I just learned how to factor quadratic equations (at least, how to factor them when they're simple and easy; I don't know how to factor when $a$ isn't equal to $1$). I'm fine at doing this, and I get that the outcome gives me the roots of the parabola. However, I would love to know how the process itself (as in the factoring and the creation of two linear equations) relates to graphing parabolas and to the general behavior of quadratic equations. Is there any visual representation of what's happening?
By the way, I am somewhat familiar with other ways to solve quadratic equations such as completing the square and using the quadratic formula, and I know how to find the vertex; I would love to know how factoring relates to these things (conceptually) as well.
Thank you, and I'm sorry for the vagueness of my question.