Final answer:
The number of protons in the nucleus is the most important factor in determining a trend across a period of the periodic table, influencing the atomic radius, ionization energies, and other properties.
Step-by-step explanation:
The most important factor in determining a trend from left to right across a period in the periodic table is the number of protons in the nucleus. As one moves from left to right across a period, protons are added to the nucleus and electrons are added to the same principal energy level. This results in an increased force of attraction between the nuclei and electrons, causing the size of atoms to decrease. This atomic radius decrease is a key periodic trend, although electron-electron repulsions can affect this trend slightly, as observed with oxygen and nitrogen.
Additionally, properties such as ionization energies generally increase from left to right due to the increasing nuclear charge, thus requiring more energy to remove an electron.