Question

X^2+y-9=0 test for symmetry

Answer 1

We have three tests of symmetry. In respect to the y-axis or in respect to the x-axis, or in respect to the origin.

For symmetry with respect to the Y-Axis, check to see if the equation is the same when we replace x with −x:

$x^2+y-9=(-x)^2+y-9=0$

Since the equation remains the same when we change the sign of the x variable, the equation is symmetric in respect to the y-axis.

For symmetry with respect to the X-Axis, check to see if the equation is the same when we replace y with −y:

$x^2+y-9\ne x^2+(-y)-9=x^2-y-9$

Since the equation does not remains the same when we change the sign of the y variable, the equation is not symmetric in respect to the x-axis.

The equation is symmetric with respect to the origin if for every point (x,y) that satisfies the equation, the point (-x, -y) also satisfies the equation.

$(-x)^2+(-y)-9=x^2-y-9\ne x^2+y-9$

Since it doesn't satisfies this condition, the equation is not symmetric with respect to the origin.