The family of bell-shaped curves occurs in probability and statistics, where it is called the normal density function.




The constant μ is called the mean and the positive constant σ is called the standard deviation. For simplicity, let€™s scale the function so as to remove the factor 1/(σ√2π) and let€™s analyze the special case where μ = 0. So we study the function f(x) = e-x2/(2σ2)
(a) Find the asymptote, maximum value, and inflection points of f.
(b) What role does σ play in the shape of the curve?
(c) Illustrate by graphing four members of this family on the same screen.

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