4.05 Integers in the coordinate plane

The coordinate plane is a two-dimensional plane used to plot points and graph lines. These points are labeled by an ordered pair of numbers of the form \left(x,\,y\right), called coordinates.

We can think of the coordinate plane as being built from two separate number lines. The first number line we lay down horizontally on the plane. This line is called the x-axis, and for every point in the plane we can talk about its x-coordinate, which is its horizontal position along this axis.

The second number line is placed in a vertical direction, perpendicular to the first number line. This vertical line is called the y-axis, and every point in the plane has a y-coordinate, which is its vertical distance along this axis.

The point at which the x-axis and the y-axis intercept is called the origin. The coordinates of the origin are \left(0,\,0\right).

The coordinates of a point are given in relation to the origin. In the image, we can see that the cat is 6 units to the right of the origin, and 2 units above the origin. So we can say the cat has the coordinates \left(6,\,2\right). The x-coordinate is 6 and the y-coordinate is 2.

Notice that the order of the numbers is important. It would be incorrect to say the cat has the coordinates \left(2,\,6\right). These coordinates refer to the point 2 units to the right of the origin, and 6 units above the origin.

Coordinates are always written with parentheses in the form \left(x,y\right) where the first number, x, is the x-coordinate and the second number, y is the y-coordinate.

Notice that the x-coordinate also tells us how far a point is from the y-axis and the y-coordinate tells us how far a point is from the x-axis.

In the image of the cat, the coordinates \left(6,\,2 \right) tell us the cat is 6 units from the y-axis and 2 units from the x-axis.

Examples

Example 1

Consider the coordinate plane shown:

a

What object has coordinates \left(1,\,4\right)?

Worked Solution

Create a strategy

Use the numbers on the axes in locating the coordinates.

Apply the idea

Start at \left(0,\,0\right). Move 1 space to the right, then 4 spaces up.

The object with coordinates \left(1,\,4\right) is a star.

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b

What object has coordinates \left(10,\,1\right)?

Worked Solution

Create a strategy

Use the numbers on the axes in locating the coordinates.

Apply the idea

Start at \left(0,\,0\right). Move 10 space to the right, then 1 space up.

The object with coordinates \left(10,\,1\right) is beach ball.

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c

What are the coordinates of the bicycle?

Worked Solution

Create a strategy

Follow the grid line up to the horizontal axis to identify the x-coordinate. Then, follow the grid line across the vertical axis to identify the y-coordinate.

Apply the idea

The number on the horizontal axis directly below the bicycle is 3 and across the vertical axis is 6. So, the coordinates are \left(3,\,6\right).

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Example 2

Plot the point \left(6,\,3\right) onto the coordinate plane.

Worked Solution

Create a strategy

Use the numbers on the axes to move on the coordinate plane.

Apply the idea

Start at \left(0,\,0\right). Plot the point 6 space to the right, then 3 spaces up.

This will be the point on the plane described by \left(6,\,3\right).

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Example 3

Consider the point with coordinates \left(9,\,4\right).

a

How far is the point from the x-axis?

Worked Solution

Create a strategy

To find the distance of a point from the x-axis, consider the y-coordinate of the point. The distance to the x-axis is measured vertically, making it equal to the absolute value of the y-coordinate.

Apply the idea

For the point \left(9,\,4\right), the y-coordinate is 4. Thus, the distance from the x-axis is \vert 4 \vert = 4 units.

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b

How far is the point from the y-axis?

Worked Solution

Create a strategy

To determine the distance of a point from the y-axis, examine the x-coordinate. This distance is measured horizontally, making it equal to the absolute value of the x-coordinate.

Apply the idea

Given the point \left(9,\,4\right), the x-coordinate is 9. Therefore, the distance from the y-axis is \vert 9\vert = 9 units.

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Example 4

Write the coordinates of the point that is 5 units to the right of \left(9,\,6\right).

Worked Solution

Create a strategy

Plot the given coordinates, then move horizontally by the required number of units.

Apply the idea

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Plot \left(9,\,6\right) on the coordinate plane and move 5 units to the right.

The new coordinates are \left(14,\,6\right).

Reflect and check

Another way to find the coordinates of the new point is by realizing that moving right will increase the x-coordinate so we need to add 5 to the x-coordinate.

\displaystyle \text{New coordinates}	\displaystyle =	\displaystyle \left(9+5\,,\,6\right)	Add 5 to 9
	\displaystyle =	\displaystyle \left(14,\,6\right)	Evaluate

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Example 5

Point A has the coordinates \left(3,\,6\right), and point B has the coordinates \left(8,6\right). What is the distance between A and B?

Worked Solution

Create a strategy

Plot the points on the coordinate plane and then count the horizontal units from point A to point B.

Apply the idea

Plot the point A and point B. The distance between them is 5 units.

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Reflect and check

Another way to find the distance between the two points is by realizing that point A and point B have the same y-coordinate, therefore the distance between them is the difference in their x-coordinate.

\displaystyle \text{Distance}	\displaystyle =	\displaystyle \left(8-3\right)	Subtract 3 from 8
	\displaystyle =	\displaystyle 5	Evaluate

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Now that we know how to graph points with positive coordinates, let's see what happens if we extend the axes of a coordinate plane in both directions.

The coordinate plane is divided into four distinct regions, called quadrants.

The 1st quadrant is on the top right. The x-coordinate and y-coordinate of a point in the 1st quadrant are both positive.

The quadrants are numbered in an counterclockwise direction:

• 2nd quadrant: x-coordinates are negative, y-coordinates are positive

• 3rd quadrant: both coordinates are negative

• 4th quadrant: x-coordinates are positive, y-coordinates are negative

Points that lie on an axis, like \left(-5,\,0\right) or \left(0,\,4\right), are not in any quadrant.

Points on the x-axis have a y-coordinate of 0.

Points on the y-axis have an x-coordinate of 0.

Examples