AP Precalculus

1.13 Course Objectives

• Identify an appropriate function type to construct a function model for a given scenario.

• Describe assumptions and restrictions related to building a function model.

1.13 Key Knowledge

• Linear functions model data sets or aspects of contextual scenarios that demonstrate roughly constant rates of change.

• Quadratic functions model data sets or aspects of contextual scenarios that demonstrate roughly linear rates of change, or data sets that are roughly symmetric with a unique maximum or minimum value.

• Geometric contexts involving area or two dimensions can often be modeled by quadratic functions. Geometric contexts involving volume or three dimensions can often be modeled by cubic functions.

• Polynomial functions model data sets or contextual scenarios with multiple real zeros or multiple maxima or minima.

• A polynomial function of degree n models data sets or contextual scenarios that demonstrate roughly constant nonzero nth differences.

• A polynomial function of degree n or less can be used to model a graph of n + 1 points with distinct input values.

• A piecewise-defined function consists of a set of functions defined over nonoverlapping domain intervals and is useful for modeling a data set or contextual scenario that demonstrates different characteristics over different intervals.

• A model may have underlying assumptions about what is consistent in the model.

• A model may have underlying assumptions about how quantities change together.

• A model may require domain restrictions based on mathematical clues, contextual clues, or extreme values in the data set.

• A model may require range restrictions, such as rounding values, based on mathematical clues, contextual clues, or extreme values in the data set.

1.14 Course Objectives

• Construct a linear, quadratic, cubic, quartic, polynomial of degree n, or related piecewise-defined function model.

• Construct a rational function model based on a context.

• Apply a function model to answer questions about a data set or contextual scenario.

1.14 Key Knowledge

• A model can be constructed based on restrictions identified in a mathematical or contextual scenario.

• A model of a data set or a contextual scenario can be constructed using transformations of the parent function.

• A model of a data set can be constructed using technology and regressions, including linear, quadratic, cubic, and quartic regressions.

• A piecewise-defined function model can be constructed through a combination of modeling techniques.

• Data sets and aspects of contextual scenarios involving quantities that are inversely proportional can often be modeled by rational functions. For example, the magnitudes of both gravitational force and electromagnetic force between objects are inversely proportional to the objects’ squared distance.

• A model can be used to draw conclusions about the modeled data set or contextual scenario, including answering key questions and predicting values, rates of change, average rates of change, and changing rates of change. Appropriate units of measure should be extracted or inferred from the given context.

Video

• The video for AP Precalculus, 1.3-1.14: Modeling with Functions will come out on Thursday, October 19, 2023.