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:

To sketch $f(x) x^3 2x^2 8x$, first determine the end behavior: since the degree is odd (3) and the leading coefficient is positive (1), the graph falls to the left ($f(x) \to \infty$ as $x \to \infty$) and rises to the right ($f(x) \to \infty$ as $x \to \infty$). Next, find the zeros by factoring: $x(x^2 2x 8) x(x 4)(x 2) 0$, giving zeros at $x 4, 0,$ and $2$, each with multiplicity 1. Finally, test the intervals: the function is negative on $(\infty, 4)$ and $(0, 2)$, and positive on $(4, 0)$ and $(2, \infty)$. Plot these intercepts and draw a smooth curve following these characteristics.