Topics
6. Functions & Relations
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United States of America
Grade 8

6.01 Represent relations

Lesson
Practice

Relations

A relation is a set of ordered pairs which represent a relationship.

For example, we can think of the names of people in a math class and their ages as ordered pairs, like \left(\text{Bob}, 13 \right). These pairs of information represent a relation.

If we chose a specific age (like 13), we could list all the names of the people who are this age. It could be one person, Bob, or it could be multiple. If a teacher wanted to look for the person who was 13 years old, that description might fit four people which means there's not one clear answer.

We can express the same relation in several different ways: as a mapping, a set of ordered pairs, an input-output table, a graph in the coordinate plane, or as an equation in terms of x and/or y that describes a graph.

A mapping diagram shows how the input values are assigned one or more output values. Consider the mapping below:

An image showing an example of a mapping of a relation. Ask your teacher for more information.

A mapping of a relation

We can write an input-output table from the mapping, making sure that each pair is represented. Remember that the the first value of a relation is an input value and the second value is the output value. The input is the value of x that is applied to the relation. The output is the y, or the answer that is received as a result of putting x into the relation. A table can be laid out horizontally (like the one shown below) or vertically.

x-1012
y2024

This also corresponds to the set of ordered pairs \{(-1, 2), (0, 0), (1, 2), (2, 4)\}, which can be graphed in the coordinate plane, as shown below.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y

A graph of the relation represents the (x, y) pairs in the coordinate plane.

Examples

Example 1

Write the relation \{(2, 2), (4, 4), (6, 3), (7, 5)\} in the table below.

x2467
y
Worked Solution
Create a strategy

Write the second coordinate of each ordered pair in the y row, below the x-value it corresponds to.

Apply the idea
x2467
y2435

Example 2

Consider the relation: \{(-9, -5), (-5, -10), (-5, -4), (-3, 7), (-2, -4), (-1, 1)\}. Represent the relation on the coordinate plane.

Worked Solution
Create a strategy

The first value of each ordered pair tells us how to move along the x-axis, while the second value tells us how to move along the y-axis.

Apply the idea
-8
-6
-4
-2
2
x
-10
-8
-6
-4
-2
2
4
6
8
y

Example 3

A relation is defined as follows: y=-4 if x is positive and y=4 if x is 0 or negative.

a

Complete the table.

x-4-3-2-101234
y
Worked Solution
Create a strategy

For each positive x-value: y=-4, otherwise y=4.

Apply the idea

When x=1, \, 2, \, 3 , \, 4: y=-4.

When x=0, \, -1, \, -2, \, -3, \, -4: y=4.

x-4-3-2-101234
y44444-4-4-4-4
b

Plot the points on the number plane.

Worked Solution
Create a strategy

Plot the points from the completed table in part (a).

Apply the idea
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
Idea summary

A relation is a set of ordered pairs which represent a relationship.

We can express the same relation in several different ways: as a mapping, a set of ordered pairs, an input-output table, a graph in the coordinate plane, or as an equation in terms of x and/or y that describes a graph.

Outcomes

8.F.A.1

Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output

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