Plant hormones act locally and don't flow around the entire organism
Auxin controls the growth at the ends of the shoots and roots
When auxins are produced, they dissolve and diffuse backward along the roots/shoots. In a shoot, the auxin stimulates growth but in a root it inhibits growth.
Auxins always accumulate on the shaded side and lower side.
Shoots are positively phototropic as they grow towards the light but negatively geotropic as they grow away from gravity.
In shoots, the auxin will accumulate in the shaded side. As auxins stimulate growth, the plant will grow faster on the shaded side. This means the shoot will curve towards the light.
In shoots the auxins stimulate growth more on the lower side, so that side will grow faster and the shoot will grow upwards away from gravity.
Roots are negatively phototropic and positively gravitopic.
In a root auxins inhibit growth so if a root is exposed to light, the upperside will grow faster and curve away from the light. This curve downwards also means the root will curve towards gravity and grow downwards.