|MadSci Network: Astronomy|
Before I answer your question, a bit of background.
The phase of the Moon is determined by the relative positions of the Sun, the Moon, and the Earth. If the Moon is between the Earth and the Sun, then we have a new Moon with the shadowed side of the Moon facing the Earth. If the Earth is between the Moon and the Sun, then we have a full Moon.
For a quarter Moon, the triangle formed by the Sun, Earth, and Moon has a right angle in it, with the Moon either preceding the Earth in its orbital path or following the Earth. A good diagram/simulation to look at can be found at: Virtual Reality Moon Phases.
There is one extra factor that determines how a phase of the Moon will appear to a person standing on the Earth. This is the person's position on the Earth. The phase shape and orientation is fixed for a given time, but the orientation of the observer is not.
A person standing at the North Pole would see an inverted Moon and phase compared to a person standing at the South Pole. A person at the Equator would see a Moon phase rotated 90 degrees compared to a person at a pole. Observers at a pole of the Earth would always see the crescent Moon with the points aligned nearly vertically with the horizon.
Observers at the Equator would always see the crescent Moon with the points aligned nearly horizontally with the horizon. The appearance is complicated by the fact that the Moon's orbit is tilted a few degree with respect to the Equator.
Since you are in Texas, which is fairly close to the Equator, you will tend to see the bottom or top half of the quarter Moon lit. If you were on holiday in Alaska, you would tend to see the quarter Moon oriented with the left or right side lit. It would be the same Moon, but you would be standing "sideways" in one place compared to the other.
I hope this helps you.
from Keith your mad scientist in Wales, UK
Try the links in the MadSci Library for more information on Astronomy.