Symmetry 2

Question

Use algebraic tests to determine whether the graph of the equation x = y³ – 4y is symmetric with respect to the x-axis, y-axis, or origin

Answer

Now a little definition first.

A point ( x,y ) is said to be symmetric with respect to the :

1) x -Axis if the point (x, -y) still lies on that curve.

2) y -Axis if the point (-x, y) still lies on that curve.

3) The origin (0,0) if the point (-x, -y) still lies on that curve.

So we just need to ‘check’ the conditions (1), (2) & (3) above for the equation given for our curve.

To check for (1), consider
x = (-y)³ – 4(-y) = – y³ + 4y is not equal in value to x = y³ – 4y and therefore its not symmetrical to the x-axis.

To check for (2), consider
(-x) = (y)³ – 4(y) = y³ – 4y implying that x = – y³ + 4y which is not equal in value to x = y³ – 4y and therefore its not symmetrical to the y-axis.

To check for (3), consider
(-x) = (-y)³ – 4(-y) = – y³ + 4y implying that x = y³ – 4y and in this case the value is equal to x = y³ – 4y and therefore it is symmetrical to the origin!.

Hence, the conclusion is the equation x = y³ – 4y is symmetric with respect to the origin and NOT with respect to the x-axis, NOR the y-axis.