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Mathematical Expression Editor
We define relations and graph examples of relations.
In the last section, we discussed plotting points using the Cartesian coordinate
system. While individual points can be useful, we often want to study collections of
points.
A relation is a collection of points of the form . If the point is in the relation, then
we say and are related.
This might seem like a strange definition, but hopefully a few examples will you see
the relationship (pun intended!) between the mathematical definitions of the words
“relation” and “related” and the way we often use these words in everyday
speech.
Metacognitive Moment What is this notation? When we write , we are usually
talking about a generic point. When we write , we have a specific point in mind. and
essentially serve as placeholders for a point such as or . We read as “ naught,
naught.”
Let’s look at the relationship between the number of chicken nuggets you can buy in
a single container at a local fast food store and the price in dollars for that container
of nuggets.
This table defines a relation because we can list these as the points , , , , and . We say
that 4 nuggets is related to and nuggets is related to . We can also represent this
relationship using a graph.
We could ask many mathematical questions about this relation. For example,
“What’s the cheapest way to buy 100 nuggets?” But for now, its enough to know it is
a relation and to know you can represent that relation using multiple representations
such as a table, a list, or a graph.
Metacognitive Moment Representations are important in math. There are often
many different ways information can be displayed or described. Common
representations include tables, graphs, formulas, or drawings. Representations help us
make sense of mathematics in different ways. Key for mathematical understanding is
making connections between different representations and being able to move or
‘translate’ between them.
It’s important to note that nothing about our definition of relation restricts what
points can be included. Assume that in our chicken nugget example above, there is a
coupon that allows you to buy chicken nuggets for and there an option to buy a
chicken nugget meal which includes chicken nuggets (and fries and a drink) for
.
We could modify the table of our relation to be:
This table still defines a relation and we can say that nuggets is related to and and .
Here is the graph of this relation.
There is a special type of relation called a function where each -coordinate is only
allowed to be related to one unique -coordinate. The relation in example 1 is a
function but the relation in example 2, which includes the coupon and a meal, is not
a function because 10 nuggets is related to more than one cost. Functions are
going to be extremely important and we will come back to them later in the
course.
Metacognitive Moment In example 2, we were given information (fries and
drink) that wasn’t used when describing the relation. This type of ‘extra’ information
is usually present when using math to make decisons or answer questions in our every
day life. One aspect of mathematics is determining what information to include or is
relevant.
The two examples we have seen so far have been a relations given by a list of points.
This does not have to be the case. Relations can contain an infinite number of points.
Some of the relations we will focus on are given by an equation relating two
variables.
Let’s consider the relation that is the collection of all points where .
Some points contained in this relation are , , , and , but these are not all the points
in this relation. Often, one of best ways to think about a relation given by an
equation is with a graph. The graph of is the circle of radius 2 centered at the
origin.
Verify algebraically that the point is on the curve .
Note that this is the same question as, “Verify that the point is a member of the
relation given by We want to show that this point satisfies the condition.
The point means that the and . If we plug these values in for and in
the equation, we want to check that both sides of the equation are equal.
A relation can also be initially given by a graph. For example, this is a relation.
We can list some of the points on this graph. It looks like and are points on this
curve, but there are many other points we cannot explicitly list.
