The knife was first seen in Bob Ogden's memory during the first lesson between Harry Potter and Albus Dumbledore. In the summer of 1925, Ogden was sent to summon Morfin to a hearing for attacking Tom Riddle Sr., a Muggle with a spell that covered his face in hives. As head of the Department of Magical Law Enforcement, visited the Gaunt shack to investigate Morfin's use of magic on a Muggle. Upon first meeting Ogden, Morfin held the knife brandished in one of his hands and with his wand making him cast a jinx that caused yellow pus to flow from Ogden's nose.
Morfin was seen using the knife again when he was threating to attack Ogden for firing a Revulsion Jinx at his father, Marvolo Gaunt when he was throttling his daughter Merope Gaunt when hearing that she was in love with the Muggle her brother attacked. It is unknown what happened to the knife when Morfin and his father were imprisoned in Azkaban, but when Morfin was released he had it in his possession again.
During Harry and Dumbledore's third lesson, the dove into Morfin's memory in which he meets his nephew (his sister's son), Tom Riddle who later became Lord Voldemort. Morfin held the knife up in his left hand and his wand on his right. Riddle entered the shack and at the very moment Morfin saw him, he thought it was his Muggle brother-in-law since Riddle resembled his father so much. He tried to attack Riddle but when he spoke in Parseltongue, Morfin stopped.
It is unknown what happened to the knife after Morfin was sent to Azkaban again after being wrongfully convicted by the Ministry of Magic for the deaths of the Riddle family where he later died, but it's possible he lost the knife forever.