It was made from his father's fang mere hours before his father died.
In the grave of Inuyasha's father, we see that one of his father's fangs (presumably the one from which the swords were made) is missing. However, when Toutousai takes one of Inuyasha's fangs to repair Tetsusaiga, he says that it'll grow back in a day. If so, then Inuyasha's father either couldn't regenerate teeth or he died before it regrew – within a day.