The graph of y^4-3 x^2+7=0 is symmetric with respect to...

The graph of $y^{4}-3 x^{2}+7=0$ is symmetric with respect to which of the following? I. the $x$ -axis II. the $y$ -axis III. the origin
(A) only I
(B) only II
(C) only III
(D) only I and II
(E) I, II, and III

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Key Concepts

Symmetry with respect to the x-axis

This concept refers to the property where the graph of an equation is unchanged if y is replaced by -y. For a graph to be symmetric with respect to the x-axis, every point (x, y) on the graph must have a corresponding point (x, -y) also on the graph, indicating that the equation remains valid when y is negated.

Symmetry with respect to the y-axis

This property means that the graph of an equation remains unchanged when x is replaced by -x. In other words, for every point (x, y) on the graph, the point (-x, y) will also lie on the graph. This happens when the equation contains x only in even powers or other functions that are even in x.

Symmetry with respect to the origin

Origin symmetry occurs when the graph is invariant under the transformation of (x, y) to (-x, -y). This means that if (x, y) satisfies the equation, then so does (-x, -y). For a graph to have origin symmetry, the equation must be such that negating both variables leaves the equation unchanged.

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