It was a rollercoaster few days for Beverly native Angie Miller on “American Idol.” On Saturday she enjoyed a triumphant return to her hometown. On Wednesday she gave three well-received performances buoyed by that trip home. And on Thursday the teen bid a tearful farewell when she was eliminated from the Fox singing competition. Buffeted by her fellow contestants Candice Glover and Kree Harrison — who advanced to next week’s finale — Miller watched a video montage of her journey from her first audition in New York City through the various stages of the competition. A cheerful presence on the show, Miller was deemed “festive” by judge Mariah Carey since she was often seen smiling and clapping her hands joyfully. Miller received high marks throughout season 12. The show’s panel of judges routinely praised the quality and power of her voice, her piano playing, and original songwriting, with Carey calling her potential limitless. After a nail-biting hour Thursday — in which the contestants watched a recap of Wednesday’s show and performances from Alicia Keys and former “Idol” contestant Lauren Alaina — Harrison, Glover, and Miller professed their love for each other before hearing the results of the voting. Miller then fought her way through a rendition of Colton Dixon’s “Never Gone,” which ended as she was embraced by family members, fellow contestants, the show’s judges, and host Ryan Seacrest. Miller’s “Idol” connection has not ended, however, as she will perform this summer on the “American Idol Live!” tour, which stops at Agganis Arena on Aug. 19. There was other “Idol’’ news Thursday. Randy Jackson, the lone remaining original judge, announced he won’t be returning next season. Jackson, 56, said in a statement that he’s leaving to focus on his record label and other business opportunities.