(A) A student claims that the overall shape and symmetry of a polynomial graph can be…
MATHEMATICS · Multi-part · AP · AP Precalculus · Polynomial and Rational Functions · Polynomial and Rational Functions
Problem
(A) A student claims that the overall shape and symmetry of a polynomial graph can be predicted from its degree and the sign of its leading coefficient.
Answer the following to examine this claim.
(i) State the symmetry property of the graph of a polynomial that contains only even-powered terms such as x2,x4,….
(ii) Explain, without using calculus, how the sign (positive or negative) of the leading coefficient influences whether such an even-degree graph possesses a global minimum or a global maximum.
(iii) Consider a polynomial that contains only odd-powered terms such as x,x3,…. Explain how its symmetry differs from the polynomial in part (i).
(iv) For the polynomial described in part (iii), discuss how the sign of its leading coefficient affects the overall rising or falling trend of the graph from left to right.