Illustrative Math Alignment: Grade 8 Unit 3

A set of resources for the TI-Nspire, TI-Nspire CX, and the Nspire App.

Here is a collection of PowerPoint presentations on selected topics in Quadratics.

Overview

Overview

The "Functions and Relations" collection from Media4Math provides a comprehensive set of definitions essential for understanding key mathematical concepts. This collection includes terms such as domain, range, function, relation, inverse function, and composite function. These definitions are crucial for students as they form the foundation for more advanced topics in mathematics, such as calculus and algebra.

Overview

Overview

Overview

Overview

Overview

Overview

Overview

This collection of 28 math examples focuses on the topic of Graphs of Rational Functions, offering a comprehensive resource for students and educators alike. The examples cover a wide range of skills and concepts related to rational functions, progressively increasing in complexity to challenge learners at various levels of understanding.

Here we gather key terms, definitions, and formulas around Measures of Central Tendency.

Overview

Linear Expressions, Equations, and Functions

In this TI Nspire tutorial, the Graph window is used to graph a linear function. This video supports the TI-Nspire Clickpad and Touchpad. This Mini-Tutorial Video includes a worksheet.

Related Resources

Video Transcripts

In this TI Nspire tutorial, the Graph window is used to graph a linear inequality in slope-intercept form using sliders. This video supports the TI-Nspire Clickpad and Touchpad. This Mini-Tutorial Video includes a worksheet.

Linear Expressions, Equations, and Functions

Linear Expressions, Equations, and Functions

Linear Expressions, Equations, and Functions

In this TI Nspire tutorial, the Graph window is used to create a slider-based graph of a linear function in slope-intercept form. This video supports the TI-Nspire Clickpad and Touchpad. This Mini-Tutorial Video includes a worksheet. .

Video Tutorial: Linear Functions: Negative Slope, Negative y-Intercept. In this video tutorial, students learn the basics of linear functions in slope-intercept form. In particular, look at the case of a linear function with a negative slope and a negative y-intercept.

Video Tutorial: Linear Functions: Negative Slope, Positive y-Intercept. In this video tutorial, students learn the basics of linear functions in slope-intercept form. In particular, look at the case of a linear function with a negative slope and a positive y-intercept.

Video Tutorial: Linear Functions: Negative Slope, Zero y-Intercept. In this video tutorial, students learn the basics of linear functions in slope-intercept form. In particular, look at the case of a linear function with a negative slope and a zero y-intercept.

Video Tutorial: Linear Functions: Positive Slope, Negative y-Intercept. In this video tutorial, students learn the basics of linear functions in slope-intercept form. In particular, look at the case of a linear function with a positive slope and a negative y-intercept.

Video Tutorial: Linear Functions: Positive Slope, Positive y-Intercept. In this video tutorial, students learn the basics of linear functions in slope-intercept form. In particular, look at the case of a linear function with a positive slope and a positive y-intercept.

Video Tutorial: Linear Functions: Positive Slope, Zero y-Intercept. In this video tutorial, students learn the basics of linear functions in slope-intercept form. In particular, look at the case of a linear function with a positive slope and a zero y-intercept.

Topic

Functions and Relations

Definition

An absolute value function is a function that contains an algebraic expression within absolute value symbols. It is defined as 

$$ f(x) = |x| $$

where |x| denotes the absolute value ofx.

Description

The absolute value function is significant in mathematics because it measures the distance of a number from zero on the number line, regardless of direction. This function is linear in nature, as it can be broken into two linear pieces: one for 

$$ x \geq 0 $$

 and one for 

$$ x < 0 $$

Topic

Functions and Relations

Definition

A composite function is a function that is formed when one function is applied to the result of another function.

Description

Composite functions are significant in mathematics because they allow the combination of two functions to form a new function. This is denoted as (f∘g)(x) = f(g(x)). Composite functions are widely used in various fields, including computer science for function composition in programming and in calculus for chain rule applications. For example, if f(x) = 2x and g(x) = x + 3, then the composite function 

(f∘g)(x) = f(g(x)) = 2(x + 3) = 2x+6

Topic

Functions and Relations

Definition

Conic sections are the curves obtained by intersecting a plane with a double-napped cone.

Description

Conic sections include circles, ellipses, parabolas, and hyperbolas, which are fundamental in the study of geometry and algebra. These shapes are described by quadratic equations and have numerous applications in physics, engineering, and astronomy. For example, the orbits of planets are ellipses, and parabolic mirrors are used in telescopes and satellite dishes. The general quadratic equation for conic sections is 

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

Topic

Functions and Relations

Definition

A constant function is a function that always returns the same value, no matter the input.

Description

The constant function is one of the simplest types of functions in mathematics, expressed as 

f(x) = c

Topic

Functions and Relations

Definition

A continuous function is a function that does not have any breaks, holes, or gaps in its domain.

Description

Continuous functions are fundamental in calculus and mathematical analysis because they allow for the application of limits, derivatives, and integrals. A function f(x) is continuous if, for every point 𝑐 c in its domain, 

lim x→c ​ f(x) = f(c)

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 𝑥1 < 𝑥2 ​

Topic

Functions and Relations

Definition

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

Topic

Functions and Relations

Definition

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

Topic

Functions and Relations

Definition

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

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 < 𝑘 < 1, it stretches. 

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)

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.

Topic

Functions and Relations

Definition

An even function is a function that satisfies the condition f(x) = f(−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

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.