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.

Math

Unit 1

9th Grade

Unit Summary

In Unit 1, 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.

Pacing: 13 instructional days (11 lessons, 1 flex day, 1 assessment day)

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Assessment

The following assessments accompany Unit 1.

Post-Unit

Use the resources below to assess student understanding of the unit content and action plan for future units.

Unit Prep

Intellectual Prep

Suggestions for how to prepare to teach this unit

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

The central mathematical concepts that students will come to understand in this unit

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

Terms and notation that students learn or use in the unit

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

Materials

The materials, representations, and tools teachers and students will need for this unit

Helpful to create graphs (if you have a Mac): Omni Graph Sketcher (free)
Desmos

Lesson Map

Topic A: Features of Functions

Lesson

Model a contextual situation graphically using appropriate scales and features.

Standards

F.IF.B.4F.IF.B.6N.Q.A.1N.Q.A.2N.Q.A.3

Lesson

Define functions and evaluate points in tables, graphs, and contextual situations using function notation.

Standards

F.IF.A.1F.IF.A.2

Lesson

Identify features of functions, including x-intercept and y-intercept, in context. Evaluate function notation in context.

Standards

F.IF.A.2F.IF.C.9

Lesson

Identify the domain and range through contextual situations, and explore domain and range on a graph. Represent domain and range with inequalities.

Standards

F.IF.A.1F.IF.B.5

Lesson

Calculate and interpret the rate of change from two points on a graph, in a situation, or in function notation.

Standards

F.IF.A.2F.IF.B.4F.IF.B.6

Lesson

Describe and sketch functions using the features of domain and range, intercepts, function behavior, and the value of the function.

Standards

F.IF.B.4

Lesson

Analyze the key features of a contextual situation and model these graphically.

Standards

F.IF.B.4F.IF.B.5F.IF.B.6N.Q.A.2

Topic B: Nonlinear Functions

Lesson

Draw quadratic functions represented contextually. Identify key features of the graph and relate to context.

Standards

A.CED.A.2F.IF.B.5F.IF.C.7.AF.LE.A.3

Lesson

Sketch an exponential function that represents a situation. Identify key features of the graph and relate to context.

Standards

A.CED.A.2F.IF.B.5F.IF.C.7.EF.LE.A.3

Lesson

Draw a graph to represent a system of functions. Identify the solution to a system represented graphically and in context.

Standards

A.REI.D.11F.IF.A.2F.IF.B.5

Lesson

Analyze functions and identify parent functions of graphs. Identify variables of a situation and the scale of the associated graph. Represent a situation in a graph, table, and description.

Standards

F.IF.B.4

Common Core Standards

Key

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

The content standards covered in this unit

Creating Equations

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

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.
High School — Number and Quantity

N.Q.A.2 — Define appropriate quantities for the purpose of descriptive modeling.
High School — Number and Quantity

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

Interpreting Functions

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).
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.
Interpreting Functions

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.
Interpreting Functions

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.
Interpreting Functions

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.
Interpreting Functions

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. 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.
Interpreting Functions

F.IF.C.7.A — Graph linear and quadratic functions and show intercepts, maxima, and minima.
Interpreting Functions

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

Linear, Quadratic, and Exponential Models

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

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

Standards covered in previous units or grades that are important background for the current unit

Expressions and Equations

Expressions and Equations

8.EE.B.5 — Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.
Expressions and Equations

8.EE.C.8 — Analyze and solve pairs of simultaneous linear equations.

Functions

Functions

8.F.A.1 — Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8.
Functions

8.F.A.2 — Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
Functions

8.F.A.3 — Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
Functions

8.F.B.4 — Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
Functions

8.F.B.5 — Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Future Standards

Standards in future grades or units that connect to the content in this unit

Creating Equations

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.
Creating Equations

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.
Creating Equations

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 Categorical and Quantitative Data

Interpreting Categorical and Quantitative Data

HSS-ID.C.7 — Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Linear, Quadratic, and Exponential Models

Linear, Quadratic, and Exponential Models

F.LE.A.1 — Distinguish between situations that can be modeled with linear functions and with exponential functions.

Reasoning with Equations and Inequalities

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.

Seeing Structure in Expressions

Seeing Structure in Expressions

A.SSE.A.1 — Interpret expressions that represent a quantity in terms of its context 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.
Seeing Structure in Expressions

A.SSE.B.3 — Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. 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.

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.

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