Plotting symbolic functions

How about this?

f[x_, a_, b_, c_] = a PDF[NormalDistribution[b, c], x]

Then

ContourPlot3D[f[x, 1, b, c], {x, -3, 3}, {b, -1, 1}, {c, 1, 2}]

Note that I am not claiming it is particularly informative!

This might be more useful?

Plot[Table[f[x, 1, b, 1], {b, -1, 1, 1/4}] // Evaluate, {x, -2, 2}, 
 PlotStyle -> ColorData[10]]

Plot[Table[f[x, 1, 1/2, c], {c, 1/2, 2, 1/4}] // Evaluate, {x, -2, 2}, 
 PlotStyle -> ColorData[10]]