Can a bad temper be caused by traumatic brain injury?

Yes: If front of the brain is hurt yes can cause rage.

Very much so: Depending on the severity and location of the traumatic brain injury personality changes can happen. A famous quote is "[mr.] gauge was no longer gauge". Moodiness, depression, reduced control of anger or other emotions, or even lack of initiation and motivation, and passivity are possible. Depression is more often left-brain injury and mania or lack of awerness - right.

Yes: This depends on the severity and location of the injury. Generally, individuals may become more irritable. Forgetful, and have difficulty concentrating. If the injury is in the frontal lobe, the person may have personality changes as well, such as emotional lability and disinhibition.

Yes: Brain injury, especially with damage to the frontal and temporal lobes can result in impulsivity, irritability and dysregulated behavior with excessive reactivity.