We define relations and graph examples of examples.
In the last section, we discussed graphing 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.
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 for that container of
nuggets.
Now this table defines a relation because we can think of these as the points , , , ,
and . We say that 4 nuggets is related to $1.99. And 40 nuggets is related to $8.99.
We can also represent this relationship using a graph.
We could ask lots of mathematical questions about this relation. One such example
might be, “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 in multiple ways
such as a table, a list, or a graph.
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 10 chicken nuggets for $3.00.
Assume there is also an option to buy a chicken nugget meal which includes 10
chicken nuggets (and fries and a drink but we don’t care about those) for
$6.49.
Then we could modify the table of our relation to be:
Now this table still defines a relation and we can say that 10 nuggets is related to
$4.49 and $3.00 and $6.49. Here is the graph of this relation.
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 be studying the most 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 intitally 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.
