Determine whether each equation has a graph that is symmetric with respect to the x-axis, the y-axis, the origin, or none of these. |x| = |y|

Verified step by step guidance

1. Identify the given equation: \(|x| = |y|\).
2. To check for symmetry with respect to the x-axis, replace \(y\) with \(-y\) in the equation and see if the equation remains unchanged.
3. To check for symmetry with respect to the y-axis, replace \(x\) with \(-x\) in the equation and see if the equation remains unchanged.
4. To check for symmetry with respect to the origin, replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation and see if the equation remains unchanged.
5. Analyze the results from the substitutions to determine the symmetry of the graph.

Key Concepts

Symmetry in Graphs

Symmetry in graphs refers to the property where a graph remains unchanged under certain transformations. For example, a graph is symmetric with respect to the x-axis if replacing y with -y yields the same equation. Similarly, it is symmetric with respect to the y-axis if replacing x with -x gives the same equation, and it is symmetric with respect to the origin if replacing both x and y with their negatives results in the same equation.

Absolute Value Functions

Absolute value functions, denoted as |x|, represent the distance of a number from zero on the number line, always yielding non-negative outputs. In the context of the equation |x| = |y|, this means that both x and y can take on positive or negative values, leading to a relationship that can be visualized in the coordinate plane, often resulting in a 'V' shape or lines that reflect symmetry.

Testing for Symmetry

To determine the symmetry of a graph, specific tests can be applied. For x-axis symmetry, substitute y with -y; for y-axis symmetry, substitute x with -x; and for origin symmetry, substitute both x and y with their negatives. If the original equation holds true after these substitutions, the graph exhibits the corresponding symmetry. This method is essential for analyzing the given equation |x| = |y|.