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EDIT: in view of the questions below, in particular Bill's one, here is the puzzling code. Manipulate and the first Plot do not work, the last Plot does. Why???:

f[x_] = a + c x^2 + d x^3;

sol = Solve[{f[0] == f0, f[1] == f1, f'[1] == ff1, f'[0] == 0}, {a, c, d}, Reals];

g[t_, f0_, f1_, ff1_] := f[t] /. sol[[1]];

Manipulate[
 Plot[g[t, f0, f1, ff1], {t, 0, 1}], {{f0, 0}, 0, 1}, {{f1, 1/2}, 0, 1}, {{ff1, 0}, -1, 0}]

Plot[g[t, 0.8, 0.9, -1/2], {t, 0, 1}]

Plot[g[t, f0, f1, ff1] /. {f0 -> 0.8, f1 -> 0.9, ff1 -> -1/2}, {t, 0, 1}]

EDIT: I believe this is not a duplicate question, because the answers to questions pointed to as similar DO NOT give practical solutions to the problems, at least not for normal Mathematica users like me (not computer science experts) who are endlessly confused by the arcane details of Set, SetDelayed, Evaluate, etc.

I thank Karsten 7 here who produced a practical, usable answer

EDIT: in view of the questions below, in particular Bill's one, here is the puzzling code. Manipulate and the first Plot do not work, the last Plot does. Why???:

f[x_] = a + c x^2 + d x^3;

sol = Solve[{f[0] == f0, f[1] == f1, f'[1] == ff1, f'[0] == 0}, {a, c, d}, Reals];

g[t_, f0_, f1_, ff1_] := f[t] /. sol[[1]];

Manipulate[
 Plot[g[t, f0, f1, ff1], {t, 0, 1}], {{f0, 0}, 0, 1}, {{f1, 1/2}, 0, 1}, {{ff1, 0}, -1, 0}]

Plot[g[t, 0.8, 0.9, -1/2], {t, 0, 1}]

Plot[g[t, f0, f1, ff1] /. {f0 -> 0.8, f1 -> 0.9, ff1 -> -1/2}, {t, 0, 1}]

EDIT: in view of the questions below, in particular Bill's one, here is the puzzling code. Manipulate and the first Plot do not work, the last Plot does. Why???: f[x_] = a + c x^2 + d x^3;

sol = Solve[{f[0] == f0, f[1] == f1, f'[1] == ff1, f'[0] == 0}, {a, c, d}, Reals];

g[t_, f0_, f1_, ff1_] := f[t] /. sol[[1]];

Manipulate[ Plot[g[t, f0, f1, ff1], {t, 0, 1}], {{f0, 0}, 0, 1}, {{f1, 1/2}, 0, 1}, {{ff1, 0}, -1, 0}]

Plot[g[t, 0.8, 0.9, -1/2], {t, 0, 1}]

Plot[g[t, f0, f1, ff1] /. {f0 -> 0.8, f1 -> 0.9, ff1 -> -1/2}, {t, 0, 1}]

Comment: trying to format the code with the 4 spaces but failing. using backtick instead

EDIT: in view of the questions below, in particular Bill's one, here is the puzzling code. Manipulate and the first Plot do not work, the last Plot does. Why???:

f[x_] = a + c x^2 + d x^3;

sol = Solve[{f[0] == f0, f[1] == f1, f'[1] == ff1, f'[0] == 0}, {a, c, d}, Reals];

g[t_, f0_, f1_, ff1_] := f[t] /. sol[[1]];

Manipulate[
 Plot[g[t, f0, f1, ff1], {t, 0, 1}], {{f0, 0}, 0, 1}, {{f1, 1/2}, 0, 1}, {{ff1, 0}, -1, 0}]

Plot[g[t, 0.8, 0.9, -1/2], {t, 0, 1}]

Plot[g[t, f0, f1, ff1] /. {f0 -> 0.8, f1 -> 0.9, ff1 -> -1/2}, {t, 0, 1}]

The following does not work:

g = a + b t;  Manipulate[Plot[g, {t, 0, 1}], {a, 0, 1}, {{b, 1/2}, 0, 1}]

can you explain why? thanks

NOTE that instead writing explicitly the function does work: Manipulate[Plot[a + b t, {t, 0, 1}], {a, 0, 1}, {{b, 1/2}, 0, 1}]

EDIT: in view of the questions below, in particular Bill's one, here is the puzzling code. Manipulate and the first Plot do not work, the last Plot does. Why???: f[x_] = a + c x^2 + d x^3;

sol = Solve[{f[0] == f0, f[1] == f1, f'[1] == ff1, f'[0] == 0}, {a, c, d}, Reals]; 

g[t_, f0_, f1_, ff1_] := f[t] /. sol[[1]];

Manipulate[ Plot[g[t, f0, f1, ff1], {t, 0, 1}], {{f0, 0}, 0, 1}, {{f1, 1/2}, 0, 1}, {{ff1, 0}, -1, 0}]

Plot[g[t, 0.8, 0.9, -1/2], {t, 0, 1}]

Plot[g[t, f0, f1, ff1] /. {f0 -> 0.8, f1 -> 0.9, ff1 -> -1/2}, {t, 0, 1}]

Comment: trying to format the code with the 4 spaces but failing. using backtick instead