• Build a rational function that describes a relationship between two quantities (F-BF.A.1$^\star$).
• Graph rational functions (A-SSE.A.1a$^\star$, F-IF.C.7d$^\star$).
• Interpret the graph of a rational function in terms of a context (F-IF.B.4$^\star$).
In this section students study simple rational functions. The emphasis is on rational functions that arise naturally out of a real-world context, and on interpreting features of their graphs in terms of that context. Students experiment with graphs using technology to learn the relationship between features of the graph and the structure of the expression defining the function.
WHAT: Students are guided through the construction of a simple rational function that models the time it takes someone to paddle upstream a certain distance as a function of the speed of the current. They interpret the vertical intercept and the vertical asymptote of the graph in terms of the context.
WHY: The purpose of this task is to introduce the idea of a vertical asymptote for a rational function and provide a context where the behavior of the function near the asymptote makes sense in terms of the context. Although this task doesn’t incorporate the entire modeling cycle, students engage in aspects of (MP.4) because they create a function to represent a given situation, and then interpret features of its graph in a context.
WHAT: Students are guided through the construction of a simple rational function that models the average cost per DVD of producing a number of DVDs as a function of the number of DVDs produced. They explain why the values of the function level off and visualize this behavior as a horizontal asymptote of the graph. They describe the domain of the function.
WHY: The purpose of this task is to give students a context where they understand numerically why a horizontal asymptote occurs and interpret the long term behavior represented by the asymptote in terms of a context. The task also provides an opportunity for students to express regularity in repeated reasoning as they construct the function by generalizing from calculations in a table (MP.8).
WHAT: Students use sliders in the online graphing calculator Desmos to make connections between expressions for rational functions and their asymptotes. This task starts with an exploration of the graphs of two functions whose expressions look very similar but whose graphs behave completely differently. At first glance this is surprising but can be explained by a closer look at the functions’ expressions.
WHY: The purpose of this task is to build on the work in the previous two activities to increase students’ understanding of the connection between expressions for rational functions and the asymptotes of their graphs. In this task there is now context and students focus purely on the structure of the expressions (MP.7).
WHAT: Students read a description of a donut shop with a pricing scheme that makes it cheaper to buy donuts in bulk. They model the cost as a linear function of the number of donuts purchased and then model the average cost per donut as a rational function of the number of donuts purchased. An extension of the activity includes using piecewise linear and piecewise rational functions to model a more complicated scheme in which the 13th donut is free.
WHY: This activity brings together several ideas from this and previous units, including building (F-BF.A.1$^\star$) and interpreting the graphs of and interpreting rational, linear and piecewise defined functions (F-IF.C.7ab$^\star$, F-IF.C.7d$^\star$(+)), and relating the domain of a function to its graph (F-IF.B.5$^\star$). Students are presented with a context and must construct and interpret a mathematical model (MP.4). By performing several computations and then generalizing with an equation, students are expressing regularity in repeated reasoning (MP.8).
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