Question

You are having a meeting with the CEO of a technology company. You have interpreted the number of laptops produced versus profit as the function P(x) = x4 − 3x3 − 8x2 + 12x + 16. Describe to the CEO what the graph looks like and, in general, how to sketch the graph without using technology. Use complete sentences, and focus on the end behaviors of the graph and where the company will break even (where P(x) = 0).

Answer 1

Factoring the equation gives P(x) = (x - 4)(x - 2)(x + 1)(x + 2).

The graph will begin at x = 0, where P(x) = 16, as this is the y-intercept of the function. From 0 <= x < 2, P(x) > 0. There are break-even points at x = 2 and x = 4, while between them, P(x) < 0. As x increases beyond x = 4, the profit P(x) increases greatly (up to infinity). The section of the graph below x = 0 is meaningless, as the company cannot produce a negative number of laptops.