Problem 31 - 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. $$y^{4}=x^{3}+6$$

Step-by-Step Solution

Answer

The graph of the given equation \(y^{4}=x^{3}+6\) is symmetric only with respect to the x-axis.

Step 1: Check for y-axis symmetry

Replace \(x\) with \(-x\) in the equation, which results in \(y^{4}=-x^{3}+6\). This does not yield the original equation, so the graph is not symmetric with respect to the y-axis

Step 2: Check for x-axis symmetry

Replace \(y\) with \(-y\) in the equation. But since \(y\) is raised to the power 4, the equation remains the same \(y^{4}=x^{3}+6\), hence the graph is symmetric with respect to the x-axis.

Step 3: Check for origin symmetry

Replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation. The equation changes to \(y^{4} = -x^{3} +6\), which does not result in the original equation, indicating that the graph is not symmetric with respect to the origin.