Question

Name and sketch the graph of each of the following equations in three-space. $$ x^{2}+y^{2}-4 z^{2}+4=0 $$

Short answer

The graph is a hyperboloid of two sheets.

Step-by-step solution

Simplify the Equation

The given equation is \( x^2 + y^2 - 4z^2 + 4 = 0 \). First, isolate the constant term by subtracting 4 from both sides. This gives: \( x^2 + y^2 - 4z^2 = -4 \).

Recognize the Form

The form \( x^2 + y^2 - 4z^2 = -4 \) resembles the standard equation of a hyperboloid of two sheets, which is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1 \). Here, \( a^2 = 1 \), \( b^2 = 1 \), and \( c^2 = 0.25 \).

Convert to Standard Form

Divide the entire equation by -4 to achieve \(-\frac{x^2}{4} - \frac{y^2}{4} + \frac{z^2}{1} = 1\). Simplifying, we rewrite the equation as \( \frac{x^2}{4} + \frac{y^2}{4} - z^2 = -1 \). Since the equation now resembles \( -\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \), switch the signs and confirm that this matches the hyperboloid of two sheets standard form.

Sketch the Graph

Since this is a hyperboloid of two sheets, visualize it as two separate curved surfaces that open along the z-axis. Each sheet is symmetric about the xz-plane and yz-plane. The point \( z = 0 \) does not exist on the surface as substitution does not satisfy the equation.

Key concepts

Three-Dimensional Graphing

Three-dimensional graphing involves visually representing equations in a 3D space, which gives us a better understanding of how the mathematical surfaces look. For this exercise, we are dealing with a hyperboloid of two sheets. This particular surface is symmetrical and defined by the equation \( \frac{x^2}{4} + \frac{y^2}{4} - z^2 = -1 \).

To sketch this graph, imagine two separate but symmetrical surfaces that do not touch the xy-plane. These surfaces open along the z-axis. Recognize that, unlike a sphere or cylinder, a hyperboloid of two sheets does not intersect the z=0 plane. This implies that the graph consists of two pieces, located above and below the centerpoint along the z-axis.

When approaching three-dimensional graphing:

• Begin by identifying the principal directions (x, y, z).
• Understand the symmetry of the graph; in this case, around the xz-plane and the yz-plane.
• Plotting points or using software can help visualize complex surfaces effectively.

Equation Simplification

Simplifying an equation is an essential skill in mathematics, especially for solving problems involving geometry in three-dimensional space. In the given exercise, the original equation \( x^2 + y^2 - 4z^2 + 4 = 0 \) was simplified step by step to uncover its geometric form.

The process involves several techniques:

• Isolate constant terms and simplify each side.
• Identify the form of the equation it resembles.
• Transform it to match standard forms, which allows easier recognition.

In simplest terms, simplifying helps in revealing the true nature of the equation and facilitates understanding when moving forward with drawing, graphing, or interpreting mathematical surfaces.

Coordinate Geometry

Coordinate geometry is the mathematical study that connects algebra to geometry through graphs and equations. It allows us to not only visualize mathematical concepts but also solve them analytically. In the context of a hyperboloid of two sheets, understanding the coordinate geometry helps in placing the graph in 3-dimensional space effectively.

Key things to remember about coordinate geometry when working on graphing or analyzing hyperboloids:

• Axes Orientation: Become familiar with the orientation of the graph's principal axis—in this case, along the z-axis.
• Symmetry: Recognize the symmetrical nature and its relevance in positioning the graph accurately.
• Equation Variables: Learn to decode variables' roles (x, y, z) in relation to real-world dimensional analysis.

Using these principles helps surface the necessary information to draw and calculate the placement of curves in a 3D field.

Conic Sections

Conic sections are figures obtained by intersecting a cone with a plane. They are foundational in understanding more complex three-dimensional figures like hyperboloids. The equation derived in the solution resembles a hyperbola, which can lead to recognizing the hyperboloid of two sheets.

For conic sections in three-dimensional problems:

• Know the basic types of conic sections: circles, ellipses, parabolas, and hyperbolas.
• Understand that hyperbolas are directly linked to hyperboloids of two sheets in 3D space when extended.
• Utilize the forms of conic equations to explore multi-dimensional graphing, recognizing which 2D conic sections resemble their 3D counterparts.

Mastering conic sections equips students with the groundwork needed to tackle complex geometric shapes and surfaces like hyperboloids.