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.
REAL WORLD TAKEAWAYS
In an auction, a higher bid has a higher probability of winning (than does a lower one) but reduces your profit.
An optimal bid can be thought of as the bid with the highest expected value.
In a two-person bid, the optimal bid is approximately half the item’s value; as more people participate, the probability of any given bid winning decreases, so the optimal bid increases.
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
This complex task is best as a culminating unit activity after students have developed formal knowledge and conceptual understanding.
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (a) Graph linear and quadratic functions and show intercepts, maxima, and minima. (b) Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. (c) Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. (d) (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. (e) Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
Write a function that describes a relationship between two quantities. (a) Determine an explicit expression, a recursive process, or steps for calculation from a context. (b) Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. (c) (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
(+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. (a) Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast- food restaurant. (b) Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.
(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
(+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.