Standard

Essential Question

Bloom’s Taxonomy Activities

Vocabulary

Pacing


F.IF.1
Understand that a function from one set (called the domain) to another set
(called the range) assigns to each element of the domain exactly one element
of the range. If f is a function
and x is an element of its domain,
then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y =f(x).

How are
domain and range related to each other?
How does f relate to the graph of y=f(x)?
Why are
tables important when considering the input and outputs of equations?

Support
the concept that the domain of a number matches to exactly one element of the
range.
 Develop
a linear function demonstrating the relationship between domain and range.

Set
Domain
Range
Element
Function
Input
Output

2 days


F.IF.2 Use
function notation, evaluate functions for inputs in their domains, and
interpret statements that use function notation in terms of a context.

How does
understanding function notation assist in creating graphs by hand?

Convert
equations in y= form to function notation.
Construct
models of linear and exponential models on paper and on a graphing
calculator.

Function
Notation

2 days


F.IF.3
Recognize that sequences are functions, sometimes defined recursively, whose
domain is a subset of integers. For
example, the Fibonacci sequence is defined recursively by f(0) = f(1), f(n+1)
= f(n) + f(n1) for n≥1.

How could
sequences be defined in function notation?

Create a
recursive sequence or formula.
 Examine
the relationship between functions and sequences.

Sequence
Subset
Recursive

2 days


F.IF.4
For a function that models a relationship between two quantities, interpret
key features of graphs and table 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 behaviors; and
periodicity.*

How do
relative maximums and relative minimums relate to where a graph is increasing
or decreasing?
What
relationships do intercepts have with the axis?

Create
two equations and graph them on paper.
Contrast
the key features of the equations and their respective graphs.

XIntercept
YIntercept
Interval
Increasing
Decreasing
Positive
Negative
Relative
Maximum
Relative
Minimum
Symmetry
End
Behavior

3 days


F.IF.5Relate
the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For
example, if the function h(n) gives the number of personhours it takes to
assemble n engines in a factory, then the positive integers would be an
appropriate domain for the function.*

Why is it
important to reread the question before providing a final answer?
How do
questions pertaining to time, people, and distance affect the domain of an
answer?

Examine
the responses to questions involving domain and word problems to discover the
quantitative relationship between what the question is asking and the
response.

Domain
Function

3 days


F.IF.6 Calculate
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.*

How do
points on a linear graph assist in determining rate of change?

Approximate
the rate of change from a given graph.
Estimate
the rate of change for a quadratic function for each interval.

Average
Rate of
Change
Interval

3 days


F.IF.7
Graph functions expressed symbolically and show key features of the graph, by
hand in simple cases and using technology for more complicated cases.*

What is
the relationship between the parts of a function and its graph?

 Examine
a function expressed symbolically to create its graph.
Judge a
given function to determine if it is best to complete manually or through
technology.

Function
Approximation

3 days


F.IF.7a
Graph linear and quadratic functions and show intercepts,
maxima,
and minim

How can
the qualities of a function expressed symbolically assist in creating its
graph?

Compare
and contrast various functions to determine the attributes of their
corresponding graphs.

Linear
Quadratic
Intercept
Maxima
Minima

3 days


F.IF.7b
Graph square root, cube root, and piecewisedefined functions, including step
functions and absolute value functions.

What is
the relationship between a graph to an integer power and a graph to a
fractional power?

Create a
poster board with a sample of each of the following graphs: square root, cube
root, and piecewisedefined functions, including step functions and absolute
value functions. Graphs may be handdrawn or created on the computer using a
program such as Microsoft Mathematics

Square
root
Cube
root
Piecewise
function
Step
function
Absolute
value function

3 days


F.IF.7c Graph
polynomial functions, identifying zeros when suitable factorizations are
available, and showing end behavior.

How can
technology support the understanding of polynomial equations?

Create a
PowerPoint slide based on a given equation; include two graphics on the
slide, 1) a handdrawn graph of the equation and 2) an electronic
representation of the polynomial

Polynomial
functions
Factorizations

3 days


F.IF.7e
Graph exponential and logarithmic functions, showing intercepts and end
behavior, and trigonometric functions, showing period, midline, and
amplitude.

What is
the relationship, if any, between the qualities of logarithmic graphs and
exponential graphs?

Create an
equation for exponential and logarithmic graphs and provide information
regarding intercepts and end behavior

Exponential
function

Logarithmic function
End
behavior

3 days


F.IF.8
Write a function defined by an expression in different but equivalent forms
to reveal and explain different properties of functions.

What
properties can be used to express functions in equivalent forms?
Why is it
beneficial to express functions in equivalent forms?

Create a
visual highlighting the different ways a function can be expressed

Function
Equivalent
forms

3 days


F.IF.8a
Use the process of factoring and completing the square in a quadratic
function to show zeros, extreme values, and symmetry of the graph, and
interpret these in terms of a context.

In what
instances would completing the square be beneficial in simplifying functions?

Distinguish
the qualities of a graph based on the result of a simplified function

Factoring
Completing
the square
Quadratic
function
Extreme
values
Symmetry

3 days


F.IF.8b
Use the properties of exponents to interpret expressions for exponential
functions. For example, identify percent
rate of change in functions such as y = (1.02)^{t}, y=(0.97)^{t},
y=(1.01)^{12t}, y=(1.2)^{t/10}, and classify them as
representing exponential growth or decay.

What are
three examples in which exponential functions are used in the real world?

Construct
a realworld example of the use of an exponential equations, share your
problem with your class, and display your work

Exponents
Exponential
function

3 days


F.IF.9
Compare 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.

What
qualities of a function expressed symbolically assist in determining the
features of the graph?

Design a
visual representation comparing two selfcreated function in four
representations: numerically in tables, graphically, verbally, and
algebraically

Properties
of functions

3 days


Standard

Essential Question

Bloom’s Taxonomy Activities

Vocabulary

Pacing


F.BF.1
Write a function that describes a relationship between two quantities.*

What are
some context clues in a word problem which assist in creating a function
which depicts the relationship between the quantities provided?

Analyze
a word problem to create a function which accurately depicts the relationship
between the two quantities.

Relationship
between functions
Sum
Difference
Increase
Divided

3 days


F.BF.1a
Determine an explicit expression, a recursive process, or steps from a
context.

What are
the benefits of creating an explicit expression for a given sequence?

Distinguish
when writing an explicit expression is more valuable than determining a
recursive process or equation.

Explicit
expression
Recursive
process
Sequence

3 days


F.BF.1b
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.

When is
it necessary to combine functions in solving a problem?

Create a
word problem where combining functions algebraically is necessary.

Standard
function

3 days


F.BF.2
Write arithmetic and geometric sequences both recursively and with an
explicit formula, use them to model situations, and translate between the two
forms.*

What is
the relationship between geometric and arithmetic sequences? In determining
equations to a high term, which form is best?

Given an
arithmetic or geometric sequence, create two formulas: one recursive and one
explicit.

Arithmetic
sequence
Geometric
sequence

3 days


F.BF.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 the
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.

How does
including an additional graph affect the graph of a function?

Develop
four scenarios in which a graph is increased or decreased by the constant, k

Constant
Translation
Transformation

1 day


F.BF.4
Find the inverse functions.

Why is
knowledge of inverse functions beneficial when considering purchasing a
house?

Choose a
house you would be interested in purchasing and research financing options,
then select which finance plan is best.

Inverse
function

1 day


F.BF.4a
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) =2x^{3} or f(x) = (x+1)/(x1) for x≠1.

What is
the relationship of domain and range in a function?

Create a
worksheet and answer key of no less than seven problems of simple functions
for peers to complete the inverse

Inverse
of a function
Domain
Range

1 day


Fbf5


Standard

Essential Question

Bloom’s Taxonomy Activities

Vocabulary

Pacing


F.LE.1
Distinguish between situations that can be modeled with linear functions and with
exponential functions.

What is
the difference between the progression of points on a linear graph and the
progression of points on an exponential graph?

Given a
situation, determine if the model created is linear or exponential based on
the context clues in the equation.

Linear
graph
Exponential
graph

3 days


F.LE.1a
Prove that linear functions grow by equal differences over equal intervals,
and that exponential functions grow by equal factors over equal intervals.

How many
points are needed to create an accurate equation for a linear function and an
accurate equation for an exponential function? Are there exceptions?

Given the
graph of a linear function, defend algebraically or graphically that the
function increases or decreases over equal intervals by equal quantities.
Given the
graph of an exponential function, prove growth occurs over equal intervals.

Interval
Equal
factor
Equal
difference

3 days


F.LE.1b
Recognize situations in which one quantity changes at a constant rate per
unit interval relative to another.

What
context clues suggest that a quantity changes at a constant rate per interval?

Analyze
several word problems relating to constant rate of change to determine which
qualities or context clues are consistent across the problems.

Constant
rate

3 days


F.LE.1c
Recognize situations in which a quantity grows or decays by a constant percent
rate per unit interval relative to another.

What
context clues suggest that a quantity changes at a constant factor per
interval?

Analyze
several word problems relating to constant factors of change to determine
which qualities or context clues are consistent across the problems.

Constant
factor

3 days


F.LE.2
Construct linear and exponential functions, including arithmetic and
geometric sequences, given a graph, a description of a relationship, or two
inputoutput pairs (including reading these from a table).

What
information is necessary to create a graphical or algebraic representation of
a described function?

Given
various expressions of information pertaining to a linear or an exponential
function, create a graphical or algebraic representation of the function.

Inputoutput
pairs

3 days


F.LE.3
Observe using graphs and tables that a quantity increasing exponentially
eventually exceeds a quantity increasing linearly, quadratically, or (more
generally) as a polynomial function.

What is
unique about an exponential function as compared to a linear or quadratic
function?

Analyze
linear, quadratic, and exponential functions to determine unique qualities
between the three types of functions.

Exponential
function
Linear
function
Quadratic
function

3 days


F.LE.4
For exponential models, express as a logarithm the solution to ab^{ct} = d where a, c, and d are numbers and the base b
is 2, 10, or e; evaluate the
logarithm using technology.

In what
instances is 2+2 not equal to 4?

Selfguide
your understanding of logs using the graphing calculator; create your own
problems and share with the class

Logarithm
Base
Exponent
Natural
log

3 days


F.LE.5
Interpret the parameters in a linear or exponential function in terms of a
context.

How does
the manipulation of the components of a function relate to its corresponding
graph?

Given a
function, manipulate various components such as slope or intercept to
determine how the function changes under the new conditions.

Parameters

1 day


Standard

Essential Question

Bloom’s Taxonomy Activities

Vocabulary

Pacing


F.TF.1
Understand radian measure of an angle as the length of the arc on the unit
circle subtended by the angle.

What is the
difference between radian measure and degree measure?

Compare the
radian measures of various unit circles and their corresponding central
angles.

Unit circle
Radian
measure
Central
angle

1 day


F.TF.2
Explain how the unit circle in the coordinate plane enables the extension of
trigonometric functions to all real numbers, interpreted as radian measures
of angles traversed counterclockwise around the unit circle.

Why is the
unit circle used to determine radian measures?

Create a
visual of a unit circle with positive and negative radian measures

Unit circle
Coordinate
plane
Trigonometric

1 day


Ftf3


Ftf4


F.TF.5
Choose trigonometric functions to model periodic phenomena with specified amplitude,
frequency, and midline.*

What
consistencies exist when analyzing trigonometric functions?

Investigate
various online trigonometry resources at http://www.homeschoolmath.net/online/trigonometry.php
to experiment with amplitude, frequency, and midline of trigonometry
functions

Sine
Cosine
Tangent
Amplitude
Frequency
Midline
Periodic

1 day


Ftf6


Ftf7


F.TF.8 Prove
the Pythagorean identity sin^{2}(θ) + cos^{2}(θ) =1 and use
it to find sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

How is
knowledge of the Pythagorean Theorem useful in determining trigonometric
identities?

Use the
Pythagorean theorem to prove sin^{2}(θ) + cos^{2}(θ) =1. Use
the identity to create at least three examples.

Pythagorean
identity

1 day

Functions
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