Algebra 1 (& GT)
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.
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.
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.09).
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: Linear vs. Exponential Growth
 Review/Rewind: Linear and Exponential Models
F.IF.A.3
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
 Check for Understanding: Explicit Formulas for Geometric Sequences  Sequences Word Problems
 Review/Rewind: Converting Recursive & Explicit Forms of Arithmetic Sequences
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.
 Check for Understanding: Modeling with Composite Functions  Equations and Inequalities Word Problems
 Review/Rewind: Constructing Exponential Models
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: Sequences Word Problems
 Review/Rewind: Converting Recursive & Explicit Forms of Arithmetic Sequences
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: Converting Arithmetic Sequences  Converting Geometric Sequences
 Review/Rewind: Constructing Exponential Models
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 of Exponentials and Polynomials
 Review/Rewind: Comparing Growth of Exponential and Quadratic Models
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.
 Check for Understanding: PointSlope Form  SlopeIntercept & Standard Form
 Review/Rewind: Graph from SlopeIntercept Equation  Graph Basic Exponential Functions
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).
 Check for Understanding: OneStep Addition and Subtraction Equations  OneStep Multiplication and Division Equations  TwoStep Equations  Equations with Parentheses  Equations with Variables on Both Sides  OneStep Inequalities  Multistep Linear Inequalities  Compound Inequalities
 Review/Rewind: MultiStep 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).
 Check for Understanding: Manipulating Formulas  Linear Equations with Unknown Coefficients
 Review/Rewind: Linear Models
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).
 Check for Understanding: Graph Interpretation Word Problems  Positive and Negative Intervals
 Review/Rewind: Interpreting Linear Tables  Interpreting Linear Graphs  End Behavior of Algebraic Models
F.IF.B.5
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
 Check for Understanding: Domain and Range from Graph  Domain of Advanced Functions
 Review/Rewind: Domain Restrictions in Linear Functions
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: Finding Average Rate of Change
 Review/Rewind: Rate of Change
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).
 Check for Understanding: Even and Odd Functions  Transforming Functions
 Review/Rewind: Horizontal Transformations of Linear Equations
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 Word Problems  Interpreting Change in Exponential Models
 Review/Rewind: Linear Functions Word Problems 1
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
 Review/Rewind: Comparing Functions
Part II: Modeling Data with Linear and Exponential Functions
Interpret linear models.
S.ID.C.9
Distinguish between correlation and causation.
 Check for Understanding: Types of Statistical Studies
 Review/Rewind: Correlation and Causality
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)
 Check for Understanding: Constructing Scatter Plots  Describing Trends in Scatter Plots
 Review/Rewind: Correlation and Causality
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)
 Check for Understanding: Eyeballing the Line of Best Fit
 Review/Rewind: Fitting a Line to Data
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.
 Check for Understanding: Estimating Slope of Line of Best Fit  Fitting Quadratic and Exponential Functions to Scatter Plots  Introduction to Residuals
 Review/Rewind: Using a Linear Regression Line to Solve Problems  Constructing an Exponential Model
Interpret linear models.
S.ID.C.8
Compute (using technology) and interpret the correlation coefficient of a linear fit.
 Review/Rewind: Correlation Coefficient Intuition
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).
 Check for Understanding: Estimating Slope of Line of Best Fit
 Review/Rewind: Line of Best Fit
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.09).
 Check for Understanding: Equivalent Systems of Equations
 Review/Rewind: Identifying Equivalent Systems of Equations
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).
 Check for Understanding: Systems of Equations with Graphing  Systems of Equations Word Problems
 Review/Rewind: Solving a System of Linear Equations
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).
 Check for Understanding: Solving Equations Graphically  Interpreting Equations Graphically
 Review/Rewind: Finding the Intersection of Two Linear Equations  Graphing a System of Linear & Exponential Equations
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.
 Check for Understanding: TwoVariable Inequalities Word Problems
 Review/Rewind: Constraining Solutions of Systems of Inequalities
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).
 Check for Understanding: Graphs of TwoVariable Inequalities  TwoVariable Inequalities from their Graphs
 Review/Rewind: Graphing TwoVariable Inequalities
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! MATH is FUN is a free online site that works on any computer or tablet without requiring any downloads. It explains a concept and then has an online practice quiz that gives students feedback. Click below to check out some of the Unit 2 concepts... 
More 4 U
What are functions and how can students represent functions in different ways? View this classroom video to learn more. Source: The Teaching Channel
