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

Question

Elijah Allen · Accepted Answer

#|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.

Madelyn Anderson · Answer

undefined To determine the symmetry of the graph of $ |y| = xy $ with respect to the x-axis, y-axis, or neither, you can substitute $ -x $ for $ x $ and $ -y $ for $ y $ separately in the equation and observe if the equation remains unchanged.

1. **Symmetry with respect to the x-axis**: If replacing $ y $ with $ -y $ yields the same equation, the graph is symmetric with respect to the x-axis.

$$ |(-y)| = x(-y) $$
$$ |-y| = -xy $$
$$ |y| = xy $$

Since the equation remains the same, the graph is symmetric with respect to the x-axis.

2. **Symmetry with respect to the y-axis**: If replacing $ x $ with $ -x $ yields the same equation, the graph is symmetric with respect to the y-axis.

$$ |y| = (-x)y $$
$$ |y| = -xy $$

The equation is not the same, so the graph is not symmetric with respect to the y-axis.

3. **Neither symmetry**: If neither of the above substitutions yields the original equation, the graph has neither x-axis nor y-axis symmetry.

In this case, the graph of $ |y| = xy $ is symmetric with respect to the x-axis but not symmetric with respect to the y-axis. Therefore, it has symmetry with respect to the x-axis only.