Writing a procedure that invokes Manipulate[]

Question

I would like to write a procedure with the signature:

Foo[vars, params]

which produces a Manipulate object. The procedure will dress up the vars into sliders of the form:

{{a, a0, "DisplayName"}, min, max, Appearance->"Labeled"}

and will produce a Plot, depending on both the params and vars. (The default value a0 and the other slider properties are generated in a particular manner from a.)

The issue I'm running into is that Manipulate requires all of its vars to be explicitly listed. I looked at the related question Proper way to handle free variables in manipulate/plot? to no avail (I couldn't get With to do what I want).

So how does one procedurally generate the commands of Manipulate? I am constrained to use Manipulate as opposed to Dynamic or other more advanced features.

---

Concrete Example:

Manipulate[
    Plot[
        c0 + c1 x + c2 x^2,
        {x,0,1}
    ],
    {{c0, 0},-1, 1},
    {{c1, 1}, 0, 2},
    {{c2, 2}, 1, 3},
]

Then the goal would be to replace the 2 with an n, i.e. to write a function Poly[n] which returns a manipulatable polynomial with coefficients c_i centered around i.

Answer 1

Please tell me if this does what you want:

poly[n_Integer?Positive] :=
 {
   Array[{{\[FormalC][#], #}, # - 1, # + 1} &, n + 1, 0],
   Sum[\[FormalC][i] \[FormalX]^i, {i, 0, n}]
 } /. {{var__}, expr_} :>
        Manipulate[Plot[expr, {\[FormalX], 0, 1}], var]

poly[4]