Question

If the new moon happens on January 15th, what shape will it be on February 6th?

Answer 1

-- From January 15 to February 6 is a period of 22 days.

-- The period of the full cycle of moon phases is 29.53 days.

-- So those dates represent (22/29.53) = 74.5% of a full cycle of phases.

-- That's almost exactly 3/4 of a full cycle, so on February 6, the moon would be almost exactly at Third Quarter.  That's the left half of a disk (viewed from the northern hemisphere).