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Chapter 1: Problem 39

Use the algebraic tests to check for symmetry with respect to both axes and the origin. $$x y^{2}+10=0$$

Short Answer

Expert verified
The function \(x y^{2}+10=0\) is symmetric with respect to the x-axis only, but not with respect to the y-axis or the origin.

Step by step solution

01

Test for symmetry with respect to the x-axis

We begin the test for symmetry with respect to the x-axis by replacing \(y\) with \(-y\) in the equation. Doing this gives us a new equation \(x (-y)^{2}+10=0\). Simplifying that equation gives us \(x y^{2}+10=0\), which is the same as our original equation. Hence, the function is symmetric with respect to the x-axis.
02

Test for symmetry with respect to the y-axis

The test for symmetry with respect to the y-axis involves replacing \(x\) with \(-x\). When we do that in the given equation, we have a new equation is \(-x y^{2}+10=0\). This is not the same as our original equation. Hence the function is not symmetric with respect to the y-axis.
03

Test for symmetry with respect to the origin

The test for symmetry with respect to the origin involves replacing \(x\) with \(-x\) and \(y\) with \(-y\). Using \(-x\) and \(-y\) in the given equation, the new equation will be \(-x(-y)^{2} + 10=0\). This simplifies to \(-x y^{2}+10=0\), which is not the same as our original equation. Therefore, the function is not symmetric with respect to the origin.

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