MCR3U (1.1) - Functions - relations, domain and range
Summary
TLDRThis 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
- π A relation is defined as an ordered pair of values, where X represents the independent variable and Y represents the dependent variable.
- π Two examples of relations discussed are: the impact of study hours on test scores and the effect of age on a person's height.
- π Relations can be displayed in four ways: tables, coordinate form, mapping diagrams, and graphs.
- π 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.
- π Mapping diagrams are a way to visually represent relations by listing independent and dependent variables and matching them up.
- 𧩠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.
- π 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.
- π The domain of a relation is the set of all possible values of the independent variable, listed from lowest to highest without repetition.
- π The range of a relation is the set of all possible values of the dependent variable, also listed from lowest to highest without repetition.
- π 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
π 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.
π 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.
π 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.
π 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.
Mindmap
Keywords
π‘Relation
π‘Independent Variable
π‘Dependent Variable
π‘Ordered Pair
π‘Mapping Diagram
π‘Coordinate Form
π‘Graph
π‘Domain
π‘Range
π‘Function
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
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