The three functions p(x)=x^4-4x^2

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

Problem

The three functions p(x)=x4−4x2,q(x)=−x4+4x2,r(x)=2xp(x)=x^4-4x^2, \quad q(x)=-x^4+4x^2, \quad r(x)=2x are to be examined on a common set of axes. (i) Find all xx-intercepts and the yy-intercept of each function. (ii) Sketch the graphs of p(x), q(x)p(x),\, q(x) and r(x)r(x) on the same coordinate plane, labelling all intercepts clearly. (iii) From your sketches, identify which of the even-degree graphs attains a global minimum and which attains a global maximum. Explain how this observation is connected to the sign of their leading coefficients. (iv) Discuss the symmetry of each graph and relate it to the degree of the corresponding polynomial.

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