Consider genus $g$ curves that admit degree $d$ covers to elliptic curves only branched at one point with a fixed ramification type. The locus of such covers forms a one parameter family $Y$ that naturally maps into the moduli space of genus $g$ curves $\bar{M}_g$.  We study the geometry of $Y$, and produce a combinatorial method by which to investigate its slope, irreducible components and genus. The results can be used to study the lower bound for slopes of effective divisors on $\bar{M}_g$.