Big Idea 4: Function families share similar graphs, behaviors, and properties.

Common Core Standards: 
N-Q.AReason Quantitatively And Use Units To Solve Problems. ( 2Define appropriate quantities for the purpose of descriptive modeling. ) 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. ) F-BF.BBuild New Functions From Existing Functions ( 3Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. ) 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. ) 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 similarities and differences among linear, quadratic, absolute value, and exponential function families based on features of their graphs or tables

    • interpret rate of change, domain and range patterns, minimums and maximums, intercepts, and symmetries for each type of function family

    • relate the rate of change and other key features of each function family to its parent function: f(x) = x (linear),  f(x) = x2  (quadratic), f(x) = |x| (absolute value) and f(x) = 2x (exponential)

  • compare functions within a family and describe transformations from the parent function

    • given a graphical representation of a parent function and another function in the same family, describe the vertical or horizontal shift and/or a dilation in the rate of change that happened (e.g., the graph shifted left 3 from the parent function)

    • compare tables of values for different functions within the same function family (the parent function and one other) (e.g., “I notice that the x-values in both tables stay the same but the corresponding  y-values are always 3 more than in the parent function’s table”)

    • create a graph or table of values for a function given the parent function and a stated transformation (NOTE: writing equations of transformations will wait until later units)

  • use linear, quadratic, exponential, piecewise or step functions to model situations

    • use the relationship between quantities in a situation to determine which function family provides the best model

      • analyze domain and range of different functions within the context of a situation to make sense of why certain functions have limitations on the domain and/or range

Instructional Routines: 
  1. What's the Missing Number?
    Contemplate then Calculate
    Supporting thinking about exponential rate of change
  2. Find the Vertex of a Parabola
    Contemplate then Calculate
    Supports using rate of change to find the vertex of a parabola
  3. Shifting Values
    Contemplate then Calculate
    Calculate a missing value on an absolute value function.
  4. Set Builder Notation
    Connecting Representations
    Chunk the graph of a function and the set builder notation associated with its domain to make connections between domain and set builder notation.
Additional Resources:
Develop conceptual understanding: 
function family, parent function, linear function, quadratic function, absolute value function, exponential function, piecewise function, step function
Supporting terms to communicate: 
function, f(x), axes, units, scale, coordinate point, ordered pair, table of values, rate of change, average rate of change, interval, increasing, decreasing, positive, negative, turning point, maximum, minimum, intercept, dependent, independent, domain, range, continuous, discrete, symmetry