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F - Building Functions

This text resource illustrates the Standards Map for the Functions Overview Building Functions domain in the Common Core State Standards.

Standards

  2. Common Core Mathematics: Functions > BF

    Building Functions (BF)

  3. Common Core Mathematics: Functions > BF > C.1

    Build a function that models a relationship between two quantities

  4. Common Core Mathematics: Functions > BF > C.1 > S.1

    Write a function that describes a relationship between two quantities.

    1. Determine an explicit expression, a recursive process, or steps for calculation from a context.
    2. 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.
    3. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
    Content Skills Learning Targets Big Ideas Essential Questions
    Functions [b]
    Functions [c]
    3 - Write (function that describes a relationship between 2 quantities)
    2 - Determine (explicit expression from a context[a])
    2 - Determine (recursive expression from a context [a])
    2 - Determine (steps for calculation from a context [a])
    3 - Use (arithmetic operations [b])
    5 - Compose (functions [c])
    3 - I can write a function that describes a relationship between 2 quantities.
    2 - I can determine explicit expression from a context [a.)
    2 - I can determine recursive expression from a context [a].
    2 - I can determine steps for calculation from a context [a].
    3 - I can use arithmetic operations [b].
    5 - I can compose functions [c].
    A. From a given relationship between two groups of numbers, one can write a function.
    B. Functions can be combined algebraically to create new functions.
    C. Functions can be combined using composition to create new functions.
    A.1 What do I need in order to write a function?
    B.1 Can I put functions together?
    C.1 What is composition?
  5. Common Core Mathematics: Functions > BF > C.1 > S.2

    Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

    Content Skills Learning Targets Big Ideas Essential Questions
    1 - Write (arithmetic sequences recursively)
    1 - Write (geometric sequences recursively)
    1 - Write (arithmetic sequences with an explicit formula)
    1 - Write (goemetric sequences with an explicit formula)
    3 - Use (sequences to model situations)
    2 - Translate (between recursive form and explicit formula)
    1 - I can write arithmetic sequences recursively.
    1 - I can write geometric sequences recursively.
    1 - I can write arithmetic sequences with an explicit formula.
    1 - Write geometric sequences with an explicit formula.
    3 - I can use sequences to model situations.
    2 - I can translate between recursive form and explicit formula.
    A. Sequences can be written in two forms, recursive and by explicit formula.
    A.1 How are sequences created?
  6. Common Core Mathematics: Functions > BF > C.2

    Build new functions from existing functions

  7. Common Core Mathematics: Functions > BF > C.2 > S.3

    Identify 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.

    Content Skills Learning Targets Big Ideas Essential Questions
    2 - Find (the value of k from a given graph)
    4 - Experiment (with cases)
    4 - Illustrate (an explanation of the effects on the graph using technology)
    1 - Recognize (even and odd functions from their graphs)
    1 - Recognize (even and odd functions from algebraic expressions)
    1 - I can identify the effect on the graph of replacing f(x) by f(x)+k for specific values of k, both positive and negative.
    1 - I can identify the effect on the graph of replacing f(x) by kf(x)for specific values of k, both positive and negative.
    1 - I can identify the effect on the graph of replacing f(x) by f(x+k)for specific values of k, both positive and negative.
    2 - I can find the value of k from a given graph.
    4 - I can experiment with cases.
    4 - I can illustrate an explanation of the effects on the graph using technology.
    1 - I can recognize even and odd functions from their graphs.
    1 - I can recognize even and odd functions from algebraic expressions.
    A. One can predict the translation of a graph by recognizing the changes in the way the formula is written.
    A.1 How do I know which way the graph will shift?
  8. Common Core Mathematics: Functions > BF > C.2 > S.4

    Find inverse functions.

    1. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 for x > 0 or f(x) = (x+1)/(x–1) for x ≠ 1.
    2. (+) Verify by composition that one function is the inverse of another.
    3. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse.
    4. (+) Produce an invertible function from a non-invertible function by restricting the domain.
    Content Skills Learning Targets Big Ideas Essential Questions
    Functions
    f(x) = c [a]
    Inverse function [c]
    Invertible function [d]
    2 - Find (inverse functions)
    2 - Solve (equation f[x]=c with inverse [a])
    1 - Write (expression for inverse [a])
    5 - Verify (by composition that a function is inverse of another function [b])
    1 - Read (values of an inverse from a graph [c])
    1 - Read (values of an inverse from a table [c])
    5 - Produce (invertible function from non-invertible function with restricted domain [d])
    2 - I can find inverse functions.
    2 - I can solve an equation f[x]=c with its inverse [a].
    1 - I can write expression for inverse [a].
    5 - I can verify by composition that a function is inverse of another function [b].
    1 - I can read values of an inverse from a graph [c].
    1 - I can read values of an inverse from a table [c].
    5 - I can produce an invertible function from non-invertible function with restricted domain [d].
    A. Functions have inverses.
    B. Inverses can be used to solve equations.
    C. Inverses can be verified by using composition.
    D. Invertible functions can be created from non-invertible functions by restricting the domain of the non-invertible function.
    A.1 When will I need to use an inverse of a function?
    B.1 How do I use an inverse to solve an equation?
    C.1 How can I make sure that I have an inverse?
    D.1 Do all functions have inverses?
  9. Common Core Mathematics: Functions > BF > C.2 > S.5

    (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

    Content Skills Learning Targets Big Ideas Essential Questions
    2 - Understand (inverse relationship between exponents and logarithms)
    3 - Use (inverse relationship to solve problems involving logarithms and exponents)
    2 - I can understand the inverse relationship between exponents and logarithms.
    3 - I can use the inverse relationship to solve problems involving logarithms and exponents.
    A. Exponential functions and logarithmic functions are inverses.
    A.1 What is the difference between exponential and logarithmic functions?

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