Section 4.7 : Symmetry
3. Determine the symmetry of each of the following equation.
\[{x^2} = 7y - {x^3} + 2\]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 –\(y\).
\[{x^2} = 7\left( { - y} \right) - {x^3} + 2\hspace{0.25in} \to \hspace{0.25in}{x^2} = - 7y - {x^3} + 2\]The resulting equation is not equivalent to the original equation (i.e. it is not same nor is it the same equation except with opposite signs on every term). Therefore, the equation is does not have 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 –\(x\).
\[{\left( { - x} \right)^2} = 7y - {\left( { - x} \right)^3} + 2\hspace{0.25in} \to \hspace{0.25in}{x^2} = 7y + {x^3} + 2\]The resulting equation is not equivalent to the original equation (i.e. it is not same nor is it the same equation except with opposite signs on every term). Therefore, the equation is does not have 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 –\(x\).
\[{\left( { - x} \right)^2} = 7\left( { - y} \right) - {\left( { - x} \right)^3} + 2\hspace{0.25in} \to \hspace{0.25in}{x^2} = - 7y + {x^3} + 2\]The resulting equation is not equivalent to the original equation (i.e. it is not same nor is it the same equation except with opposite signs on every term). Therefore, the equation does not have symmetry about the origin.