Manipulate[domain={-1,1};
nodes=Table[domain[[1]]+i*(domain[[2]]-domain[[1]]),{i,0,1,1/n}];

f[x_]:=1/(1+25x^2);
points=Table[{nodes[[i]],f[nodes[[i]]]},{i,1,Length[nodes]}];
Show[Plot[f[x],{x,domain[[1]],domain[[2]]},PlotRange->{{-1,1},{-.5,1}},ImageSize->Full],Plot[InterpolatingPolynomial[points,x],{x,domain[[1]],domain[[2]]}],Graphics[{PointSize[Large],Point[points]}]],{{n,1},1,7}]

Manipulate[domain={-1,1};
x0=Mean[{domain[[1]],domain[[2]]}];
nodes=Table[x0+i*(domain[[2]]-domain[[1]])/n,{i,0,n/2}];
nodes=Join[nodes,Table[x0-i*(domain[[2]]-domain[[1]])/n,{i,1,n/2}]];

f[x_]:=1/(1+25x^2);
points=Table[{nodes[[i]],f[nodes[[i]]]},{i,1,Length[nodes]}];
Show[Plot[f[x],{x,domain[[1]],domain[[2]]},PlotRange->{{-1,1},{-.5,1}},ImageSize->Full],Plot[InterpolatingPolynomial[points,x],{x,domain[[1]],domain[[2]]}],Graphics[{PointSize[Large],Point[points]}]],{{n,1},1,7}]