Assuming your name was not on the property deed prior to the quit-claim, your mom gifted the property to you. So your cost basis, is whatever your mom's cost basis was, plus the cost of any property improvements you paid for after you owned it.
So your cost basis is what your mom paid for it, plus the cost of any improvements your mom paid for while she owned it, plus the cost of any improvements you paid for while you owned it. So if you sold it for more than your final cost basis, you pay taxes on the difference.
The usual rule, for a gift, is that the recipient's basis is the giver's basis (what you mother paid for it). But there is an exception for the gift of her home, where she retained the right to live there ("life estate"). (seehttp://www.njelderlawestateplanning.com/2010/02/articles/estate-and-inheritance-tax/life-estates-estate-tax-and-inheritance-tax-consequences/ which states in part "If you give away an asset and keep a life estate in that asset..... the cost basis of the house is "stepped-up" to the value of the house on date of death [IRC 2036]")