Reduced-Dose Intensity-Modulated Radiation Therapy With or Without Cisplatin in Treating Patients With Advanced Oropharyngeal Cancer

PRIMARY OBJECTIVES: -To select the arm(s) achieving a 2-year progression-free survival rate of \>= 85% without unacceptable swallowing toxicity at 1 year. SECONDARY OBJECTIVES: * To determine patterns of failure (locoregional relapse versus distant) and survival -(overall and progression-free) at 6 months and 2 years. * To determine acute toxicity profiles at the end of radiation therapy and at 1 and 6 months. * To determine late toxicity profiles at 1 and 2 years. * To determine patient-reported swallowing outcomes at 6 months and 1 and 2 years. * To determine the predictive value of 12-14 week, post-treatment fludeoxyglucose F 18 (FDG)-positron emission tomography (PET)/computed tomography (CT) for locoregional control and progression free survival (PFS) at 2 years. * To determine the predictive value of blood and tissue biomarkers for disease outcomes at 2 years. * To determine swallowing recovery per videofluoroscopy imaging at 2 years. After completion of study treatment, patients are followed at 1 and 3 months then every 3 months for 2 years, every 6 months for 3 years, and then annually thereafter.