**Unformatted text preview: **New York State Common Core Mathematics Curriculum
ALGEBRA II • MODULE 3 Table of Contents1 Exponential and Logarithmic Functions
Module Overview .................................................................................................................................................. 3
Topic A: Real Numbers (N-RN.A.1, N-RN.A.2, N-Q.A.2, F-IF.B.6, F-BF.A.1a, F-LE.A.2) ...................................... 11
Lesson 1: Integer Exponents ................................................................................................................... 14
Lesson 2: Base 10 and Scientific Notation .............................................................................................. 30
1 1 Lesson 3: Rational Exponents—What are 22 and 23 ? ........................................................................... 43
Lesson 4: Properties of Exponents and Radicals ................................................................................... 62
Lesson 5: Irrational Exponents—What are 2√2 and 2 ? ....................................................................... 75
Lesson 6: Euler’s Number, .................................................................................................................. 86
Topic B: Logarithms (N-Q.A.2, A-CED.A.1, F-BF.A.1a, F-LE.A.4)........................................................................ 102
Lesson 7: Bacteria and Exponential Growth ......................................................................................... 105
Lesson 8: The “WhatPower” Function.................................................................................................. 116
Lesson 9: Logarithms—How Many Digits Do You Need? ..................................................................... 128
Lesson 10: Building Logarithmic Tables ................................................................................................ 137
Lesson 11: The Most Important Property of Logarithms ..................................................................... 149
Lesson 12: Properties of Logarithms .................................................................................................... 160
Lesson 13: Changing the Base .............................................................................................................. 175
Lesson 14: Solving Logarithmic Equations ............................................................................................ 193
Lesson 15: Why Were Logarithms Developed? .................................................................................... 207
Mid-Module Assessment and Rubric ................................................................................................................ 225
Topics A through B (assessment 1 day, return 1 day, remediation or further applications 4 days)
Topic C: Exponential and Logarithmic Functions and their Graphs (F-IF.B.4, F-IF.B.5, F-IF.C.7e,
F-BF.A.1a, F-BF.B.3, F-BF.B.4a, F-LE.A.2, F-LE.A.4).............................................................................. 250
Lesson 16: Rational and Irrational Numbers ........................................................................................ 253
Lesson 17: Graphing the Logarithm Function....................................................................................... 266
1Each lesson is ONE day, and ONE day is considered a 45-minute period. Module 3: © 2015 Great Minds. eureka-math.org
ALG II-M3-TE-1.3.0-08.2015 Exponential and Logarithmic Functions 1
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview M3 ALGEBRA II Lesson 18: Graphs of Exponential Functions and Logarithmic Functions ............................................ 284
Lesson 19: The Inverse Relationship Between Logarithmic and Exponential Functions...................... 298
Lesson 20: Transformations of the Graphs of Logarithmic and Exponential Functions ....................... 316
Lesson 21: The Graph of the Natural Logarithm Function ................................................................... 339
Lesson 22: Choosing a Model ............................................................................................................... 354
Topic D: Using Logarithms in Modeling Situations (A-SSE.B.3c, A-CED.A.1, A-REI.D.11, F-IF.A.3,
F-IF.B.6, F-IF.C.8b, F-IF.C.9, F-BF.A.1a, F-BF.A.1b, F-BF.A.2, F-BF.B.4a, F-LE.A.4, F-LE.B.5) ............... 366
Lesson 23: Bean Counting..................................................................................................................... 369
Lesson 24: Solving Exponential Equations ............................................................................................ 386
Lesson 25: Geometric Sequences and Exponential Growth and Decay ............................................... 407
Lesson 26: Percent Rate of Change ...................................................................................................... 423
Lesson 27: Modeling with Exponential Functions ................................................................................ 441
Lesson 28: Newton’s Law of Cooling, Revisited ................................................................................... 462
Topic E: Geometric Series and Finance (A-SSE.B.4, F-IF.C.7e, F-IF.C.8b, F-IF.C.9, F-BF.A.1b, F.BF.A.2,
F-LE.B.5) ............................................................................................................................................... 474
Lesson 29: The Mathematics Behind a Structured Savings Plan .......................................................... 476
Lesson 30: Buying a Car ........................................................................................................................ 499
Lesson 31: Credit Cards ........................................................................................................................ 513
Lesson 32: Buying a House ................................................................................................................... 528
Lesson 33: The Million Dollar Problem ................................................................................................. 542
End-of-Module Assessment and Rubric ............................................................................................................ 554
Topics C through E (assessment 1 day, return 1 day, remediation or further applications 4 days) Module 3: © 2015 Great Minds. eureka-math.org
ALG II-M3-TE-1.3.0-08.2015 Exponential and Logarithmic Functions 2
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Module Overview NYS COMMON CORE MATHEMATICS CURRICULUM M3 ALGEBRA II Algebra II • Module 3 Exponential and Logarithmic Functions
OVERVIEW
In this module, students synthesize and generalize what they have learned about a variety of function
families. They extend the domain of exponential functions to the entire real line (N-RN.A.1) and then extend
their work with these functions to include solving exponential equations with logarithms (F-LE.A.4). They use
appropriate tools to explore the effects of transformations on graphs of exponential and logarithmic
functions. They notice that the transformations of a graph of a logarithmic function relate to the logarithmic
properties (F-BF.B.3). Students 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
judgments about the domain over which a model is a good fit. The description of modeling as “the process of
choosing and using mathematics and statistics to analyze empirical situations, to understand them better,
and to make decisions” is at the heart of this module. In particular, through repeated opportunities in
working through the modeling cycle (see page 61 of CCSS-M), students acquire the insight that the same
mathematical or statistical structure can sometimes model seemingly different situations.
This module builds on the work in Algebra I, Modules 3 and 5, where students first modeled situations using
exponential functions and considered which type of function would best model a given real-world situation.
The module also introduces students to the extension standards relating to inverse functions and composition
of functions to further enhance student understanding of logarithms.
Topic E is a culminating project spread out over several lessons where students consider applying their
knowledge to financial literacy. They plan a budget, consider borrowing money to buy a car and a home,
study paying off a credit card balance, and finally, decide how they could accumulate one million dollars. Focus Standards
Extend the properties of exponents to rational exponents.
N-RN.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
1 of rational exponents. For example, we define 53 to be the cube root of 5 because we want
1 3
�53 � =5 1 3
3 to � � 1 3 hold, so �53 � must equal 5. N-RN.A.2 2 Rewrite expressions involving radicals and rational exponents using the properties of
exponents.
2Including expressions where either base or exponent may contain variables Module 3: © 2015 Great Minds. eureka-math.org
ALG II-M3-TE-1.3.0-08.2015 Exponential and Logarithmic Functions 3
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Module Overview NYS COMMON CORE MATHEMATICS CURRICULUM M3 ALGEBRA II Reason quantitatively and use units to solve problems.
N-Q.A.2 3 Define appropriate quantities for the purpose of descriptive modeling. ★ Write expressions in equivalent forms to solve problems.
A-SSE.B.3 4 Choose and produce an equivalent form of an expression to reveal and explain properties of
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the quantity represented by the expression.
c. Use the properties of exponents to transform expressions for exponential functions. For
1 12 example the expression 1.15 can be rewritten as �1.1512 � ≈ 1.01212 to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
A-SSE.B.4 5 Derive the formula for the sum of a finite geometric series (when the common ratio is not
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1), and use the formula to solve problems. For example, calculate mortgage payments. Create equations that describe numbers or relationships.
A-CED.A.1 6 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
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functions. Represent and solve equations and inequalities graphically.
A-REI.D.11 7 Explain why the -coordinates of the points where the graphs of the equations = ()
and = () intersect are the solutions of the equation () = (); find the solutions
approximately, e.g., using technology to graph the functions, make tables of values, or find
successive approximations. Include cases where () and/or () are linear, polynomial,
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rational, absolute value, exponential, and logarithmic functions. Understand the concept of a function and use function notation.
F-IF.A.3 8 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) = (1) = 1, ( + 1) = () + ( − 1) for ≥ 1. 3This standard is assessed in Algebra II by ensuring that some modeling tasks (involving Algebra II content or securely held content
from previous grades and courses) require students to create a quantity of interest in the situation being described (i.e., this is not
provided in the task). For example, in a situation involving periodic phenomena, students might autonomously decide that amplitude
is a key variable in a situation and then choose to work with peak amplitude.
4Tasks have a real-world context. As described in the standard, there is an interplay between the mathematical structure of the
expression and the structure of the situation, such that choosing and producing an equivalent form of the expression reveals
something about the situation. In Algebra II, tasks include exponential expressions with rational or real exponents.
5This standard includes using the summation notation symbol.
6Tasks have a real-world context. In Algebra II, tasks include exponential equations with rational or real exponents, rational functions,
and absolute value functions.
7In Algebra II, tasks may involve any of the function types mentioned in the standard.
8This standard is Supporting Content in Algebra II. This standard should support the Major Content in F-BF.2 for coherence. Module 3: © 2015 Great Minds. eureka-math.org
ALG II-M3-TE-1.3.0-08.2015 Exponential and Logarithmic Functions 4
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Module Overview NYS COMMON CORE MATHEMATICS CURRICULUM M3 ALGEBRA II Interpret functions that arise in applications in terms of the context.
F-IF.B.4 9 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;
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symmetries; end behavior; and periodicity. F-IF.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() gives the number of person-hours
it takes to assemble engines in a factory, then the positive integers would be an
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appropriate domain for the function. F-IF.B.69 Calculate and interpret the average rate of change of a function (presented symbolically or
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as a table) over a specified interval. Estimate the rate of change from a graph. Analyze functions using different representations.
F-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in
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simple cases and using technology for more complicated cases.
e. F-IF.C.8 10 Graph exponential and logarithmic functions, showing intercepts and end behavior, and
trigonometric functions, showing period, midline, and amplitude. 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 = (1.02) , = (0.97) , = (1.01)12 , = (1.2)10, and classify them as representing exponential growth or
decay.
F-IF.C.9 11 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. Build a function that models a relationship between two quantities.
F-BF.A.1 ★ Write a function that describes a relationship between two quantities.
a. 9Tasks Determine an explicit expression, a recursive process, or steps for calculation from a
context. 12 have a real-world context. In Algebra II, tasks may involve polynomial, exponential, logarithmic, and trigonometric functions. include knowing and applying = and = �1 + � . 11In Algebra II, tasks may involve polynomial, exponential, logarithmic, and trigonometric functions.
12Tasks have a real-world context. In Algebra II, tasks may involve linear functions, quadratic functions, and exponential functions.
10Tasks Module 3: © 2015 Great Minds. eureka-math.org
ALG II-M3-TE-1.3.0-08.2015 Exponential and Logarithmic Functions 5
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. NYS COMMON CORE MATHEMATICS CURRICULUM Module Overview M3 ALGEBRA II b. F-BF.A.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.13 Write arithmetic and geometric sequences both recursively and with an explicit formula, use
them to model situations, and translate between the two forms.★ Build new functions from existing functions.
F-BF.B.3 14 Identify the effect on the graph of replacing () by () + , (), (), and ( + )
for specific values of (both positive and negative); find the value of 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. F-BF.B.4 Find inverse functions.
a. Solve an equation of the form () = for a simple function that has an inverse and
write an expression for the inverse. For example, () = 2 3 or
() = ( + 1)/( − 1) for ≠ 1. Construct and compare linear, quadratic, and exponential models and solve problems.
F-LE.A.2 15 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
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these from a table). F-LE.A.4 16 For exponential models, express as a logarithm the solution to = where , , and ★
are numbers and the base is 2, 10, or ; evaluate the logarithm using technology. Interpret expressions for functions in terms of the situation they model.
F-LE.B.5 17 ★ Interpret the parameters in a linear or exponential function in terms of a context. Foundational Standards
Use properties of rational and irrational numbers.
N-RN.B.3 Explain why the sum or product of two rational numbers is rational; that the sum of a
rational number and an irrational number is irrational; and that the product of a nonzero
rational number and an irrational number is irrational. 13Combining functions also includes composition of functions.
Algebra II, tasks may involve polynomial, exponential, logarithmic, and trigonometric functions. Tasks may involve recognizing
even and odd functions.
15In Algebra II, tasks include solving multi-step problems by constructing linear and exponential functions.
16Students learn terminology that logarithm without a base specified is base 10 and that natural logarithm always refers to base e.
17Tasks have a real-world context. In Algebra II, tasks include exponential functions with domains not in the integers.
14In Module 3: © 2015 Great Minds. eureka-math.org
ALG II-M3-TE-1.3.0-08.2015 Exponential and Logarithmic Functions 6
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Module Overview NYS COMMON CORE MATHEMATICS CURRICULUM M3 ALGEBRA II Interpret the structure of expressions.
A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see 4 − 4
as ( 2 )2 − ( 2 )2, thus recognizing it as a difference of squares that can be factored as
( 2 − 2 )( 2 + 2 ). Create equations that describe numbers or relationships.
A-CED.A.2 Create equations in two or more variables to represent relationships between quantities;
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graph equations on coordinate axes with labels and scales. A-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving
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equations. For example, rearrange Ohm’s law = to highlight resistance . Represent and solve equations and inequalities graphically.
A-REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions
plotted in the coordinate plane, often forming a curve (which could be a line). Understand the concept of a function and use function notation.
F-IF.A.1 Understan...

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