Question

Classify the periodic trends according to whether they increase or decrease from left to right:
Ionization energy, metallic character, electronegativity, and atomic radius.

Answer 1

left to right (The increasing number of protons are able to attract the few added electrons leading to a higher effective nuclear charge, meaning that the force of the added protons are a lot stronger than the force of the few added electrons. This lead to a smaller atom, which in turn leads to a higher ionization energy.) Electronegativity also increases going left because the atoms try to establish a full valence shell, which is much more stable (ex, Chlorine has 5 electrons in its p orbital, thus it is strongly wants to gain an extra electron in order to have a full shell. Sodium on the other hand has only 1 electron in its s orbital and thus wants to lose it to have a full p orbital.)

Top to bottom: The build up of extra orbitals (example, s orbitals) leads to a larger atom. This is different from going left to right in that taking Sodium and Potassium for example, the difference between the two is a whole p and s orbital. These two extra shells make the Potassium atom much bigger than Sodium. While there is an added number of protons, the electrons in this case are able to overcome it with a large number of shells leading to shielding effects of the outer electrons (allowing them to "float away") and electron-electron repulsion between the two electrons in each sub-levels
This in effect leads to a lower ionization energy as it is easier to remove the outer electrons. This also leads to a lower electronegativity as the protons do not attract outside electrons that are far away as much as they are shielded by the many orbital levels.

Summary. Ionization energy increases left to right and bottom to top. Electronegativity increases left to right and bottom to top. Atomic radius increases right to left and top to bottom.

My apologies, I do not quite understand what you mean by metallic character. Let me research that more and come back with a follow up.