How do you determine whether the graph of #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|=xy# is neither symmetric w.r.t. #x# axis nor w.r.t. #y# axis

If a function is symmetric w.r.t. #x#-axis, then if #(x_1,y_1)# satisfies the equation, so does #(x_1,-y_1)#

If #(x_1,y_1)# satisfies equation then #|y_1|=x_1y_1#

and for #(x_1,-y_1)#, #|-y_1|=|y_1|# and #x_1*(-y_1)=-x_1y_1#

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

If a function is symmetric w.r.t. #y#-axis, then if #(x_1,y_1)# satisfies the equation, so does #(-x_1,y_1)#

If #(x_1,y_1)# satisfies equation then #|y_1|=x_1y_1#

and for #(-x_1,y_1)#, #|y_1|=|y_1|# and #(-x_1)*y_1=-x_1y_1#

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