When you see a half-moon in the morning sky, it's a waning moon in its last quarter. A moon in the last quarter phase rises in the middle of the night and sets in the middle of the day.
This photo was taken around 9 a.m. on the day of the moon's last quarter. This was about 5 hours after the exact moment of the half-moon; since the moon waned a bit over those 5 hours, you see just slightly less than a half-moon in the photo.
