Question

If the graph of the function $y=f\left(x\right)$ is symmetrical about the line $x=2$, then

• A. $f(x+2)=f(x-2)$
• B. $f(x-2)=f(2-x)$
• C. $f\left(x\right)=f(-x)$
• D. $f\left(x\right)=-f(-x)$

Answer 1

The correct option is B $f(x-2)=f(2-x)$

When a graph is symmetric about $x=2$, the values of $f\left(x\right)$ at equal distance from $x=2$ on either side are equal.
If we translate the origin to $(2,0)$, in the new coordinate system, for the function we can write $f\left(X\right)=f(-X)$.
The relation between new and old coordinates are as follows:
$x=X+2$ and $y=Y+0$ (as shown in figure)
Hence, in terms of $x,y$ we can write
$f(x-2)=f(2-x)$