Final answer:
The bold-faced word 'appearing' starts a participial phrase that functions as an adjective and describes the campaign.
Step-by-step explanation:
The type of phrase illustrated by the bold-faced word 'appearing' in the sentence 'appearing to be the underdog his campaign had taken a positive turn late in the election' is a participial phrase. Participial phrases begin with a past or present participle and act as adjectives, giving more information about a noun. In this case, 'appearing' serves as the present participle of the verb 'appear,' and the phrase 'appearing to be the underdog' describes the campaign.