

Unit 3 Algebra II 

Unit
Topic and Length: 45
days In this unit, students extend their
knowledge of linear, quadratic, polynomial, rational, radical and
trigonometric functions learned in Algebra I and the first half of Algebra
II, and use them in modeling situations.
The focus for this unit is on exponential and logarithmic functions. In Unit 3 students synthesize and
generalize what they have learned about a variety of function families. They extend their work with exponential
functions to include solving exponential equations with logarithms. They
explore the effects of transformations on graphs of diverse functions,
including functions arising in an application, in order to abstract the
general principle that transformations on a graph always have the same effect
regardless of the type of the underlying function. They identify appropriate types of
functions to model a situation, they adjust parameters to improve the model,
and they compare models by analyzing appropriateness of fit and making
judgements about the domain over which a model is a good fit. (Mathematics
Appendix A, p.3642, with adjustments) ·
Extend
the properties of exponents to rational exponents. ·
Reason
quantitatively and use units to solve problems. ·
Write
expressions in equivalent forms to solve problems. ·
Create
equations that describe numbers or relationships. ·
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 the 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. 

Standards: NRN.A.1 Explain
how the definition of the meaning of rational exponents follows from
extending the properties of integer exponents to those values, allowing for a
notation for radicals in terms of rational exponents. For example, we define 5^{1/3 }to be
the cube root of 5 because we want (5^{1/3})^{3} = 5^{(1/3)3}
to hold, so (5^{1/3})^{3} must equal 5. NRN.A.2
Rewrite
expressions involving radicals and rational exponents using the properties of
exponents.
NQ.A.2
Define
appropriate quantities for the purpose of descriptive modeling.
ASSE.B.3c 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})^{12t }≈
1.012^{12t} to reveal the approximate equivalent monthly interest
rate if the annual rate is 15%.
ASSE.B.4 Derive the formula
for the sum of a finite geometric series (when the common ratio is not 1),
and use the formula to solve problems. For example, calculate mortgage
payments.★
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 x‐coordinates 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 f(x) and/or g(x) are linear, polynomial, rational, absolute
value, exponential, and logarithmic functions.★
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 f(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.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.7e Graph functions expressed symbolically
and show key features of the graph, by hand in simple cases and using
technology for more complicated cases.★ e. Graph exponential and logarithmic
functions, showing intercepts and end behavior, and trigonometric functions,
showing period, midline, and amplitude.
FIF.C.8b Write a function defined by an
expression in different but equivalent forms to reveal and explain different
properties of the function. b. 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.
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.
FBF.A.1a Write a function that describes a
relationship between two quantities.★ a. Determine an
explicit expression, a recursive process, or steps for calculation from a
context.
FBF.A.1b Write a function that describes a
relationship between two quantities.★ 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.
FBF.A.2 Write arithmetic and geometric
sequences both recursively and with an explicit formula, use them to model
situations, and translate between the two forms.★ 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.
FBFB.4a Find inverse functions. 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) =2x^{3} or
f(x) = (x+1)/(x‐1) for x ≠ 1.
FLE.A.2 Construct linear and exponential
functions, including arithmetic and geometric sequences, given a graph, a
description of a relationship, or two input‐output pairs (include reading these from a
table).
FLE.A.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.
FLE.B.5 Interpret the parameters in a linear,
quadratic, 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: NRN.A.1 ·
I can define radical notation as a convention used to
represent rational exponents. ·
I can explain the properties of operations of rational
exponents as an extension of the properties of integer exponents. ·
I can explain how radical notation, rational exponents,
and properties of integer exponents relate to one another. NRN.A.2 ·
I can, using the
properties of exponents, rewrite a radical expression as an expression with a
rational exponent. ·
I can, using the
properties of exponents, rewrite an expression with a rational exponent as a
radical expression. NQ.A.2 ·
I can define descriptive
modeling. ·
I can determine
appropriate quantities for the purpose of descriptive modeling. ASSE.B.3c ·
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. ASSE.B.4 ·
I can find the first
term in a geometric sequence given at least two other terms. ·
I can define a
geometric series as a series with a constant ratio between successive terms. ·
I can use the formula S
= ((a (1r^{n}))/ (1r)) to solve problems. ACED.A.1 ·
I can solve equations
specified in this standard in one variable. ·
I can solve
inequalities specified in this standard in one variable. ·
I can describe the
relationships between the quantities in the problem (for example, how the
quantities are changing or growing with respect to each other); express these
relationships using mathematical operations to create an appropriate equation
or inequality to solve. ·
I can create equations
and inequalities specified in this standard in one variable and use them to
solve problems. ·
I can create equations
and inequalities specified in this standard in one variable to model
realworld situations. ·
I can compare and
contrast problems that can be solved by the different types of equations
specified in this standard. AREI.D.11 ·
I can approximate or
find the solutions to a system involving any of the function types listed in
this standard. ·
I can explain why the
solution to a system involving any of the function types listed in this
standard will occur at the point(s) of intersection. 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 identify the key
features from a table or graph of polynomial,
exponential, logarithmic, and trigonometric functions. ·
I can sketch functions
that model key feature behavior. FIF.B.6 ·
I can calculate the
average rate of change of a given interval for polynomial,
exponential, logarithmic, and trigonometric functions. ·
I can demonstrate that
the average rate of change of a nonlinear function is different for
different intervals. FIF.C.7e ·
I can graph polynomial,
exponential, logarithmic, and trigonometric functions. ·
I can describe key
features of polynomial, exponential, logarithmic, and trigonometric
functions. FIF.C.8b ·
I can identify how key
features of an exponential function relate to characteristics in a realworld
context. ·
I can write and
classify realworld problems as an exponential growth or decay. ·
I can use and applying A = Pe^{rt} and A = P (1 + r/n)^{nt}^{ }. FIF.C.9 ·
I can compare key
features of two representations of functions (polynomial, exponential,
logarithmic, and trigonometric). 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 (involving polynomial,
exponential, logarithmic, and trigonometric functions). FBF.A.1b ·
I can combine two
functions (all functions studied) using the operations of addition,
subtraction, multiplication, and division. ·
I can evaluate the
domain of the combined function (all functions studied). ·
I can build standard
functions (all functions studied) to represent relevant
relationships/quantities given a realworld situation or mathematical
process. ·
I can determine which
arithmetic operation should be performed to build the appropriate combined
function (all functions studied) given a realworld situation or mathematical
process. ·
I can relate the
combined function (all functions studied) to the context of the problem. FBF.A.2 ·
I can identify
arithmetic and geometric patterns in given sequences. ·
I can generate
arithmetic and geometric sequences from recursive and explicit formulas. ·
I can, given an
arithmetic or geometric sequence in recursive form, translate into the
explicit formula. ·
I can, given an
arithmetic or geometric sequence as an explicit formula, translate into the
recursive form. ·
I can use given and
constructed arithmetic and geometric sequences, expresses both recursively
and with explicit formulas, to model reallife situations. ·
I can determine the
recursive rule given arithmetic and geometric sequences. ·
I can determine the
explicit formula given arithmetic and geometric sequences. ·
I can justify the
translation between the recursive form and explicit formula for arithmetic
and geometric sequences. FBF.B.3 ·
I can perform the tasks
below which may involve simple radical, rational, polynomial, exponential,
logarithmic, and trigonometric functions. ·
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. ·
I can identify
transformations of a function on a graph. ·
I can describe the
effects of transformations on parent functions. FBFB.4a ·
I can define inverse
function (this extends to simple rational, simple radical, and simple exponential functions). ·
I can 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 (this extends to simple rational, simple radical, and
simple exponential functions). FLE.A.2 ·
I can recognize
arithmetic sequences can be expressed as linear functions. ·
I can recognize
geometric sequences can be expressed 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.4 ·
I can use the
properties of logs. ·
I can describe the key
features of logs. ·
I can use logarithmic
form to solve exponential models. ·
I can recognize that
logarithm without a base specified is base 10 and that natural logarithm
always refers to base e.
·
I can recognize and use logarithms as
solutions for exponentials. 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. ·
I can interpret the
parameters of exponential functions with domains not in the integers. 

Enduring Understandings: ·
A function is a
relationship between variables in which each value of the input variable is
associated with a unique value of the output variable. ·
Various mathematical
situations require various functions to model the data. ·
The basic modeling
cycle can be used to shed light on mathematical structures. ·
Adjusting the
parameters of a situation can improve the model. ·
Solving exponential
and logarithmic equations and graphing exponential and logarithmic functions
can be approached by hand and by using technology. ·
Comparing the speed
at which a function increases can help determine which type of function best
models the data. ·
They
will comprehend the meaning of a logarithm of a number and know when to use
logarithms to solve exponential functions. ·
Transformations on a
graph always have the same effect regardless of the type of underlying
function. ·
Transforming
functions is essential to the interpretation of data. ·
It is important to be
able to translate easily among the equation of a function, its graph, its
verbal representation, and its numerical representation. 
Essential Questions: ·
How are exponents and
logarithms related? ·
What is the
relationship between exponential functions and logarithms? ·
How do you model a
quantity that changes regularly over time by the same percentage? ·
How can the modeling
cycle help determine which type of function best models a given set of data? ·
How can data be used
to determine the function that fits it best? ·
What can be learned
from studying models of exponential growth and decay? ·
How can studying
families of functions be beneficial to interpreting graphs and data? ·
What are some of the
characteristics of the graph of an exponential function? ·
What are some of the
characteristics of the graph of a logarithmic function? ·
How can you use
properties of exponents to derive properties of logarithms? ·
How can you write a
rule for the nth term of a
sequence? ·
How can you recognize
an arithmetic sequence from its graph? ·
How can you recognize
a geometric sequence from its graph? 
