Transformation of random variables vs shift of functions
I think you've actually answered your own question by observing that the pdf of the random variable (RV) $Y=X-a$ is given by $f_Y(y)=f_X(y+a)$, which corresponds to a left shift for $a&gt;0$. This is also immediately clear from the fact that by subtracting $a$ from $X$, the mean of the transformed RV is also shifted to $m_Y=m_X-a$.
Shifting a general function $f(t)$ is just the same, and you actually did exactly that with the pdf of the original RV to obtain the pdf of the transformed RV. For a shifted function $f(t-t_0)$, the origin (i.e., the value $f(0)$) is shifted to the value $t=t_0$. So for $t_0&gt;0$ this corresponds to a shift to the right, and for $t_0&lt;0$ this corresponds to a shift to the left.