We know that is defined in terms of the exponential function as follows
 
 

By rewriting the function as
 
 

we can analyze the signs which this function takes for various values of x.
 
 

So, since is always positive, and the denominator (=  2)  of is always positive, we have that
 
 

So, the graph of the function is negative when x < 0, the graph of the function is positive when x > 0, and the graph equals 0 when x = 0 (plug x = 0 into the exponential form of ).
 

Now, we take the derivative of and obtain
 
 

If we analyze the signs for this function which can be rewritten as
 
 

and note that both factors in the numerator are positive for all values of x we obtain that
 
 

Thus, the derivative of is positive for all values of x, and so the graph is always increasing.
 

Finally, we take the second derivative of .  But the second derivative is
 
 

and so the same analysis for signs applies for the second derivative as did for the function itself.  That is, for x < 0, the second derivative is negative and so the graph is concave down, and for x > 0, the second derivative is positive and so the graph is concave up.