Hybrid Atomic Orbitals

In order to orient an atomic orbital in the correct region of space to bond in a particular geometry, we must do some math on the atomic orbitals. Orbitals are not "real" in a physical sense; they are mathematical functions that describe the behavior of an electron. Thus we can add, subtract, multiply or divide them. This leads to "hybrid AOs" with the same average energy as the underlying canonical AOs, but oriented in different particular directions. The number we come up with depends on what we start with.
(Note: the functions provided are qualitative only; a full treatment requires a normalization term.)


Tetrahedral

We need 4 AOs to point to the corners of a tetrahedron. By adding/subtracting the 2s and all three 2p AOs (4 orbitals in = 4 orbitals out), we achieve this:

Ψ1 = s - px + py - pz
Ψ1 = s + px - py - pz
Ψ1 = s + px + py + pz
Ψ1 = s - px - py + pz

Trigonal

A trigonal geometry needs to point to 3 directions defining a plane. We can get that from any two p orbitals plus an s orbital:
Ψ1 = s + px - py
Ψ1 = s + 2*py
Ψ1 = s - px - py

The last canonical AO, pz, remains unhybridized (and has a higher energy than the hybrid AOs).

Linear

The linear geometry requires a single p orbital, added to or subtracted from the s orbital:
Ψ1 = s + px
Ψ1 = s - px

The remaining p orbitals are again unhybridized and of higher energy.