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

Apr 19, 2018

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

#### Explanation:

If a function is symmetric w.r.t. $x$-axis, then if $\left({x}_{1} , {y}_{1}\right)$ satisfies the equation, so does $\left({x}_{1} , - {y}_{1}\right)$

If $\left({x}_{1} , {y}_{1}\right)$ satisfies equation then $| {y}_{1} | = {x}_{1} {y}_{1}$

and for $\left({x}_{1} , - {y}_{1}\right)$, $| - {y}_{1} | = | {y}_{1} |$ and ${x}_{1} \cdot \left(- {y}_{1}\right) = - {x}_{1} {y}_{1}$

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

If a function is symmetric w.r.t. $y$-axis, then if $\left({x}_{1} , {y}_{1}\right)$ satisfies the equation, so does $\left(- {x}_{1} , {y}_{1}\right)$

If $\left({x}_{1} , {y}_{1}\right)$ satisfies equation then $| {y}_{1} | = {x}_{1} {y}_{1}$

and for $\left(- {x}_{1} , {y}_{1}\right)$, $| {y}_{1} | = | {y}_{1} |$ and $\left(- {x}_{1}\right) \cdot {y}_{1} = - {x}_{1} {y}_{1}$

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