How do you determine whether the graph of $| y | = x y$ $absy=xy$ is symmetric with respect to the x axis, y axis or neither?

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Answer 1 · 2018-04-19T16:32:30

$| y | = x y$ $|y|=xy$ is neither symmetric w.r.t. $x$ $x$ axis nor w.r.t. $y$ $y$ axis

If a function is symmetric w.r.t. $x$ $x$ -axis, then if $(x_{1}, y_{1})$ $(x_1,y_1)$ satisfies the equation, so does $(x_{1}, - y_{1})$ $(x_1,-y_1)$

If $(x_{1}, y_{1})$ $(x_1,y_1)$ satisfies equation then $| y_{1} | = x_{1} y_{1}$ $|y_1|=x_1y_1$

and for $(x_{1}, - y_{1})$ $(x_1,-y_1)$ , $| - y_{1} | = | y_{1} |$ $|-y_1|=|y_1|$ and $x_{1} \cdot (- y_{1}) = - x_{1} y_{1}$ $x_1*(-y_1)=-x_1y_1$

Hence $| y | = x y$ $|y|=xy$ is not symmetric w.r.t. $x$ $x$ axis.

If a function is symmetric w.r.t. $y$ $y$ -axis, then if $(x_{1}, y_{1})$ $(x_1,y_1)$ satisfies the equation, so does $(- x_{1}, y_{1})$ $(-x_1,y_1)$

If $(x_{1}, y_{1})$ $(x_1,y_1)$ satisfies equation then $| y_{1} | = x_{1} y_{1}$ $|y_1|=x_1y_1$

and for $(- x_{1}, y_{1})$ $(-x_1,y_1)$ , $| y_{1} | = | y_{1} |$ $|y_1|=|y_1|$ and $(- x_{1}) \cdot y_{1} = - x_{1} y_{1}$ $(-x_1)*y_1=-x_1y_1$

Hence $| y | = x y$ $|y|=xy$ is not symmetric w.r.t. $y$ $y$ axis.

How do you determine whether the graph of absy=xy|y|=xyabsy=xy is symmetric with respect to the x axis, y axis or neither?