Common Core High School Functions Teaching and Learning Resources
Domain: BF: Building Functions
Select the cluster for resources on this page:
BFA: Build a function that models a relationship between two quantities
Standards:
 BFA.1. Write a function that describes a
relationship between two quantities.
 BFA.1.a. Determine an explicit expression,
a recursive process, or steps for calculation from a context.
 BFA.1.b. 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.
 BFA.1.c. (+) 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.
 BFA.2. Write arithmetic and geometric
sequences both recursively and with an explicit formula, use them to
model situations, and translate between the two forms.
Technologyenhanced investigations:
Mathwords.com: Math Dictionary:
Key vocabulary for this domain. Use with BFA, BFB:
Study.com: Algebra:
Functions: These are video
lessons on functions, each of which includes sample problems and a short quiz.
Use the following with BFA.1:
Study.com: Algebra:
Sequences and Series: This chapter contains seven video
lessons, each of which includes sample problems and a short quiz.
Videos define sequences and series, how to find and classify arithmetic
and geometric sequences, summation notation and mathematical series, and
how to calculate arithmetic and geometric series with practice problems
on these latter calculations provided. There is a
chapter multiple choice test following the last video.
Use all to address BFA.2.
LearnZillion:
 Lesson set:
Write a
function that describes a linear relationship between two
quantities. Five video lessons: Identify linear
relationships by exploring rectangle perimeter, define a
recursive process by exploring patterns, define an explicit
expression by exploring patterns, compare explicit expressions
and recursive processes, connect linear relationships, explicit
expressions, and recursive processes. Aligns with BFA.1
and BFA.1.a.
 Lesson set:
Write a
function that describes an exponential relationship between two
quantities. Five video lessons: Determine if a
relationship between two quantities is exponential, writing an
explicit formula for an exponential relationship, writing an
explicit formula for the graph of an exponential relationship,
writing an exponential formula for reallife problems, writing a
recursive formula for an exponential relationship. Aligns
with BFA.1 and BFA.1.a.
 Lesson set:
Combine
standard function types using arithmetic operations.
Five video lessons: Combining functions by adding and
subtracting, multiplying, dividing; creating a new function from
the graphs of two functions, applications of combining
functions. Aligns with BFA.1 and BFA.1.b.
Southeast Comprehensive Center of SEDL: Video:
Build a function that models a relationship between two
quantities: This video aligns with standard BFA.1.a. View online or download.
MIT BLOSSOMS: Video lesson with additional teacher
and learner resources. Description is from the video summary.
Amazing Problems: Arithmetic and Geometric Sequences: "This lesson
aims to introduce arithmetic sequences and geometric sequences
to secondary school students. It uses two wonderful problems to
do that and it explains the laws of those two kinds of
sequences." Aligns with IFA.3, BFA.1.a, and BFA.2.
NCTM Illuminations:

Function Matching:
Students see the graph and determine parameters of the function.
The graph might be linear, a polynomial (up to degree 4),
exponential, logarithmic, or periodic (sine, cosine, or
tangent). You can choose the type of graph to generate, or
display graphs randomly. Use with BFA.1.

Trout Pond:
This investigation illustrates the use of iteration, recursion and
algebra to model and analyze a changing fish population. Graphs,
equations, tables, and technological tools are used to investigate the
effect of varying parameters on the longterm population. NCTM's
tool:
Affine Recurrence Plotter can also be used with this activity.
Purple Math:
 Composition
of Functions: Composing functions that are sets of points,
composing functions with other functions, word problems using
composition, inverse functions and composition. Use with
BFA.1.c.
 Number Patterns
finding the next number in a sequence, including common differences and
recursions. Use with BFA.2.
 Sequences
and Series: Terminology and notation, basic examples, arithmetic
and geometric sequences, arithmetic series, finite and infinite
geometric series. Use with BFA.2.
Shodor Interactivate:
Introduction to Arithmetic and Geometric Sequences: Lesson
introduces learners to these concepts.
Sequencer is the virtual manipulative for students to explore
further through producing sequences by varying the starting number,
multiplier, and addon. The numbers in the sequence are displayed
on a graph, and they are also listed below the graph. Use with
BFA.2.
Teaching Channel Video:
Ferris Wheel: Trigonometric Functions. Lesson objectives:
Model how a trigonometric function describes the relationship of a
Ferris wheel rider as the wheel spins at a constant rate with
relationship to the height of the rider from the ground. Discern
the relationship between the given measure and the period, phase, offset
and amplitude of a cosine function.
Questions for learners to consider are included. Use with BFA.1.b,
BFB.3, and TFA.1.
Wolfram Demonstrations Project:
Download the free Wolfram CDF player to interact with the following
manipulatives:

US Common Core State Standard Math HSFBF.A.1: 29 manipulatives.
Among those: Combining functions, composition of functions,
exponential decay, continuous exponential growth, composition
and transformations, iterating linear functions, iterating
rational functions, Fibonacci numbers and the Golden Ratio, and
more.

US Common Core State Standard Math HSFBF.A.2:
9 manipulatives. Among those: Plot of a geometric sequence and
its partial sums, summing an arithmetic progression, sum of a
geometric series, arithmetic sequences, arithmetic series, and
more.
Multiple Choice:
Constructedresponse:
Algebra Lab:
 Arithmetic Sequences:
Lesson and practice problems with writing recursive and explicit
formulas. Aligns with BFA.2.
 Geometric Sequences:
Lesson and practice problems with writing recursive and explicit
formulas. Aligns with BFA.2.
Khan Academy: Practice questions with videos.
MATHguide.com:

Arithmetic Sequences: Students learn how to identify arithmetic
sequences, calculate the nth term in an arithmetic sequence, find
the number of terms in an arithmetic sequence and find the sum of an
arithmetic sequence. Each lesson has a quizmaster. Fill in and check answers.
Aligns with BFA.2.

Geometric Sequences: Students learn how to identify geometric
sequences, calculate the nth term in a geometric sequence, find the
number of terms in a geometric sequence and find the sum of a
geometric sequence. Each lesson has a quizmaster. Fill in and check answers.
Aligns with BFA.2.
ACT Academy and OpenEd:
Performance tasks:
Illustrative Mathematics:
Functions:
Inside Mathematics: MARS Tasks:

Coffee:
Students use a chart to solve simultaneous equations.
This task aligns with function standard BFA.1 and algebra standard REIC.6.

How old are they?: Students form expressions and form and solve
an equation to solve an age problem. This task aligns with
function standards BFA.1 and BFB.4.

Conference tables: Students find and extend a number pattern in a practical
context and find and use a rule or formula. This task aligns with function standards BFA.1 and BFB.4.
Mathematics Assessment Project: Standards: High School: Functions:
https://www.map.mathshell.org/stds.php?standardid=1448 Choose from the following:
Mathematics Common Core Toolbox:
PARCC Prototype Project: Under High School Tasks, select:
 Cellular growth: "Students write geometric
sequences both recursively and with an explicit formula and use
the two representations to model situations. In this task,
students consider the use of one or more representations to
solve a realworld problem and choose the type of sequence to
represent a situation." This fourpart task aligns with function
standards BFA.2 and LEA.2 and mathematical practice standards
MP.2 and MP.4.
 Golf balls in water: "Students model
approximately linear data with a linear function. To do so, they
must use contextual information and construct representations of
linear relationships as well as interpret parameters of the
linear models in the context. It makes use of common sense
understandings of volume as it relates to changing the
dimensions of a glass jar; thus, the mathematics is presented in
a context that is novel, yet has multiple entry points for
students." This threepart task aligns with function standards BFA.1,
LEA.2, LEB.5, and mathematical practice standards MP.1, MP.2
and MP.4.
 Isabella's credit card: Students see a page resembling a
spreadsheet. They "consider a financial context requiring multiple
calculations for each month ... Students create a recursive process for
calculating values within the context." They "calculate values and
complete the table for the second and third months. ... the task allows for the use
of either the recursive process or a description of steps that
lead to a solution." Finally, they "extend their use of the
recursive process or sequence of steps to find additional
values. They identify the important values and interpret those
values within the context to answer the questions." This
threepart task aligns with function
standards BFA.1, IFB.4, and mathematical practice standards MP.2, MP.4 and MP.5.
 Rabbit populations: "Students analyze
the situation (rabbit population growth) and create a
mathematical model that represents it. Students are then
presented with a second rabbit population that is slightly
different and asked to compare the models for the two
populations. This requires students to identify which function
parameters and key features of the graph will change based on
the adjustment in context. Students then use the two models
together to answer mathematical questions." This threepart task aligns
with function standards BFA.1, LEA.2, LEB.5, and mathematical
practice standard MP.4.
Mathematics Vision Project,
Secondary 1 Student Edition:

Module
3: Arithmetic and Geometric Sequences:
This module contains 10 classroom tasks. Function standards
addressed in Module 2 include LEA.1, LEA.2, LEB.5, and BFA.1.
Task 1: Growing Dots; Task 2: Growing, Growing Dots; Task 3: Scott’s
Workout; Task 4: Don’t Break the Chain; Task 5: Something to Chew On;
Task 6: Chew On This; and Task 7: What Comes Next? What Comes Later?
align to function standard BFA.1.

Module
4: Linear and Exponential Functions:
This module contains 10 classroom tasks. Function standards
addressed in Module 3 include IFA.3, IFC.7, LEA.1, LEA.2, LEA.3,
LEB.5, and BFA.1, BFA.2. Task 6: Growing, Growing, Gone aligns
to function standards BFA.1, BFA.2.

Module
5: Features of Functions:
This module contains nine classroom tasks. Function standards
addressed in Module 4 include IFA.1, IFA.2, IFA.3, and IFB.4,
IFB.5, IFC.7, and BFA.1. Task 5: Pooling It Together; and Task
6: Interpreting Functions align to standard BFA.1.

Module
7: Connecting Algebra and Geometry:
This module contains six classroom tasks. Task 4: Training Day;
Task 5: Training Day Part II; and Task 6: Shifting Functions each
address the function standards BFA.1, BFB.3, and IFC.9. The
other three tasks address geometry standards GPEB.4, GPEB.5, and
GPEB.7.
Mathematics Vision Project,
Secondary
2 Student Edition:
 Module
1: Quadratic Functions: This module contains 7 classroom tasks. Function standards
addressed in Module 1 include those in BF and LE.
Tasks 16 align to function standard BFA.1.
Mathematics Vision Project,
Secondary
3 Student Edition:
NCTM's Reasoning and Sense Making Task Library: Over the Hill includes the task overview, teacher notes for its use
in the classroom, and student activity sheet. Aligns with function standard BFA.1b,
algebra standards CEDA.2, REIC.6, and mathematical practice standards 1, 4, 5, and 7.
WeTeachNYC, developed by the New York City Department of Education: High School Algebra:
Aussie Fir Tree: Unit Overview: “The Aussie Fir Tree task is a
culminating task for a 23 week unit on algebra that uses the
investigation of growing patterns as a vehicle to teach students to
visualize, identify and describe real world mathematical relationships.
Students who demonstrate mastery of the unit are able to solve the
Aussie Fir Tree task in one class period.” Function standards addressed include
IFA.3, BFA.1 and BFA.1.a. Algebra standards include CEDA.1,
REIB.4 and REIB.4.a. Mathematical Practices standards include MP.1, MP.3,
MP.4, MP.6, and MP.7.
Back to Back to top
BFB: Build new functions from existing functions
Standards:
 BFB.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.
 BFB.4. Find inverse functions.
 BFB.4.a. 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 x^{3}
or f(x) = (x+1)/(x–1) for x ≠ 1.
 BFB.4.b. (+) Verify by composition that
one function is the inverse of another.
 BFB.4.c. (+) Read values of an inverse
function from a graph or a table, given that the function has an inverse.
 BFB.4.d. (+) Produce an invertible
function from a noninvertible function by restricting the domain.
 BFB.5. (+) Understand the inverse relationship between exponents and
logarithms and use this relationship to solve problems involving logarithms and exponents.
Technologyenhanced investigations:
Concord Consortium STEM resources: Mathematics: Each of the
following includes an applet, student activity, teacher lesson plan, and
student assessment.
Study.com: Algebra:
Functions: These are video lessons, each of which includes sample
problems and a short quiz. Use the following with BFB.3 and
BFB.4:
Teaching Channel Video:
Ferris Wheel: Trigonometric Functions. Lesson objectives:
Model how a trigonometric function describes the relationship of a
Ferris wheel rider as the wheel spins at a constant rate with
relationship to the height of the rider from the ground. Discern
the relationship between the given measure and the period, phase, offset
and amplitude of a cosine function.
Questions for learners to consider are included. Use with BFA.1.b,
BFB.3, and TFA.1.
LearnZillion:
 Lesson set:
Manipulate
linear functions. Eight video lessons: Vertically,
horizontally translate linear equations; reflect linear
equations; apply horizontal, vertical scaling transformations to
linear equations; identify the value of k in translations,
scaling transformation of linear equations; apply knowledge of
transformations to linear equations. Aligns with BFB.3.
 Lesson set:
Manipulate
quadratic functions. Six video lessons: Vertically,
horizontally translate quadratic equations; understand vertical,
horizontal scaling of quadratic equations; even and odd
functions, application of quadratic function transformations.
Aligns with BFB.3.
Saltire Software: Common Core Nuggets: There are four applets
addressing
vertical
and horizontal translations and dilations in graphing, which align with
BFB.3.
Study.com: Algebra:
Exponentials and Logarithms: There are eight video lessons in this
chapter, each of which includes sample problems and a short quiz.
Videos address the basic definitions of exponentials and logarithms,
exponential growth and decay, how to evaluate logarithms, logarithm
properties and practice problems for those properties, and how to solve
exponential and logarithmic equations. There is
a chapter multiple choice test following the last video Use
all the videos with BFB.5.
A. Dendane: Analyze Math: Select the following:
 Even and Odd
Functions: Applet and practice problems for investigating eight
given functions designed to help learners understand
even and odd functions graphically and analytically. Aligns with
BFB.3.
 Find Inverse of Rational Functions: A video illustrating how
to find the inverse of rational functions. Aligns with
BFB.4.
 Logarithm Functions: Tutorial with definition, examples, and two applets.
The first applet helps learners to explore properties of
logarithms of the form f(x)= a log_{B}( b( x + c)
) + d by changing parameters a, b, c, d, and base B. The
second applet can be used to explore the logarithm function and
its inverse, the exponential function, by changing values of
parameters c and d and base B in f(x)= log_{B}( x + c) +
d. Aligns with BFB.
MIT BLOSSOMS: Video lessons with additional teacher
and learner resources. Descriptions are from the video summaries.
 The Quadratic Equation: It’s Hip to Be Squared: "The lesson begins with three
quadratics and their graphs (three parabolas): y = x^{2} 
2x + (0 or 1 or 2). The prerequisite or corequisite is some working
experience with algebra, like factoring x^{2} 2x into
x(x2). The objective is to connect four things: the formula for
y, the graph of y (a parabola), the roots of y and the minimum
or maximum of y." Aligns with Algebra standards SSEB.3.a,
CEDA.2, and REIB.4.b and Function standards IFB.4, IFC.7.a,
IFC.8.a, BFB.3.
 Flu Math Games:
"This video lesson shows students that math can play a role in
understanding how an infectious disease spreads and how it can
be controlled." Additional simulations are included.
Aligns with Algebra standards SSEB.3.c and REIA.1; Function
standards IFC.8.b, BFB.4.a, and LEA1.(a, c); and Statistics
and Probability standards IDB.6.a, ICA.1, ICB.4, CPA.2, and
MDA.1.
Thinking Mathematics:
Math Open Reference:
Graphical Function Explorer: Graph up
to three functions on the same set of axes using this free online
graphing tool.
National Library of Virtual Manipulatives: Scroll down to Algebra (912) and select
Function Transformations:
Explore how simple transformations affect the graph of a function.
Purple Math: Use with BFB.
 Function Transformations:
Translations
 Inverse
Functions: Definition and Drawing an Inverse from a graph;
Is the inverse a function?; Finding inverses; Proving inverses
Shodor Interactivate:
Function Flyer:
This activity allows the manipulation of the constants and coefficients
in any function thereby encouraging the user to explore the effects on
the graph of the function by changing those numbers. Good for
studying about translations and dilations.
Wolfram Demonstrations Project:
Download the free Wolfram CDF player
to interact with the following manipulatives:

US Common Core State Standard Math HSFBF.B.3:
24 manipulatives. Among those: Transforming parent
functions, a library of functions with transformations,
compositions and transformations, graphing functions with four
parameters, and more, including the following:
 Quadratic in Vertex Form: Learners manipulate the parameters of
the quadratic in symmetryvertex form to see the affect on the graph
(dilation, translation horizontally and/or vertically). Aligns
with BFB.3.
 Even and Odd Functions: This demonstration allows learners
to explore the definitions of even and odd functions by
selecting graphs of each type to view. Aligns with BFB.3.
 Function Transformations and Inverses: This is a
comprehensive manipulative in which learners can choose f(x) (e.g.,
linear, quadratic, cubic, absolute value, square root,
piecewise, exponential, logarithmic, sine, cosine, tangent,
inverse sine, inverse cosine, inverse tangent) and set parameters a, b, c, and d to create
g(x)=a f(b(x + c))+d. The inverse graph can be shown, which may or
may not be a function. Aligns with BFB.3, BFB.4, and BFB.5.

US Common Core State Standard Math HSFBF.B.4:
6 manipulatives. Among those: Function identification
game, domain and range of inverse trig functions, inverse
composition rule, and the next two.

US Common Core State Standard Math HSFBF.B.5:
5 manipulatives. Among those: Swing the logarithm curve
around (1,0), calculating integer logarithms in different bases,
rules for logarithms, elementary transcendental functions and
their inverses, multiplication by adding logs.
Multiple Choice:
Constructedresponse:
Algebra Lab:
Inverse Functions:
Lesson and practice problems finding inverse functions and verifying
that two functions are inverses of each other.
Khan Academy: Practice questions with videos.
Performance tasks:
Georgia Standards of Excellence Framework:
PreCalculus Unit
2: Trigonometric Functions. This unit
contains two tasks: Right Triangles and the Unit Circle, and Inverse
Trigonometric Functions. Tasks align to BFB.4, BFB.4.d, TFA.3,
TFA.4, TFB.6, TFB.7 and are also related to TFA.2, TFB.5, and
IFC.7.e.
Illustrative Mathematics:
Functions:
Inside Mathematics MARS Tasks:

How old are they?:
Students form expressions and form and solve
an equation to solve an age problem. Aligns with
BFA.1 and BFB.4.

Conference tables:
Students find and extend a number pattern in
a practical context and find and use a rule or formula. Aligns with BFA.1 and BFB.4.
Mathematics Common Core Toolbox:
PARCC Prototype Project: Under High School Tasks, select:
 Transforming graphs of quadratic functions:
"Students use the innovative graphical interaction to experiment
and create the image of the transformed function, providing a
dynamic technological replacement for paperandpencil
sketching. While the technology is not essential to solving the
problem, it provides a support for students to interact with
function transformations and has significant implications for
future item development—a next iteration of this task could ask
students to manipulate the graph using the sliders and submit
their answers through the interactive tool instead of filling in
the blanks." This twopart task aligns with function standard
BFB.3 and mathematical practice standards MP.3 and MP.7.
Mathematics Assessment Project: Standards: High School: Functions:
https://www.map.mathshell.org/stds.php?standardid=1448
Mathematics Vision Project,
Secondary 1 Student Edition:

Module
7: Connecting Algebra and Geometry: This module contains
six classroom tasks. Task 4: Training Day; Task 5: Training
Day Part II; and Task 6: Shifting Functions each address the function standards
BFA.1, BFB.3, and IFC.9. The other three
tasks address geometry standards GPEB.4, GPEB.5, and GPEB.7.
Mathematics Vision Project,
Secondary
2 Student Edition:

Module 4: More Functions, More Features: This module
contains 7 classroom tasks. Function standards
addressed in Module 4 include IFC.7b within tasks 14;
BFB.4 in tasks 56; and IFB.4 in task 7.
Mathematics Vision Project,
Secondary
3 Student Edition:
NCTM's Reasoning and Sense Making Task Library:
Tidal Waves includes the task overview, teacher notes for its use
in the classroom, and student activity sheet. Aligns with BFB.3,
TFB.5 and mathematical practice standards 1 and 7.
Back to top Common Core Math:
Intro  HS Functions Domain:
IF  BF  LE  TF 