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Illustrative Math-Media4Math Alignment

 

 

Illustrative Math Alignment: Grade 8 Unit 3

Functions and Volume

Lesson 3: Equations for Functions

Use the following Media4Math resources with this Illustrative Math lesson.

Thumbnail Image Title Body Curriculum Topic
Definition--Functions and Relations Concepts--Decreasing Function Definition--Functions and Relations Concepts--Decreasing Function Decreasing Function

Topic

Functions and Relations

Definition

A decreasing function is a function where the value of the function decreases as the input increases.

Description

Decreasing functions are important in mathematics because they describe scenarios where an increase in one variable leads to a decrease in another. This is mathematically represented as 

f(x1) > f(x2) for any š‘„1 < š‘„2 ā€‹

Relations and Functions
Definition--Functions and Relations Concepts--Dependent Variable Definition--Functions and Relations Concepts--Dependent Variable Dependent Variable

Topic

Functions and Relations

Definition

A dependent variable is a variable whose value depends on one or more other variables.

Relations and Functions
Definition--Functions and Relations Concepts--Discontinuous Function Definition--Functions and Relations Concepts--Discontinuous Function Discontinuous Function

Topic

Functions and Relations

Definition

A discontinuous function is a function that has one or more points where it is not continuous.

Relations and Functions
Definition--Functions and Relations Concepts--Discrete Functions Definition--Functions and Relations Concepts--Discrete Functions Discrete Functions

Topic

Functions and Relations

Definition

Discrete functions are functions that are defined only for a set of discrete points.

Special Functions
Definition--Functions and Relations Concepts--Distorting a Function Horizontally Definition--Functions and Relations Concepts--Distorting a Function Horizontally Distorting a Function Horizontally

Topic

Functions and Relations

Definition

Distorting a function horizontally involves stretching or compressing the graph of the function along the x-axis.

Description

Horizontal distortions of functions are significant because they alter the input values while maintaining the overall shape of the graph. This is mathematically represented as 

f(kx)

where k is a constant. If k > 1, the function compresses horizontally, and if 0 < š‘˜ < 1, it stretches. 

Relations and Functions
Definition--Functions and Relations Concepts--Distorting a Function Vertically Definition--Functions and Relations Concepts--Distorting a Function Vertically Distorting a Function Vertically

Topic

Functions and Relations

Definition

Distorting a function vertically involves stretching or compressing the graph of the function along the y-axis.

Description

Vertical distortions of functions are significant because they alter the output values while maintaining the overall shape of the graph. This is mathematically represented as 

kf(x)

Relations and Functions
Definition--Functions and Relations Concepts--Domain of a Function Definition--Functions and Relations Concepts--Domain of a Function Domain of a Function

Topic

Functions and Relations

Definition

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

Relations and Functions
Definition--Functions and Relations Concepts--Even Function Definition--Functions and Relations Concepts--Even Function Even Function

Topic

Functions and Relations

Definition

An even function is a function that satisfies the condition f(x) = f(āˆ’x) for all x in its domain.

Description

Even functions are important in mathematics because they exhibit symmetry about the y-axis. This property is useful in various fields, including physics and engineering, where symmetry simplifies analysis and problem-solving. For example, the function 

f(x)=x2

Relations and Functions
Definition--Functions and Relations Concepts--Function Definition--Functions and Relations Concepts--Function Function

Topic

Functions and Relations

Definition

A function is a relation that uniquely associates each element of a set with exactly one element of another set.

Relations and Functions
Definition--Functions and Relations Concepts--Function Machine Definition--Functions and Relations Concepts--Function Machine Function Machine

Topic

Functions and Relations

Definition

A function machine is a conceptual tool used to understand how functions work by visualizing inputs and outputs.

Description

The function machine is a useful educational tool that helps students grasp the concept of functions by visualizing the process of converting inputs into outputs. It emphasizes the idea that a function takes an input, processes it according to a specific rule, and produces an output. For example, if the function is 

f(x) = x + 2

Relations and Functions
Definition--Functions and Relations Concepts--Function Notation Definition--Functions and Relations Concepts--Function Notation Function Notation

Topic

Functions and Relations

Definition

Function notation is a way to represent functions in the form f(x), where f denotes the function and x denotes the input variable.

Description

Function notation is a standardized way to write functions, making it easier to understand and communicate mathematical relationships. It is widely used in algebra, calculus, and other branches of mathematics. For example, the notation 

f(x) = x2 

Relations and Functions
Definition--Functions and Relations Concepts--Function Table Definition--Functions and Relations Concepts--Function Table Function Table

Topic

Functions and Relations

Definition

A function table is a table that lists input values and their corresponding output values for a given function.

Description

Function tables are useful tools in mathematics for organizing and analyzing the relationship between inputs and outputs of a function. They help in visualizing how a function behaves and in identifying patterns. For example, a function table for 

f(x) = 2x + 1

Relations and Functions
Definition--Functions and Relations Concepts--Graphs of Functions Definition--Functions and Relations Concepts--Graphs of Functions Graphs of Functions

Topic

Functions and Relations

Definition

Graphs of functions are visual representations of the relationship between input values and output values of a function.

Description

Graphs of functions are essential tools in mathematics for visualizing how a function behaves. They provide a clear picture of the relationship between the input and output values, making it easier to analyze and interpret the function. For example, the graph of 

f(x) = x2 

Relations and Functions
Definition--Functions and Relations Concepts--Graphs of Relations Definition--Functions and Relations Concepts--Graphs of Relations Graphs of Relations

Topic

Functions and Relations

Definition

Graphs of relations are visual representations of the relationship between two sets of values, not necessarily functions.

Relations and Functions
Definition--Functions and Relations Concepts--Increasing Function Definition--Functions and Relations Concepts--Increasing Function Increasing Function

Topic

Functions and Relations

Definition

An increasing function is a function where the value of the function increases as the input increases.

Relations and Functions
Definition--Functions and Relations Concepts--Independent Variable 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 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(fāˆ’1(x)) = x.

Functions and Their Inverses
Definition--Functions and Relations Concepts--Odd Function 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(āˆ’x) = āˆ’f(x) for all x in its domain.

Relations and Functions
Definition--Functions and Relations Concepts--Parametric Equations 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 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 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 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) = nā‹…f(nāˆ’1) with f(0)=1 

Relations and Functions
Definition--Functions and Relations Concepts--Reflecting a Function 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 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 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 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 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 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 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 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 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: Creating an Exercise Chart INSTRUCTIONAL RESOURCE: Algebra Application: Creating an Exercise Chart INSTRUCTIONAL RESOURCE: Algebra Application: Creating an Exercise Chart

In this Algebra Application, students develop a linear mathematical model based on the maximum heart rate during exercise based on age. Using this model, students investigate heart rate for moderate and vigorous workouts. The math topics covered include: Mathematical modeling, Linear functions, Families of functions, Computational thinking. The culminating activity is for students to create a simple program (using either a spreadsheet or a programming language like Python) to develop an exercise chart. This is a great back-to-school activity for middle school or high school students. A relevant real-world application allows them to review math concepts.

Applications of Linear Functions and Slope-Intercept Form
INSTRUCTIONAL RESOURCE: Algebra Application: Linear Functions: Circumference vs. Diameter INSTRUCTIONAL RESOURCE: Algebra Application: Linear Functions: Circumference vs. Diameter INSTRUCTIONAL RESOURCE: Algebra Application: Linear Functions: Circumference vs. Diameter

In this Algebra Application, students study the direction between diameter and circumference of a circle. Through measurement and data gathering students analyze the line of best fit and explore ways of calculating pi. The math topics covered include: Mathematical modeling, Linear functions, Data gathering and analysis, Ratios, Direct variation. This is a great back-to-school activity for middle school or high school students. This is also a great crossover activity that ties algebra and geometry.

Applications of Linear Functions, Applications of Ratios, Proportions, and Percents and Applications of Circles
INSTRUCTIONAL RESOURCE: Tutorial: Analyzing Graphs of Quadratic Functions in Factored Form 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 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 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
PowerPointPresentations.jpg Instructional Resource: Applications of Linear Functions: Circumference vs. Diameter

In this Slide Show, apply concepts of linear functions to the context of circumference vs. diameter.

Note: The download is a PPT file.

Related Resources

To see the complete collection of Tutorials on this topic, click on this link: https://bit.ly/3g0P3cN
Applications of Linear Functions
PowerPointPresentations.jpg Instructional Resource: Applications of Linear Functions: Cost vs. Time

In this Slide Show, apply concepts of linear functions to the context of cost vs. time data and graphs.

Note: The download is a PPT file.

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To see the complete collection of Tutorials on this topic, click on this link: https://bit.ly/3g0P3cN
Applications of Linear Functions
PowerPointPresentations.jpg Instructional Resource: Applications of Linear Functions: Cricket Chirps

In this Slide Show, apply concepts of linear functions to the context of cricket chirps.

Note: The download is a PPT file.

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To see the complete collection of Tutorials on this topic, click on this link: https://bit.ly/3g0P3cN
Applications of Linear Functions
PowerPointPresentations.jpg Instructional Resource: Applications of Linear Functions: Distance vs. Time

In this Slide Show, apply concepts of linear functions to the context of distance vs. time data and graphs.

Note: The download is a PPT file.

Related Resources

To see the complete collection of Tutorials on this topic, click on this link: https://bit.ly/3g0P3cN
Applications of Linear Functions
PowerPointPresentations.jpg Instructional Resource: Applications of Linear Functions: Hooke's Law

In this Slide Show, apply concepts of linear functions to the context of Hooke's law.

Note: The download is a PPT file.

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To see the complete collection of Tutorials on this topic, click on this link: https://bit.ly/3g0P3cN
Applications of Linear Functions and Proportions
PowerPointPresentations.jpg Instructional Resource: Applications of Linear Functions: Momentum and Impulse

In this Slide Show, apply concepts of linear functions to the context of momentum and impulse.

Note: The download is a PPT file.

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Applications of Linear Functions
PowerPointPresentations.jpg Instructional Resource: Applications of Linear Functions: Perimeter of a Rectangle

In this Slide Show, apply concepts of linear functions to the context of rectangular perimeter.

Note: The download is a PPT file.

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To see the complete collection of Tutorials on this topic, click on this link: https://bit.ly/3g0P3cN
Applications of Linear Functions
PowerPointPresentations.jpg Instructional Resource: Applications of Linear Functions: Saving Money

In this Slide Show, apply concepts of linear functions to the context of saving money.

Note: The download is a PPT file.

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To see the complete collection of Tutorials on this topic, click on this link: https://bit.ly/3g0P3cN
Applications of Linear Functions
PowerPointPresentations.jpg Instructional Resource: Applications of Linear Functions: Speed and acceleration

In this Slide Show, apply concepts of linear functions to the context of speed and acceleration.

Note: The download is a PPT file.

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To see the complete collection of Tutorials on this topic, click on this link: https://bit.ly/3g0P3cN
Applications of Linear Functions
PowerPointPresentations.jpg Instructional Resource: Applications of Linear Functions: Temperature Conversion

In this Slide Show, apply concepts of linear functions to the context of converting Celsius and Fahrenheit temperatures.

Note: The download is a PPT file.

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To see the complete collection of Tutorials on this topic, click on this link: https://bit.ly/3g0P3cN
Applications of Linear Functions
INSTRUCTIONAL RESOURCE: Desmos Tutorial: Linear Functions in Slope Intercept Form 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 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: Desmos Tutorial: Matching Coordinates to Absolute Value Functions INSTRUCTIONAL RESOURCE: Desmos Tutorial: Matching Coordinates to Absolute Value Functions INSTRUCTIONAL RESOURCE: Desmos Tutorial: Matching Coordinates to Absolute Value Functions

In this Slide Show, use the Desmos graphing calculator to explore absolute value functions. 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.

Special Functions
INSTRUCTIONAL RESOURCE: Desmos Tutorial: Matching Coordinates to Linear Functions INSTRUCTIONAL RESOURCE: Desmos Tutorial: Matching Coordinates to Linear Functions INSTRUCTIONAL RESOURCE: Desmos Tutorial: Matching Coordinates to Linear Functions

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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.

Applications of Linear Functions