Illustrative Math Alignment: Grade 8 Unit 3

Functions and Volume

Lesson 1: Inputs and Outputs

Use the following Media4Math resources with this Illustrative Math lesson.

Title	Body	Curriculum Topic
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: 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: Desmos Tutorial: Linear Functions in Standard Form	INSTRUCTIONAL RESOURCE: Desmos Tutorial: Linear Functions in Standard Form In this Slide Show, use the Desmos graphing calculator to explore linear equations in standard 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.	Standard Form
INSTRUCTIONAL RESOURCE: Tutorial: Linear Equations in Standard Form	INSTRUCTIONAL RESOURCE: Tutorial: Linear Equations in Standard Form In this Slide Show, learn how to convert from linear equations in standard form to 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.	Standard Form
INSTRUCTIONAL RESOURCE: Math Examples 15	INSTRUCTIONAL RESOURCE: Math Examples--Exponential Functions in Tabular and Graph Form This set of tutorials provides 54 examples of exponential functions in tabular 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.	Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
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: Math Examples 29	INSTRUCTIONAL RESOURCE: Math Examples--Logarithmic Functions in Tabular and Graph Form This set of tutorials provides 120 examples of logarithmic functions in tabular 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.	Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
INSTRUCTIONAL RESOURCE: Math Examples 41	INSTRUCTIONAL RESOURCE: Math Examples--Quadratic Functions in Tabular and Graph Form This set of tutorials provides 37 examples of quadratic functions in tabular 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.	Quadratic Equations and Functions
INSTRUCTIONAL RESOURCE: Math Examples 45	INSTRUCTIONAL RESOURCE: Math Examples--Rational Functions in Tabular and Graph Form This set of tutorials provides 28 examples of rational functions in tabular 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.	Rational Expressions and Rational Functions and Equations
INSTRUCTIONAL RESOURCE: Math Examples 47	INSTRUCTIONAL RESOURCE: Math Examples--Secant Functions in Tabular and Graph Form This set of tutorials provides 65 examples of secant functions in tabular 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.	Trigonometric Functions
INSTRUCTIONAL RESOURCE: Math Examples 48	INSTRUCTIONAL RESOURCE: Math Examples--Sine Functions in Tabular and Graph Form This set of tutorials provides 65 examples of sine functions in tabular 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.	Trigonometric Functions
INSTRUCTIONAL RESOURCE: Math Examples 52	INSTRUCTIONAL RESOURCE: Math Examples--Square Root Functions in Tabular and Graph Form This set of tutorials provides 40 examples of square root functions in tabular 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.	Radical Functions and Equations
INSTRUCTIONAL RESOURCE: Math Examples 56	INSTRUCTIONAL RESOURCE: Math Examples--Tangent Functions in Tabular and Graph Form This set of tutorials provides 65 examples of tangent functions in tabular 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.	Trigonometric Functions
INSTRUCTIONAL RESOURCE: Math Examples 58	MATH EXAMPLES--Absolute Value Functions in Tabular and Graph Form This set of tutorials provides 40 examples of absolute value functions in tabular and graph form. NOTE: The download is a PPT file.	Special Functions
INSTRUCTIONAL RESOURCE: Math Examples 6	INSTRUCTIONAL RESOURCE: Math Examples--Cosecant Functions in Tabular and Graph Form This set of tutorials provides 65 examples of cosecant functions in tabular 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.	Trigonometric Functions
INSTRUCTIONAL RESOURCE: Math Examples 7	INSTRUCTIONAL RESOURCE: Math Examples--Cosine Functions in Tabular and Graph Form This set of tutorials provides 65 examples of cosine functions in tabular 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.	Trigonometric Functions
INSTRUCTIONAL RESOURCE: Math Examples 8	INSTRUCTIONAL RESOURCE: Math Examples--Cube Root Functions in Tabular and Graph Form This set of tutorials provides 40 examples of cube root functions in tabular 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.	Polynomial Functions and Equations
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: 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: 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 Crossword Puzzle--Linear Functions	Interactive Crossword Puzzle--Linear Functions This interactive crossword puzzle tests knowledge of key terms on the topic of linear functions. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Applications of Linear Functions
Interactive Math Game--Algebra Jeopardy: Linear Functions	Interactive Math Game--Algebra Jeopardy: Linear Functions Use this Jeopardy-style math game to review linear functions. This Jeopardy game has a bank of hundreds of questions for thousands of unique game experiences. To see the complete set of Math Games and Puzzles, click on this link. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Graphs of Linear Functions and Slope-Intercept Form
Interactive Math Game--DragNDrop--Equation of a Line from Two Points	Interactive Math Game--DragNDrop Math--Equation of a Line from Two Points In this drag-and-drop game, match the coordinates with the equation of the line that crosses these points. This game generates thousands of different equation combinations, offering an ideal opportunity for skill review in a game format. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Point-Slope Form, Slope-Intercept Form and Standard Form
Interactive Math Game--DragNDrop--Evaluating Linear Functions	Interactive Math Game--DragNDrop Math--Evaluating Linear Functions In this drag-and-drop game, match f(x) with f(a) for a given value of a. This game generates thousands of different equation combinations, offering an ideal opportunity for skill review in a game format. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Applications of Linear Functions
Interactive Math Game--DragNDrop--Evaluating Quadratic Functions	Interactive Math Game--DragNDrop Math--Evaluating Quadratic Functions In this drag-and-drop game, match f(x) with f(a) for a given value of a. This game generates thousands of different equation combinations, offering an ideal opportunity for skill review in a game format. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Quadratic Equations and Functions
Interactive Math Game--DragNDrop--Linear Functions	Interactive Math Game--DragNDrop Math--Linear Functions In this drag-and-drop game, match the linear function in slope-intercept form with its description. This game generates thousands of different equation combinations, offering an ideal opportunity for skill review in a game format. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Applications of Linear Functions
Interactive Math Game--DragNDrop--Linear Functions in Standard Form	Interactive Math Game--DragNDrop Math--Linear Functions in Standard Form In this drag-and-drop game, match the linear function in standard form with its slope-intercept counterpart. This game generates thousands of different equation combinations, offering an ideal opportunity for skill review in a game format. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Standard Form
Interactive Math Game--DragNDrop--The Point-Slope Form	Interactive Math Game--DragNDrop Math--The Point-Slope Form In this drag-and-drop game, match the linear equation in point-slope form with its slope-intercept counterpart. This game generates thousands of different equation combinations, offering an ideal opportunity for skill review in a game format. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Point-Slope Form
Interactive Math Game--DragNDrop--The Slope Formula	Engaging Math Activities & Games For Learning Slope In this drag-and-drop game, match the coordinates to the slope of the line that passes through the two points. This game generates thousands of different equation combinations, offering an ideal opportunity for skill review in a game format. Improve math skills and have fun at the same time with our selection of interactive games and activities that teach the concept of slope. These entertaining games make learning enjoyable and engaging, helping young learners understand this mathematical process better.	Slope
Interactive Math Game--Equation Burst: Linear Functions	Interactive Math Game--Equation Burst: Linear Functions Use this math game to review linear functions in slope-intercept form. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Slope-Intercept Form
Interactive Math Game--Memory Game: Linear Functions	Interactive Math Game--Memory Game: Linear Functions Use this math game to review Linear Functions. This is a Memory-style game in which students must remember the location of pairs of identical images. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Applications of Linear Functions
Interactive Math Game--Treasure Math!	Interactive Math Game--Treasure Math! Use this math game to review graphs of linear functions in slope-intercept form. Students are shown a "treasure map" coordinate grid and several numbered "gems". Find the right combination of gems to identify the graph of the equation shown and score 50 points; otherwise, lose 20 points. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Slope-Intercept Form
Interactive Word Search Puzzle--Linear Functions, Puzzle 1	Interactive Word Search Puzzle--Linear Functions, Puzzle 1 Solve an interactive word search puzzle on the topic of Linear Equations and Functions. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Applications of Linear Functions
Interactive Word Search Puzzle--Linear Functions, Puzzle 2	Interactive Word Search Puzzle--Linear Functions, Puzzle 2 Solve an interactive word search puzzle on the topic of Linear Equations and Functions. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Applications of Linear Functions
Interactive Word Search Puzzle--Linear Functions, Puzzle 3	Interactive Word Search Puzzle--Linear Functions, Puzzle 3 Solve an interactive word search puzzle on the topic of Linear Equations and Functions. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Applications of Linear Functions
Interactive Word Search Puzzle--Linear Functions, Puzzle 4	Interactive Word Search Puzzle--Linear Functions, Puzzle 4 Solve an interactive word search puzzle on the topic of Linear Equations and Functions. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Applications of Linear Functions
Interactive Word Search Puzzle--Linear Functions, Puzzle 5	Interactive Word Search Puzzle--Linear Functions, Puzzle 5 Solve an interactive word search puzzle on the topic of Linear Equations and Functions. This is part of a collection of math games and interactives. To see the complete collection of the games, click on this link. Note: The download is the teacher's guide. Related Resources To see additional resources on this topic, click on the Related Resources tab.	Applications of Linear Functions
Math Example--Exponential Concepts--Exponential Functions in Tabular and Graph Form: Example 1	Math Example--Exponential Concepts--Exponential Functions in Tabular and Graph Form: Example 1 Topic Exponential Functions Description This example demonstrates how to create a table of x-y coordinates and graph the exponential function y = 1 * 2x. The table lists coordinates for x values from -2 to 2, showing how the y values grow exponentially. The corresponding graph visually represents this rapid growth, with points plotted at (-2, 0.25), (-1, 0.5), (0, 1), (1, 2), and (2, 4).	Graphs of Exponential and Logarithmic Functions
Math Example--Exponential Concepts--Exponential Functions in Tabular and Graph Form: Example 10	Math Example--Exponential Concepts--Exponential Functions in Tabular and Graph Form: Example 10 Topic Exponential Functions Description This example demonstrates the creation of a table of x-y coordinates and the graphing of the exponential function y = 1 * 2-0.3x. The image provides both a table of values and a graph plotted on a coordinate plane, with points labeled for x values ranging from -2 to 2. This representation helps students visualize the behavior of a decreasing exponential function with a fractional exponent.	Graphs of Exponential and Logarithmic Functions
Math Example--Exponential Concepts--Exponential Functions in Tabular and Graph Form: Example 11	Math Example--Exponential Concepts--Exponential Functions in Tabular and Graph Form: Example 11 Topic Exponential Functions Description This example illustrates the process of creating a table of x-y coordinates and graphing the exponential function y = -0.4 * 2-0.6x. The image presents both a table of values and a graph plotted on a coordinate plane, with points labeled for x values from -2 to 2. This representation helps students visualize the behavior of a decreasing exponential function with a negative coefficient and a fractional exponent.	Graphs of Exponential and Logarithmic Functions
Math Example--Exponential Concepts--Exponential Functions in Tabular and Graph Form: Example 12	Math Example--Exponential Concepts--Exponential Functions in Tabular and Graph Form: Example 12 Topic Exponential Functions Description This example demonstrates the creation of a table of x-y coordinates and the graphing of the exponential function y = -0.5 * 2-x. The image provides both a table of values and a graph plotted on a coordinate plane, with points labeled for x values ranging from -2 to 2. This representation helps students visualize the behavior of a decreasing exponential function with a negative coefficient and a negative exponent.	Graphs of Exponential and Logarithmic Functions