[1] Graydon graduated from the Classical course at Butler University in 1878 and began teaching Greek at a school in Indianapolis replacing her own professor, John O. Hopkins, after his death in November 1877.

In 1883 she gained her master's degree from Indiana University Bloomington and was subsequently appointed as the assistant professor of Latin and Greek there.

[1] Graydon's teaching career at Indiana was cut short after her relationship with Lemuel Moss, then president of the university, was exposed by a group of students who spied on the couple through peep holes drilled into the attic above her seminar room.

After 1892, she spent several years as a private tutor for the children of naturalist John Muir, whom she had known since childhood through her aunt Catharine Merrill, the second female university professor in the United States.

She remained in this position until 1907 when she returned to Butler University to take up a chair in English, named after her by-then deceased aunt Catharine Merrill.