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How much should you bid in an auction? From Sotheby’s to eBay, auctions are a popular way to buy everything from paintings to baby clothes. Of course, the goal of an auction isn’t just to win the item; it’s to pay as little as possible for it.

In this lesson, students create polynomial functions to model the expected value of a given bid and determine the optimal amount someone should bid in any auction.

### Students will

• Write and graph a quadratic equation to model a real-world situation
• Write and graph higher-order polynomial equations to describe a real-world situation
• Calculate and model expected value in a real-world context

### Before you begin

The most important prerequisite for this lesson is an understanding of how to use values and probabilities to calculate an expected value. One of the expected value functions turns out to be quadratic, so students will have to find the vertex in order to maximize expected profits. If they can do this algebraically, that’s great, but they will also have the opportunity to find a maximum graphically. In fact, as they move on to higher-order polynomials, they will have to find local maxima graphically, unless you’re using this as a calculus lesson on maximization...which is a great idea.