Final answer:
The characteristic of all living organisms demonstrated by the roots growing downwards while the shoot grows upwards and the flower turns towards the sun is tropism. Specifically, the roots exhibit gravitropism by growing downward in response to gravity, and the shoot exhibits phototropism by growing towards the light.
Step-by-step explanation:
The characteristic of all living organisms that is demonstrated by the roots of a sunflower plant growing downwards while the shoot grows upwards and the flower turns towards the sun is tropism.
Tropism is the ability of an organism to turn or move in response to a stimulus in the environment. In this case, the roots exhibit gravitropism by growing downward in response to gravity, while the shoot exhibits phototropism by growing towards the light.
These tropisms allow the plant to optimize their growth and survival by positioning their roots for water and nutrients and their shoots for sunlight, which are essential for their growth and photosynthesis.