Questions: Test the equation for symmetry. (Select all that apply.) x=3 y^4-8 y^2 The graph of the equation is symmetric with respect to the x-axis. The graph of the equation is symmetric with respect to the y-axis. The graph of the equation is symmetric with respect to the origin.

Solution Steps

To test the equation for symmetry, we need to check the following:

1. Symmetry with respect to the x-axis: Replace \( y \) with \(-y\) in the equation and see if the equation remains unchanged.
2. Symmetry with respect to the y-axis: Replace \( x \) with \(-x\) in the equation and see if the equation remains unchanged.
3. Symmetry with respect to the origin: Replace \( x \) with \(-x\) and \( y \) with \(-y\) in the equation and see if the equation remains unchanged.

To determine if the graph is symmetric with respect to the x-axis, we substitute \( y \) with \(-y\) in the equation \( x = 3y^4 - 8y^2 \). The equation becomes: \[ x = 3(-y)^4 - 8(-y)^2 = 3y^4 - 8y^2 \] Since the equation remains unchanged, the graph is symmetric with respect to the x-axis.

To determine if the graph is symmetric with respect to the y-axis, we substitute \( x \) with \(-x\) in the equation \( x = 3y^4 - 8y^2 \). The equation becomes: \[ -x = 3y^4 - 8y^2 \] This equation is not equivalent to the original equation, so the graph is not symmetric with respect to the y-axis.

To determine if the graph is symmetric with respect to the origin, we substitute \( x \) with \(-x\) and \( y \) with \(-y\) in the equation \( x = 3y^4 - 8y^2 \). The equation becomes: \[ -x = 3(-y)^4 - 8(-y)^2 = 3y^4 - 8y^2 \] This equation is not equivalent to the original equation, so the graph is not symmetric with respect to the origin.

Final Answer