MAFS.912.F-IF.3.7Archived Standard

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.
General Information
Subject Area: Mathematics
Grade: 912
Domain-Subdomain: Functions: Interpreting Functions
Cluster: Level 2: Basic Application of Skills & Concepts
Cluster: 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.

Date Adopted or Revised: 02/14
Date of Last Rating: 02/14
Status: State Board Approved - Archived
Assessed: Yes
Test Item Specifications
    Assessed with:

    MAFS.912.F-IF.3.8


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This benchmark is part of these courses.
1200310: Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200320: Algebra 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200330: Algebra 2 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200340: Algebra 2 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200370: Algebra 1-A (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200380: Algebra 1-B (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
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1201300: Mathematical Analysis Honors (Specifically in versions: 2014 - 2015, 2015 - 2022 (course terminated))
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2000360: Anatomy and Physiology Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2001350: Astronomy Solar/Galactic (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2020910: Astronomy Solar/Galactic Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2000320: Biology 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2000330: Biology 2 Honors (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2022, 2022 and beyond (current))
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2000370: Botany (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2003340: Chemistry 1 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2003350: Chemistry 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2003360: Chemistry 2 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
2001320: Earth/Space Science Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
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Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.

Related Resources

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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

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

Graphing a Quadratic Function:

Students are asked to graph a quadratic function and answer questions about the intercepts, maximum, and minimum.

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

Lesson Plans

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.

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

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.

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

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.

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 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.

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.

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Original Student Tutorials

Graphing Linear Functions Part 1: Table of Values:

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Type: Original Student Tutorial

Graphing Quadratic Functions:

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Type: Original Student Tutorial

Perspectives Video: Experts

Jumping Robots and Quadratics:

<p>Jump to it and learn more about how quadratic equations are&nbsp;used in robot navigation problem solving!</p>

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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.

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Asymptotic Behavior in Shark Growth Research:

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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.

Type: Problem-Solving Task

Graphs of Quadratic Functions:

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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.

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Graphs of Power Functions:

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You will learn how the parent function for a quadratic function is affected when f(x) = x2.

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Graphing Quadractic Functions in Vertex Form:

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Graphing Quadratic Equations:

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Unit/Lesson Sequences

Sample Algebra 1 Curriculum Plan Using CMAP:

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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

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Quadratic Functions: Workshop 4:

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Type: Unit/Lesson Sequence

Video/Audio/Animation

Graphing Lines 1:

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Virtual Manipulatives

Slope Slider:

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Type: Virtual Manipulative

Graphing Equations Using Intercepts:

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Type: Virtual Manipulative

Graphing Lines:

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Type: Virtual Manipulative

Data Flyer:

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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

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.

Type: Worksheet

MFAS Formative Assessments

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.

Graphing a Quadratic Function:

Students are asked to graph a quadratic function and answer questions about the intercepts, maximum, and minimum.

Graphing a Rational Function:

Students are asked to graph a rational function with the use of technology and identify key features 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 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 Root Functions:

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

Original Student Tutorials Mathematics - Grades 9-12

Graphing Linear Functions Part 1: Table of Values:

Learn how to graph linear functions by creating a table of values based on the equation in this interactive tutorial.

This is part 1 of a series of tutorials on linear functions.

Graphing Quadratic Functions:

Follow as we discover key features of a quadratic equation written in vertex form in this interactive tutorial.

Student Resources

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

Original Student Tutorials

Graphing Linear Functions Part 1: Table of Values:

Learn how to graph linear functions by creating a table of values based on the equation in this interactive tutorial.

This is part 1 of a series of tutorials on linear functions.

Type: Original Student Tutorial

Graphing Quadratic Functions:

Follow as we discover key features of a quadratic equation written in vertex form in this interactive tutorial.

Type: Original Student Tutorial

Perspectives Video: Experts

Jumping Robots and Quadratics:

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Type: Perspectives Video: Expert

Mathematically Exploring the Wakulla Caves:

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Type: Perspectives Video: Expert

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Graphs of Power Functions:

This task requires students to recognize the graphs of different (positive) powers of x.

Type: Problem-Solving Task

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

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.

Type: Tutorial

Graphing Exponential Equations:

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

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 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 benchmark.

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Graphs of Power Functions:

This task requires students to recognize the graphs of different (positive) powers of x.

Type: Problem-Solving Task

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