Big Idea:

Big Idea 1

Rate of change determines other characteristics of a function.

1 week

Evidence of Understanding

  • analyze the rate of change and initial value on a function’s table or graph and use them to describe characteristics about the function
    • determine the intercepts exactly (for integers) or approximately, from a table or graph
    • justify whether given values belong in the domain or range of the function
    • estimate the rate of change from a linear, exponential, piecewise, step, or absolute value graph
      • use the rate of change to describe patterns for different function families
    • analyze how transformations impact the function’s graph and table (emphasis on dilations to the rate of change and/or translations of the initial value)


  • compare linear and exponential function families using rate of change
    • distinguish situations and number sequences (esp. arithmetic and geometric) that are best modeled by linearexponential or other functions
      • recognize and apply vocabulary to describe linear or exponential rate of change (Example: decay, compound, fixed rate, etc.)
    • recognize when quantities changes at a constant rate relative to one another
    • show that a quantity increasing exponentially eventually exceeds a quantity increasing linearly
    • analyze examples and nonexamples to define linear functions as growing by equal differences over equal intervals, and exponential functions growing by equal factors over equal intervals


  • use a function’s rate of change to predict future states/generate next steps 
    • use the rate of change to determine specific output or input values (given the other) from a situation, graph, table, or number sequence
    • calculate and describe the meaning of the average rate of change in relation to context
      • recognize when the average rate of change is zero and describe its meaning
    • describe the rate of change patterns using words for recursive and explicit forms
    • accurately graph a linear or exponential function and label its intercepts given at least three coordinate pairs and justify reasoning


Develop conceptual understanding with these terms:

rate of change, slope, initial value, y intercept, x intercept, arithmetic sequence, geometric sequence, linear, exponential, recursive

Support students in using these terms:

function, average rate of change, interval, dependent, independent, input, output, domain, range, integer, continuous, discrete, function family

Core Resource
A core resource supports multiple days of instruction.
  • Comparing Rates of Change
    A multiple-day resource that supports students in understanding and comparing rates of change.
    Comparing Rates of Change

    Students start by creating definitions of linear and exponential functions by analyzing examples and non-examples of these types of functions. Students then use what they have learned about linear and exponential functions to solve word problems related to these function types. What is not covered in this resource is representing piecewise, step, or absolute functions as graphs, calculating average rate of change, and writing explicit and recursive descriptions of functions.

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