A1.F.2.1  Distinguish between linear and nonlinear (including exponential) functions arising from realworld and mathematical situations that are represented in tables, graphs, and equations. Understand that linear functions grow by equal intervals and that exponential functions grow by equal factors over equal intervals.
In a Nutshell
Students will distinguish and describe functions as linear or nonlinear functions. Using their knowledge acquired during this unit the students will determine whether linear equations are functions. Students will use tables, graphs, and equations to identify characteristics of nonlinear functions. Realworld situations are useful for students to understand the equal intervals that functions tend to grow. Graphing and creating tables for various equations can show how exponential functions grow as equal factors over equal intervals. Parent functions of other graphs will be introduced in order to determine the differences of linear and nonlinear.
Student Actions

Teacher Actions

 Develop accurate and appropriate procedural fluency by creating tables, writing equations, and graphing functions to determine linear and nonlinear relationships.
 Develop a Deep and Flexible Conceptual Understanding by exploring linear and nonlinear functions to understand the relationship between the independent and dependent variables.
 Students will Develop Mathematical Reasoning by recognizing every input having exactly one output is essential in a function. Students must understand that changing the input leads to a change in the output.


Use and connect mathematical representations by engaging students to connect the concepts of linear and nonlinear functions to their equations, graphs, tables, and charts.

Pose purposeful questions and examples to connect students to realworld situations by using examples of several different functions as they explore the similarities and differences of those functions.
 Implement tasks that promote reasoning and problem solving byhaving students interpret the mathematical relationship of tables, graphs, and equations of linear or nonlinear functions.

Key Understandings

Misconceptions


Students will understand there is a relationship between input and output and then represent this relationship in function notation.

Students will recognize if a function is linear it will take the form of f(x)=mx+b.

Students will identify a linear function by using tables, graphs, or equations.

Students will determine the outcome of a graph as linear or nonlinear.
 Students will determine a parent function or its graph as linear or nonlinear.


Students may confuse horizontal and vertical lines and write their equations incorrectly.

Students may forget the definition of a function when writing the equation in function notation.

Students may try to write a function for an equation of a vertical line; such as x=2 written incorrectly as f(x) =2.

Students may have a misunderstanding about the notation f and f(x).

Students believe that f is the only letter that can be used to represent a function rule.

When students see f(x), they initially think it means f times x.

When writing a linear function, students sometimes leave the equation as y=mx+b instead of using f(x) for y.

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