The United States Constitution guarantees the right to freedom of speech in the First Amendment:
Congress shall make no law ... abridging the freedom of speech, or of the press.
In one way, this seems unequivocal, but subsequent legal decisions have invoked what is termed in philosophy "the harm principle." In other words, it means that liberties may only be infringed when freedom to do some act will harm others. For example, one is not free to commit murder or steal.
A student researching the boundaries of free speech with respect to speech denigrating the LGBTQIA+ community should address several issues. The first is where free speech can be transformed into bullying. Cyberbullying of LGBTQIA+ students can be just as harmful as physical bullying, and schools need to protect students from bullying in all forms.
A second issue one can look at is the difference between expressing an opinion and inciting violence. If X says, "I do not like gay people," X is expressing an opinion. If X says, "All gay people should be forcibly converted," X is inciting violence. If someone organizes a lynching, that is a conspiracy to commit violence.
Thus, in writing an essay about this topic, a student's main consideration should be thinking about the different types of speech involved and the degree of harm done by those types of speech.