Total alkali (sodium plus potassium expressed as oxides) is a fair proxy for the alkali or A-to-P modal dimension of the QAP diagram, and silica (total silicon as SiO2) is a fair proxy for the quartz or Q direction. Geologists usually use the TAS classification because it's more consistent. As igneous rocks evolve during their time beneath the Earth's crust, their compositions tend to move upward and rightward on this diagram. The diagonal line running from basalt to phonolite roughly marks the division between quartz-bearing and quartz-free rocks.
Trachyte and trachydacite are distinguished by their quartz content versus total feldspar, the vertical dimension on the QAP diagram. Trachyte has less than 20 percent Q, trachydacite has more.
The division between foidite, tephrite and basanite is dashed because it takes more than just alkali versus silica to classify them. All three are without any quartz or feldspars (instead they have feldspathoid minerals), tephrite has less than 10 percent olivine, basanite has more, and foidite is predominantly feldspathoid.