Families of Functions

11 Components
Initial Task
Sorting Functions
Big Ideas
Big Idea 1 Big Idea 2 Big Idea 3 Big Idea 4 Big Idea 5
Formative Assessment Lesson
Functions & Situations
Re-engagement
End of Unit Assessments
End of Unit Assessment for Unit 1 Performance Task for Unit 1 End of Unit Assessment Teacher Resources ALG II Unit 1 Find aligned Regents questions with Quiz Banker
A2 U1: Families of Functions
Browse Components
Initial Task
Sorting Functions
Big Ideas
Big Idea 1 Big Idea 2 Big Idea 3 Big Idea 4 Big Idea 5
Formative Assessment Lesson
Functions & Situations
Re-engagement
End of Unit Assessments
End of Unit Assessment for Unit 1 Performance Task for Unit 1 End of Unit Assessment Teacher Resources ALG II Unit 1 Find aligned Regents questions with Quiz Banker
Big Idea:

Big Idea 3

Functions can be represented in multiple, equivalent ways.

1 week
See Standards
Common Core: Major Standards:
F-IF.B.4

Interpret functions that arise in applications in terms of the context. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F-IF.B.6

Interpret functions that arise in applications in terms of the context. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

F-BF.A.1a

Build a function that models a relationship between two quantities. Determine an explicit expression, a recursive process, or steps for calculation from a context.

F-BF.A.1b

Build a function that models a relationship between two quantities.  Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

Common Core: Supporting Standards:
F-IF.C.9

Analyze functions using different representations. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

F-IF.C.7c

Analyze functions using different representations. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

F-LE.A.2

Linear, Quadratic, & Exponential Models: Construct and compare linear, quadratic, and exponential models and solve problems. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Common Core: Additional Standards:
F-LE.B.5

Interpret expressions for functions in terms of the situation they model. Interpret the parameters in a linear or exponential function in terms of a context.

Evidence of Understanding

  • analyze equivalent representations of the same function
    • identify characteristics of a function across different representations, especially end behavior, symmetry, critical points (including distinguishing mins and maxs as relative or absolute), zeros, rate of change, and increasing/decreasing/positive/negative intervals
      • Example: recognize increasing output values in the table correspond to an increasing interval on the graph, etc.
      • relate features of a parabola to features of its quadratic equation
      • relate the zeros of factorable polynomial functions to its equation (Note: this supports the Remainder Theorem in the Unit 4)
    • articulate and compare the advantages of different representations, including graphs, tables of values, equations, written descriptions, symbolic notations, etc.
  • create equivalent representations for a polynomial, exponential, piecewise or step function
    • identify missing characteristics for one representation using features from a different representation of the same function
    • create an equivalent representation given a graph, table of values, or situation
      • justify why both representations are equivalent
      • correctly label all important parts of each representation and describe how these labels translate across equivalent representations
      • use rates of change to generate equivalent representations
  • compare properties of two functions each represented in a different way
    • use general characteristics of function families to distinguish and compare the functions
    • describe characteristics that are similar and different
      • Example: identify which function has a larger maximum, more zeros, etc.

Develop conceptual understanding:

end behavior, symmetry, critical point, zero, rate of change, polynomial function

Supporting terms to communicate:

domain, range, interval, intercepts, roots, relative/absolute min and max, linear, quadratic, exponential, logarithmic, square root, cube root, polynomial, rational, periodic

Core Resource

There is no Core Resource for this Big Idea.

Consider using the Instructional Routines linked below for teaching towards this Big Idea.

Instructional Routine: Contemplate then Calculate
These tasks are embedded within the instructional routine called Contemplate then Calculate. COMING SOON!
    Instructional Routine: Connecting Representations
    These tasks are embedded within the instructional routine called Connecting Representations.