Q: I cannot turn airplay on using the icon or display preferences- not given as an option. Help please

Try here >  Use AirPlay to display video from your Mac on an HDTV - Apple Support

What model is your Mac, it needs to be from 2011 or later.

Are Mac and Apple Tv on the same network?

Is AirPlay enabled on Apple TV?

Also, did you check the firewall settings in that reference document?

Have you tried restarting Apple TV and the network router? The magic that Apple devices use to find one another can be a little finicky. Did AirPlay ever work for you? What OS X version are you running? What version Apple TV is it? See Identify your Apple TV model - Apple Support if not sure. If it's ATV 4, what tvOS?

I also have this problem.  I did not have this problem until I updated my OS X to Sierra.  I am have tried everything suggested and even talked to Apple Support for ten hours at two different times.  I am not sure what the fix is.  I am still unable to have my MacBook air and apple tv talk to one another on the same network.