Problem 21 - 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. $$y^{2}=x^{2}+6$$

Step-by-Step Solution

Answer

The graph of the equation \(y^{2}=x^{2}+6\) is symmetric with respect to the \(y\)-axis, the \(x\)-axis, and the origin.

Step 1: Check for Y-axis symmetry

Replace \(x\) with \(-x\) in the equation \(y^{2}=x^{2}+6\) which gives \(y^{2}=(-x)^{2}+6\). This simplifies to \(y^{2}=x^{2}+6\) which is the original equation. Hence, the graph is symmetric about the \(y\)-axis.

Step 2: Check for X-axis symmetry

Replace \(y\) with \(-y\) in the equation \(y^{2}=x^{2}+6\) which gives \((-y)^{2}=x^{2}+6\). This simplifies to \(y^{2}=x^{2}+6\) which is the original equation. So, the graph is symmetric about the \(x\)-axis.

Step 3: Check for Origin symmetry

Replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation \(y^{2}=x^{2}+6\) which gives \((-y)^{2}=(-x)^{2}+6\). This simplifies to \(y^{2}=x^{2}+6\) which is the original equation. Therefore, the graph is also symmetric with respect to the origin.