A road profile in a design software is represented

MATHEMATICS · Multi-part · AP · AP Precalculus · Polynomial and Rational Functions · Polynomial and Rational Functions

Problem

A road profile in a design software is represented, for large values of xx, by the polynomial r(x)=0.5x6+c1x4+c2x2+c3r(x)=0.5x^{6}+c_1x^{4}+c_2x^{2}+c_3, where c1,c2,c3∈Rc_1,c_2,c_3\in\mathbb{R}. Without calculating the constants, answer the following: (i) State whether the graph of r(x)r(x) is symmetric about the yy-axis, symmetric about the origin, or neither. Give a brief justification. (ii) Determine the direction in which the road profile rises or falls as one moves far to the east (x→+∞x\to+\infty) and far to the west (x→−∞x\to-\infty). (iii) If the leading coefficient 0.50.5 is replaced by −0.5-0.5, describe precisely how the answers in part (ii) will change.

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