Final answer:
The phase of the Moon after the first quarter where the visible illuminated portion is growing and more than half is seen is called the waxing gibbous. This phase precedes the full moon, where the entire face of the Moon visible from Earth is lit up, making it a notable feature of the night sky.
Step-by-step explanation:
The phase of the Moon after the first quarter, when the sunlit portion is increasing and more than half is illuminated, is known as the waxing gibbous phase. During the week after the first quarter, the amount of the illuminated hemisphere of the Moon that is visible from Earth continues to grow. This occurs because the Moon is on its eastward orbit around Earth, gradually showing more of its sunlit side until it reaches the full moon phase, where the entire face of the Moon that is visible from Earth is illuminated.
As it transitions from the first quarter to a full moon, the Moon undergoes a transformation. After the first quarter (position C in the lunar cycle), when exactly half of the Moon's visible side is illuminated, it moves into the waxing (growing) gibbous phase (position D). The term gibbous is derived from the Latin gibbus, meaning hump. Eventually, the Moon arrives at its full phase (position E), where it is opposite the Sun in the sky, rising at sunset and setting at sunrise, being fully illuminated throughout the night.
In this phase, the Moon becomes a prominent feature in the night sky, aiding in activities that occur under the cover of darkness. Observing the Moon through binoculars during different phases provides varying levels of topographic detail due to changes in lighting and shadowing, with the first or third quarter phase allowing for more detailed study of the Moon's surface features.