Building Functions, Clarified

So, I really appreciate all the thoughtful input on this previous post. I started commenting, but the comment got real big.

To give some background, I'm writing a unit flow for an introduction to quadratics unit. The big things I'd like students to remember from this unit for a long time are:

• some situations that can be modeled with quadratics, i.e. falling objects, triangular numbers, area
• compare/contrast with linear and exponential (which come before this). how can we tell if given information (situation, table, graph, 3 points) can be modeled with linear, exponential, or quadratic?
• what information can easily be obtained from each of the forms (vertex, standard, factored), given an equation
• what form (vertex, standard, factored) is it easiest to write an equation in, given different information
• why the graphs look like that (why symmetrical? why a U-shape? why does it have a max or min?)

What I DON'T want to do in this unit is algebraically convert between equivalent forms, or solve quadratics with various techniques.

The big question is still, how do you introduce things that are more complicated than ax²?

Option 1: Okay, kids, come up with a rule that models something like {(0,3), (1, 4), (2,7) (3,12)}

or

Option 2: y=a(x-h)²+k with some sliders, mess around on Desmos and see how a, h, and k make it different from y=x². Now we'll also mess around with y=a(x-p)(x-q) and y=ax²+bx+c.

The room seems to be divided on this, but leaning toward option 2.

Pretty Big Ideas for Intermediate (Highschoolish) Mathematics

I am just jumping into this for fun. I think they all capture the idea of a big idea or pretty big idea, but feel free to argue.

variable - we can work with quantities, and learn things about them and draw conclusions from them, even when we don't know what they are.
inverse operations - operations can be undone (and sometimes they can't, at least not uniquely) and this is useful for solving all kinds of problems.
functions - Different kinds of rules that map a set of numbers to another set of numbers follow certain patterns.
transformations - rules can be changed in systematic and useful ways.
equivalence - How do we know when things are the same? How do we know when they are not the same?
proof  - Usually a big feature of a geometry class, but I'd argue at least as important in algebra. How do we know for sure that something is true?