(a)

Normal density function is .

Where  is mean and positive constant  is called standard deviation.

Let the function in special case by removing the factor  and considering .

Therefore, the function is .

Horizontal asymptote :

Therefore  is the horizontal asymptote of the function.

Maximum value:

Consider .

Differentiate on each side with respect to .

Find the critical points by equating the derivatve to zero.

The function has the maximum value at , Since .

Maximum value is .

Inflection points:

.

Differentiate  on each side with respect to .

Find the inflection points by equating  to zero.

Substitute  in the function.

Inflection point is .

(b)

The curve equation is .

Property : , stretch the graph of  the function  horizontally by a factor of .

stretch the graph of  the function  horizontally by a factor of , for .

(c)

Graph :

Graph the curve  for .

Observe the graph :

stretch horizontally as  increases.

 is the horizontal asymptote of the function.

Maximum value is .

Inflection point is .

stretch the graph of  the function  horizontally by a factor of , for .

stretch horizontally as  increases.