Without graphing, 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. See Examples 3 and 4. x^2+y^2=12

Verified step by step guidance

1. Identify the type of equation: The given equation \(x^2 + y^2 = 12\) is a circle centered at the origin with radius \(\sqrt{12}\).
2. Test for symmetry with respect to the x-axis: Replace \(y\) with \(-y\) in the equation to see if the equation remains unchanged. Substitute to get \(x^2 + (-y)^2 = 12\), which simplifies to \(x^2 + y^2 = 12\). The equation remains unchanged, indicating symmetry with respect to the x-axis.
3. Test for symmetry with respect to the y-axis: Replace \(x\) with \(-x\) in the equation to see if the equation remains unchanged. Substitute to get \((-x)^2 + y^2 = 12\), which simplifies to \(x^2 + y^2 = 12\). The equation remains unchanged, indicating symmetry with respect to the y-axis.
4. Test for symmetry with respect to the origin: Replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation to see if the equation remains unchanged. Substitute to get \((-x)^2 + (-y)^2 = 12\), which simplifies to \(x^2 + y^2 = 12\). The equation remains unchanged, indicating symmetry with respect to the origin.
5. Conclude the symmetries: Since the equation is unchanged for all three tests, the graph of the equation is symmetric with respect to the x-axis, the y-axis, and the origin.

Key Concepts

Symmetry with respect to the x-axis

An equation is symmetric with respect to the x-axis if replacing y with -y yields an equivalent equation. This means that for every point (x, y) on the graph, the point (x, -y) will also be on the graph. This type of symmetry indicates that the graph is mirrored across the x-axis.

Symmetry with respect to the y-axis

An equation is symmetric with respect to the y-axis if replacing x with -x results in an equivalent equation. This implies that for every point (x, y) on the graph, the point (-x, y) will also be present. Such symmetry suggests that the graph is mirrored across the y-axis.

Symmetry with respect to the origin

An equation is symmetric with respect to the origin if replacing both x with -x and y with -y yields an equivalent equation. This means that for every point (x, y) on the graph, the point (-x, -y) will also be on the graph. This type of symmetry indicates that the graph has rotational symmetry of 180 degrees around the origin.