MCR3U (1.1) - Functions - relations, domain and range

Summary

This video introduces the concept of relations, focusing on independent and dependent variables. It explains how to represent relations in four different ways: tables, coordinate pairs, mapping diagrams, and graphs. Using examples, the video demonstrates how study hours impact test scores and how age affects height. It also defines domain and range, emphasizing the importance of listing values from lowest to highest without repetition. The video concludes with a teaser for the next lesson on functions, which will build upon the examples discussed.

Takeaways

• 🔑 A relation is defined as an ordered pair of values, where X represents the independent variable and Y represents the dependent variable.
• 📚 Two examples of relations discussed are: the impact of study hours on test scores and the effect of age on a person's height.
• 📊 Relations can be displayed in four ways: tables, coordinate form, mapping diagrams, and graphs.
• 📈 In coordinate form, the independent variable is listed first, followed by the dependent variable, as in (2, 55) for 2 hours studied and a score of 55.
• 📝 Mapping diagrams are a way to visually represent relations by listing independent and dependent variables and matching them up.
• 🧩 In mapping diagrams, variables are listed in order from lowest to highest value without repetition, even if they appear more than once in the data.
• 📉 Graphs are used to plot the points from the relations, with the independent variable on the x-axis and the dependent variable on the y-axis.
• 📌 The domain of a relation is the set of all possible values of the independent variable, listed from lowest to highest without repetition.
• 🌟 The range of a relation is the set of all possible values of the dependent variable, also listed from lowest to highest without repetition.
• 📚 The concepts of domain and range will be further explored in future videos, with a focus on understanding functions.

Q & A

What is a relation in the context of mathematics?

A relation is an ordered pair of X values (independent variable) and Y values (dependent variable). The X values represent the independent variable, and the Y values represent the dependent variable.

In the first example, what is the independent and dependent variable?

In the first example, the number of hours studied is the independent variable, and the final mark on the test is the dependent variable.

How can relations be represented?

Relations can be represented in four ways: a table of values, a list of coordinates, a mapping diagram, and a graph.

What is a mapping diagram and how is it used?

A mapping diagram is a visual representation where the independent and dependent variables are listed in boxes, and lines are drawn to connect related values. Variables are listed in order, and each variable appears only once.

How do you represent a relation in a graph?

In a graph, the independent variable is plotted on the x-axis, and the dependent variable is plotted on the y-axis. Points are plotted based on the coordinate pairs derived from the table of values.

What is the domain of a relation?

The domain is the set of all values of the independent variable in a relation, listed from lowest to highest, without repetition.

What is the range of a relation?

The range is the set of all values of the dependent variable in a relation, listed from lowest to highest, without repetition.

Why is it important to list variables in order in mapping diagrams?

Listing variables in order (from lowest to highest) helps maintain clarity and organization in the representation of relationships between variables, ensuring accuracy.

How does domain and range relate to mapping diagrams?

Both domain and range are similar to mapping diagrams, as they list the values of the independent (domain) and dependent (range) variables in order without repeating any value.

What concept will be covered in the next video following this one?

The next video will cover the concept of functions, which is an important mathematical concept related to relations and will be tested in the course.

Outlines

00:00

📝 Introduction to Relations: Independent and Dependent Variables

The video begins by defining what a relation is, explaining it as an ordered pair of X (independent) and Y (dependent) values. The speaker gives two examples: one showing how a final mark on a test is affected by hours studied, and another demonstrating how age influences height. It emphasizes identifying independent and dependent variables in each relation and introduces the concept of showing relations in different forms such as tables, coordinates, mapping diagrams, and graphs. The example of a student studying 2 hours and achieving a final mark of 55 is used to illustrate how relations can be represented in coordinate form.

05:02

📊 Exploring Relations Through Tables, Coordinates, and Mapping Diagrams

This section dives deeper into the ways to show relations. The speaker explains how to convert relations into mapping diagrams by listing independent and dependent variables in order and matching them, highlighting the importance of not repeating values and maintaining a sequential order. The concept is further illustrated by mapping examples, where values like 165 and 167.5 are linked to a single independent variable (age 10). This helps demonstrate how relations can have multiple dependent variables, but each independent variable is only listed once.

📉 Graphical Representation of Relations

The video transitions to showing relations through graphs, using the examples from the previous sections. It explains how the independent variable is plotted on the X-axis and the dependent variable on the Y-axis. The speaker reiterates the importance of presenting relations in four ways: tables, coordinate pairs, mapping diagrams, and graphs. By plotting the table of values from both examples, viewers can visually understand how these relations translate into graph form. This reinforces the interconnectedness of the different methods of representation.

📖 Introduction to Domain and Range

The speaker introduces the concepts of domain and range, defining the domain as the set of values for the independent variable, and the range as the set of values for the dependent variable. Using the earlier examples, the speaker explains how to list domain and range values in order, without repetition, even if a variable appears more than once in the relation. The section emphasizes the importance of writing out values in order, preparing viewers for future discussions on functions in upcoming videos.

Keywords

💡Relation

A relation in mathematics refers to a set of ordered pairs that show a connection between two variables. In the video, the relation between the independent variable (e.g., hours studied) and the dependent variable (e.g., final mark) is demonstrated through examples. Understanding how one variable affects another is crucial for analyzing relations.

💡Independent Variable

The independent variable is the variable that is changed or controlled in a mathematical relationship. In the video, the independent variable in the first example is the number of hours studied, which affects the final mark on the test. It is important because it helps to determine how it influences the dependent variable.

💡Dependent Variable

The dependent variable is the outcome or response that depends on the independent variable. In the first example from the video, the final mark is the dependent variable because it depends on the hours studied. This concept helps in understanding how changes in the independent variable impact the results.

💡Ordered Pair

An ordered pair consists of two elements, where the first element represents the independent variable and the second represents the dependent variable. In the video, examples like (2, 55) represent the relation between 2 hours studied (independent) and a final mark of 55 (dependent). It is fundamental in graphing and mapping relations.

💡Mapping Diagram

A mapping diagram is a visual representation of a relation, where independent variables are listed in one box and dependent variables in another, with lines connecting related values. The video shows how mapping diagrams can simplify relations by listing variables in order and only once, making the relationships clearer.

💡Coordinate Form

Coordinate form is another method to represent relations where independent and dependent variables are shown as points on a graph. The video demonstrates how to use coordinates like (2, 55) to visualize relations, providing a clearer understanding of how variables are linked.

💡Graph

A graph is a visual representation of data showing the relationship between two variables. In the video, graphs are used to plot the independent variable on the x-axis and the dependent variable on the y-axis. This helps in visualizing the trend or connection between variables, such as hours studied and final marks.

💡Domain

The domain of a relation is the set of all possible values for the independent variable. In the video, the domain is derived from the list of hours studied or ages in the examples. Listing the domain helps to understand the possible range of input values in a relation.

💡Range

The range is the set of all possible values for the dependent variable in a relation. In the video, the range corresponds to the final marks or heights depending on the example. Understanding the range is key to knowing the possible outcomes based on the independent variable.

💡Function

A function is a special type of relation where each input (independent variable) has exactly one output (dependent variable). The video hints at the upcoming explanation of functions, which is an important concept for understanding how structured relationships between variables operate in mathematics.

Highlights

Introduction to relations, domain, and range, with a simple definition of relations as ordered pairs of X (independent) and Y (dependent) variables.

Example 1: Demonstrates how final marks on a test depend on the number of hours studied (independent variable).

Example 2: Shows the relationship between a person's age (independent variable) and their height (dependent variable).

Key concept: Identifying which is the independent variable and which is the dependent variable in a relation.

Relations can be represented in multiple ways, including in tables, where independent and dependent variables are paired.

Introduction to coordinate form, where ordered pairs of independent and dependent variables are plotted (e.g., (2, 55)).

Introduction to mapping diagrams as a way to visually connect independent and dependent variables, ensuring the variables are listed in order.

Example of a mapping diagram where a variable appears twice but is only listed once in the diagram, matched with two dependent variables.

Mapping diagrams require variables to be listed in order from lowest to highest values.

Graphing relations is another method of visualizing data, with the independent variable on the x-axis and the dependent variable on the y-axis.

Four ways to represent a relation: Table, coordinates, mapping diagram, and graph.

Introduction to domain and range: The domain is the set of independent variable values, and the range is the set of dependent variable values.

Domain and range are similar to mapping diagrams, where values are listed from lowest to highest, without repetition.

Example of a domain where a value appears twice but is only written once in the domain list.

Preview of next video: Explanation of functions using the same examples, a key concept that will be tested.

Transcripts

yo what's up guys in this video we're

going to talk about relations domain and

range and to begin let's start the video

off with a simple definition of what a

relation is and if you remember to

previous grades it's just basically an

ordered pair of X values remember the X

values represent the independent

variable that we're working with and Y

values which is the dependent variable

so let's show two examples of certain

relations so the first example that

we're dealing with is we're seeing how

is the final mark on a test affected by

the number of hour studied before and

then in the second relation we're

seeing how does the age of a

person affect their height now the first

thing that you always want to check with

relations Is which is the in independent

variable and which is the dependent

variable so in this specific relation

here obviously our final Mark is

dependent on the hours that we study so

the final

Mark is the dependent

variable and the hour studied is the

independent

variable and then in our second relation

the height is dependent on on the age so

the height is the dependent

variable and the age is the

independent

variable now there are multiple ways to

show a relation so we started off by

showing these two relations in tables

but you can also show them in coordinate

forms so for example in this case case

our first student that we tested they uh

they studied 2 hours and got a final

Mark of 55 so the independent variable

is this two and the dependent variable

is this 55 well we can put that as a

coordinate right the independent

variable comes first of two and the

dependent variable comes second of 55

and then we did that with the rest of

the students we took all of these points

and then we put them in coordinate form

same thing here we took these point

points here and put them in coordinate

form now a third way to show these

relations Is through something called a

mapping diagram and a mapping diagram is

very simple you just basically list out

the

independent and dependent variables in

these boxes and then you just match them

up so the two with the 55 the three with

the 58 etc etc now the thing to remember

about mapping diagrams is that when you

put your variables in these boxes

they have to be in

order and you only list them out once

and that's better shown through this

relation so notice here's our mapping

diagram of this relation notice how this

10 here this independent variable

appears twice but when we put it in the

mapping diagram we only put it once but

we match it to two different dependent

variables of 165 and 167.5

and then notice how in both mapping

diagrams I listed all of the variables

in order from lowest value to highest

value that's what I mean by order so for

example in uh in this relation the

170 came after the

172.5 or the 162.5 came after the 165

but when I put them in the mapping

diagram I listed them in order from

lowest to highest and the fourth and

final way to show a relation is through

a graph I had to erase the other three

ways just to give myself some extra room

but just add this to your notes this is

the fourth way so I took the table of

values from both examples and just

plotted the points now remember the

independent variable goes on the xais so

the hours and the age respectively and

then the dependent variable goes on the

Y AIS so the mark in example one the

final Mark in percentages and the height

in example two so in both examples we

took a relation and we showed it in four

different ways first way was a table of

values second way was a list of

coordinates the Third Way was a mapping

diagram and then the fourth way was a

graph now before finishing off the video

I want to go over two more Concepts and

that's domain and range now domain and

range is something that we're going to

be getting into a lot more detail in

future sections and future videos but

for now let's just start off with the

basic definitions so the domain is the

set of values of the independent

variable in a relation and the range is

the set of values of the dependent

variable in a relation so going back to

our two

examples if we were asked to find the

domain and range of each of them it

would be listed out like this

so it's actually sort of similar to the

mapping diagrams so what you do is you

list out all the points from lowest to

highest in order and you don't repeat

any variables that appear twice so for

example this 10 here notice how this x

value of 10 repeats twice there are two

coordinates that have an x value of 10

but when we write the

domain it only appears once so again

when you're writing the domain and range

out it's always from lowest to highest

and you always write out the values just

once now in the next video I'm going to

go over a concept called a function and

that's what this course is called

functions it's a very important concept

you'll be getting tested on it for sure

and the way I'm going to explain it is

I'm going to use the same examples so

it's going to be a continuation of this

video