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|

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|.