The correct name for a system of roots that grows as one large vertical root with smaller lateral offshoots is a tap root system.

Here's a breakdown:

* Tap root: The large, main vertical root that grows straight down.

* Lateral roots: The smaller roots that branch off horizontally from the tap root.

This type of root system is common in many dicotyledonous plants, such as carrots, dandelions, and oaks.