3 things my students struggle with are domain, range, the connections of a quadratic to it's parts.
I introduce domain as a king, and what he can control. Given a word problem, then generalizing to a function, what is in control (x) of the person/thing who is beginning the action. The range is connected to the outcome (a king's domain is the range of his power), and what are viable outcomes. They don't always have to be numeric, they can be object oriented.
With a quadratic, I have a project where a motorcylist is trying to jump over object. I give the initial height and velocity, the students have to figure out the end of his jump, the height, and write an equation that describes the path. It connects the parts of a quadracit