Linear Functions and Applications

Students review and extend the Algebra 1 skills of graphing, manipulating, and describing solutions in order to deepen their understanding of modeling situations using linear functions.

Unit Summary

In Unit 1, Linear Functions and Applications, students review and extend the Algebra 1 skills of graphing, manipulating, and describing solutions to linear functions to deepen their understanding of modeling situations using linear functions. In this unit, students review concepts, such as using multiple representations, inverse, constraints, and systems, that are essential for studying polynomial, rational, exponential, and logarithmic functions, and trigonometric functions in later units through linear functions, a familiar and basic parent function.

Unit 1 begins with students translating between representations of linear functions to identify the strengths of each representation and to highlight features of linear functions. Inverse of linear functions is studied through contextual situations, to ensure that students grasp the symmetry between a function and its inverse, as well as connect the meaning of each variable and its role (dependent or independent) in the concept of inverse. Topics in this unit continue through systems—a concept that is very familiar to students. The focus in this section of the unit is again contextual as well as procedural to allow students to be fully fluent in solving systems algebraically, graphically, with three variables, and with absolute value functions, an application of a linear function. Students review piecewise functions through a linear lens to bolster their facility with constraints and analysis of functions in preparation for nonlinear piecewise functions. Students who are taking the pre-calculus portion of this course will continue on to compose functions within and outside of context to model and identify solutions to real-life scenarios.

As Algebra 2 progresses, students will draw on the concepts from this unit to find the inverse of functions, restrict domains to allow a function to be invertible, operate with various functions, model with functions, identify solutions to systems of functions graphically and algebraically, and analyze functions for their value and behavior. Mastering the skills in this unit will allow for students to have a solid framework to build upon when studying other functions. 

Assessment

This assessment accompanies Unit 1 and should be given on the suggested assessment day or after completing the unit.

Unit Prep

Intellectual Prep

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Internalization of Standards via the Unit Assessment:

  • Take unit assessment. Annotate for: 
    • Standards that each question aligns to
    • Purpose of each question: spiral, foundational, mastery, developing
    • Strategies and representations used in daily lessons
    • Relationship to Essential Understandings of unit 
    • Lesson(s) that assessment points to

Internalization of Trajectory of Unit:

  • Read and annotate “Unit Summary."
  • Notice the progression of concepts through the unit using “Unit at a Glance.”
  • Do all target tasks. Annotate the target tasks for: 
    • Essential understandings
    • Connection to assessment questions 

Essential Understandings

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  • A linear relationship can be represented contextually, graphically, in a table of values, and algebraically. By manipulating a linear function algebraically, including finding the inverse of a linear function, relationships of the linear function are highlighted. 
  • Contextual situations often have limitations that need to be represented by constraints on the domain or the range of the equation to accurately represent the situation. 
  • Systems of functions are used to show when two or more functions have the same value in contextual and non-contextual situations. 
  • Piecewise functions model situations where more than one function is required, but those functions happen in sequence rather than simultaneously as in systems. Piecewise functions will, by definition, require constraints to describe when one function starts and another begins. 
  • Functions can be composed where an output from one function is used as the input of another function to describe situations where operations are nested to describe relationships.

Vocabulary

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domain restriction constraint inverse functions
inverse notation $${f^{-1}(x)}$$ invertible functions extraneous solution(s)

Unit Materials, Representations and Tools

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  • Graphical, algebraic, contextual, and tabular representations
  • Desmos or graphing calculator to visualize graphs 

Lesson Map

Topic A: Features of Linear Functions

Topic B: Systems of Functions and Constraints

Topic C: Piecewise Functions

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

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Building Functions
  • F.BF.A.1.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.

  • F.BF.A.1.C — Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

  • F.BF.B.4.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) =2 x3 or f(x) = (x+1)/(x—1) for x ? 1.

  • F.BF.B.4.B — Verify by composition that one function is the inverse of another.

  • F.BF.B.4.C — Read values of an inverse function from a graph or a table, given that the function has an inverse.

Creating Equations
  • A.CED.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.

  • A.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

  • A.CED.A.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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

  • A.CED.A.4 — Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.

Interpreting Functions
  • 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.

  • 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; relative maximums and minimums; symmetries; end behavior; and periodicity. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

  • 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(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

  • F.IF.C.7.B — Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

  • F.IF.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.

Reasoning with Equations and Inequalities
  • A.REI.C.5 — Prove that, given a system of two 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.

  • 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.

  • 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).

  • A.REI.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. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

  • A.REI.D.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 as the intersection of the corresponding half-planes.

Seeing Structure in Expressions
  • A.SSE.A.1.A — Interpret parts of an expression, such as terms, factors, and coefficients.

Foundational Standards

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Arithmetic with Polynomials and Rational Expressions
  • A.APR.A.1

Creating Equations
  • A.CED.A.1

  • A.CED.A.2

  • A.CED.A.4

Expressions and Equations
  • 7.EE.B.4.B

  • 8.EE.C.8

Functions
  • 8.F.A.1

  • 8.F.A.2

  • 8.F.B.4

Interpreting Functions
  • F.IF.A.2

  • F.IF.B.4

  • F.IF.C.7.A

  • F.IF.C.9

Reasoning with Equations and Inequalities
  • A.REI.A.1

  • A.REI.B.3

  • A.REI.C.5

  • A.REI.C.6

  • A.REI.D.10

Future Standards

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Building Functions
  • F.BF.A.1

  • F.BF.B.4

Interpreting Functions
  • F.IF.B.4

  • F.IF.B.5

  • F.IF.C.8

  • F.IF.C.9

Reasoning with Equations and Inequalities
  • A.REI.A.2

Standards for Mathematical Practice

  • CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

  • CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

  • CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

  • CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

  • CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

  • CCSS.MATH.PRACTICE.MP6 — Attend to precision.

  • CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

  • CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.