Problem 26 - 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. $$x^{3}-y^{2}=5$$

Step-by-Step Solution

Answer

The graph of the equation \(x^3-y^2=5\) is symmetric with respect to the \(x\)-axis.

Step 1: Symmetry with respect to the y-axis

To test the equation for symmetry with respect to the \(y\)-axis, replace \(x\) with \(-x\). If the equation remains the same, then the graph is symmetric with respect to the \(y\)-axis. Replace \(x\) with \(-x\) in the equation \(x^3-y^2=5\) to get \((-x)^3-y^2=5\) or \(-x^3-y^2=5\). This is not the same as the original equation, so the graph is not symmetric with respect to the \(y\)-axis.

Step 2: Symmetry with respect to the x-axis

To test the equation for symmetry with respect to the \(x\)-axis, replace \(y\) with \(-y\). If the equation remains the same, then the graph is symmetric with respect to the \(x\)-axis. Replace \(y\) with \(-y\) in the equation \(x^3-y^2=5\) to get \(x^3-(-y)^2=5\) or \(x^3-y^2=5\). This is the same as original equation, therefore the graph is symmetric about the \(x\)-axis.

Step 3: Symmetry with respect to the origin

In order to be symmetric with respect to the origin, the equation must satisfy the condition that if \(x\) is replaced with \(-x\) and \(y\) is replaced with \(-y\), the equation remains the same. Apply these replacements to \(x^3-y^2=5\) to get \((-x)^3-(-y)^2=5\), which simplifies to \(-x^3-y^2=5\). This does not correspond to the original equation, so the graph is not symmetric about the origin.