What is a logarithm? The answer is more simple than meets the eye. Essentially, a logarithm is the “backwards” version of an exponent and we will explore it that way. For many reasons we’ll discuss on class, there is a variety of connections between logarithms and exponential functions. Find some helpful videos below to build meaning around what a logarithm is, the properties it holds and how to use them to solve exponent problems.
F.BF.B.4  Find the inverse functions. 
F.BF.B.5  Understand the inverse relationship between exponents and logarithms and use this relationship to solve problem involving logarithms and exponents. 
F.IF.C.7.e  Graph exponential and logarithmic functions, showing intercepts and end behavior. 
F.LE.A.4  For exponential models, express as a logarithm the solution to abct= d where a,c, and d are numbers and the base b is 2, 10 or e; evaluate the logarithm using technology. 
This unit will be focusing on exponential functions. We will start by reviewing exponent rules, looking at rational exponents (exponents that have fraction exponents), then we will use function notation to make data tables and graph those. There are MANY applications to exponents, many used in everyday life. We will explore how to create exponential expressions based on contexts to answer real world questions.
F.IF.A.2  Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. 
A.SSE.B.3.c  Use the properties of exponents to transform expressions for exponential functions.

F.IF.C.8.b  Use the properties of exponents to interpret expressions for exponential functions. 
F.BF.A.1  Write a function that describes a relationship between two quantities. 
Day 1: Exponent Laws Discovery Activity
Day 2: Drug Filtering Worksheet
Day 3/4: Bees Gone Crazy Task – Exponential Functions
Day 4: Who is right, who is wrong (answers)
]]>Standard  Language 
A.CED.2  Create equations in two or more variables to present relationships between quantities; graph equations on coordinate axes with labels and scales. 
A.REI.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). 
A.REI.12  Graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables ad the intersection of the corresponding halfplanes. 
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. 
S.ID.1  Represent data with plots on the real number line (dot plots, histograms, and box plots). 
S.ID.2  Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. 
S.ID.3  Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). 
S.ID.6c  Fit a linear function for a scatter plot that suggests a linear association. 
S.ID.7  Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. 
Data can be messy, but there are patterns everywhere! When we look at a scatter plot, all we see is a bunch of dots…can we make any conclusions from these things. Sometimes, the answer is clearly YES…we can use data to create equations (sometimes just by drawing a graphical picture) and we can use that information to make claims about what is likely to happen in the future based on this trend. We will extend our understanding of functions by collecting a lot of inputs and noticing they have a common additive growth over time. Looking at patterns, we will find that complicated things can be made more simple by considering the average change that happens over time. Math is incredibly powerful at using patterns to predict the future.
Lorum Ipsum
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Standard 
Language 
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, the 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). 
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. 
F.IF.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. 
F.IF.5 
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. 
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. 
F.IF.9 
Compare the properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). 
F.BF.1b 
Combine standard function types using arithmetic operations. 
Functions take an input (we call this the domain)…the function “does something” and we get an output (we call this the range). Anything that has a predictable output is called a function, if it doesn’t the input and output are relational, but NOT a function. Understanding what functions are, how they behave and what is considered possible inputs or possible outputs are essential to mathematical thinking. Essentially, we are generalizing possibilities for what could be…and making formal mathematical statements about these possibilities. To make things more complex, we should be able to consider multiple functions and combine them together…the benefit of this is that similar operations only need to be completed once and matters can be made simple. The purpose of learning information in this unit is to understand that functions can make our life more simple, learn how to use functions, learn the “possibilities” of different functions and combine functions to make very complicated things more simple.
Lorum Ipsum
]]>Standard  Language 
A.SSE.3c  Use the properties of exponents to transform expressions for exponential functions. 
F.BF.4  Find the inverse functions. 
F.BF.5  Understand the inverse relationship between exponents and logarithms and use this relationship to solve problem involving logarithms and exponents. 
F.IF.7  Graph exponential and logarithmic functions, showing intercepts and end behavior. 
F.IF.8  Use the properties of exponents to interpret expressions for exponential functions. 
F.LE.4  For exponential models, express as a logarithm the solution to abct= d where a,c, and d are numbers and the base b is 2, 10 or e; evaluate the logarithm using technology. 
F.BF.1  Write a function that describes a relationship between to quantities. 
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 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 function from their graphs and algebraic expressions for them. 
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. 
Exponential functions are the most powerful in the world and they are used in every day life all of the time. Loans and credit cards…thank exponential functions, nuclear weapons…again thank exponential functions. As we explore exponents, we will use real world situations to model using exponential functions, then we will learn how to think of exponential functions “backwards” by considering the inverse of exponents. By using exponential functions and their opposites (which are called logarithms), we can answer some very powerful questions like, “How many hours before the Ebola virus infects all of humanity?” We will use graphs, tables, functions and models to explore exponential and logarithmic functions.
Standard  Language 
A.CED.2  Create equations in two or more variables to represent relationships between quantities; graph equation on coordinate axes with label and scales. 
A.CED.3  Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. 
A.REI.5  Prove that, given a system of two equations in two variables, replacing on equation by the sum of that equation and a multiple of the other produces a system with the same solutions. 
A.REI.6  Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 
A.REI.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). 
A.REI.11  Explain why the xcoordinates of the points where the graphs of the equation 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, rational, absolute value, exponential, and logarithmic functions. 
A.REI.12  Graph the solutions to a linear inequality in two variables as a halfplane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding halfplanes. 
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. 
Data can be messy, but there are patterns everywhere! When we look at a scatter plot, all we see is a bunch of dots…can we make any conclusions from these things. Sometimes, the answer is clearly YES…we can use data to create equations (sometimes just by drawing a graphical picture) and we can use that information to make claims about what is likely to happen in the future based on this trend. We will extend our understanding of functions by collecting a lot of inputs and noticing they have a common additive growth over time. Looking at patterns, we will find that complicated things can be made more simple by considering the average change that happens over time. Math is incredibly powerful at using patterns to predict the future.
Along with all of the other resources on this page, the following document can support in your relearning of the Unit 2 materials for the Unit 2 Test Retake. The last day to retake the Unit 2 Assessment is on December 13th, 2019.
Algebra 2 Unit 2 Practice work for Retake
Standard 
Language 
A.CED.1 
Create equations and inequalities in one variable and use them to solve problems. 
A.CED.3 
Represent constraints by equations or inequalities and by systems of equations and /or inequalities, and interpret solutions as viable or nonviable options in a modeling context. 
A.CED.4 
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. 
A.REI.1 
Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 
A.REI.3 
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 
Our world has balance, well it should, and we learn about equality through the balance of equal and opposites. Learning to solve an equation or an inequality is understanding balance and equal opposites. We can use equations and inequalities to solve for things we don’t know yet based on the simple premise that the two sides are balanced. We will expand and generalize on some common truths we will learn throughout the unit.
Standard 
Language 
A.SSE.1 
Interpret expressions that represent a quantity in terms of its context. 
A.SSE.1.a 
Interpret parts of an expression, such as terms, factors, and coefficients. 
A.SSE.2 
Use the structure of an expression to identify ways to rewrite it. 
A.APR.1 
Understand that polynomials form a system analogous to the integers, namely they are closed under the operation of addition, subtraction, and multiplication; Add, subtract, multiply polynomials. 
N.RN.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. 
Numbers tell a story about the world they are placed into. We need a common language to talk about these numbers and symbols. We need to build a common language and understand what basic expectations can be known when a math sentence (expression) is displayed. Expressions and numbers (rational & irrational) interact with one another in specific ways, mathematicians hold specific expectations for how these sets of numbers and symbols interact with one another, and generalizations can be made to simplify how much we need to think about these relationships. While seemingly tricky at first, these numbers and symbols always behave in the same ways…we need to understand those ways of interacting.
Standard 
Language 
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, the 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). 
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. 
F.IF.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. 
F.IF.5 
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. 
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. 
F.BF.1 
Combine standard function types using arithmetic operations. 
Functions take an input (we call this the domain)…the function “does something” and we get an output (we call this the range). Anything that has a predictable output is called a function, if it doesn’t the input and output are relational, but NOT a function. Understanding what functions are, how they behave and what is considered possible inputs or possible outputs are essential to mathematical thinking. Essentially, we are generalizing possibilities for what could be…and making formal mathematical statements about these possibilities. To make things more complex, we should be able to consider multiple functions and combine them together…the benefit of this is that similar operations only need to be completed once and matters can be made simple. The purpose of learning information in this unit is to understand that functions can make our life more simple, learn how to use functions, learn the “possibilities” of different functions and combine functions to make very complicated things more simple.
I look forward to an exciting year with you.
Mr. Germanis