3.01 Review: Plot points and represent relations

A pair of coordinates describes a point's position away from the origin. A negative coordinate indicates a direction of left or down from the origin. A positive coordinate indicates a direction of right or up from the origin.

When we plot a pair of coordinates, we draw them on a coordinate plane. For example, to plot the point \left(4, -2\right), we would start at the origin, move 4 spaces to the right and 2 spaces down before plotting the point.

To name coordinates, we write the horizontal value, then the vertical value that a point is away from the origin. Remember, points are written as ordered pairs \left(x, y\right). We can always remember this because x comes before y in the alphabet, so we should do the same in our ordered pair.

Examples

Example 1

Consider the points A and B.

-4

-3

-2

-1

1

2

3

4

x

-4

-3

-2

-1

1

2

3

4

y

a

Write the coordinates of the plotted points.

Worked Solution

Create a strategy

Count the spaces the points lie from the origin and consider the direction from the origin.

Apply the idea

Point A lies 4 units to the right of the origin. Point B lies 1.5 units to the left of the origin.

The coordinates are: A \left(4, 0\right) and B \left(-1.5, 0\right).

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b

Which axis do points A and B lie on?

Worked Solution

Create a strategy

The horizontal axis is called the x-axis and the vertical axis is called the y-axis.

Apply the idea

Both points lie on the horizontal axis or x-axis.

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Example 2

Consider the point \left(6, -8\right).

a

Plot the point on the coordinate plane.

Worked Solution

Create a strategy

The first value of the 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

The x-coordinate of 6 tells us to move 6 spaces to the right of the origin, and the y-coordinate of -8 tells us to move 8 spaces down from the origin.

-3

-2

-1

1

2

3

4

5

6

7

8

9

x

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

y

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b

In which quadrant does the point \left(6, -8\right) lie?

Worked Solution

Create a strategy

The coordinate plane is divided into four quadrants by the x-axis and y-axis. Refer to the image below:

Apply the idea

Looking at the plotted point in the coordinate plane, we can say that \left(6, -8\right) lies in quadrant 4.

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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 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.

In an input-output table 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.

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

Examples