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A1.F.2.1 - Distinguish between linear and nonlinear (including exponential) functions arising from real-world 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. Real-world 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

  • Students develop accurate and appropriate procedural fluency by creating tables, writing equations, and graphing functions to determine linear and nonlinear relationships.
  • Students develop a deep and flexible conceptual understanding by exploring linear and nonlinear functions to understand the relationship between the independent and dependent variables.

  • Students explore functions to determine whether they are linear or nonlinear and justify their reasoning mathematically to their teacher and peers.


  • Engage students with many different mathematical representations 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 real-world 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 by having students interpret the mathematical relationship between tables, graphs, and equations of  linear or nonlinear functions.

  • Encourage productive struggle as students work through real-world situations to determine whether a function is linear or nonlinear.


Key Understandings


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

  • Students will distinguish between linear or nonlinear functions by using tables, graphs, or equations.

  • Students will determine a parent function or its graph as linear or nonlinear.

  • Students understand that linear functions change by equal intervals while exponential functions increase by equal factors over equal intervals.

  • Students will mix up the input and the output and incorrectly identify a relationship as linear when it is nonlinear.

  • If a function is not in slope-intercept form, the students will not recognize it as a linear function.

  • When using tables to identify functions, students will not consider the rate of change and will look only to patterns.

    • Ex: Given this table, students may see a non-linear pattern because they only consider the numbers in the table and do not calculate the rate of change, when in fact the change  is an equal interval

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