Why does $f(-x+a)$ shift to the right?

$f(x+a)$ ($a>0$) represents a shift of $f(x)$ by $a$ units to the left , but $f(-x+a)$ represents a shift of $f(-x)$ to the right by $a$ units. Can someone explain to me why this happens? I thought that $f(-x+a) = f(-(x-a))$ (that is, shift $f(x)$ $a$ units to the right, and then reflect in the $y$-axis), but this is clearly incorrect, too. I have a feeling that this has an obvious explanation, but I cannot see it. All help appreciated.

Solution 1:

1. If you have $g(x)=f(-x)$, then $f(-x+a) = g(x-a)$.

2. the graph of $g(x-a)$ is the graph of $g(x)$, shifted to the right by $a$.

So there is no contradiction.

Solution 2:

It's shifting to the right because even if you have FIRST translated to the left by $a$ unit, THEN you reflect on the y-axis (because of the (-1) factor of x). so the LEFT translation becomes a RIGHT translation, due to this reflection.

For example:

$f(x + a)$ : You are translating to the left (by $a$ units).

$f(-x + a)$ : You are translating to the left (by $a$ units), THEN, reflecting it. The result is indeed a RIGHT translation.