Section 4.7 : Symmetry
2. Determine the symmetry of each of the following equation.
\[\frac{{{y^2}}}{4} = 1 + \frac{{{x^2}}}{9}\]Show All Steps Hide All Steps
Start SolutionLet’s first check for symmetry about the \(x\)-axis. To do this we need to replace all the \(y\)’s with –\(y\).
\[\frac{{{{\left( { - y} \right)}^2}}}{4} = 1 + \frac{{{x^2}}}{9}\hspace{0.25in} \to \hspace{0.25in}\frac{{{y^2}}}{4} = 1 + \frac{{{x^2}}}{9}\]The resulting equation is identical to the original equation and so is equivalent to the original equation. Therefore, the equation is has symmetry about the \(x\)-axis.
Show Step 2Next, we’ll check for symmetry about the \(y\)-axis. To do this we need to replace all the \(x\)’s with –\(x\).
\[\frac{{{y^2}}}{4} = 1 + \frac{{{{\left( { - x} \right)}^2}}}{9}\hspace{0.25in} \to \hspace{0.25in}\frac{{{y^2}}}{4} = 1 + \frac{{{x^2}}}{9}\]The resulting equation is identical to the original equation and so is equivalent to the original equation. Therefore, the equation is has symmetry about the \(y\)-axis.
Show Step 3Finally, a check for symmetry about the origin. For this check we need to replace all the \(y\)’s with –\(y\) and to replace all the \(x\)’s with –\(x\).
\[\frac{{{{\left( { - y} \right)}^2}}}{4} = 1 + \frac{{{{\left( { - x} \right)}^2}}}{9}\hspace{0.25in} \to \hspace{0.25in}\frac{{{y^2}}}{4} = 1 + \frac{{{x^2}}}{9}\]The resulting equation is identical to the original equation and so is equivalent to the original equation. Therefore, the equation is has symmetry about the origin.