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?)

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

^{2}?

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=x

^{2}. Now we'll also mess around with y=a(x-p)(x-q) and y=ax

^{2}+bx+c.

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