Functions, Graphs and Features

Students are introduced to the main features of functions that they will learn throughout the year, providing students with a conceptual understanding of how functions are used to model various situations.

Unit Summary

In Unit 1: Functions, Graphs, and Features, students are introduced to all of the main features of functions they will learn throughout the year through basic graphical modeling of contextual situations. Students will learn function notation and use this to analyze and express features of functions represented in graphs and contextually. Students will use the tools of domain and range, rates of change, intercepts, and where a function is changing to describe contextual situations. 

Unit 1 begins with a review of how to sketch a function from a contextual situation and then introduces function notation and features of functions, such as domain and range, intercepts, and rate of change. Students will learn that to describe the features of functions, they will need to identify the intervals over which the function is behaving in a certain manner. These intervals are described using inequalities in Algebra 1. As the unit progresses, students apply these features of functions to new parent functions—quadratic and exponential—as well as to systems previously learned in 8th grade, and model and analyze situations in these new parent functions. Students will be expected to translate features between the representations of graphs, tables, situations, and, in cases of some linear functions, equations. 

As Algebra 1 progresses, students will apply this introduction to analyzing functions throughout the year, from Statistics to each of the functions studied more in-depth. Students will also expand their understanding of systems of functions beyond just linear systems to include thinking about systems of linear and quadratic equations, linear and exponential equations, etc. Skills learned in this unit will be revisited throughout Algebra 1, in Algebra 2, and in AP Calculus. This unit will provide students with a solid conceptual understanding of how functions can be used to model and interpret functions.

Unit Prep

Essential Understandings

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  • Functions can be used to model everyday situations. Features of functions such as domain, range, rate of change, intercepts, points of change, and shape of the graph can all be used to provide details about what is happening, how it is happening, and when it is happening. 
  • Features of a function can be represented in tables, graphs, equations, and descriptions. Each of these representations will help you to “see” features of the function in a different way. 
  • To effectively model a situation, the quantities represented in the situation need to be adequately represented in the model. Identifying variables, choosing a scale, and keeping track of units during the solution of a problem are important to the function having meaning.

Vocabulary

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function independent variable dependent variable
interval function notation "f of x" x-intercept/y-intercept
system of functions parent function domain
range average rate of change/rate of change quadratic functions
parabola exponential functions solution

Unit Materials, Representations and Tools

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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 

Assessment

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

Lesson Map

Topic A: Features of Functions

Topic B: Nonlinear Functions

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

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Creating Equations
  • 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.

High School — Number and Quantity
  • N.Q.A.1 — Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

  • N.Q.A.2 — Define appropriate quantities for the purpose of descriptive modeling.

  • N.Q.A.3 — Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Interpreting Functions
  • F.IF.A.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, then 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.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.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. 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.A — Graph linear and quadratic functions and show intercepts, maxima, and minima.

  • F.IF.C.7.E — Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

  • 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.

Linear, Quadratic, and Exponential Models
  • F.LE.A.3 — Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Reasoning with Equations and Inequalities
  • 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.

Foundational Standards

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Expressions and Equations
  • 8.EE.B.5

  • 8.EE.C.8

Functions
  • 8.F.A.1

  • 8.F.A.2

  • 8.F.A.3

  • 8.F.B.4

  • 8.F.B.5

Future Standards

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Creating Equations
  • A.CED.A.1

  • A.CED.A.3

  • A.CED.A.4

Interpreting Categorical and Quantitative Data
  • HSS-ID.C.7

Linear, Quadratic, and Exponential Models
  • F.LE.A.1

Reasoning with Equations and Inequalities
  • A.REI.C.5

Seeing Structure in Expressions
  • A.SSE.A.1

  • A.SSE.B.3

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.