Big Idea 2: Equivalent representations of a function highlight different properties.

Common Core Standards: 
F-IF.AUnderstand The Concept Of A Function And Use Function Notation ( 2Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context., 3Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. ) 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., 6Calculate 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-IF.CAnalyze Functions Using Different Representations ( 7Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases., 7aGraph linear and quadratic functions and show intercepts, maxima, and minima., 7bGraph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions., 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. ) F-LE.BInterpret Expressions For Functions In Terms Of The Situation They Model ( 5Interpret the parameters in a linear or exponential function in terms of a context. )
Evidence of Understanding: 
  • identify and justify equivalent representations for a quadratic function
    • describe a quadratic function from the factored, standard, and vertex form of its equation
      • Ex: factored form quickly shows the roots, the average of those roots can determine the axis of symmetry
    • use specific features of a quadratic function to prove representations are equivalent (table, graph, function rule in factored form, standard form, or vertex form)
      • Ex: suppose (3, 9) is the vertex, explain 9 is the y coordinate of the vertex on the graph, 9 is the highest output value in the table and the only output value not repeated
      • Ex: suppose (3, 9) is the vertex, show x = 3 is the axis of symmetry in vertex form and 3 is in the middle of -1 and 7 which are the roots in factored form, etc.


  • create equivalent representations for a quadratic function
    • given an equation, use technology to graph a parabola and identify its vertex and intercepts (approximately for non-integer values)
      • justify the scale and orientation of a function’s graph using its intercepts and vertex and can adjust the window of the graphing calculator appropriately
    • without technology, graph a function rule in factored form, standard form, or vertex form
    • use intercepts and/or the vertex from a table or graph to create and justify a function rule in factored form, standard form, or vertex form
    • NOTE:  algebraic conversion between forms of a quadratic equation is part of Unit 6   


Core Resource: 
  1. Equivalent Representations
    During this week students create and study different representations of quadratic functions in order to make connections between the different representations. Some of the parts of these activities could easily be turned into Connecting Representations task instead of a worksheet.
Instructional Routines: 
  1. Graphs and Equations
    Connecting Representations
    Use the structure of a set of graphs of parabolas and a set of quadratic equations to make connections related to the direction the graph opens and the y-intercept of the graph.
  2. Quadratic Function Forms
    Connecting Representations
    Students will match quadratic functions in standard, vertex or factored form to their corresponding graphs. Students will make connections between key features of the graphs to elements of the functions in each form.
  3. Colorful Growing Sequences
    Connecting Representations
    Use the structure of a set, including the color, of growing sequences to make matches to corresponding quadratic functions.
  4. Quadratic Sequence #2
    Contemplate then Calculate
    Chunk a visual pattern and connect counting to finding area to find shortcuts for finding the next item in a visual quadratic sequence.
Additional Resources:
Develop conceptual understanding: 
factor, factored form, standard form, vertex form
Supporting terms to communicate: 
function, quadratic, parabola, y intercept, axis of symmetry, reflection, turning point, maximum, minimum, vertex, x intercept, root, average, domain, range, equivalent