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# Test for symmetry with respect to each axis and to the origin. $y=\left|x^{3}+x\right|$

Solution

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We have $y= \left| x^3+x\right|$

$\color{#4257b2}\text{First: we'll test the equation for symmetry with respect to the x-axis}$:

$(-y)= \left| x^3+x\right|$( Replace y by -y)

$\Rightarrow y=- \left| x^3+x\right|$

The result is not an equivalent equation.

Therefore the graph is not symmetric with respect to the x-axis

$\color{#4257b2}\text{Second: we'll test the equation for symmetry with respect to the y-axis}$:

$y= \left| (-x)^3-x\right|$( Replace x by -x)

$\Rightarrow y= \left| -(x^3+x)\right| = \left| x^3+x\right|$

The result is an equivalent equation.

Therefore the graph is symmetric with respect to the y-axis

$\color{#4257b2}\text{Third: we'll test the equation for symmetry with respect to the origin}$:

$-y= \left|(- x)^3-x\right|$( Replace y by -y and x by -x)

$\Rightarrow -y= \left| x^3+x\right|$

The result is not an equivalent equation.

Therefore the graph is not symmetric with respect to the origin

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