It's like a bell curve; there's a high point in the middle with both sides dropping off symmetrically away from it and then flattening out as you move away from the drop offs.
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A Guassian function has a top in the middle and it's ends reach until infinity but the graph never touches the x axis. The location of the top depends on the parameters used.
It has no special name - other than a normal (or Gaussian) distribution graph.
yes, h=1/sigma(standard deviation)
the graph is called a line
A sine graph!