Algebra 1
Unit 2: Linear and Exponential Relationships
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 functions.
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What will my child learn?
Students will:
Part I: Representing Linear and Exponential Functions
Construct and compare linear and exponential models and solve problems.
F.LE.A.1
Distinguish between situations that can be modeled with linear functions and with exponential functions (SAT® Content  PSDA.04).
a. Prove that linear functions grow by equal differences over equal intervals; and that exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity change at a constant rate per unit interval relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
Understand the concept of a function and use function notation.
F.IF.A.3
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
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. Determine an explicit expression, a recursive process, or steps for calculation from a context.
Build a function that models a relationship between two quantities.
F.BF.A.2
Write geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
Construct and compare linear and exponential models and solve problems.
F.LE.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) (SAT® Content  PAM.01).
F.LE.A.3
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasingly linearly or exponentially.
Analyze linear and exponential functions using different representations.
F.IF.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 (SAT® Content  PAM.13).
a. Graph linear and exponential functions and show intercepts, maxima, and minima.
b. Graph exponential functions, showing intercepts and end behavior.
Solve equations and inequalities in one variable
A.REI.B.3
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters (SAT® Content  HOA.01  HOA.02).
Create equations that describe numbers or relationships.
A.CED.A.4
Rearrange linear formulas to highlight a quantity of interest, using the same reasoning as in solving equations (SAT® Content  PAM.11).
Interpret linear and exponential functions that arise in applications in terms of a context.
F.IF.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, and end behavior (SAT® Content  HOA.06  PSDA.06).
F.IF.B.5
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
Interpret linear and exponential functions that arise in applications in terms of a context.
F.IF.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.
Build new functions from existing functions.
F.BF.B.3
Identify the effect of 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 values of k given the graphs. Experiment with cases that illustrate an explanation of the effects on the graph using technology. Focus on vertical translations of graphs of linear and exponential functions. Relate the vertical translation of a linear function to its yintercept (SAT® Content  HOA.06).
Interpret expressions for functions in terms of the situations they model.
F.LE.B.5
Interpret the parameters in a linear or exponential function in terms of a context. Limit exponential functions to those of the form f(x) = b^x + k
Analyze linear and exponential functions using different representations.
F.IF.C.9
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions) (SAT® Content  HOA.06).
Part II: Modeling Data with Linear and Exponential Functions
Interpret linear models.
S.ID.C.9
Distinguish between correlation and causation.
Investigate patterns of association in bivariate data.
8.SP.A.1
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. (Grades 7 and 8 only)
8.SP.A.2
Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. (Grades 7 and 8 only)
Summarize, represent, and interpret data on quantitative variables.
S.ID.B.6
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related (SAT® Content  PSDA.05).
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models.
b. Informally assess the fit of a linear function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.
Interpret linear models.
S.ID.C.8
Compute (using technology) and interpret the correlation coefficient of a linear fit.
S.ID.C.7
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data (SAT® Content  HOA.04).
Part III. Systems of Equations and Inequalities
Solve systems of equations.
A.REI.C.5
Prove that, given a system of equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions (SAT® Content  HOA.08).
A.REI.C.6
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. (Note: Extend this standard to include three equations and three unknowns.) (SAT® Content  HOA.07  HOA.08).
Represent and solve equations and inequalities graphically.
A.REI.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 f(x) and/or g(x) are linear and exponential functions (SAT® Content  HOA.08  HOA.09).
Create equations that describe numbers or relationships.
A.CED.A.3
Represent constraints by linear equations or inequalities, and by systems of linear equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
Represent and solve equations and inequalities graphically.
A.REI.D.12
Graph the solutions to a linear inequality in two variables as a halfplane (excluding the boundary in the case of strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding halfplanes (SAT® Content  HOA.09).
Students will:
Part I: Representing Linear and Exponential Functions
Construct and compare linear and exponential models and solve problems.
F.LE.A.1
Distinguish between situations that can be modeled with linear functions and with exponential functions (SAT® Content  PSDA.04).
a. Prove that linear functions grow by equal differences over equal intervals; and that exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity change at a constant rate per unit interval relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
 Check for Understanding: Understanding Linear and Exponential Models
 Enrichment Tasks: Boiling Water  Choosing an Appropriate Growth Model
Understand the concept of a function and use function notation.
F.IF.A.3
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
 Background Info.
 Check for Understanding: Defining Sequences as Functions  Modeling with Sequences
 Enrichment Task: Snake on a Plane
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. Determine an explicit expression, a recursive process, or steps for calculation from a context.
 Background Info.
 Check for Understanding: Modeling with Composite Functions  Modeling with OneVariable Equations and Inequalities
 Enrichment Tasks: Summer Intern  Skeleton Tower
Build a function that models a relationship between two quantities.
F.BF.A.2
Write geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
 Check for Understanding: Modeling with Sequences
 Enrichment Tasks: The Random Walk  Cell Phones
Construct and compare linear and exponential models and solve problems.
F.LE.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) (SAT® Content  PAM.01).
 Check for Understanding: Constructing Linear and Exponential Functions
 Enrichment Tasks: Rumors  Algae Blooms
F.LE.A.3
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasingly linearly or exponentially.
 Check for Understanding: Comparing Growth Rates of Exponentials and Polynomials
 Enrichment Tasks: Exponential Growth Versus Polynomial Growth  Exponential Growth Versus Linear Growth I
Analyze linear and exponential functions using different representations.
F.IF.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 (SAT® Content  PAM.13).
a. Graph linear and exponential functions and show intercepts, maxima, and minima.
b. Graph exponential functions, showing intercepts and end behavior.
 Background Info.
 Check for Understanding: Converting Between PointSlope and SlopeIntercept Form  Graphing Parabolas in All Forms
 Enrichment Tasks: Identifying Graphs of Functions  Modeling London's Population
Solve equations and inequalities in one variable
A.REI.B.3
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters (SAT® Content  HOA.01  HOA.02).
 Background Info.
 Check for Understanding: OneStep Equations with Addition and Subtraction  OneStep Equations with Multiplication and Division  TwoStep Equations  Multistep Equations with Distribution  Equations with Variables on Both Sides  OneStep Inequalities  Multistep Linear Inequalities  Compound Inequalities
 Enrichment Tasks: Reasoning with Linear Inequalities
Create equations that describe numbers or relationships.
A.CED.A.4
Rearrange linear formulas to highlight a quantity of interest, using the same reasoning as in solving equations (SAT® Content  PAM.11).
 Background Info.
 Check for Understanding: Manipulating Formulas  Solving Equations in terms of a Variable
 Enrichment Task: Equations and Formulas
Interpret linear and exponential functions that arise in applications in terms of a context.
F.IF.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, and end behavior (SAT® Content  HOA.06  PSDA.06).
 Background Info.
 Check for Understanding: Interpreting Features of Functions  Positive and Negative Parts of Functions
 Enrichment Tasks: Telling a Story with Graphs  Model Airplane Acrobatics
F.IF.B.5
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
 Background Info.
 Check for Understanding: Domain and Range from Graph  Domain of a Function
 Enrichment Tasks: Average Cost  The Canoe Trip, Variation 2
Interpret linear and exponential functions that arise in applications in terms of a context.
F.IF.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.
 Check for Understanding: Average Rate of Change
 Enrichment Tasks: Temperature Change  High School Gym
Build new functions from existing functions.
F.BF.B.3
Identify the effect of 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 values of k given the graphs. Experiment with cases that illustrate an explanation of the effects on the graph using technology. Focus on vertical translations of graphs of linear and exponential functions. Relate the vertical translation of a linear function to its yintercept (SAT® Content  HOA.06).
 Background Info.
 Check for Understanding: Even and Odd Functions  Shifting and Reflecting Functions
 Enrichment Tasks: Medieval Archer  Identifying Even and Odd Functions
Interpret expressions for functions in terms of the situations they model.
F.LE.B.5
Interpret the parameters in a linear or exponential function in terms of a context. Limit exponential functions to those of the form f(x) = b^x + k
 Check for Understanding: Comparing Linear Functions Applications  Modeling with Exponential Functions
 Enrichment Tasks: Carbon 14 Dating in Practice I  Illegal Fish
Analyze linear and exponential functions using different representations.
F.IF.C.9
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions) (SAT® Content  HOA.06).
 Check for Understanding: Comparing Features of Functions
 Enrichment Task: Throwing Baseballs
Part II: Modeling Data with Linear and Exponential Functions
Interpret linear models.
S.ID.C.9
Distinguish between correlation and causation.
 Background Info.
 Check for Understanding: Types of Statistical Studies
 Enrichment Tasks: Golf and Divorce  High Blood Pressure
Investigate patterns of association in bivariate data.
8.SP.A.1
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. (Grades 7 and 8 only)
 Background Info. & Guided Practice
 Check for Understanding: Constructing Scatter Plots  Interpreting Scatter Plots
 Enrichment Tasks: Animal Brains  Hand Span and Height
8.SP.A.2
Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. (Grades 7 and 8 only)
 Background Info.
 Check for Understanding: Estimating the Line of Best Fit
 Enrichment Tasks: Bird's Eggs  Laptop Battery Charge
Summarize, represent, and interpret data on quantitative variables.
S.ID.B.6
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related (SAT® Content  PSDA.05).
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models.
b. Informally assess the fit of a linear function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.
 Background Info.
 Check for Understanding: Analyzing Residual  Linear Models of Bivariate Data
 Enrichment Tasks: Coffee and Crime  Used Subaru Foresters I
Interpret linear models.
S.ID.C.8
Compute (using technology) and interpret the correlation coefficient of a linear fit.
 Background Info.
Check for Understanding: Skill coming soon
 Enrichment Task: Coffee Crime
S.ID.C.7
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data (SAT® Content  HOA.04).
 Background Info.
 Check for Understanding: Linear Models of Bivariate Data
 Enrichment Tasks: Texting and Grade II  Used Subaru Foresters II
Part III. Systems of Equations and Inequalities
Solve systems of equations.
A.REI.C.5
Prove that, given a system of equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions (SAT® Content  HOA.08).
 Background Info. & Guided Practice
 Check for Understanding: Graphically Understanding Solution Methods to Systems of Equations
 Enrichment Task: Solving Two Equations in Two Unknowns
A.REI.C.6
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. (Note: Extend this standard to include three equations and three unknowns.) (SAT® Content  HOA.07  HOA.08).
 Background Info.
 Check for Understanding: Graphing Systems of Equations  Systems of Equations Word Problems
 Enrichment Tasks: Find A System  Pairs of Whole Numbers
Represent and solve equations and inequalities graphically.
A.REI.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 f(x) and/or g(x) are linear and exponential functions (SAT® Content  HOA.08  HOA.09).
 Background Info.
 Check for Understanding: Intersecting Functions  Systems of Nonlinear Equations
 Enrichment Task: Ideal Gas Law)
Create equations that describe numbers or relationships.
A.CED.A.3
Represent constraints by linear equations or inequalities, and by systems of linear equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
 Background Info.
 Check for Understanding: Modeling constraints with TwoVariable Inequalities
 Enrichment Tasks: Fishing Adventures 3  Writing Constraints
Represent and solve equations and inequalities graphically.
A.REI.D.12
Graph the solutions to a linear inequality in two variables as a halfplane (excluding the boundary in the case of strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding halfplanes (SAT® Content  HOA.09).
 Background Info.
 Check for Understanding: Graphing and Solving Linear Inequalities  Graphs of Inequalities in Two Variables
 Enrichment Tasks: Solution Sets  Fishing Adventures 3
What are some signs of student mastery?
Representing Linear and Exponential Functions

Tools & Technology
Desmos is a free online graphing calculator that works on any computer or tablet without requiring any downloads. A FREE Desmos iPad app is available too! 
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What are functions and how can students represent functions in different ways? View this classroom video to learn more. Source: The Teaching Channel
