Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

What condition leads to a graph that is symmetric with respect to the following? (a) \(x z\) -plane (b) \(y\) -axis (c) \(x\) -axis

Step-by-Step Solution

Answer

Reflecting points appropriately for each symmetry condition ensures the graph remains the same.

Step 1: Understanding Symmetry with respect to the xz-plane

A graph is symmetric with respect to the \(xz\)-plane if for every point \((x, y, z)\) on the graph, the point \((x, -y, z)\) is also on the graph. This means that if you reflect any point over the \(xz\)-plane (which means changing \(y\) to \(-y\)), the new point should also be part of the graph.

Step 2: Understanding Symmetry with respect to the y-axis

A graph is symmetric with respect to the \(y\)-axis if for every point \((x, y, z)\) on the graph, the point \((-x, y, z)\) is also on the graph. This indicates that reflecting any point over the \(y\)-axis (which involves changing \(x\) to \(-x\)) results in another point that is still on the graph.

Step 3: Understanding Symmetry with respect to the x-axis

A graph is symmetric with respect to the \(x\)-axis if for every point \((x, y, z)\) on the graph, the point \((x, y, -z)\) is on that graph. This means that if any point is reflected over the \(x\)-axis (where you change \(z\) to \(-z\)), the resulting point remains part of the graph.

Key Concepts

xz-plane symmetry

Have you ever imagined looking at a 3D graph where any point could "mirror" itself across a flat plane? This plane is what we call the "xz-plane." When a graph is symmetric with respect to the xz-plane, it means that reflections over this plane are perfectly captured within the graph.
To picture this, any time you reflect a point \(x, y, z\) to \(x, -y, z\) it must still lie on the graph. Thus, the reflection occurs by flipping the sign of the \(y\) coordinate while keeping the \(x\) and \(z\) parts the same.

• This symmetry reflects all points across a flat, vertical plane that cuts through the origin along the x- and z-axes.
• Imagine laying the graph on a tabletop and flipping everything above the xz-plane downward, and everything below upward; if the graph looks identical, it’s symmetric with respect to the xz-plane.

Understanding this symmetry can help simplify complex graphs by visually reducing them into more manageable sections.

y-axis symmetry

Imagine slicing a 3D structure with a plane that runs directly through the front, dividing it into left and right halves. This is akin to y-axis symmetry!
This symmetry involves reflecting a graph across the y-axis, which means for any point \(x, y, z\), flipping it to become \(-x, y, z\). If the graph remains unchanged upon reflection, it has y-axis symmetry.

• This concept helps in identifying shapes and forms that are even and balanced when split in half vertically.
• Visualizing this, think of a face - split it down the middle and observe if one side mirrors the other, that's y-axis symmetry!
• Practical applications include simplifying models or equations by considering only one side and mirroring it.

Grasping the y-axis symmetry concept can help greatly in analyzing systems or equations due to their simplicity and innate balance.

x-axis symmetry

When we talk about x-axis symmetry, imagine a reflection over a horizontal line that slices through the middle of a 3D graph. This involves creating a mirror image across the x-axis.
For any point \(x, y, z\), the reflection is \(x, y, -z\). This means if such a reflected point remains on the graph, it enjoys x-axis symmetry.

• This type of symmetry is vital for recognizing balanced behaviors or shapes in plots or models.
• Visualize this like flipping objects upside down, yet maintaining its aesthetics or properties.
• Applying this concept simplifies your analysis, especially in physics or engineering, when you only need to consider one half of a system.

Understanding x-axis symmetry is about recognizing when flipping things upside down continues to reveal the same design or structure.