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

(a) Replace \(y\) with \(-y\). (b) Replace \(x\) with \(-x\). (c) Replace \(y\) with \(-y\).

Step 1: Identify the Symmetric Condition for the xy-plane

To determine symmetry with respect to the \(xz\)-plane, replace \(y\) with \(-y\) in the equation of the graph. If the equation remains unchanged, the graph is symmetric with respect to the \(xz\)-plane.

Step 2: Identify the Symmetric Condition for the y-axis

For symmetry about the \(y\)-axis, replace \(x\) with \(-x\) in the equation of the graph. If the equation looks the same after substitution, it means the graph is symmetric with respect to the \(y\)-axis.

Step 3: Identify the Symmetric Condition for the x-axis

To check for symmetry about the \(x\)-axis, replace \(y\) with \(-y\) in the function's equation. If the equation doesn't change after this alteration, then the graph is symmetric about the \(x\)-axis.

Key Concepts

Graph Symmetry

Graph symmetry refers to a situation where a graph looks the same on both sides of a particular line, plane, or point. It's like folding a piece of paper and finding out that both sides match up perfectly. This concept is essential in understanding how graphs behave and how they can be manipulated.

• Visual Interpretation: Just like a mirror image, a graph that is symmetric will look identical on both sides of the axis or plane.
• Equations Involved: For finding symmetry analytically, we alter variables in the equations. For example, replacing variables like changing a sign or swapping them to test for symmetry.

Graph symmetry makes it easier to predict the behavior of functions and equations in mathematics and physics, helping in simplifying problems by reducing redundant calculations.

Coordinate Plane Symmetry

When we talk about symmetry in a coordinate plane, we're referring to how a graph may reflect or match over an axis or a particular plane.- yz-plane symmetry involves replacing the variable that is not involved in your equation, say replacing \( x \) with \( -x \).- Symmetry around the x-axis means replacing \( y \) with \( -y \). If your resulting equation remains unchanged, this symmetry exists.- Symmetry around the y-axis also requires alteration, such as replacing \( x \) with \( -x \).Ensuring a graph is symmetric in the coordinate plane allows us to understand various physical phenomena where symmetry is applicable, such as physics engine behaviors or structural forces in civil engineering.

Equation Transformations

Equation transformations are essential operations that modify a function or equation to test for symmetry and other properties. These transformations help us to determine how a graph behaves under certain conditions.- Reflection over an axis: If you swap one variable's sign, such as \( y \) to \( -y \), it reflects over the x-axis.- Reflection over the origin: Change both \( x \) to \( -x \) and \( y \) to \( -y \). A symmetric graph about the origin remains unchanged.With these transformations, students can explore and understand the effects these changes have on their graphs. Such explorations can lead to identifying inherent symmetries in larger systems of equations, aiding in their simplification and understanding.