Question

Name and sketch the graph of each of the following equations in three-space. $$ 9 x^{2}-y^{2}+9 z^{2}-9=0 $$

Short answer

The surface is a hyperboloid of one sheet centered along the y-axis, stretched along the y-axis.

Step-by-step solution

Rearranging the Equation

Given the equation is \(9x^2 - y^2 + 9z^2 - 9 = 0\). We first rearrange it to make it look like a standard form for identifying the surface type. Add 9 to both sides to get \(9x^2 - y^2 + 9z^2 = 9\).

Dividing through by 9

We divide every term in the equation \(9x^2 - y^2 + 9z^2 = 9\) by 9, resulting in \(x^2 - \frac{y^2}{9} + z^2 = 1\).

Analyzing the Equation

The equation \(x^2 - \frac{y^2}{9} + z^2 = 1\) can be rewritten as \(\frac{x^2}{1} - \frac{y^2}{9} + \frac{z^2}{1} = 1\). This is in the form of a hyperboloid of one sheet, which is represented by \(\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\).

Determining Characteristics

From the equation \(\frac{x^2}{1} - \frac{y^2}{9} + \frac{z^2}{1} = 1\), it can be seen that the hyperboloid has a vertical axis of symmetry along the y-axis. The constants in the denominators indicate the scaling: stretching along the y-axis by the factor of 3.

Sketching the Graph

To sketch the graph, imagine a hyperboloid of one sheet with its axis along the y-direction. It is circular in cross-section perpendicular to the y-axis and is contracted in the y-axis direction compared to x and z directions.

Key concepts

Hyperboloid of One Sheet

A hyperboloid of one sheet is a fascinating and important type of surface in three-dimensional space. It has a distinct shape that resembles a cooling tower or a sports trophy. This surface can be described using a mathematical equation of the form: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \). The equation indicates that the hyperboloid consists of one continuous surface without any breaks.
In practical terms, you can think of it as a smooth, saddle-like shape but extended to form a curved tube or tunnel. Understanding this form helps in visualizing complex structures in higher dimensions, as well as in architecture and physics where these shapes are commonly applied.

Equation Rearrangement

Rearranging equations is a crucial step in graphing and identifying mathematical surfaces in three-dimensional graphs. For the given equation \(9x^2 - y^2 + 9z^2 - 9 = 0\), the first step is to move all terms except the variable terms to the other side. This is crucial for simplifying and recognizing the standard form.
After adding 9 to both sides, the equation becomes \(9x^2 - y^2 + 9z^2 = 9\). The next step involves dividing all terms by 9, resulting in \(x^2 - \frac{y^2}{9} + z^2 = 1\). This places the equation in a recognizable form similar to the standard equation of a hyperboloid of one sheet, making it easier to work with.

Graph Sketching

Graph sketching in three-space involves visualizing and illustrating the three-dimensional surface described by an equation. For a hyperboloid of one sheet, the equation \( \frac{x^2}{1} - \frac{y^2}{9} + \frac{z^2}{1} = 1 \) provides the blueprint for sketching.
To begin, identify the axis and note that this surface is circular in cross-section perpendicular to the y-axis. Imagine slices of the graph in planes parallel to the xz-plane and notice that they look like expanding and contracting circles. The symmetry and proportions, informed by the coefficients in the equation, guide this sketching. This visualization helps in understanding complex surfaces and their properties in relation to each axis.

Axis of Symmetry

The axis of symmetry is a crucial feature for understanding and sketching three-dimensional graphs like the hyperboloid of one sheet. From the rearranged and simplified equation \( \frac{x^2}{1} - \frac{y^2}{9} + \frac{z^2}{1} = 1 \), the axis of symmetry can be identified easily.
In this equation, the y-axis is the axis of symmetry. This means the figure is symmetrical with respect to rotations around the y-axis. Any slice perpendicular to the y-axis results in similar shapes. The equation also indicates that compared to x and z, the y-dimension is stretched more. This symmetry fosters a deeper understanding of how changes in one dimension may affect the entire shape. Recognizing this helps in both the theoretical and practical applications of three-dimensional graphing.