I'd like to see far more emphasis on understanding rate of change, from 8th grade algebra on; bake the concepts of calculus in to our whole secondary school math curriculum. The world is described by differential equations, and linear and exponential functions are the easiest differential models to understand and solve. That's why I want to start there.
I like exponential functions as a topic for fairly naive students because the idea of rate of change proportional to the quantity is natural and easy to motivate -- the idea that both interest, and populations without predators or resource limitations change in that fashion makes sense. All you need is for each rabbit to be equally likely to have a baby, regardless of the size of the rabbit population, to very quickly get to the exponential equations.
It's harder to come up with a situation that's obviously modeled with linear rate of change, and the connection between linear change and quadratic equation is not as easy to understand as the way exponentials arise from population growth. Yes, of course I know that's how you get from acceleration to velocity to position, and that you can go from constant acceleration to quadratic position with basic geometry, and no need for calculus machinery. I think that would be a great topic to teach, somewhere in the HS math sequence -- in fact, I think it's a great way to motivate interest in polynomials. I just think it's a crappy place to start.
Currently, we teach quadratics and polynomials long before we teach the physics connections. While ninth graders have problems which involve balls following parabolic trajectories, they have no idea why those trajectories are parabolas, instead of any other arbitrary equation, so there's precious little connection actually being made between the math and the real world. That's what people (non-STEM people) remember from algebra, and that's what I want to get away from.
To be clear, I think working with polynomials, factoring, and the quadratic equation should be taught, just not with the emphasis that we currently give it as the cornerstone topic in first year algebra. Treat them like the trig identities: important math tools you meet along the road to calculus. Just don't make them the way that road starts.