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. Recall the tests for symmetry of a graph: - Symmetry about the x-axis: Replace \(y\) with \(-y\) and check if the equation remains unchanged. - Symmetry about the y-axis: Replace \(x\) with \(-x\) and check if the equation remains unchanged. - Symmetry about the origin: Replace \(x\) with \(-x\) and \(y\) with \(-y\) simultaneously and check if the equation remains unchanged.
2. Start with the given equation: \(|x| = |y|\).
3. Test for symmetry about the x-axis by replacing \(y\) with \(-y\): \(|x| = |-y|\). Since the absolute value of \(-y\) is the same as \(|y|\), the equation remains \(|x| = |y|\), so the graph is symmetric about the x-axis.
4. Test for symmetry about the y-axis by replacing \(x\) with \(-x\): \(|-x| = |y|\). Since the absolute value of \(-x\) is the same as \(|x|\), the equation remains \(|x| = |y|\), so the graph is symmetric about the y-axis.
5. Test for symmetry about the origin by replacing \(x\) with \(-x\) and \(y\) with \(-y\): \(|-x| = |-y|\). This simplifies to \(|x| = |y|\), so the graph is symmetric about the origin as well.

Key Concepts

Symmetry in Graphs

Symmetry in graphs refers to how a graph mirrors itself across a line or point. Common symmetries include the x-axis, y-axis, and origin. Identifying symmetry helps understand the shape and properties of the graph without plotting every point.

Absolute Value Function

The absolute value function, denoted |x|, measures the distance of x from zero on the number line, always yielding a non-negative result. It affects graph shape by reflecting negative inputs to positive outputs, often creating V-shaped graphs or symmetric patterns.

Testing Symmetry Algebraically

To test symmetry algebraically, replace variables with their negatives and check if the equation remains unchanged. For x-axis symmetry, replace y with -y; for y-axis, replace x with -x; for origin, replace both x and y with their negatives.