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

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.

General 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
Domain-Subdomain: Functions: Interpreting Functions

## Related Standards

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MAFS.912.F-IF.3.AP.7a
Select a graph of a function that displays its symbolic representation (e.g., f(x) = 3x + 5).
MAFS.912.F-IF.3.AP.7b
Locate the key features of linear and quadratic equations.
MAFS.912.F-IF.3.AP.8a
Write or select an equivalent form of a function [e.g., y = mx + b, f(x) = y, y – y1 = m(x – x1), Ax + By = C].
MAFS.912.F-IF.3.AP.8b
Describe the properties of a function (e.g., rate of change, maximum, minimum, etc.).
MAFS.912.F-IF.3.AP.9a
Compare the properties of two functions.

## Related Resources

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

## Assessments

Sample 4 - High School Algebra 2 State Interim Assessment:

This is a State Interim Assessment for 9th-12th grade.

Type: Assessment

Sample 2 - High School Algebra 2 State Interim Assessment:

This is a State Interim Assessment for 9th-12th grades.

Type: Assessment

Sample 4 - High School Algebra 1 State Interim Assessment:

This is a State Interim Assessment for 9th-12th grades.

Type: Assessment

Sample 1 - High School Algebra 2 State Interim Assessment:

This is a State Interim Assessment for 9th-12th grades.

Type: Assessment

Sample 3 - High School Algebra 1 State Interim Assessment:

This is a State Interim Assessment for 9th-12th grades.

Type: Assessment

Sample 2 - High School Algebra 1 State Interim Assessment:

This is a State Interim Assessment for 9th-12th grades.

Type: Assessment

Sample 1 - High School Algebra 1 State Interim Assessment:

This is the State Interim Assessment for high school.

Type: Assessment

## 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.

Type: Educational Software / Tool

## 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.

Type: Formative Assessment

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

Type: Formative Assessment

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.

Type: Formative Assessment

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.

Type: Formative Assessment

Graphing Root Functions:

Students are asked to graph two root functions and answer questions about the domain, maxima, and minima.

Type: Formative Assessment

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.

Type: Formative Assessment

Graphing a Step Function:

Students are asked to graph a step function, state the domain of the function, and name any intercepts.

Type: Formative Assessment

Type: Formative Assessment

Exponential Functions - 2:

Students are asked to identify the percent rate of change of a given exponential function.

Type: Formative Assessment

Exponential Functions - 1:

Students are asked to identify the percent rate of change of a given exponential function.

Type: Formative Assessment

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.

Type: Formative Assessment

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.

Type: Formative Assessment

Launch From a Hill:

Students are asked to factor and find the zeros of a polynomial function given in context.

Type: Formative Assessment

## 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.

Type: Lesson Plan

How High Can I Go?:

Students will work on graphing quadratic equations; identifying the axis of symmetry, maximum/minimum, vertex, and roots. Students will work in pairs and will get to move around the room matching equations with given graphs.

Type: Lesson Plan

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 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.

Type: Lesson Plan

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.

Type: Lesson Plan

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).

Type: Lesson Plan

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.

Type: Lesson Plan

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.

Type: Lesson Plan

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.

Type: Lesson Plan

Taming the Behavior of Polynomials:

This lesson will cover sketching the graphs of polynomials while in factored form without the use of a calculator.

Type: Lesson Plan

Dancing Polynomials/Graph Me Baby:

Dancing Polynomials is designed to lead students from the understanding that the equation of a line produces a linear pattern to the realization that using an exponent greater than one will produce curvature in a graph and that further patterns emerge allowing students to predict what happens at the end of the graph. Using graphing calculators, students will examine the patterns that emerge to predict the end behavior of polynomial functions. They will experiment by manipulating equations superimposed onto landmarks in the shape of parabolas and polynomial functions. An end behavior song and dance, called "Graph Me Baby" will allow students to become graphs in order to physically understand the end behavior of the graph.

Type: Lesson Plan

This lesson introduces students to the graph of the quadratic parent function. It provides a note taking sheet for students to organize their learning of basic transformations to the parent function. There is a "FUN" cut and paste activity for students to match graphs with verbal descriptions and their equations.

Type: Lesson Plan

Parts and more Parts-- Parabola Fun:

This is an entry lesson into quadratic functions and their shapes. Students see many 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.

Type: Lesson Plan

This lesson covers quadratic translations as they relate to vertex form of a quadratic equation. Students will predict what will happen to the graph of a quadratic function when more than one constant is in a quadratic equation. Then, the students will graph quadratic equations in vertex form using their knowledge of the translations of a quadratic function, as well as describe the translations that occur. Students will also identify the parent function of any quadratic function as .

Type: Lesson Plan

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 own knowledge of problems in the unit, as well as the teacher evaluation the mastery of the problems in the unit. The students receive immediate feedback on their own work and review while the teacher works the problems correcting any errors or misconceptions that the student had while working the problem. 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.

Type: Lesson Plan

This is an introductory lesson to graphing quadratic equations. This lesson uses graphing technology to illustrate the differences between quadratic equations and linear equations. In addition, it allows students to identify important parts of the quadratic equation and how each piece changes the look of the graph.

Type: Lesson Plan

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.

Type: Lesson Plan

## Original Student Tutorial

The graph of a quadratic equation is called a parabola [puh-ra-bow-luh]. The key features we will focus on in this tutorial are the vertex (a maximum or minimum extreme) and the direction of its opening. You will learn how to examine a quadratic equation written in vertex form in order to distinguish each of these key features.

Type: Original Student Tutorial

## Perspectives Video: Experts

Type: Perspectives Video: Expert

Birdsong Series: Mathematically Modeling Birdsong:

Richard Bertram discusses his mathematical modeling contribution to the Birdsong project that helps the progress of neuron and ion channel research.

Type: Perspectives Video: Expert

Using Mathematics to Optimize Wing Design:

Nick Moore discusses his research behind optimizing wing design using inspiration from animals and how they swim and fly.

Type: Perspectives Video: Expert

Mathematically Exploring the Wakulla Caves:

The tide is high!  How can we statistically prove there is a relationship between the tides on the Gulf Coast and in a fresh water spring 20 miles from each other?

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiast

Asymptotic Behavior in Shark Growth Research:

Fishery Scientist from Florida State University discusses his new research in deep sea sharks and the unusual behavior that is found when the data is graphed.

Type: Perspectives Video: Professional/Enthusiast

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.

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.

Type: Tutorial

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.

Type: Tutorial

This tutorial will help the learners to graph the equation of the quadratic function using the coordinates of the vertex of a parabola adn its x- intercepts.

Type: Tutorial

Graphing Exponential Equations:

This tutorial will help you to learn about the exponential functions by graphing various equations representing exponential growth and decay.

Type: Tutorial

Power of a Power Property:

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

Type: Tutorial

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.

Type: Tutorial

## 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 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

Type: Unit/Lesson Sequence

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.

Type: Unit/Lesson Sequence

## Video/Audio/Animations

Solving Quadratic Equations using Square Roots:

This video will demonstrate how to solve a quadratic equation using square roots.

Type: Video/Audio/Animation

This video gives a more in-depth look at graphing quadratic functions than previously offered in Quadratic Functions 1.

Type: Video/Audio/Animation

Graphing Lines 1:

Khan Academy video tutorial on graphing linear equations: "Algebra: Graphing Lines 1"

Type: Video/Audio/Animation

## 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.

Type: Virtual Manipulative

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.

Type: Virtual Manipulative

Graphing Lines:

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative

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.

Type: Virtual Manipulative

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.

Type: Virtual Manipulative

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.

Type: Virtual Manipulative

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).

Type: Virtual Manipulative

## Worksheet

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.

Type: Worksheet

## Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

## Original Student Tutorial

The graph of a quadratic equation is called a parabola [puh-ra-bow-luh]. The key features we will focus on in this tutorial are the vertex (a maximum or minimum extreme) and the direction of its opening. You will learn how to examine a quadratic equation written in vertex form in order to distinguish each of these key features.

Type: Original Student Tutorial

## Perspectives Video: Experts

Type: Perspectives Video: Expert

Mathematically Exploring the Wakulla Caves:

The tide is high!  How can we statistically prove there is a relationship between the tides on the Gulf Coast and in a fresh water spring 20 miles from each other?

Type: Perspectives Video: Expert

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.

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.

Type: Tutorial

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.

Type: Tutorial

This tutorial will help the learners to graph the equation of the quadratic function using the coordinates of the vertex of a parabola adn its x- intercepts.

Type: Tutorial

Graphing Exponential Equations:

This tutorial will help you to learn about the exponential functions by graphing various equations representing exponential growth and decay.

Type: Tutorial

Power of a Power Property:

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

Type: Tutorial

## Video/Audio/Animations

Solving Quadratic Equations using Square Roots:

This video will demonstrate how to solve a quadratic equation using square roots.

Type: Video/Audio/Animation

This video gives a more in-depth look at graphing quadratic functions than previously offered in Quadratic Functions 1.

Type: Video/Audio/Animation

Graphing Lines 1:

Khan Academy video tutorial on graphing linear equations: "Algebra: Graphing Lines 1"

Type: Video/Audio/Animation

## 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.

Type: Virtual Manipulative

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.

Type: Virtual Manipulative

Graphing Lines:

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative

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.

Type: Virtual Manipulative

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.

Type: Virtual Manipulative

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.

Type: Virtual Manipulative

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).

Type: Virtual Manipulative

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.

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.

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.

## Tutorial

Power of a Power Property:

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

Type: Tutorial

## Video/Audio/Animation

Graphing Lines 1:

Khan Academy video tutorial on graphing linear equations: "Algebra: Graphing Lines 1"

Type: Video/Audio/Animation

## Virtual Manipulative

Graphing Lines:

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative