  Let $X$ be a proper geodesic metric space. We give a new construction of the
Morse Boundary that realizes its points as equivalence classes of functions on
$X$ which behave similar to the "distance to a point" function. When $G=\langle
S \rangle $ is a finitely generated group and $X=Cay(G,S)$, we use this
construction to give a symbolic presentation of the Morse boundary as a space
of "derivatives" on $Cay(G,S).$ The collection of such derivatives naturally
embeds in the shift space $\mathcal{A}^G$ for some finite set $\mathcal{A}.$
