Modeling with rational functions
• Graph rational functions and use the graphs to solve problems (A-REI.D.11$^\star$).
• Express rational functions in different forms to see different aspects of the situation they model (A-APR.D.6).
• Solve simple rational equations and understand why extraneous roots can arise (A-REI.A.2).
In the previous section students encountered simple rational functions. Here, they extend that work. They consider contexts which can be modeled with rational functions. They rewrite simple rational expressions in different forms to see different aspects of the context, and they find approximate solutions using graphical methods to rational equations that arise from the context. They also solve simple rational equations algebraically and study how and why extraneous solutions may arise.
WHAT: Students are given an equation that models the pressure in a container as a function of the volume of the container. They are prompted to make sense of the equation in context, sketch a graph of it, and use the graph to find the volume corresponding to a given pressure.
WHY: The purpose of this task is to provide students with an authentic example of a situation modeled by a rational function. The task shows the usefulness of graphing the function and estimating a solution to an equation from the graph.
WHAT: Students are given a rational expression for a car’s combined fuel efficiency in terms of its city and highway gas mileage. Students evaluate the expression for a specific car, then rewrite the expression in a more useful form for finding an approximation when the city fuel economy is large relative to the difference between highway and city fuel economy.
WHY: The purpose of this task is to provide a context where students manipulate an expression that has meaning, rather than being an abstract object, and where the manipulation has a purpose (MP.7). The task also provides an opportunity for quantitative reasoning (MP.2).
WHAT: Students are presented a rational equation, where the apparent solution arrived at algebraically does not work when substituted back in. They are prompted to explain why.
WHY: The goal of this task is to examine how extraneous solutions can arise when solving rational equations. In the context of this unit it would make sense to study the equation graphically as recommended in the solution.