A statement by Izabela Rzeszowska's mother Jolanta Gartska, read by a family liasion officer, said the family was "heart broken" over their deaths.
Damian Rzeszowski armed himself with kitchen knives and stabbed his father-in-law Marek Gartska, 56, wife Izabela Rzeszowska, 30, daughter Kinga, five, and two-year-old son Kacper, at their Jersey home on August 14 last year.
The 31-year-old Polish builder also turned on his wife's friend, Marta De La Haye, 34, and her five-year-old daughter Julia, who were at the house for a barbecue.
Rzeszowski was cleared of murder but found guilty of manslaughter through diminished responsibility at the Royal Court in the Channel Island's capital St Helier.
The court heard that Rzeszowski's marriage had been under strain after his wife confessed to a two-month affair with another man. Rzeszowski claims that on the day of the killings he heard voices which said "Kill, kill" and cannot remember the attacks.
Izabela Rzeszowska and Marek Gartska's family said that they had been "devastated by an enormous tragedy". Izabela's mother Jolanta Gartska, brother Krzysztof Gartska and sister-in-law Agnieszka Gartska, said in a statement that the tragedy was even worse because they had lost children and grandchildren.