Amanda Rogers was a female Q born in 2351 in Topeka, Kansas, on Earth's North American continent. Unlike other Q, she was conceived and raised as a Human into adulthood when she joined the Q Continuum and consequently retained a moral code.
Amanda was further raised by adoptive parents, two Starfleet marine biologists. As such, Amanda did not know about her true Q nature and lived a Human life. However, she occasionally experienced objects come in and out of existence based on her desires. She never had it explained and simply lived with it, but tried not to let it affect her normal life. Increasingly, however, she gained the attention of the other Q, who became concerned about whether she was fully Q and potentially a danger if not brought into the Continuum.
Later in her teens, she had three dogs, which her parents called her "zoo". She wanted a dozen, but her mother said "enough is enough". Upon leaving home, she noted that she would miss them. At the time of her departure, her parents had been reassigned to the Bilaren system. (TNG: "True Q")
Her transcript of academic accomplishments included honors work in neurobiology, plasma dynamics, and eco-regeneration. Her paper on eco-regeneration later helped familiarize her on the use of barystatic filters.
By 2369, she was accepted to Starfleet Academy and selected out of hundreds of applicants for an internship on the USS Enterprise-D. She traveled to Starbase 112 where, on stardate 46192.3, she transferred to the Enterprise-D to begin her internship. Doctor Beverly Crusher acted as her mentor, and arranged for her to be able to work in all the major departments.
For her first task, she tested a number of tricorders that were to be packed in supply containers and designated for the relief effort on Tagra IV. Her job was to ensure that each of their readouts worked correctly, thereafter she assisted Nurse Alyssa Ogawa with tranferring them to the shuttlebay.
About six months before arriving on the Enterprise, Amanda found her unexplained Q nature began to emerge, where according to her, "things that I would wish for would just suddenly appear. I thought I was going crazy." When the Continuum discovered this, they sent their "expert in humanity", Q, to investigate.
After a litter of puppies appeared in her quarters in private, Q covertly tested her on her abilities. Q first staged a container to fall on Commander Riker, which she secretly prevented, then he caused a warp core breach, which she successfully contained and reversed.
Thereafter, Q explained his intentions to test and train Rogers to the Enterprise crew, Amanda was informed of her true nature and was introduced to Q by Jean-Luc Picard and Beverly Crusher. She was shocked at the scope of what she was told, and, initially, decided to stay on the ship and learn not only her Enterprise duties but also her Q abilities.
However, when she tried to mix them, by performing an experiment with those abilities without the due diligence it required, she realized they may not be able to coexist. Furthermore, she acted on a crush she had on Riker by taking him to a romantic fantasy only to realize it wasn't real after he pointed it out.
While this was going on, Picard decided to investigate Rogers' parents' death and found out the truth. He confronted Q about it and told Rogers. Q then gave Rogers the same choice her parents were given.
She initially decided to stay and refrain from using her powers, but then the Enterprise-D received a distress call from Commander Riker, who was organizing the Enterprise-D's relief mission on Tagra IV. A reactor was overloading, threatening thousands of people. When La Forge told them that he could not fix the problem and meltdown was imminent, Rogers intervened and the problem seemed to resolve itself. Even the planet's significant atmospheric contamination was restored to its natural state.
Acknowledging her true nature, Amanda told Q that she would go with him, but that first, she would need to visit her adoptive parents and explain these events. She told Q, sternly, that he would have to be patient with her as she struggled to do this. Before leaving Rogers asked Dr. Crusher if she might come back one day to see her and then left the Enterprise-D with Q. (TNG: "True Q")
Amanda, along with Q (Junior), are the only children of Q known to exist.
As an in-joke, episode writer René Echevarria named her "Samantha" in early drafts of the script, after the main character in Bewitched. Rick Berman insisted that this be changed. (Star Trek: The Next Generation Companion (2nd ed., p. 222))
In a 2011 interview, d'Abo commented, "Amanda had a vulnerability I related to because she was an orphan and in the episode was clearly at a stage in her life where she had some pretty important choices to make about her future. She had a strength and a wisdom about her. She was searching for the truth and searching for her own sense of truth. Though she was impressionable at times, it was inevitable to me through playing her that through her own self discovery she'd know which choices to make and what path to take." 
During the 0 Crisis, during which the USS Enterprise-E attempted to penetrate the galactic barrier, the female Q told Doctor Crusher that Amanda Rogers didn't "count" as a true child of the Q as she was conceived in a purely Human fashion, whereas "Junior" was conceived in the Q Continuum itself following "The Q and the Grey".
Amanda Rogers appears again in The Eternal Tide – referred to as "Q" by other Q – where she is shown to have formed a sibling-like relationship with "Junior". She is subsequently erased from history when trying to determine the source of an anomaly that is preventing her and Junior from traveling into the future by accessing an alternate timeline where USS Voyager was able to deal with the anomaly on its original trip home (before the events of "Endgame" changed history) only to be erased from existence as the resolution of the threat in that timeline also erased the Q from existence. Only Junior is shown to retain any memory of Amanda's existence.
An alternate fate for Amanda is presented in the miniseries Star Trek: The Q Conflict; after Amanda assists Wesley Crusher in giving the four captains the tools needed to end Q's conflict with other omnipotent races, part of the final truce sees Amanda and Wesley being stripped of their powers, Q assuring Picard at the end of the series that Amanda has been sent to begin a career in biomedical sciences.