函数 y = f (x)的图像关于点A (a ,b)对称的充要条件是 f (x) + f (2a－x) = 2b

证明：（必要性）设点P(x ,y)是y = f (x)图像上任一点,∵点P( x ,y)关于点A (a ,b)的对称点P‘（2a－x,2b－y）也在y = f (x)图像上,∴ 2b－y = f (2a－x)
即y + f (2a－x)=2b故f (x) + f (2a－x) = 2b,必要性得证.
（充分性）设点P(x0,y0)是y = f (x)图像上任一点,则y0 = f (x0)
∵ f (x) + f (2a－x) =2b∴f (x0) + f (2a－x0) =2b,即2b－y0 = f (2a－x0) .
故点P‘（2a－x0,2b－y0）也在y = f (x) 图像上,而点P与点P‘关于点A (a ,b)对称,充分性得征.