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.
$\mid y\mid =-x$

Verified step by step guidance

1. Rewrite the given equation: \(|y| = -x\).
2. Recall the symmetry tests: - For symmetry about the x-axis, replace \(y\) with \(-y\) and check if the equation remains unchanged. - For symmetry about the y-axis, replace \(x\) with \(-x\) and check if the equation remains unchanged. - For symmetry about the origin, replace \(x\) with \(-x\) and \(y\) with \(-y\) and check if the equation remains unchanged.
3. Test for x-axis symmetry: Replace \(y\) with \(-y\) in \(|y| = -x\). Since \(|y| = |-y|\), the equation becomes \(|y| = -x\), which is the same as the original equation. So, the equation is symmetric with respect to the x-axis.
4. Test for y-axis symmetry: Replace \(x\) with \(-x\) in \(|y| = -x\). The equation becomes \(|y| = -(-x) = x\). This is not the same as the original equation, so it is not symmetric about the y-axis.
5. Test for origin symmetry: Replace \(x\) with \(-x\) and \(y\) with \(-y\). The equation becomes \(|-y| = -(-x)\), which simplifies to \(|y| = x\). This is not the same as the original equation, so it is not symmetric about the origin.

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. Testing symmetry involves substituting variables (e.g., replacing y with -y for x-axis symmetry) and checking if the equation remains unchanged.

Absolute Value Function

The absolute value function, denoted |y|, represents the non-negative value of y regardless of its sign. It affects the graph by reflecting negative y-values to positive, which influences symmetry and the shape of the graph.

Domain and Range Restrictions

Understanding the domain and range is crucial, especially when absolute values and negative signs are involved. For example, |y| is always non-negative, so an equation like |y| = -x restricts x to values where the right side is non-negative, impacting the graph's existence and symmetry.