Algebra 2 Unit 1: Families of Functions

Unit Driving Question 

What makes each family of functions unique?

Essential Questions 

1. How can arithmetic operations on functions be represented by a graph and related to an algebraic process?

2. What are the similarities and differences between functions and within a family of functions?

3. What families of functions also have inverses that are functions?

4. How are functions used to represent real-world problems?

  Launch Task

Big Ideas for Development Lessons 

Closure & Assessment 

1-2 Lessons 

4 Weeks (approximately 1 week per big idea) 

1 Week 

• Unit 1 Hook-Polynomial Patterns.

Click on the links below to see each Big Idea's Lesson Overview (includes links to teacher notes and student activities) 

1. Operations on functions can be represented with both algebraic and graphical models.
2. Families of functions share similar behaviors, characteristics and properties.
3. Two functions are inverses if the function f maps x to y and the inverse of f maps y to x.
4. Functions within the same family are made by applying transformations onto the parent function.

• Formative Assessment Instructions for two card sorts.
• Formative Assessment Reference Equation Cards
• Card Sort 1
• Card Sort 2  (after Big Idea 2 lessons)
• Re-engagement Activity (not provided)
• Unit Assessment 

Big Idea 1: Operations on functions can be represented with both algebraic and graphical models.

OAS-M: A2.A.2.2, A2.F.2.1 

Lessons and Additional Activities

Big Idea 1 Lessons 1-3 Overview (includes links to teacher notes and student activities)

Big Idea 1, Lessons 1-3 Lesson Plans

Evidence of Understanding  

Recognize and perform arithmetic operations on polynomials and rational expressions.

• Form conjectures and make generalizations about the resulting graph when arithmetic operations are performed.

• Explore and describe the resulting polynomial by graphing the original functions and the resulting function.

Simplify polynomials expressions properly.

• Make generalizations about how to simplify polynomials. (Division restricted to non-reducible forms.)

Recognize when a rational expression is not in simplest form and simplify it properly.

• Discover and predict the domain of rational functions using various means.

• Use algebraic means and other tools such as graphing calculator or desmos to explore the behavior of rational functions.

Big Idea 2: Families of functions share similar behaviors, characteristics and properties.

Lessons and Additional Activities

Big Idea 2 Lessons 1-3 Overview (includes links to teacher notes and student activities)

Big Idea 2, Lessons 1-3 Lesson Plans

Evidence of Understanding  

Use end behavior and the shape of a graph to identify and compare function families.

• Conjecture the effect of the leading coefficient on function families.

• Notice the effect a leading coefficient has on a function by using technology to explore various functions in a family and describe the pattern.

• Form conjectures and generalizations about function families based on the interval behavior of the graph.

• Use the characteristics of functions to describe real world situations.

• Discover and predict end behavior of function families using the shape of the graph and its rate of change.

• Generalize end behaviors of specific families of functions using various means such as graphs, tables, equations, and technology.

• Discover how symmetry can help identify types of function families.

Use critical points to identify and compare function families.

• Locate zeros of a function from a graph or a table.

• Use tools such as graphing calculator or desmos activities.

• Make the connection that the zeros (x-intercepts, roots) of a function can separate positive and negative intervals.

• Use a variety of tools such as graphs, tables, equations, and/or technology (graphing calculator, desmos, etc.) to discover and explore the concept of zeros and positive and negative intervals.

• Describe intervals using interval notation, inequalities and/or equations, and verbiage.

• Recognize and justify how maximum and minimum points separate the intervals where a function increases and decreases.

• Describe intervals using interval notation, inequalities and/or equations, and verbiage.

• Determine whether maximum and minimum points are relative or absolute by using the end behavior of a function.

• Technology tools such as graphing calculators and desmos can be used to solidify this concept.

Use the end behavior, shape of the graph and critical points to identify and compare function families.

• Create a general graph for a function family when given end behavior, shape of the graph, and critical points.

• Describe and generalize function family characteristics using end behavior, shape of the graph, and critical points.

• Analyze the meaning of critical points and key characteristics of a function in a real world situation.

Big Idea 3: Two functions are inverses if the function f maps x to y and the inverse of f maps y to x.

Lessons and Additional Activities

Big Idea 3 Lessons 1-4 Overview (includes links to teacher notes and student activities)

Big Idea 3, Lessons 1-4 Lesson Plans

Evidence of Understanding

Determine the composition of two functions. 

• Find the composition of functions graphically and algebraically.

Use the composition of two functions to prove or disprove that two functions are inverses.

• Justify and explain why the domain of a function is the range of its inverse and vice versa.

• Recognize and justify families of functions that are inverses of each other.

Determine the inverse of a function using a graph or a table.

• Recognize and verify that two functions are inverses by analyzing points on a table or graph for the function and its inverse.

• Compare and contrast the shape of a function and its inverse.

• Use a variety of tools, to include technology, to help students discover general characteristics of a function’s inverse.

• Build a function’s inverse by identifying points on a function’s graph or table of values.

• Explore one-to-oneness and the need for domain restrictions.

• Verify that two functions are inverses by their graphs and the line y = x.

• Explain why two functions that are inverses reflect over the line y = x.

Determine the algebraic equation for the inverse of a function and be able to apply it to real world scenarios.

• Create the inverse of a function algebraically.

• Justify that function you have created is the inverse in multiple ways. (i.e. using composition of the two functions, graphically using y=x, with a table of values or other means.)

Big Idea 4: Functions within the same family are made by applying transformations onto the parent function.

Lessons and Additional Activities

Big Idea 4 Lessons 1-2 Overview (includes links to teacher notes and student activities)

Big Idea 4, Lessons 1-2 Lesson Plans

Evidence of Understanding  

Describe transformations within a function family. 

• Compare functions within the same family and explain transformations from the parent function.

• Develop generalizations and conjectures about how each transformation affects the original function (ie. translations, dilations, and reflections). Use technology such as graphing calculators or desmos to discover and validate your findings.

• Write a verbal description and/or a formula describing each type of transformation.

• Create a new function when given a parent function and transformation details.

• Display the new function in a variety of forms. (ie. table of values, graph, equation, etc.)

Discover and describe how transformations affect critical points and characteristics of functions in the same family. 

• Generalize how each transformation affects intercepts, maximums, minimums, end behavior, asymptotes, domain and range.

• Conjecture and verbally describe what is preserved with each transformation and what changes.

• Include changes in orientation, rate of change, distances, etc by targeting the critical points on the original graph and comparing to the critical points on the transformed graph.

• Use visuals such as technology to help explore, engage, solidify, and justify student findings.

• Describe the effect transformations have on real world scenarios.

• Describe how the shape and characteristics of the graph are affected or not affected.