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Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

Standard 3 : Analyze functions using different representations. (Algebra 1 - Supporting Cluster) (Algebra 2 - Supporting Cluster)Archived
Cluster Standards

This cluster includes the following benchmarks.

Visit the specific benchmark webpage to find related instructional resources.

  • MAFS.912.F-IF.3.7 : Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. ★

    1. Graph linear and quadratic functions and show intercepts, maxima, and minima.
    2. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
    3. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. 
    4. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. 
    5. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift.
  • MAFS.912.F-IF.3.8 : Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
    1. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
    2. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = , y = , y = , y = , and classify them as representing exponential growth or decay.
  • MAFS.912.F-IF.3.9 : 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.

Cluster Information
Number:
MAFS.912.F-IF.3
Title:
Analyze functions using different representations. (Algebra 1 - Supporting Cluster) (Algebra 2 - Supporting Cluster)
Type:
Cluster
Subject:
Mathematics - Archived
Grade:
912
Domain-Subdomain
Functions: Interpreting Functions
Cluster Access Points

This cluster includes the following Access Points.

Cluster Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

Original Student Tutorials
Educational Software / Tool
  • Free Graph Paper: A variety of graph paper types for printing, including Cartesian, polar, engineering, isometric, logarithmic, hexagonal, probability, and Smith chart.

Formative Assessments
  • Graphing a Rational Function: Students are asked to graph a rational function with the use of technology and identify key features of the graph.

  • Comparing Quadratics: Students are asked to compare two quadratic functions, one given by a table and the other by a function.

  • Comparing Linear and Exponential Functions: Students are given a linear function represented by an equation and an exponential function represented by a graph in a real-world context and are asked to compare the rates of change of the two functions.

  • Comparing Linear Functions: Students are given two linear functions, one represented by a graph and the other by an equation, and asked to compare their intercepts in the context of a problem.

  • Graphing Root Functions: Students are asked to graph two root functions and answer questions about the domain, maxima, and minima.

  • Graphing an Exponential Function: Students are asked to graph an exponential function and to determine if the function is an example of exponential growth or decay, describe any intercepts, and describe the end behavior of the graph.

  • Graphing a Step Function: Students are asked to graph a step function, state the domain of the function, and name any intercepts.

  • Graphing a Quadratic Function: Students are asked to graph a quadratic function and answer questions about the intercepts, maximum, and minimum.

  • Exponential Functions - 2: Students are asked to identify the percent rate of change of a given exponential function.

  • Exponential Functions - 1: Students are asked to identify the percent rate of change of a given exponential function.

  • Graphing a Linear Function: Students are asked to graph a linear function and to find the intercepts of the function as well as the maximum and minimum of the function within a given interval of the domain.

  • A Home for Fido: Students are asked to rewrite a quadratic function in an equivalent form by completing the square and to use this form to identify the vertex of the graph and explain its meaning in context.

  • Launch From a Hill: Students are asked to factor and find the zeros of a polynomial function given in context.

Lesson Plans
  • The Towers of Hanoi: Experiential Recursive Thinking:
    This lesson is about the Towers of Hanoi problem, a classic famous problem involving recursive thinking to reduce what appears to be a very large and difficult problem into a series of simpler ones.  The learning objective is for students to begin to understand recursive logic and thinking, relevant to computer scientists, mathematicians and engineers.   The lesson is experiential, in that each student will be working with her/his own Towers of Hanoi manipulative, inexpensively obtained.  There is no formal prerequisite, although some familiarity with set theory and functions is helpful.  The last three sections of the lesson involve some more formal concepts with recursive equations and proof by induction, so the students who work on those sections should probably be level 11 or 12 in a K-12 educational system.  The lesson has a Stop Point for 50-minute classes, followed by three more segments that may require a half to full additional class time.  So the teacher may use only those segments up to the Stop Point, or if two class sessions are to be devoted to the lesson, the entire set of segments.  Supplies are modest, and may be a set of coins or some washers from a hardware store to assemble small piles of disks in front of each student, each set of disks representing a Towers of Hanoi manipulative.  Or the students may assemble before the class a more complete Towers of Hanoi at home, as demonstrated in the video.  The classroom activities involve attempting to solve with hand and mind the Towers of Hanoi problem and discussing with fellow students patterns in the process and strategies for solution.

  • How High Can I Go?: Students will graph quadratic equations, and identify the axis of symmetry, the maximum/minimum point, the vertex, and the roots. Students will work in pairs and will move around the room matching equations with given graphs.

  • Stop That Arguing: Students will explore representing the movement of objects and the relationship between the various forms of representation: verbal descriptions, value tables, graphs, and equations. These representations include speed, starting position, and direction. This exploration includes brief direct instruction, guided practice in the form of a game, and independent practice in the form of a word problem. Students will demonstrate understanding of this concept through a written commitment of their answer to the word problem supported with evidence from value tables, graphs, and equations.

  • Linear Functions Representations: Students will compare properties of linear functions when presented in different representations. Students work as pairs within groups to analyze and confirm which representations are best suited for different needs. This lesson focuses on linear functions.

  • Representing Polynomials: This lesson unit is intended to help you assess how well students are able to translate between graphs and algebraic representations of polynomials. In particular, this unit aims to help you identify and assist students who have difficulties in recognizing the connection between the zeros of polynomials when suitable factorizations are available, and graphs of the functions defined by polynomials as well as recognizing the connection between transformations of the graphs and transformations of the functions obtained by replacing f(x) by f(x + k), f(x) + k, -f(x), f(-x).
  • Functions and Everyday Situations: This lesson unit is intended to help you assess how well students are able to articulate verbally the relationships between variables arising in everyday contexts, translate between everyday situations and sketch graphs of relationships between variables, interpret algebraic functions in terms of the contexts in which they arise and reflect on the domains of everyday functions and in particular whether they should be discrete or continuous.
  • Forming Quadratics: This lesson unit is intended to help you assess how well students are able to understand what the different algebraic forms of a quadratic function reveal about the properties of its graphical representation. In particular, the lesson will help you identify and help students who have the following difficulties in understanding how the factored form of the function can identify a graph's roots, how the completed square form of the function can identify a graph's maximum or minimum point, and how the standard form of the function can identify a graph's intercept.
  • Forced To Learn: Using inquiry techniques, students, working in groups, are asked to design and conduct an experiment to test Newton's Second Law of Motion. Upon being provided with textbooks, rulers, measuring tapes, mini-storage containers, golf balls, marbles, rubber balls, steel balls, and pennies they work cooperatively to implement and revise their hypotheses. With limited guidance from the teacher, students are able to visualize the direct relationships between force and mass; force and acceleration; and the inverse relationship between mass and acceleration.

  • Parts and more Parts-- Parabola Fun: This is an entry lesson into quadratic functions and their shapes. Students see some real-life representations of parabolas. This lesson provides important vocabulary associated with quadratic functions and their graphs in an interactive manner. Students create a foldable and complete a worksheet using their foldable notes.

  • Leap Frog Review Game: In this lesson students will demonstrate their knowledge of limits, graphing, and exact trig limits evaluated using substitution. The students will play a game in which they evaluate their knowledge of problems in the unit while it serves as a formative assessment for the teacher. The students receive immediate feedback on their work while the teacher works the problems, correcting errors or misconceptions. This lesson gives the student a power review of the concepts in the unit because the timing is determined by the teacher. All students are engaged and focused while playing this game. Giving students access to the PowerPoint of the game after the lesson provides a good study tool for the students.

  • Exponential Graphing Using Technology: This lesson is teacher/student directed for discovering and translating exponential functions using a graphing app. The lesson focuses on the translations from a parent graph and how changing the coefficient, base and exponent values relate to the transformation.

Perspectives Video: Experts
Perspectives Video: Professional/Enthusiast
Problem-Solving Tasks
  • Finding Parabolas through Two Points: This problem-solving task challenges students to find all quadratic functions described by given equation and coordinates, and describe how the graphs of those functions are related to one another.

  • Which Function?: The task addresses knowledge related to interpreting forms of functions derived by factoring or completing the square. It requires students to pay special attention to the information provided by the way the equation is represented as well as the sign of the leading coefficient, which is not written out explicitly, and then to connect this information to the important features of the graph.

  • Throwing Baseballs: This task could be used for assessment or for practice. It allows students to compare characteristics of two quadratic functions that are each represented differently, one as the graph of a quadratic function and one written out algebraically. Specifically, students are asked to determine which function has the greatest maximum and the greatest non-negative root.

  • Springboard Dive: The problem presents a context where a quadratic function arises. Careful analysis, including graphing of the function, is closely related to the context. The student will gain valuable experience applying the quadratic formula and the exercise also gives a possible implementation of completing the square.

  • Graphs of Quadratic Functions: Students compare graphs of different quadratic functions, then produce equations of their own to satisfy given conditions.

    This exploration can be done in class near the beginning of a unit on graphing parabolas. Students need to be familiar with intercepts, and need to know what the vertex is. It is effective after students have graphed parabolas in vertex form (y=a(x–h)2+k), but have not yet explored graphing other forms.

  • Graphs of Power Functions: This task requires students to recognize the graphs of different (positive) powers of x.

  • Kimi and Jordan: In the middle grades, students have lots of experience analyzing and comparing linear functions using graphs, table, symbolic expressions, and verbal descriptions. In this task, students may choose a representation that suits them and then reason from within that representation.

Tutorials
  • Graphs and Solutions of Functions in Quadratic Equations: You will learn how the parent function for a quadratic function is affected when f(x) = x2.

  • Graphing Quadractic Functions in Vertex Form: This tutorial will help the students to identify the vertex of a parabola from the equation, and then graph the parabola.

  • Graphing Quadratic Equations: This tutorial helps the learners to graph the equation of a quadratic function using the coordinates of the vertex of a parabola and its x- intercepts.

  • Graphing Exponential Equations: This tutorial will help you to learn about exponential functions by graphing various equations representing exponential growth and decay.

  • Power of a Power Property: This tutorial demonstrates how to use the power of a power property with both numerals and variables.

  • Linear Functions: In this tutorial, "Linear functions of the form f(x) = ax + b and the properties of their graphs are explored interactively using an applet." The applet allows students to manipulate variables to discover the changes in intercepts and slope of the graphed line. There are six questions for students to answer, exploring the applet and observing changes. The questions' answers are included on this site. Additionally, a tutorial for graphing linear functions by hand is included.
Unit/Lesson Sequences
  • Sample Algebra 1 Curriculum Plan Using CMAP:
    This sample Algebra 1 CMAP is a fully customizable resource and curriculum-planning tool that provides a framework for the Algebra 1 Course. The units and standards are customizable and the CMAP allows instructors to add lessons, worksheets, and other resources as needed. This CMAP also includes rows that automatically filter and display Math Formative Assessments System tasks, E-Learning Original Student Tutorials and Perspectives Videos that are aligned to the standards, available on CPALMS.

    Learn more about the sample Algebra 1 CMAP, its features and customizability by watching the following video:

    Using this CMAP

    To view an introduction on the CMAP tool, please .

    To view the CMAP, click on the "Open Resource Page" button above; be sure you are logged in to your iCPALMS account.

    To use this CMAP, click on the "Clone" button once the CMAP opens in the "Open Resource Page." Once the CMAP is cloned, you will be able to see it as a class inside your iCPALMS My Planner (CMAPs) app.

    To access your My Planner App and the cloned CMAP, click on the iCPALMS tab in the top menu.

    All CMAP tutorials can be found within the iCPALMS Planner App or at the following URL: http://www.cpalms.org/support/tutorials_and_informational_videos.aspx

  • Quadratic Functions: Workshop 4: Lesson 1 of two lessons requires students to explore quadratic functions by examining the family of functions described by y = a (x - h)squared+ k. In Lesson 2 students explore quadratic functions by using a motion detector known as a Calculator Based Ranger (CBR) to examine the heights of the different bounces of a ball. Students will represent each bounce with a quadratic function of the form y = a (x - h)squared + k. Background information, resources, references and videos of the lessons are included. Students work in teams of four.
Video/Audio/Animations
Virtual Manipulatives
  • Slope Slider: In this activity, students adjust slider bars which adjust the coefficients and constants of a linear function and examine how their changes affect the graph. The equation of the line can be in slope-intercept form or standard form. This activity allows students to explore linear equations, slopes, and y-intercepts and their visual representation on a graph. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

  • Graphing Equations Using Intercepts: This resource provides linear functions in standard form and asks the user to graph it using intercepts on an interactive graph below the problem. Immediate feedback is provided, and for incorrect responses, each step of the solution is thoroughly modeled.

  • Graphing Lines: Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the slope and various types of line equations can be explored.

  • Data Flyer: Using this virtual manipulative, students are able to graph a function and a set of ordered pairs on the same coordinate plane. The constants, coefficients, and exponents can be adjusted using slider bars, so the student can explore the affect on the graph as the function parameters are changed. Students can also examine the deviation of the data from the function. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

  • Function Flyer: In this online tool, students input a function to create a graph where the constants, coefficients, and exponents can be adjusted by slider bars. This tool allows students to explore graphs of functions and how adjusting the numbers in the function affect the graph. Using tabs at the top of the page you can also access supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

  • Curve Fitting: With a mouse, students will drag data points (with their error bars) and watch the best-fit polynomial curve form instantly. Students can choose the type of fit: linear, quadratic, cubic, or quartic. Best fit or adjustable fit can be displayed.

  • Equation Grapher: This interactive simulation investigates graphing linear and quadratic equations. Users are given the ability to define and change the coefficients and constants in order to observe resulting changes in the graph(s).

Worksheet
  • Quadratic Functions: This worksheet gives students one place to show all transformations (reflections, vertical stretches/compressions, and translations) for the quadratic function. The worksheet also has a place for domain and range for each transformation.