If a function is even (i.e. symmetric about the y-axis), then: , i.e. if you plug in -x to your function, you'll get the same function if you plugged in x.
If a function is odd (i.e. symmetric about the origin), then: , i.e. if you plug in -x to your function, then you'll get the negative of f(x).
Show us your work and we'll point out any errors.
As for the circle, given , the centre is given by (note the minus signs in front of them in the equation). For example, has centre