Adel:
Before trying to get things to work in Manipulate, I suggest getting them to work with the minimum amount of code first.
Since you want to graph functions, I suggest sticking with Plot[]. Here is an example:
f[x_?NumericQ, n_Integer] := n * x;
Plot[{f[x, 2], f[x, 3], f[x, 4]}, {x, 0, 10},
PlotRange -> {{0, 10}, {0, 20}}]
Keep in mind that Plot[] has many options to customize the look.
Have a great and safe holiday.
