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Chapter 1: Problem 19

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. $$x=y^{2}+6$$

Short Answer

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

Step by step solution

01

Check for Symmetry with respect to the Y-axis

Replace \(x\) with \(-x\) in the equation \(x=y^{2}+6\), the new equation becomes \(-x = y^{2} + 6\), which is not the same as the original. Therefore, the graph of the equation is not symmetric with respect to the y-axis.
02

Check for Symmetry with respect to the X-axis

Replace \(y\) with \(-y\) in the equation \(x=y^{2}+6\), the equation becomes \(x=(-y)^{2}+6\). After simplifying, we get \(x = y^{2} + 6\), which is the same as the original equation. Therefore, the graph of the equation is symmetric with respect to the x-axis.
03

Check for Symmetry with respect to the Origin

Replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation \(x=y^{2}+6\), the new equation becomes \(-x = (-y)^{2} + 6\), which is not the same as the original equation. Therefore, the graph of the equation is not symmetric with respect to the origin.

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Most popular questions from this chapter

Graph \(y_{1}=x^{2}-2 x, y_{2}=x,\) and \(y_{3}=y_{1} \div y_{2}\) in the same [-10,10,1] by [-10,10,1] vicwing rectangle. Then use the TRACE l feature to trace along \(y_{3}\). What happens at \(x=0 ?\) Explain why this occurs.

Will help you prepare for the material covered in the next section. Solve for \(y: 3 x+2 y-4=0\)

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