Illustrative Math Alignment: Grade 8 Unit 3

Functions and Volume

Lesson 6: Even More Graphs of Functions

Use the following Media4Math resources with this Illustrative Math lesson.

Title	Body	Curriculum Topic
Definition--Functions and Relations Concepts--Independent Variable	Independent Variable Topic Functions and Relations Definition An independent variable is a variable that represents the input or cause and is manipulated to observe its effect on the dependent variable.	Relations and Functions
Definition--Functions and Relations Concepts--Inverse Function	Inverse Function Topic Functions and Relations Definition An inverse function is a function that reverses the effect of the original function, such that f(f−1(x)) = x.	Functions and Their Inverses
Definition--Functions and Relations Concepts--Odd Function	Odd Function Topic Functions and Relations Definition An odd function is a function that satisfies the condition f(−x) = −f(x) for all x in its domain.	Relations and Functions
Definition--Functions and Relations Concepts--Parametric Equations	Parametric Equations Topic Functions and Relations Definition Parametric equations are a set of equations that express the coordinates of the points of a curve as functions of a variable called a parameter.	Relations and Functions
Definition--Functions and Relations Concepts--Piecewise Functions	Piecewise Functions Topic Functions and Relations Definition Piecewise functions are functions defined by multiple sub-functions, each applying to a certain interval of the domain.	Special Functions
Definition--Functions and Relations Concepts--Range of a Function	Range of a Function Topic Functions and Relations Definition The range of a function is the set of all possible output values (y-values) that the function can produce. Description Understanding the range of a function is crucial in mathematics because it defines the scope of possible outputs. The range is determined by the function's rule and the domain. For example, the range of the function f(x) = x2	Relations and Functions
Definition--Functions and Relations Concepts--Recursive Function	Recursive Function Topic Functions and Relations Definition A recursive function is a function that calls itself in its definition. Description Recursive functions are important in mathematics and computer science because they provide a way to solve problems by breaking them down into simpler sub-problems. They are defined by a base case and a recursive case. For example, the factorial function f(n) = n⋅f(n−1) with f(0)=1	Relations and Functions
Definition--Functions and Relations Concepts--Reflecting a Function	Reflecting a Function Topic Functions and Relations Definition Reflecting a function involves flipping the graph of the function over a specified axis. Description Reflecting functions is significant in mathematics because it helps in understanding the symmetry and transformations of functions. A function can be reflected over the x-axis or y-axis, changing its orientation. For example, reflecting the function f(x) = x2	Relations and Functions
Definition--Functions and Relations Concepts--Relation	Relation Topic Functions and Relations Definition A relation is a set of ordered pairs, where each element from one set is paired with an element from another set.	Relations and Functions
Definition--Functions and Relations Concepts--Step Function	Step Function Topic Functions and Relations Definition A step function is a function that increases or decreases abruptly at certain points, creating a series of steps.	Special Functions
Definition--Functions and Relations Concepts--Translating a Function	Translating a Function Topic Functions and Relations Definition Translating a function involves shifting the graph of the function horizontally, vertically, or both, without changing its shape. Description Translating functions is significant in mathematics because it helps in understanding how functions behave under shifts. A function can be translated horizontally by adding or subtracting a constant to the input, and vertically by adding or subtracting a constant to the output. For example, translating the function f(x) = x2 horizontally by 2 units results in	Relations and Functions
Definition--Functions and Relations Concepts--Vertical Line Test	Vertical Line Test Topic Functions and Relations Definition The vertical line test is a method used to determine if a graph represents a function by checking if any vertical line intersects the graph more than once. Description The vertical line test is important in mathematics because it helps in identifying whether a given graph represents a function. If a vertical line intersects the graph at more than one point, then the graph does not represent a function. This test is used in various fields, including computer science for validating functions in programming and in mathematics for analyzing graphs. For example, the graph of	Relations and Functions
Definition--Functions and Relations Concepts--Visual Models of Functions	Visual Models of Functions Topic Functions and Relations Definition Visual models of functions are graphical representations that illustrate the relationship between input and output values of functions. Description Visual models of functions are important in mathematics because they provide a clear and intuitive way to understand how functions behave. These models include graphs, tables, and diagrams that show the relationship between input and output values. For example, the graph of f(x) = x2	Relations and Functions
Definition--Functions and Relations Concepts--Visual Models of Relations	Visual Models of Relations Topic Functions and Relations Definition Visual models of relations are graphical representations that illustrate the relationship between two sets of values, not necessarily functions.	Relations and Functions
Definition--Functions and Relations Concepts--Visualization of the Dependent Variable	Visualization of the Dependent Variable Topic Functions and Relations Definition Visualization of the dependent variable involves graphically representing the outcomes or responses that depend on the independent variable.	Relations and Functions
Definition--Functions and Relations Concepts--Visualization of the Independent Variable	Visualization of the Independent Variable Topic Functions and Relations Definition Visualization of the independent variable involves graphically representing the variable that is manipulated to observe its effect on the dependent variable.	Relations and Functions
INSTRUCTIONAL RESOURCE: Algebra Application: Accident Investigation	INSTRUCTIONAL RESOURCE: Algebra Application: Accident Investigation In this Algebra Application, students develop a quadratic mathematical model for calculating the speed of a car based on the length of its skid marks. Using this model, students investigate the corresponding parabola. The math topics covered include: Mathematical modeling, Quadratic functions in standard form, Square root formula, Data analysis. The culminating activity is for students to write a program, using either a spreadsheet or a computer language like Python, that can calculate the speed of a car, given the skid mark length and coefficient of friction as the two inputs.	Applications of Quadratic Functions and Quadratic Equations and Functions
INSTRUCTIONAL RESOURCE: Algebra Application: Fireworks Displays	INSTRUCTIONAL RESOURCE: Algebra Application: Fireworks Displays In this Algebra Application, students develop a quadratic mathematical model based for the path of fireworks. Using this model, students investigate the properties of parabolas, including orientation, y-intercept, and vertex. The math topics covered include: Mathematical modeling, Quadratic functions in standard, vertex, and factored form, Curve fitting, Solving equations. The culminating activity is for students to plan a fireworks displays that mathematically uses the properties of quadratics restricted to the first quadrant. Given the coordinates of the vertices, what are the equations of parabolas that fit the given parameters.	Applications of Quadratic Functions and Quadratic Equations and Functions
INSTRUCTIONAL RESOURCE: Tutorial: Analyzing Graphs of Quadratic Functions in Factored Form	INSTRUCTIONAL RESOURCE: Tutorial: Analyzing Graphs of Quadratic Functions in Factored Form This slide show provides 8 examples of quadratic functions in factored form and analyzes their graphs. This is part of a collection of tutorials on a variety of math topics. To see the complete collection of these resources, click on this link. Note: The download is a PPT file. Library of Instructional Resources To see the complete library of Instructional Resources , click on this link.	Quadratic Equations and Functions
INSTRUCTIONAL RESOURCE: Tutorial: Analyzing Graphs of Quadratic Functions in Standard Form	INSTRUCTIONAL RESOURCE: Tutorial: Analyzing Graphs of Quadratic Functions in Standard Form This slide show provides 18 examples of quadratic functions in standard form and analyzes their graphs. This is part of a collection of tutorials on a variety of math topics. To see the complete collection of these resources, click on this link. Note: The download is a PPT file. Library of Instructional Resources To see the complete library of Instructional Resources , click on this link.	Quadratic Equations and Functions
INSTRUCTIONAL RESOURCE: Tutorial: Analyzing Graphs of Quadratic Functions in Vertex Form	INSTRUCTIONAL RESOURCE: Tutorial: Analyzing Graphs of Quadratic Functions in Vertex Form This slide show provides 8 examples of quadratic functions in vertex form and analyzes their graphs. This is part of a collection of tutorials on a variety of math topics. To see the complete collection of these resources, click on this link. Note: The download is a PPT file. Library of Instructional Resources To see the complete library of Instructional Resources , click on this link.	Quadratic Equations and Functions
INSTRUCTIONAL RESOURCE: Desmos Tutorial: Linear Functions in Slope Intercept Form	INSTRUCTIONAL RESOURCE: Desmos Tutorial: Linear Functions in Slope Intercept Form In this Slide Show, use the Desmos graphing calculator to explore linear equations in slope intercept form. To see the complete collection of Desmos Resources click on this link. Note: The download is a PPT file. This is part of a collection of Desmos tutorials on a variety of math topics. To see the complete collection of these resources, click on this link. Library of Instructional Resources To see the complete library of Instructional Resources , click on this link.	Graphs of Linear Functions and Slope-Intercept Form
INSTRUCTIONAL RESOURCE: Math Examples 19	INSTRUCTIONAL RESOURCE: Math Examples--Graphing Linear Functions in Slope-Intercept Form The complete set of 13 examples that make up this set of tutorials. This is part of a collection of math examples for a variety of math topics. To see the complete collection of these resources, click on this link. Note: The download is a PPT file. Library of Instructional Resources To see the complete library of Instructional Resources , click on this link.	Applications of Linear Functions and Graphs of Linear Functions
INSTRUCTIONAL RESOURCE: Math Examples 23	INSTRUCTIONAL RESOURCE: Math Examples--Graphs of Rational Functions The complete set of 28 examples that make up this set of tutorials. This is part of a collection of math examples for a variety of math topics. To see the complete collection of these resources, click on this link. Note: The download is a PPT file. Library of Instructional Resources To see the complete library of Instructional Resources , click on this link.	Rational Functions and Equations
INSTRUCTIONAL RESOURCE: Math Examples 27	INSTRUCTIONAL RESOURCE: Math Examples--Linear Functions in Tabular and Graph Form This set of 21 tutorials provides different examples of linear functions in function table and graph form. This is part of a collection of math examples for a variety of math topics. To see the complete collection of these resources, click on this link. Note: The download is a PPT file. Library of Instructional Resources To see the complete library of Instructional Resources , click on this link.	Applications of Linear Functions
INSTRUCTIONAL RESOURCE: Nspire App Tutorial: Graphing a Scatterplot	In this Slide Show, learn how graph data in a scatterplot using the graph window. This presentation requires the use of the TI-Nspire iPad App. Note: the download is a PPT.	Graphs of Linear Functions, Slope-Intercept Form and Data Analysis
INSTRUCTIONAL RESOURCE: Nspire App Tutorial: Graphs of Absolute Value Functions	In this Slide Show, absolute value functions are graphed, including graphs centered at the origin, graphs displaced along the x-axis, and graphs displaced along the y-axis. This presentation requires the use of the TI-Nspire iPad App. Note: the download is a PPT.	Special Functions
INSTRUCTIONAL RESOURCE: Nspire App Tutorial: Graphs of Linear Functions	In this Slide Show, linear functions are graphed. This presentation requires the use of the TI-Nspire iPad App. Note: the download is a PPT.	Applications of Linear Functions
INSTRUCTIONAL RESOURCE: Nspire App Tutorial: Graphs of Quadratic Functions in Standard Form with Sliders	In this Slide Show, learn how to graph quadratic functions in standard form using sliders for the values of a, b, and c. This presentation requires the use of the TI-Nspire iPad App. Note: the download is a PPT.	Applications of Quadratic Functions and Graphs of Quadratic Functions
INSTRUCTIONAL RESOURCE: Nspire App Tutorial: Graphs of Quadratic Functions in Vertex Form with Sliders	In this Slide Show, learn how to graph quadratic functions in vertex form using sliders for the values of h and k. This presentation requires the use of the TI-Nspire iPad App. Note: the download is a PPT.	Applications of Quadratic Functions and Graphs of Quadratic Functions
INSTRUCTIONAL RESOURCE: Nspire App Tutorial: Slope-Intercept Form with Sliders	In this Slide Show, linear functions in slope-intercept form are graphed using sliders. This presentation requires the use of the TI-Nspire iPad App. Note: the download is a PPT.	Graphs of Linear Functions and Slope-Intercept Form
INSTRUCTIONAL RESOURCE: TI-Nspire CX Activity: Graphs of Quadratic Functions	In this Slide Show, learn how to graph quadratic functions in standard form using the TI-Nspire CX. Note: The download is a PPT file.	Applications of Quadratic Functions and Graphs of Quadratic Functions
INSTRUCTIONAL RESOURCE: Tutorial: The Slope-Intercept Form	INSTRUCTIONAL RESOURCE: Tutorial: The Slope-Intercept Form In this Slide Show, learn about the slope-intercept form. This is part of a collection of tutorials on a variety of math topics. To see the complete collection of these resources, click on this link. Note: The download is a PPT file. Library of Instructional Resources To see the complete library of Instructional Resources , click on this link.	Graphs of Linear Functions and Slope-Intercept Form
Interactive Quiz: Graphs of Linear Functions, Quiz 01, Level 1		Graphs of Linear Functions
Interactive Quiz: Graphs of Linear Functions, Quiz 01, Level 2	Interactive Quiz: Graphs of Linear Functions, Quiz 01, Level 2 This is part of a collection of math quizzes on the topic of graphs of linear functions. Some quizzes are interactive and some are in PDF format. To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key). Related Resources To see additional resources on this topic, click on the Related Resources tab. Quiz Library To see the complete collection of Quizzes, click on this link.	Graphs of Linear Functions
Interactive Quiz: Graphs of Linear Functions, Quiz 01, Level 3	Interactive Quiz: Graphs of Linear Functions, Quiz 01, Level 3 This is part of a collection of math quizzes on the topic of graphs of linear functions. Some quizzes are interactive and some are in PDF format. To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key). Related Resources To see additional resources on this topic, click on the Related Resources tab. Quiz Library To see the complete collection of Quizzes, click on this link.	Graphs of Linear Functions
Interactive Quiz: Graphs of Linear Functions, Quiz 02, Level 1	Interactive Quiz: Graphs of Linear Functions, Quiz 02, Level 1 This is part of a collection of math quizzes on the topic of graphs of linear functions. Some quizzes are interactive and some are in PDF format. To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key). Related Resources To see additional resources on this topic, click on the Related Resources tab. Quiz Library To see the complete collection of Quizzes, click on this link.	Graphs of Linear Functions
Interactive Quiz: Graphs of Linear Functions, Quiz 02, Level 2	Interactive Quiz: Graphs of Linear Functions, Quiz 02, Level 2 This is part of a collection of math quizzes on the topic of graphs of linear functions. Some quizzes are interactive and some are in PDF format. To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key). Related Resources To see additional resources on this topic, click on the Related Resources tab. Quiz Library To see the complete collection of Quizzes, click on this link.	Graphs of Linear Functions
Interactive Quiz: Graphs of Linear Functions, Quiz 02, Level 3	Interactive Quiz: Graphs of Linear Functions, Quiz 02, Level 3 This is part of a collection of math quizzes on the topic of graphs of linear functions. Some quizzes are interactive and some are in PDF format. To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key). Related Resources To see additional resources on this topic, click on the Related Resources tab. Quiz Library To see the complete collection of Quizzes, click on this link.	Graphs of Linear Functions
Interactive Quiz: Graphs of Linear Functions, Quiz 03, Level 1	Interactive Quiz: Graphs of Linear Functions, Quiz 03, Level 1 This is part of a collection of math quizzes on the topic of graphs of linear functions. Some quizzes are interactive and some are in PDF format. To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key). Related Resources To see additional resources on this topic, click on the Related Resources tab. Quiz Library To see the complete collection of Quizzes, click on this link.	Graphs of Linear Functions
Interactive Quiz: Graphs of Linear Functions, Quiz 03, Level 2	Interactive Quiz: Graphs of Linear Functions, Quiz 03, Level 2 This is part of a collection of math quizzes on the topic of graphs of linear functions. Some quizzes are interactive and some are in PDF format. To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key). Related Resources To see additional resources on this topic, click on the Related Resources tab. Quiz Library To see the complete collection of Quizzes, click on this link.	Graphs of Linear Functions
Interactive Quiz: Graphs of Linear Functions, Quiz 03, Level 3	Interactive Quiz: Graphs of Linear Functions, Quiz 03, Level 3 This is part of a collection of math quizzes on the topic of graphs of linear functions. Some quizzes are interactive and some are in PDF format. To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key). Related Resources To see additional resources on this topic, click on the Related Resources tab. Quiz Library To see the complete collection of Quizzes, click on this link.	Graphs of Linear Functions
Interactive Quiz: Graphs of Linear Functions, Quiz 04, Level 1	Interactive Quiz: Graphs of Linear Functions, Quiz 04, Level 1 This is part of a collection of math quizzes on the topic of graphs of linear functions. Some quizzes are interactive and some are in PDF format. To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key). Related Resources To see additional resources on this topic, click on the Related Resources tab. Quiz Library To see the complete collection of Quizzes, click on this link.	Graphs of Linear Functions
Interactive Quiz: Graphs of Linear Functions, Quiz 04, Level 2	Interactive Quiz: Graphs of Linear Functions, Quiz 04, Level 2 This is part of a collection of math quizzes on the topic of graphs of linear functions. Some quizzes are interactive and some are in PDF format. To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key). Related Resources To see additional resources on this topic, click on the Related Resources tab. Quiz Library To see the complete collection of Quizzes, click on this link.	Graphs of Linear Functions
Interactive Quiz: Graphs of Linear Functions, Quiz 04, Level 3	Interactive Quiz: Graphs of Linear Functions, Quiz 04, Level 3 This is part of a collection of math quizzes on the topic of graphs of linear functions. Some quizzes are interactive and some are in PDF format. To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key). Related Resources To see additional resources on this topic, click on the Related Resources tab. Quiz Library To see the complete collection of Quizzes, click on this link.	Graphs of Linear Functions
Interactive Quiz: Graphs of Linear Functions, Quiz 05, Level 1	Interactive Quiz: Graphs of Linear Functions, Quiz 05, Level 1 This is part of a collection of math quizzes on the topic of graphs of linear functions. Some quizzes are interactive and some are in PDF format. To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key). Related Resources To see additional resources on this topic, click on the Related Resources tab. Quiz Library To see the complete collection of Quizzes, click on this link.	Graphs of Linear Functions
Interactive Quiz: Graphs of Linear Functions, Quiz 05, Level 2	Interactive Quiz: Graphs of Linear Functions, Quiz 05, Level 2 This is part of a collection of math quizzes on the topic of graphs of linear functions. Some quizzes are interactive and some are in PDF format. To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key). Related Resources To see additional resources on this topic, click on the Related Resources tab. Quiz Library To see the complete collection of Quizzes, click on this link.	Graphs of Linear Functions
Interactive Quiz: Graphs of Linear Functions, Quiz 05, Level 3	Interactive Quiz: Graphs of Linear Functions, Quiz 05, Level 3 This is part of a collection of math quizzes on the topic of graphs of linear functions. Some quizzes are interactive and some are in PDF format. To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key). Related Resources To see additional resources on this topic, click on the Related Resources tab. Quiz Library To see the complete collection of Quizzes, click on this link.	Graphs of Linear Functions
Interactive Quiz: Graphs of Linear Functions, Quiz 06, Level 1	Interactive Quiz: Graphs of Linear Functions, Quiz 06, Level 1 This is part of a collection of math quizzes on the topic of graphs of linear functions. Some quizzes are interactive and some are in PDF format. To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key). Related Resources To see additional resources on this topic, click on the Related Resources tab. Quiz Library To see the complete collection of Quizzes, click on this link.	Graphs of Linear Functions
Interactive Quiz: Graphs of Linear Functions, Quiz 06, Level 2	Interactive Quiz: Graphs of Linear Functions, Quiz 06, Level 2 This is part of a collection of math quizzes on the topic of graphs of linear functions. Some quizzes are interactive and some are in PDF format. To see the complete quiz collection on this topic, click on this link. Note: The download is the PDF version of the quiz (with answer key). Related Resources To see additional resources on this topic, click on the Related Resources tab. Quiz Library To see the complete collection of Quizzes, click on this link.	Graphs of Linear Functions