A curve passes through the origin and is symmetric about the origin
MATHEMATICS · Multi-part · AP · AP Precalculus · Polynomial and Rational Functions · Polynomial and Rational Functions
Problem
A curve passes through the origin and is symmetric about the origin. As x takes large positive values, y becomes large positive; for large negative x, y becomes large negative. The curve crosses the x-axis only at the origin.
(i) Decide whether the degree of the simplest polynomial that could model this curve should be even or odd. Give a reason.
(ii) Determine the sign (positive or negative) of the leading coefficient of this polynomial. Justify your answer.
(iii) Write the simplest possible algebraic form f(x)=kxn (where k=0 and n is a positive integer) that satisfies all the given conditions, and explain why no simpler polynomial can represent the same graph.