Overview (Big Ideas): As an extension of their work with various function families, students will begin to investigate the graphs of simple inverse functions, focusing on the inverses of linear functions and other cases, such as square root functions. Students should be able to identify when an inverse is not a function and simple cases in which the domain can be easily restricted in order to create an inverse function. This should lead into the exploration of the graphical representations of other models, to include square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Students will be able to graph these functions by hand in simple cases and by using technology in more complex cases. In all cases, students should identify key features of their graphs. Students should identify parent functions and describe or sketch the effects of simple transformations on those parent functions.

Enduring Understandings: 1. Functions are single-valued mappings from one set – the domain of the function – to another – its range. 2. Functions apply to a wide range of situations. They do not have to be described by any specific expressions or follow a regular pattern. They apply to cases other than those of “continuous variation.” For example, sequences are functions. 3. Under appropriate conditions, functions have inverses. The logarithmic functions are inverses of the exponential functions. The square root function is the inverse of the squaring functions.

Source: Cooney, T., Beckmann, S., & Lloyd, G. (2010). Developing Essential Understanding of Functions Grades 9-12. Reston, VA: The National Council of Teachers of Mathematics, Inc.

Essential Questions:

How can functions be compared, graphically, to their inverses?

How might you determine if the inverse of a function will result in another function? Can domain restrictions be applied that will allow a function to exist?

What scenarios might require models other than linear, exponential or quadratic?

What are the key features to other models, and what do they mean in the context of the problem?

How can the key features of a model be used to sketch a graph?

What are the general effects of transformations on models?

How can piecewise-defined functions be graphed? In what situations might these functions be applied?

What are the most common piecewise-defined functions, and when can they be used?

Common Misconceptions:

Students struggle to sketch graphs with an appropriate level of accuracy.

o Students may find it difficult to determine which model should be used to represent a given situation.

o The interpretation of key features in the context of a problem may be difficult for some students.

o Piecewise functions, particularly step functions, are difficult to interpret and graph.

Modeling and Other FunctionsOverview (Big Ideas):As an extension of their work with various function families, students will begin to investigate the graphs of simple inverse functions, focusing on the inverses of linear functions and other cases, such as square root functions. Students should be able to identify when an inverse is not a function and simple cases in which the domain can be easily restricted in order to create an inverse function.

This should lead into the exploration of the graphical representations of other models, to include square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Students will be able to graph these functions by hand in simple cases and by using technology in more complex cases. In all cases, students should identify key features of their graphs. Students should identify parent functions and describe or sketch the effects of simple transformations on those parent functions.

Enduring Understandings:1. Functions are single-valued mappings from one set – the domain of the function – to another – its range.

2. Functions apply to a wide range of situations. They do not have to be described by any specific expressions or follow a regular pattern. They apply to cases other than those of “continuous variation.” For example, sequences are functions.

3. Under appropriate conditions, functions have inverses. The logarithmic functions are inverses of the exponential functions. The square root function is the inverse of the squaring functions.

Source:Cooney, T., Beckmann, S., & Lloyd, G. (2010). Developing Essential Understanding of Functions Grades 9-12. Reston, VA: The National Council of Teachers of Mathematics, Inc.

Essential Questions:Common Misconceptions: