Illustrative Math Alignment: Grade 8 Unit 3

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.

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

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 

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

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 

Topic

Functions and Relations

Definition

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

Topic

Functions and Relations

Definition

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

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.

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.

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.

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.

Topic

Functions and Relations

Definition

Piecewise functions are functions defined by multiple sub-functions, each applying to a certain interval of the domain.

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 

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 

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

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.

Topic

Functions and Relations

Definition

A step function is a function that increases or decreases abruptly at certain points, creating a series of steps.

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 

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 

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 

Topic

Functions and Relations

Definition

Visual models of relations are graphical representations that illustrate the relationship between two sets of values, not necessarily functions.

Topic

Functions and Relations

Definition

Visualization of the dependent variable involves graphically representing the outcomes or responses that depend on 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.

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.

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.

This slide show provides 8 examples of quadratic functions in factored form and analyzes their graphs.

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This slide show provides 18 examples of quadratic functions in standard form and analyzes their graphs.

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This slide show provides 8 examples of quadratic functions in vertex form and analyzes their graphs.

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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.

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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.

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In this Slide Show, learn how to convert from linear equations in standard form to slope intercept form.

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This set of tutorials provides 54 examples of exponential functions in tabular and graph form.

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The complete set of 13 examples that make up this set of tutorials.

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The complete set of 28 examples that make up this set of tutorials.

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This set of 21 tutorials provides different examples of linear functions in function table and graph form.

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This set of tutorials provides 120 examples of logarithmic functions in tabular and graph form.

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This set of tutorials provides 37 examples of quadratic functions in tabular and graph form.

Library of Instructional Resources

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This set of tutorials provides 28 examples of rational functions in tabular and graph form.

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This set of tutorials provides 65 examples of secant functions in tabular and graph form.

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This set of tutorials provides 65 examples of sine functions in tabular and graph form.

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This set of tutorials provides 40 examples of square root functions in tabular and graph form.

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This set of tutorials provides 65 examples of tangent functions in tabular and graph form.

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This set of tutorials provides 40 examples of absolute value functions in tabular and graph form. NOTE: The download is a PPT file.

This set of tutorials provides 65 examples of cosecant functions in tabular and graph form.

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This set of tutorials provides 65 examples of cosine functions in tabular and graph form.

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This set of tutorials provides 40 examples of cube root functions in tabular and graph form.

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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.

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.

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.

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.