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:

The function factors as f(x) = x(x + 4)(x - 2), so its zeros are -4, 0, and 2, each with multiplicity 1. Since it is an odd-degree function with a positive leading coefficient, the graph falls to the left and rises to the right. It is negative on (-∞, -4) and (0, 2), and positive on (-4, 0) and (2, ∞). The graph crosses the x-axis at each zero.