Yes a child
that is 17 can be claimed so long as you provided more than half the support of the child.
For details on all the qualifications see A "Qualifying Child
In general, to be a taxpayer's qualifying child, a person must satisfy four tests:
Relationship - the taxpayer's child or stepchild (whether by blood or adoption), foster child, sibling or stepsibling, or a descendant of one of these.
Residence - has the same principal residence as the taxpayer for more than half the tax
year. Exceptions apply, in certain cases, for children of divorced or separated parents, kidnapped children, temporary absences, and for children who were born or died during the year.
Age - must be under the age of 19 at the end of the tax year, or under the age of 24 if a full-time student for at least five months of the year, or be permanently and totally disabled at any time during the year.
Support - did not provide more than one-half of his/her own support for the year.
I hope this helps for explaining the qualifications to claim a dependent when the child is a 17 yeara old.