Model richer contexts with quadratic functions
• Fit functions to verbal descriptions and graphs using key features (F-IF.C.7a$^\star$, F-BF.B.3).
• Solve modeling problems (F-IF.B.4$^\star$, F-LE.A.1, F-LE.A.2).
Students are now ready to start applying their knowledge of quadratic functions. They draw on their ability to construct expressions for functions to meet a given purpose. They identify intercepts and vertices in order to write functions that match a given situation. The work in this section prepares students for the work on solving quadratic equations in one variable but does not require students to find exact solutions to them.
WHAT: In this scaffolded exploration, the path of an arrow is modeled with a quadratic function. With the path initially given the arrow does not make it over a wall. Students must make sense of a diagram and an equation and determine where the archer could stand in order for the arrow to clear the wall, and for which equations of paths the arrow clears the wall (MP1).
WHY: The questions are deceptively simple, but answering them requires a thorough understanding of changing the expression for a quadratic function with translations, and making connections between the graph, the equation, and the situation.
WHAT: In this activity, students investigate the population growth of three cities and evaluate linear, quadratic and exponential models, using technology to find the models.
WHY: This activity ties the work of this section with the work of Sections 4 and 5, exposing students to more authentic situations where different mathematical models could be chosen and need to be evaluated. It prepares students for the full extent of the modeling cycle (MP4).
WHAT: Students are presented with photographs of only part of the path of a tennis ball, and must decide whether the ball will hit a trash can in each.
WHY: This task lends itself to an open lesson format where the students need to investigate, gather data, and model in order to answer the question (F-IF.B.4$^\star$, F-IF.C.7a$^\star$, MP2, MP4).