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Magazine Chapter 9: Problem 38
Use the tests for symmetry to decide whether the graph of each relation is symmetric with respect to the \(x\) -axis, the y-axis, or the origin. More than one of these symmetries, or none of them, may apply. $$ x y=-6 $$
Short Answer
Expert verified
The graph is symmetric with respect to the origin.
Step by step solution
01
- Test for Symmetry with respect to the x-axis
To test for symmetry with respect to the x-axis, replace every occurrence of y in the equation with -y. If the equation remains unchanged, then it is symmetric with respect to the x-axis. For the given equation, starting with y x = -6replace y with -y to get x (-y) = -6 which simplifies to -xy = -6 This simplifies to xy = 6.Since this is not the same as the original equation, the graph is not symmetric with respect to the x-axis.
02
- Test for Symmetry with respect to the y-axis
To test for symmetry with respect to the y-axis, replace every occurrence of x in the equation with -x. If the equation remains unchanged, then it is symmetric with respect to the y-axis. For the given equation, starting with xy = -6replace x with -x to get (-x)y = -6 which simplifies to -xy = -6 Simplifies to xy = 6.Since this is not the same as the original equation, the graph is not symmetric with respect to the y-axis.
03
- Test for Symmetry with respect to the origin
To test for symmetry with respect to the origin, replace x with -x and y with -y. If the equation remains unchanged, then it is symmetric with respect to the origin. For the given equation, starting with xy = -6replace x with -x and y with -y to get (-x)(-y) = -6 which simplifies to xy = -6Since this is the same as the original equation, the graph is symmetric with respect to the origin.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Symmetry with Respect to the x-axis
Understanding symmetry with respect to the x-axis is a valuable skill in algebra. The general rule is simple. Replace every occurrence of \( y \) in the equation with \( -y \). If the equation remains unchanged, the graph is symmetric with respect to the x-axis. Here鈥檚 a breakdown:
- Start with the given equation.
- Replace \( y \) with \( -y \).
- Simplify the new equation.
- Replace \( y \) with \(-y \) to get \( x(-y) = -6 \).
- This simplifies to \(-xy = -6 \).
- Simplify further to get \( xy = 6 \), which is not the same as the original equation.
Symmetry with Respect to the y-axis
Testing for symmetry with respect to the y-axis follows a similar method. Here, replace every occurrence of \( x \) in the equation with \( -x \). If the resulting equation is identical to the original, then the graph is symmetric with respect to the y-axis. Let's simplify this:
- Take the original equation.
- Replace \( x \) with \(-x \).
- Simplify the new equation received.
- Replace \( x \) with \(-x \) to get \( (-x)y = -6 \).
- This simplifies to \(-xy = -6 \).
- Thus, we get \( xy = 6 \).
Symmetry with Respect to the Origin
Testing for symmetry with respect to the origin requires replacing both \( x \) with \(-x \) and \( y \) with \(-y \). If the modified equation remains unchanged, then there is symmetry with respect to the origin. Here's how this is done:
- Take the original equation.
- Replace \( x \) with \(-x \) and \( y \) with \(-y \).
- Simplify the resulting equation.
- Replace \( x \) with \(-x \) and \( y \) with \(-y \).
- It becomes \((-x)(-y) = -6 \).
- This simplifies to \( xy = -6 \).
Algebraic Equations
An algebraic equation is a mathematical statement which uses variables, constants, and algebraic operations like addition, subtraction, and multiplication. Equations can often help us understand relationships between different quantities.
The given equation \[ xy = -6 \] is an example of a simple algebraic equation.
Here are key points:
The given equation \[ xy = -6 \] is an example of a simple algebraic equation.
Here are key points:
- An equation represents a relation between variables.
- It helps predict the behavior of variables under different conditions.
- Solving equations often involves simplifying them to easiest form.
Graphing Relations
Graphing relations is a vital concept in understanding the graphical behavior of algebraic equations. A relation between two variables can be visualized on a coordinate plane to help interpret their relationship.
Important aspects include:
Important aspects include:
- Plotting points based on the equation.
- Determining the type of symmetry (if any).
- Understanding shifts, stretches, and reflections on the graph.
- Each point (x, y) indicates values that satisfy the equation.
- Graphical representation helps us see patterns or symmetry.
- Identifying the symmetry simplifies the understanding of the graph.
