Problem 25 - Precalculus Essentials

Question

Determine whether the graph of each equation is symmetric with respect to the \(y\) -axis, the \(x\) -axis, the origin, more than one of these, or none of these. $$x^{2}-y^{3}=2$$

Step-by-Step Solution

Answer

The graph of the equation \(x^{2}-y^{3}=2\) is symmetric only with respect to the y-axis.

Step 1: Symmetry with respect to the y-axis

Replace \(x\) with \(-x\) in the equation, the new equation becomes \((-x)^{2}-y^{3}=2\), simplifying this gives \(x^{2}-y^{3}=2\). The new equation is identical to the initial equation. Therefore, the graph is symmetric with respect to the y-axis.

Step 2: Symmetry with respect to the x-axis

Replace \(y\) with \(-y\) in the original equation. The new equation becomes \(x^{2}-(-y)^{3}=2\), simplifying this gives \(x^{2}+y^{3}=2\), which is not identical to the original equation. Hence the graph is not symmetric with respect to the x-axis.

Step 3: Symmetry with respect to the origin

Replace \(x\) and \(y\) with \(-x\) and \(-y\) in the original equation. The new equation becomes \((-x)^{2}-(-y)^{3}=2\), simplifying this gives \(x^{2}+y^{3}=2\), which is not identical to the original equation. Hence the graph is not symmetric with respect to the origin.