Why does the federal government subsidize state programs in these policy areas?

States can choose to accept federal mandates; as a result, the federal government provides funding for certain programs.
States are required to provide certain programs; as a result, they receive federal funding based on state tax revenue.
Federal mandates require states to provide certain programs; as a result, the federal government provides a portion of the funding.
Federal mandates require states to provide certain programs; as a result, states receive equal funding for all programs.