Graphing Polynomial Functions Answers
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Explain how to sketch a graph of the function f(x) = x3 + 2x2 – 8x. Be sure to include end-behavior, zeroes, and intervals where the function is positive and negative.
Answer:
Factor f(x) = x(x + 4)(x - 2). The zeroes are -4, 0, and 2, each with multiplicity 1, so the graph crosses the x-axis at each one. Since the function has odd degree and a positive leading coefficient, its left end falls toward negative infinity and its right end rises toward positive infinity. The function is negative on (-infinity, -4) and (0, 2), and positive on (-4, 0) and (2, infinity). Use these points and signs to draw a smooth cubic curve.
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Did your explanation include the following?
A
The degree of the function is odd and the leading coefficient is positive – so the function goes to negative infinity as x goes to negative infinity and to positive infinity as x goes to positive infinity.B
The zeroes are –4, 0, and 2, all with multiplicity 1.C
The function is negative from negative infinity to –4 and from 0 to 2.D
The function is positive from –4 to 0 and from 2 to infinity.
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