$
• $b^2 = 9$ implies $b = 3$
• $c^2 = 16$ implies $c = 4$

Thus, the hyperboloid opens along the $z$-axis and has elliptical cross-sections along planes perpendicular to the $z$-axis.

Key Features of the Graph:

• Symmetry: The hyperboloid is symmetric about the $x$, $y$, and $z$-axes.
• Shape: The graph consists of one continuous surface that narrows near the origin and expands away from it.
• Cross-Sections:
  • The cross-sections parallel to the $xy$-plane are ellipses.
  • The cross-sections parallel to the $xz$-plane or $yz$-plane are hyperbolas.

Would you like me to generate a 3D graph of this hyperboloid?

Related Questions:

1. What is the difference between a hyperboloid of one sheet and a hyperboloid of two sheets?
2. How do you determine which axis a hyperboloid opens along?
3. What are the applications of hyperboloids in real-world structures?
4. Can a hyperboloid of one sheet be related to other types of quadratic surfaces like ellipsoids or paraboloids?
5. How do changes in coefficients (like 4, 9, and 16) affect the shape and size of a hyperboloid?

Tip:

When graphing 3D surfaces, it's helpful to examine cross-sections along different planes to understand the structure better!