Polynomial coefficients
Polynomial coefficients is a theme within functional relations between variables. Links to relevant activities and resources are on the right hand side of this page. 
The list below shows how different ways of writing quadratics show different features of the function:
 the yintercept is visible clearly as c in the form: y = ax^{2 }+bx+c but the roles of a and b are obscure;

the xintercepts are visible as roots p and q in the factorised form
y = k(xp)(xq) but the yintercept is not obvious; changing the value of k is interesting, particularly with software; 
the turning point (probably the most obvious visual feature of quadratics) is most visible as (s,t) in the completed square form:
y = k (xs)^{2} + t. The displacement of +s in the xdirection appears in
(xs) which many students find a counterintuitive representation.
Working with quadratic functions helps students learn that relation of the shape of the graph to the parameters (coefficients) needs interpretation, and also to move away from the assumption that everything is linear and onetoone.
Quadratics as a 'bridge'
It could be argued that exponential and trigonometric functions have more obvious 'real world' applications and quadratics seem too abstract for students. But quadratics are still calculable, and so provide a bridge between students’ intuitive assumptions about graphs and the world of functions as tools to describe complex relations.