If f(x) is symmetric about (a, b)

Then the function values of equidistant x on both sides of a are equal in size and opposite in sign.

That is, f(a-x)=-f(a+x)

When x=a-t

The sign of the parameter f(t)=-f(2a-t) does not affect the function, i.e.

f(x)=-f(2a-x)

The y-axis direction is the same.

b-y=-(b+y)

And y=f(x)

b-y=-[b-f(2a-x)]

2b-y=f(2a-x)

2、

If f(x) is symmetric about x=m, then there is

f(m-x)=f(m+x)

Derive from both sides, get

-f'(m-x)=f'(m+x)

According to the definition, it means that f'(x) is symmetrical about the center of point (m, 0).

3、

Like the first question, f(x) is symmetrical about the center of (m, n).

That is, the x-axis direction is symmetrical about m, the y-axis direction is symmetrical about n, and the negative sign is opposite.

have

f(m-x)=-f(m+x)

n-y=-(n+y)

Derive x from the above two equations respectively.

f'(m-x)=f'(m+x)

N=0 (meaningless)

By definition, f'(x) is symmetric about x = m.