

Unit 2 Linear
& Exponential Functions Algebra
I 


Unit
Length and Description: 35
days In earlier grades, students define,
evaluate, and compare functions, and use them to model relationships between
quantities. In this unit, students will learn function notation and develop
the concepts of domain and range. They move beyond viewing functions as
processes that take inputs and yield outputs and start viewing functions as
objects in their own right. They explore many examples of functions,
including sequences; they interpret functions given graphically, numerically,
symbolically, and verbally, translate between representations, and understand
the limitations of various representations. They work with functions given by
graphs and tables, keeping in mind that, depending upon the context, these
representations are likely to be approximate and incomplete. Their work
includes functions that can be described or approximated by formulas as well
as those that cannot. When functions describe relationships between
quantities arising from a context, students reason with the units in which
those quantities are measured. Students explore systems of equations and
inequalities, and they find and interpret their solutions. Students build on
and informally extend their understanding of integer exponents to consider
exponential functions. They compare and contrast linear and exponential
functions, distinguishing between additive and multiplicative change. They
interpret arithmetic sequences as linear functions and geometric sequences as
exponential function. (Mathematics Appendix A, p.19) ·
Represent
and solve equations and inequalities graphically ·
Understand
the concept of a function and use function notation ·
Interpret
functions that arise in applications in terms of a context. ·
Analyze
functions using different representations. ·
Build
a function that models a relationship between two quantities ·
Build
new functions from existing functions. ·
Construct
and compare linear, quadratic, and exponential models and solve problems. ·
Interpret
expressions for functions in terms of the situation they model. ·
Write
expressions in equivalent forms to solve problems. ·
Create
equations that describe numbers or relationships. 

Standards: ACED.A.1
Create equations and inequalities in one variable and use them to solve
problems. Include equations arising
from linear and quadratic functions, and simple rational and exponential
functions.^{★} AREI.D.11
Explain why the 𝑥coordinates of the points where the graphs of the equations 𝑦=𝑓(𝑥) and 𝑦=𝑔(𝑥) intersect are the solutions of the equation 𝑓(𝑥)=𝑔(𝑥); find the solutions approximately, e.g., using technology to graph
the functions, make tables of values, or find successive approximations.
Include cases where (𝑥) and/or (𝑥) are linear, polynomial, rational, absolute value, exponential, and
logarithmic functions.^{★} ASSE.B.3 Choose and produce an equivalent form of an
expression to reveal and explain properties of the quantity represented by
the expression.^{★} c. Use the properties of exponents to transform expressions for
exponential functions. For example the
expression 1.15^{t} can be rewritten as (1.15^{1/12})12^{t}≈1.01212t
to reveal the approximate equivalent monthly interest rate if the annual rate
is 15%. FBF.A.1
Write a function that describes a relationship between two quantities.^{★}
FBF.B.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. FIF.A.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 (𝑥) denotes the output of 𝑓
corresponding to the input 𝑥. The graph of 𝑓 is the graph of the equation 𝑦=(𝑥). FIF.A.2
Use function notation, evaluate functions for inputs
in their domains, and interpret statements that use function notation in
terms of a context. FIF.A.3
Recognize that sequences are functions, sometimes defined recursively, whose
domain is a subset of the integers. For
example, the Fibonacci sequence is defined recursively by (0)= 𝑓(1)=1, 𝑓(𝑛+1)=𝑓(𝑛)+𝑓(𝑛−1) for 𝑛≥1. FIF.B.4
For 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.^{★} FIF.B.5
Relate 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.^{★} FIF.B.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.^{★} FIF.C.7
Graph functions expressed symbolically and show key features of the graph, by
hand in simple cases and using technology for more complicated cases.^{★}
FIF.C.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. FLE.A.1
Distinguish between situations that can be modeled with linear functions and
with exponential functions.^{★}
FLE.A.2
Construct linear and exponential functions, including arithmetic and
geometric sequences, given a graph, a description of a relationship, or two
inputoutput pairs (include reading these from a table).^{★} FLE.A.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.^{★} FLE.B.5
Interpret the parameters in a linear or exponential function in terms of a
context.^{★} Focus Standards of Mathematical Practice: MP.1 Make
sense of problems and persevere in solving them. MP.2 Reason
abstractly and quantitatively. MP.3
Construct viable arguments and critique the reasoning of others. MP.4 Model
with mathematics. MP.5 Use
appropriate tools strategically. MP.6 Attend
to precision. MP.7 Look for
and make use of structure. MP.8 Look for
and express regularity in repeated reasoning. Instructional
Outcomes: Full Development of the Major Clusters, Supporting
Clusters, Additional Clusters and Mathematical Practices for this unit could
include the following instructional outcomes: ACED.A.1 ·
I
can solve quadratic equations in one variable ·
I
can solve quadratic inequalities in one variable ·
I
can create quadratic equations and inequalities in one variable and use them
to solve problems ·
I
can create quadratic equations and inequalities in one variable to model
realworld situations AREI.D.11 ·
I
can recognize and use function notation to represent linear and exponential
equations ·
I
can recognize that if (x1 , y1) and (x2 , y2) share the same location in the
coordinate plane that x1 = x2 and y1 = y2 ·
I
can recognize that f(x) = g(x) means that there may be particular inputs of f
and g for which the outputs of f and g are equal ·
I
can explain why the xcoordinates of the points where the graph of the
equations y=f(x) and y=g(x) intersect are the solutions of the equations f(x)
= g(x). (Include cases where f(x) and/or g(x) are linear and exponential
equations) ·
I
can approximate/find the solution(s) using an appropriate method for example,
using technology to graph the functions, make tables of values or find
successive approximations (Include cases where f(x) and/or g(x) are linear
and exponential equations) ASSE.B.3.c ·
I
can use the properties of exponents to transform simple expressions for
exponential functions ·
I
can use the properties of exponents to transform expressions for exponential
functions ·
I
can choose and produce an equivalent form of an exponential expression to
reveal and explain properties of the quantity represented by the original
expression ·
I
can explain the properties of the quantity or quantities represented by the
transformed exponential expression FBF.A.1a ·
I
can define “explicit function” and “recursive process” ·
I
can write a function that describes a relationship between two quantities by
determining an explicit expression, a recursive process, or steps for
calculation from a context FBF.B.3 ·
I
can identify the effect a single transformation will have on the function
(symbolic or graphic) ·
I
can use technology to identify effects of single transformations on graphs of
functions ·
I
can graph a given function by replacing f(x) with f(x)+k, kf(x),
f(kx), or f(x+k) for
specific values of k (both positive and negative) ·
I
can describe the differences and similarities between a parent function and
the transformed function ·
I
can find the value of k, given the graphs of a parent function, f(x), and the
transformed function: f(x)+k, kf(x), f(kx), or f(x+k) ·
I
can recognize even and odd functions from their graphs and from their
equations ·
I
can experiment with cases and illustrate an explanation of the effects on the
graph using technology FIF.A.1 ·
I
can identify the domain and range of a function ·
I
can determine if a relation is a function ·
I
can determine the value of the function with proper notation (i.e. f(x) = y,
the y value is the value of the function at a particular value of x) ·
I
can evaluate functions for given values of x FIF.A.2 ·
I
can identify mathematical relationships and express them using function
notation ·
I
can define a reasonable domain, which depends on the context and/or
mathematical situation, for a function focusing on linear and exponential
functions ·
I
can evaluate functions at a given input in the domain, focusing on linear and
exponential functions ·
I
can interpret statements that use functions in terms of real world
situations, focusing on linear and exponential functions FIF.A.3 ·
I
can recognize that sequences are functions, sometimes defined recursively,
whose domain is a subset of the integers. For example, the Fibonacci sequence
is defined by f(0)=f(1)=1, f(n+1) = f(n) + f(n1) for n≥1 FIF.B.4 ·
I
can define and recognize the key features in tables and graphs of linear and
exponential functions: intercepts; intervals where the function is
increasing, decreasing, positive, or negative, and end behavior ·
I
can identify whether the function is linear or exponential, given its table
or graph ·
I
can interpret key features of graphs and tables of function in the terms of
the contextual quantities the function represents ·
I
can sketch graphs showing key features of a function that models a
relationship between two quantities from a given verbal description of the
relationship FIF.B.5 ·
I
can, given the graph or a verbal/written description of a function, identify
and describe the domain of the function ·
I
can identify an appropriate domain based on the unit, quantity , and type of
the function it describes ·
I
can relate the domain of the function to its graph and, where applicable, to
the quantitative relationship it describes ·
I
can explain why a domain is appropriate for a given situation FIF.B.6 ·
I
can recognize slope as an average rate of change ·
I
can calculate the average rate of change of a function (presented
symbolically or as a table) over a specified interval ·
I
can estimate the rate of change from a linear or exponential graph ·
I
can interpret the average rate of change of a function (presented
symbolically or as a table) over a specified interval FIF.C.7a ·
I
can graph linear functions by hand in simple cases or using technology for
more complicated cases and show/label intercepts of the graph FIF.C.9 ·
I
can identify types of functions based on verbal, numerical, algebraic, and
graphical descriptions and state key properties (e.g. intercepts, growth
rates, average rates of change, and end behaviors) ·
I
can differentiate between exponential and linear functions using a variety of
descriptors (graphically, verbally, numerically, and algebraically) ·
I
can use a variety of function representations algebraically, graphically,
numerically in tables, or by verbal descriptions) to compare and contrast
properties of two functions FLE.A.1 ·
I
can recognize that linear functions grow by equal differences over equal
intervals ·
I
can recognize that exponential functions grow by equal factors over equal
intervals ·
I
can distinguish between situations that can be modeled with linear functions
and with exponential functions to solve mathematical and realworld problems ·
I
can prove that linear functions grow by equal differences over equal
intervals ·
I
can prove that exponential functions grow by equal factors over equal
intervals ·
I
can recognize situations in which one quantity changes at a constant rate per
unit (equal differences) interval relative to another to solve mathematical
and realworld problems ·
I
can recognize situations in which a quantity grows or decays by a constant
percent rate per unit (equal factors) interval relative to another to solve
mathematical and realworld problems FLE.A.2 ·
I
can recognize arithmetic sequences can be expressed as linear functions ·
I
can recognize geometric sequences can be expresses as exponential functions ·
I
can construct linear functions, including arithmetic sequences, given a
graph, a description of a relationship, or two inputoutput pairs (include
reading these from a table) ·
I
can construct exponential functions, including geometric sequences, given a
graph, a description of relationship, or two inputoutput pairs (include
reading these from a table) ·
I
can determine when a graph, a description of a relationship, or two
inputoutput pairs (include reading these from a table) represents a linear
or exponential function in order to solve problems FLE.A.3 ·
I
can informally define the concept of “end behavior” ·
I
can compare tables and graphs of linear and exponential functions to observe
that a quantity increasing exponentially exceeds all others to solve
mathematical and realworld problems FLE.B.5 ·
I
can recognize the parameters in a linear or exponential function including:
vertical and horizontal shifts, vertical and horizontal dilations ·
I
can recognize rate of change and intercept as “parameters” in linear or
exponential functions ·
I
can interpret the parameters in a linear or exponential function in terms of
a context 

Enduring
Understandings: ·
Algebra
is the language of symbols, operations, and relationships that allow us to
understand many real world phenomena. ·
There
are many times in real life when one quantity depends on another. ·
Real
world situations can be represented symbolically, numerically, verbally, and
graphically. ·
New
functions can be created from parent functions through a series of
transformations. ·
Real
world phenomena often involve nonlinear relationships. ·
Nonconstant
rates of change denote nonlinear relations. ·
Mathematical
rules that reflect recurring patterns facilitate efficiency in problem
solving. ·
When
analyzing linear and exponential functions, different representations may be
used based on the situation presented. ·
Algebraic
representations can be used to generalize patterns in mathematics. ·
Relationships
between quantities can be represented symbolically, numerically, graphically,
and verbally in the exploration of real world situations. ·
The
characteristics of radical functions and its representations are useful in
solving realworld problems. ·
Families
of functions exhibit properties and behaviors that can be recognized across
representations. ·
Functions
can be transformed, combined, and composed to create new functions in
mathematical and real world situations. ·
Mathematical
functions are relationships that assign each member of one set to a unique
member of another set and the relationship is recognizable across
representations. 
Essential
Questions: ·
What
are some ways I can represent the relationship between quantities? ·
When
quantities share a relationship, how can I use that relationship to determine
reasonable values for the quantities? ·
How
do I know when one quantity depends on another ·
How
can the characteristics of a function be determined from a graph or a given
function? ·
Why
are relations and functions represented in multiple ways? ·
How
are properties of function and functional operations useful? ·
What
types of reallife situations involve nonconstant rates of change? ·
How
do linear and nonlinear relationships differ? How are they the same? ·
How
can I use patterns to establish relationships that will help make decisions
in reallife situations? ·
How
can a new function be created from an existing function? ·
What
characteristics of problems would determine how to model the situation and
develop a problem solving strategy? ·
How
do radical functions model realworld problems and their solutions? ·
How
are expressions involving radicals and exponents related? ·
How
can an equation, table, and graph be used to analyze the rate of change and
other applicable information, related to a realworld problem and the
representative function? ·
How
can a given function be represented graphically, within a table, by an
equation, and in the realworld? ·
What
connections can be made between various functions and various representations
of functions? 
