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Chapter 1: Problem 20

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. $$x=y^{2}-2$$

Short Answer

Expert verified
The graph of the equation \(x = y^{2} - 2\) is symmetric with respect to the x-axis only.

Step by step solution

01

Test for Symmetry about the y-axis

Replace \(x\) with \(-x\) in the equation \(x = y^{2} - 2\). This results in \(-x = y^{2} - 2\). This is not the same as the original equation, so the graph of the equation is not symmetric with respect to the y-axis.
02

Test for Symmetry about the x-axis

Replace \(y\) with \(-y\) in the equation \(x = y^{2} - 2\). This results in \(x = (-y)^{2} - 2 = y^{2} - 2\). This is the same as the original equation, so the graph of the equation is symmetric with respect to the x-axis.
03

Test for Symmetry about the Origin

Replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation \(x = y^{2} - 2\). This results in \(-x = (-y)^{2} - 2\). This is not the same as the original equation, so the graph of the equation is not symmetric with respect to the origin.

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