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.