Symmetry 3

Question

Use algebraic tests to determine whether the graph of the equation 4x² – 9y² = 1 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
4(x)² – 9(-y)² = 4x² – 9y² = 1; and thus this point lies on the curve since it satisfies the equation!

Similarly, for ( 2 ),
4(-x)² – 9(y)² = 4x² – 9y² = 1; and thus this point lies on the curve since it satisfies the equation!

For (3),
4(-x)² – 9(-y)² = 4x² – 9y² = 1; and thus this point lies on the curve since it satisfies the equation!

Thus we have shown that our graph for the curve 4x² – 9y² = 1 is symmetrical for all three x – axis, y – axis and the origin!