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Chapter 2: 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 given equation \(x=y^{2}-2\) is symmetric with respect to the x-axis but not symmetric with respect to the y-axis and the origin.

Step by step solution

01

Check for y-axis symmetry

Replace \(x\) with \(-x\) in the given equation and simplify. The new equation is \(-x=y^{2}-2\), which is not the same as the original equation \(x=y^{2}-2\). So, the graph is not symmetric with respect to the y-axis.
02

Check for x-axis symmetry

Replace \(y\) with \(-y\) in the original equation and simplify. The new equation is \(x=(-y)^{2}-2\) which simplifies to \(x=y^{2}-2\). This is the same as the original equation, therefore, the graph is symmetric about the x-axis.
03

Check for origin symmetry

Replace \(x\) with \(-x\) and \(y\) with \(-y\) in the original equation and simplify. The resulting equation is \(-x=(-y)^{2}-2\), which simplifies to \(-x=y^{2}-2\). The changed equation is different from the original equation \(x=y^{2}-2\), so the graph is not symmetric with respect to the origin.

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