A vehicle control apparatus, wherein the vehicle control apparatus: - receives a plurality of pieces of relative position information acquired at different timings by an external world recognition apparatus and each indicating a position of a preceding vehicle relative to a subject vehicle, the preceding vehicle being a target that the subject vehicle follows ahead thereof; - stores the received plurality of pieces of relative position information; - calculates an approximation equation of an N-th degree function (N is an integer equal to or greater than one in this case) as a movement trajectory of the preceding vehicle from at least two pieces of relative position information in each of a plurality of retrospective ranges set as ranges to which the vehicle control apparatus should date back from newest relative position information in a history of the stored plurality of pieces of relative position information, and calculates a coefficient of a predetermined degree in each approximation equation, and - in the calculation of the coefficient of the predetermined degree, - uses an approximation equation calculated from the at least two pieces of relative position information acquired when setting the retrospective range to the same range or a narrower range compared to when calculating a coefficient of a relatively low degree, when calculating a coefficient of a relatively high degree, and - uses an approximation equation calculated from the at least two pieces of relative position information acquired when setting the retrospective range to a narrower range compared to when calculating a coefficient of a lowest degree, at least when calculating a coefficient of a highest degree; and - outputs an instruction according to the calculated coefficient of each degree to a steering control apparatus of the subject vehicle. - receives a plurality of pieces of relative position information acquired at different timings by an external world recognition apparatus and each indicating a position of a preceding vehicle relative to a subject vehicle, the preceding vehicle being a target that the subject vehicle follows ahead thereof; - stores the received plurality of pieces of relative position information; - calculates an approximation equation of an N-th degree function (N is an integer equal to or greater than one in this case) as a movement trajectory of the preceding vehicle from at least two pieces of relative position information in each of a plurality of retrospective ranges set as ranges to which the vehicle control apparatus should date back from newest relative position information in a history of the stored plurality of pieces of relative position information, and calculates a coefficient of a predetermined degree in each approximation equation, and - in the calculation of the coefficient of the predetermined degree, - uses an approximation equation calculated from the at least two pieces of relative position information acquired when setting the retrospective range to the same range or a narrower range compared to when calculating a coefficient of a relatively low degree, when calculating a coefficient of a relatively high degree, and - uses an approximation equation calculated from the at least two pieces of relative position information acquired when setting the retrospective range to a narrower range compared to when calculating a coefficient of a lowest degree, at least when calculating a coefficient of a highest degree; and - uses an approximation equation calculated from the at least two pieces of relative position information acquired when setting the retrospective range to the same range or a narrower range compared to when calculating a coefficient of a relatively low degree, when calculating a coefficient of a relatively high degree, and - uses an approximation equation calculated from the at least two pieces of relative position information acquired when setting the retrospective range to a narrower range compared to when calculating a coefficient of a lowest degree, at least when calculating a coefficient of a highest degree; and - outputs an instruction according to the calculated coefficient of each degree to a steering control apparatus of the subject vehicle.