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