Hey there adamfromtempe,
Thanks for reaching out to the Apple Support Communities.
So we understand the issue correctly, you're having issues with using AirPlay to mirror your MacBook display on your Apple TV (3rd generation) is that correct?
If so, give the steps listed in the following resource a try that can help with troubleshooting this issue: If screen mirroring or streaming isn't working on your AirPlay-compatible device - Apple Support
Try these steps first
After each step, see if you still need help.
1. Make sure that your AirPlay-compatible devices are turned on and near each other.
2. Check that the devices are updated to the latest software and are on the same Wi-Fi network.
3. Restart the devices that you want to use with AirPlay or screen mirroring.
If you're prompted to enter a password, you might have restrictions turned on, or you might need to adjust your AirPlay password settings.
From the information you've provided, we do see that there is a macOS update available for your device. Since updating can often bring the latest bug fixes, we would recommend updating and test the issue further.
For help with updating the macOS, check out this next link here: Update macOS on Mac - Apple Support
Use Software Update
Before following these steps, it's a good idea to back up your Mac.
1. From the Apple menu in the corner of your screen, choose System Settings or System Preferences.
- If you chose System Settings, click General on the left side of the window. Then click Software Update on the right.
- If you chose System Preferences, click Software Update in the window.
2. Software Update then checks for new software.
- If Software Update finds new software, click the button to install it. The button might be named Update Now, Upgrade Now, Install Now, or Restart Now, for example. You might then be asked to enter your administrator password.
- If Software Update says that your Mac is up to date, then no new software is currently available for your Mac model. Software Update shows only software that is compatible with your Mac.
Keep us updated on the results.
Have a great day!
