Big Idea 2: Functions can be represented in multiple, equivalent ways.

Common Core Standards: 
N-Q.AReason Quantitatively And Use Units To Solve Problems. ( 1Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays., 3Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. ) F-IF.AUnderstand The Concept Of A Function And Use Function Notation ( 1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x)., 2Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. ) F-IF.BInterpret Functions That Arise In Applications In Terms Of The Context ( 4For 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., 5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. ) F-IF.CAnalyze Functions Using Different Representations ( 9Compare 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-LE.AConstruct And Compare Linear, Quadratic, And Exponential Models And Solve Problems ( 1Distinguish between situations that can be modeled with linear functions and with exponential functions., 1bRecognize situations in which one quantity changes at a constant rate per unit interval relative to another., 3Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. )
Evidence of Understanding: 


  • justify why a table of values and graph are equivalent representations for the same function
    • e.g., map coordinate points on a graph to specific pairs of values in a table, recognize increasing output values in the table correspond to an increasing interval on the graph, etc.
    • describe when a graph may be more useful than a table of values (and vice versa)


  • create equivalent mathematical representations for linear, quadratic, exponential, or step functions
    • analyze information from a graph or situation, represent the key features in an organized table of values, and justify why both representations are equivalent
      • use function notation to represent coordinate points and describe the relationship between the independent and dependent variables
    • create a graph that models a situation and justify its characteristics (or vice versa)
    • create a situation or graph that accurately represents a function's table of values
      • create graphs with a scale other than 1 and use the domain and range to justify choices
    • understand how to label all important parts of tables and graphs (input, output, axes, title, lines extended, etc.) and how these labels translate across equivalent representations
      • e.g., the label for the axes corresponds to the units in the situation, the scale of the graph corresponds to quantity changes in the table, lines are extended according to the situation, etc.


  • recognize functions and non-functions from tables, mappings, graphs, etc.
    • identify and justify the domain and range of a function from a situation, table, or graph
      • recognize when using interval or set notation is useful and efficient for representing the domain and range of a function with a large number of values
    • utilize the Vertical Line Test as a visual way to determine if a given graph is a function or not
    • analyze various representations and notice that functions are single-valued mappings from the domain of the function to its range and an input has at most one output.
Core Resource: 
  1. Interpreting Information
    Multiple-day resource that helps students interpret and make deeper sense of intercepts, mins, maxs, domain, range, and other key information from multiple representations.
Instructional Routines: 
  1. Triangles
    Contemplate then Calculate
    Supports finding the next item in a pattern
  2. Legos
    Contemplate then Calculate
    Supports finding the next item in a pattern
  3. X-squares
    Contemplate then Calculate
    Supports finding the next item in a pattern
  4. Restricted Domains
    Connecting Representations
    Students will connect linear functions with restricted domains with their graph representations.
  5. Function or Not?
    Connecting Representations
    Use the verbal structure of a set of definitions of a function/non-function and match these definitions to a set of visuals of different types.
Additional Resources:
Develop conceptual understanding: 
table of values, scale, domain, range, function, relation, vertical line test, function notation, f(x)
Supporting terms to communicate: 
axes, units, coordinate point, ordered pair, interval, increasing, decreasing, positive, negative, turning point, maximum, minimum, intercept, dependent, independent, domain, range, continuous, discrete, interval and set-builder notation