How do you graph #y=sec (x/2)#?

The first step to graphing a reciprocal function is to expand the function with the definition of the reciprocal function. In this case:

#sectheta=1/costheta#

So, to put it in the actual function:

#sec(x/2)=1/cos(x/2)#

Now, graph the wave of #cos(x/2)# with a dotted line. The graph starts at #(0,1)# and has a period of #(2pi)/(1/2)# or #4pi#. (drawn in green)

![https://www.desmos.com/calculator]()

Next, mark a dot on the points where the wave reaches a maximum or a minimum (blue in the picture). Also, mark asymptotes (vertical lines) wherever the wave crosses the #x#-axis (red):

![https://www.desmos.com/calculator]()

Lastly, you can draw the actual function. To do this, you have to draw figures that look like parabolas between the red lines. If the blue point in that section is above the #x#-axis, then the "parabola" is also above the #x#-axis. If it is below, then the "parabola" is below also. In the end, it looks like this (in purple):

![https://www.desmos.com/calculator]()

Here's just the #sec(x/2)# graph:

![https://www.desmos.com/calculator]()

I created a Desmos website with all of this information:
https://www.desmos.com/calculator/uquf0kxbeo