The family were at home in Whiterock Gardens when gunshots were fired through the window.
The attack took place at about 10pm on Wednesday and police are treating it as a hate crime.
The family, who are believed to be members of the Travelling community, were not injured.
A number of windows were broken at the house, where they are believed to have been living for less than a week.
Police have appealed for anyone with information to contact them on 101.
COMMENT RULES: Comments that are judged to be defamatory, abusive or in bad taste are not acceptable and contributors who consistently fall below certain criteria will be permanently blacklisted. The moderator will not enter into debate with individual contributors and the moderator’s decision is final. It is Belfast Telegraph policy to close comments on court cases, tribunals and active legal investigations. We may also close comments on articles which are being targeted for abuse. Problems with commenting? firstname.lastname@example.org