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1) the degree of the function is odd, so the ends of the graph continue in opposite directions. because the leading coefficient is positive, the left side of the graph continues down the coordinate plane and the right side continues upward.

the mandate of heaven or tian ming (chinese: 天命; pinyin: tiānmìng; wade–giles: t'ien-ming) is a chinese political and religious doctrine used since ancient times to justify the rule of the king or emperor of china. according to this belief, heaven (天, tian)—which embodies the natural order and will of the universe—bestows the mandate on a just ruler of china, the "son of heaven" of the "celestial empire". if a ruler was overthrown, this was interpreted as an indication that the ruler was unworthy, and had lost the mandate. it was also a common belief among citizens that natural disasters such as famine and flood were signs of heaven's displeasure with the ruler, so there would often be revolts following major disasters as citizens saw these as signs that the mandate of heaven had been withdrawn.[1]

a brief flow chart describing the dynastic cycles in imperial china on claiming to withdrawing the mandate of heaven

the mandate of heaven does not require a legitimate ruler to be of noble birth, depending instead on the just and able performance of the rulers and their heirs. dynasties such as the han and ming dynasties were founded by men of common origins. the concept is in some ways similar to the european concept of the divine right of kings; however, unlike the european concept, it does not in theory confer an unconditional right to rule, despite this being exactly the case in practicality. the mandate would in theory be a preoccupation in a ruler's lifetime, when he would hold onto the mandate and live according to heavens. intrinsic to the concept of the mandate of heaven was the right of rebellion against an unjust ruler. chinese historians interpreted a successful revolt as evidence that heaven had withdrawn its mandate from the ruler. throughout chinese history, times of poverty and natural disasters were often taken as signs that heaven considered the incumbent ruler unjust and thus in need of replacement. the mandate of heaven was often invoked by philosophers and scholars in china as a way to curtail the abuse of power by the ruler. while each dynasty was not the same, they each had a lineage that passed on the prospective ruler by order of generational descent or their priority of birth. many emperors during the imperial times would optimize to have many sons who could be candidates to fill the position after the current ruler has died. in addition heaven was thought to be of how a ruler's works and performance was, which reflected upon how favorable they would be to heaven.