Franklin played college football at Texas A&M University where he set the record for the longest collegiate field goal of 65 yards on October 16, 1976 against Baylor University. This record was short-lived when Ove Johansson—on the same day—kicked a 69-yard field goal. However, Franklin also kicked a 64-yard field goal in the same game and remains the only college player to kick two 60+ yard field goals in one game.
Franklin finished his collegiate career with 18 NCAA records, including most career field goals (56), career kick scoring (291 points), most field goals of 50 yards or longer in a career (16), most field goals made in a three year career (45) and longest average for field goals made in a career (39.5 yards). He was a consensus All-American pick as a sophomore in 1976 and earned All-American honors again as a senior in 1978. He was a Second-team All-America selection as a junior, in 1977.
He is listed as a "Legend of the Sun Bowl" by the Sun Bowl Association. He is also a member of the Sun Bowl's 75th Anniversary All-Sun Bowl Team.
Franklin was drafted by the Philadelphia Eagles in the third round of the 1979 NFL Draft. He was credited with the fourth longest field goal (at the time) in NFL history during his rookie year with the Eagles, a 59-yarder at Dallas on November 12, 1979. Franklin played in Super Bowl XV and kicked a 30-yard field goal.
On February 21, 1984, Franklin was traded to the New England Patriots for a sixth-round draft choice in the 1985 NFL Draft. He led the NFL in scoring (140 points) and field goals made (32) in 1986 and was selected to represent the AFC in the Pro Bowl. He played in Super Bowl XX and kicked a 36-yard field goal. The 32 field goals he made in 1986 was a Patriots single-season franchise record until Stephen Gostkowski broke the record in 2008.