Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

The graph of an equation in \(x, y\), and \(z\) is symmetric with respect to the \(x y\) -plane if replacing \(z\) by \(-z\) results in an equivalent equation. What condition leads to a graph that is symmetric with respect to each of the following? (a) \(y z\) -plane (b) \(z\) -axis (c) origin

Step-by-Step Solution

Answer

(a) Replace \\(x\\) by \\(-x\\), (b) Replace \\(x\\) and \\(y\\) by \\(-x\\) and \\(-y\\), (c) Replace all with negatives.

Step 1: Analyze symmetry with respect to the yz-plane

To determine symmetry with respect to the yz-plane, consider the role of reflecting points across this plane. Any point \(x, y, z\) becomes \(-x, y, z\). Thus, the equation is symmetric with respect to the yz-plane if replacing \(x\) with \(-x\) results in an equivalent equation.

Step 2: Analyze symmetry with respect to the z-axis

Symmetry with respect to the z-axis involves rotating around this axis and thus changing both \(x\) and \(y\). Specifically, the transformation in this case is \(-x, -y, z\). An equation is symmetric about the z-axis if this transformation does not change the form of the equation.

Step 3: Analyze symmetry with respect to the origin

Symmetry with respect to the origin involves inverting all coordinates, meaning \(x, y, z\) is transformed to \(-x, -y, -z\). The equation is symmetric with respect to the origin if replacing all coordinates with their negatives results in the same equation.

Key Concepts

Symmetry and the yz-plane

In the realm of three-dimensional geometry, symmetry can play a vital role in understanding and visualizing equations. When discussing symmetry with respect to the yz-plane, we focus on how the graph of an equation behaves when reflected across this plane.

The yz-plane is where the x-coordinate is equal to zero. To determine if an equation is symmetric concerning the yz-plane, we perform a specific test: we replace every "x" in the equation with "-x."

If this transformation results in an equivalent equation, meaning the equation looks the same in its form and it describes the same graph, then the graph is considered symmetric with the yz-plane.

• This type of symmetry is akin to looking at an object in the mirror that lies in the yz-plane.
• The object appears unaltered from the original view when mirrored.
• Any changes when replacing "x" with "-x" indicate an absence of yz-plane symmetry.

Understanding yz-plane symmetry is crucial for solving, graphing, and interpreting three-dimensional equations.

Z-axis Symmetry

Examining z-axis symmetry requires slightly different steps than other symmetries because here, the symmetry revolves around a line, the z-axis, rather than a plane.

The z-axis is located where both the x and y-coordinates are zero. For z-axis symmetry, we rotate points around this axis, transforming \(x, y, z\) into \(-x, -y, z\).
If the original equation remains unchanged after this transformation, it can be confirmed that the equation is symmetric about the z-axis.

To better visualize:

• Imagine holding an object and spinning it around its vertical center line (the z-axis).
• If the object looks the same after a 180-degree spin (or any equivalent series), it can be said to have z-axis symmetry.
• In mathematical terms, this means replacing both coordinates "x" and "y" by their negatives.

The concept of rotation is central here, distinguishing it from merely mirroring across a plane. Z-axis symmetry gives insight into rotational properties of graphs in three-dimensions.

Origin Symmetry

Origin symmetry is an intriguing aspect where all coordinates are simultaneously inverted, offering a complete flip around the center of the three-dimensional space, known as the origin.

Testing for symmetry with the origin involves changing every coordinate in a point \(x, y, z\) to its negative counterpart, \(-x, -y, -z\).

After performing this transformation, if the equation is unchanged, it exhibits origin symmetry.

This testing can be visualized as:

• Thinking of a globe, origin symmetry is like flipping it to see the hemispheres reversed in every direction.
• This transformation is a simultaneous reflection across all three coordinate planes: xy, yz, and zx.
• When applied in equations, it considers visualizing the symmetry within its sub-elements in 3D space.

Origin symmetry is a powerful concept that shows complete inversion and helps in deep understanding the transformations of 3D figures and equations.