Problem 29 - 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. $$x^{2} y^{2}+3 x y=1$$

Step-by-Step Solution

Answer

The graph of the equation \(x^{2} y^{2}+3 x y=1\) is symmetric with respect to the origin.

Step 1: Test for y-axis symmetry

Replace \(x\) with \(-x\) in the equation. The original equation is \(x^{2} y^{2}+3 x y=1\). After replacement, the equation becomes \( (-x)^{2} y^{2}+3 (-x) y=1\), which simplifies to \(x^{2} y^{2}-3 x y=1\). This is not the same as the original equation, so the graph is not symmetric with respect to the y-axis.

Step 2: Test for x-axis symmetry

Replace \(y\) with \(-y\) in the equation. The original equation is \(x^{2} y^{2}+3 x y=1\). After replacement, the equation becomes \(x^{2} (-y)^{2}+3 x (-y)=1\), which simplifies to \(x^{2} y^{2}-3 x y=1\). This is not the same as the original equation, so the graph is not symmetric with respect to the x-axis.

Step 3: Test for origin symmetry

Replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation. The original equation is \(x^{2} y^{2}+3 x y=1\). After replacements, the equation becomes \( (-x)^{2} (-y)^{2}+3(-x)(-y)=1\), which simplifies to \(x^{2} y^{2}+3 x y=1\). This is the same as the original equation, so the graph is symmetric with respect to the origin.