Coordinates in Four Quadrants

You’ve probably come across coordinates in some form before. Maybe you’ve seen a map, and been asked to locate the position of something, or maybe you’ve had practice reading from a line graph and not even realised you were using coordinates. And what did those coordinate systems look like? Most likely they looked like this, a number line counting from zero upwards, and a number line counting from zero going out to the right.

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Last Updated: 01.07.2022
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These number lines are known as axes and together, these two number lines create what is known as the coordinate plane. The coordinate plane is a useful tool, as we can describe any location (point) on it with a number on the horizontal axis, and a number on the vertical axis.

coordinates in four quadrants simple positive coordinate system studysmarter A simple coordinate plane with two axes - StudySmarter Originals

Well, the coordinate plane doesn’t stop there. In fact, this simple coordinate plane isn’t the whole story at all, you could even say it’s only a quarter of the story...

So what exactly do we mean when we talk about coordinates in four quadrants?

Meaning of Coordinates in Four Quadrants

The simple coordinate plane you are used to seeing is only one of the four 2-dimensional coordinate quadrants. What do we mean by quadrants? Well, start by drawing the simple coordinate plane axes we are familiar with. It should look something like this.

coordinates in four quadrants drawing four quadrant coordinate system studysmarter Simple coordinate plane - StudySmarter Originals

Now, extend the vertical line downwards toward the bottom of the page. Done that? Good!

Now extend the horizontal line backwards, toward the left side of the page. You should be left with something like this.

coordinates in four quadrants drawing a four quadrant coordinate system studysmarter Beginnings of a four-quadrant coordinate plane - StudySmarter Originals

So what’s happened to our simple coordinate system?

Well, by extending each line we have actually revealed more of the coordinate plane. We can see that the simple system consisting of two perpendicular positive number lines was actually just one of four separate areas created by our axes. Each of these four areas is called a quadrant, from the Latin quadrans meaning four.

From here, all we need to do to understand our four-quadrant coordinate plane is to label each of the axes with numerical values. If we keep zero in the same place on each axis, where the two axes cross, then it makes sense that to the left along the horizontal axis, we count down from zero: –1, –2, –3 ... and do the same going vertically downward from zero on the vertical axis: –1, –2, –3...

coordinates in four quadrants system studysmarter Four-quadrant coordinate plane - StudySmarter Originals

So what exactly do we mean by coordinates in four quadrants? Well...

Coordinates in four quadrants are coordinates plotted in a coordinate system in which both axes extend in both positive and negative directions creating four quadrants.

But how exactly do we graph coordinates in four quadrants? Let's take a look!

Graphing Coordinates in Four Quadrants

As in any two-axis coordinate plane, any point is made up of an $x$ -coordinate and $y$ -coordinate, in the form $(x, y)$ . In a four-quadrant coordinate system,

any point in the upper-right quadrant, the first quadrant, will have both a positive $x$ -coordinate and positive $y$ -coordinate.

and any point in the upper-left quadrant, the second quadrant, will have a negative $x$ -coordinate, and positive $y$ -coordinate.

On the other hand,

any point in the lower-left quadrant, the third quadrant, will have both a negative $x$ -coordinate and negative $y$ -coordinate;

and any point in the lower-right quadrant, the fourth quadrant, will have a positive $x$ -coordinate, and negative $y$ -coordinate.

Coordinates in four quadrants examples of coordinates in four quadrants studysmarter Coordinates in four-quadrants example - StudySmarter Originals

(1)

Plot the point $(4, - 5)$ on a four-quadrant coordinate plane.

Solution:

The point $(4, - 5)$ has a positive $x$ -coordinate and negative $y$ -coordinate, therefore it will be in the lower-right quadrant.

coordinates in four quadrants plotting points example studysmarter

(2)

Plot the point $(- 2, 6)$ on a four-quadrant coordinate plane.

Solution:

The point $(- 2, 6)$ has a negative $x$ -coordinate and positive $y$ -coordinate, therefore it will be in the upper-left quadrant.

coordinates in four quadrants plotting points example studysmarter

(3)

Plot the point $(4, 8)$ on a four-quadrant coordinate plane.

Solution:

The point $(4, 8)$ has a positive $x$ -coordinate and positive $y$ -coordinate, therefore it will be in the upper-right quadrant.

coordinate in four quadrants studysmarter

(4)

Plot the point $(- 3, - 10)$ on a four-quadrant coordinate plane.

Solution:

The point $(- 3, - 10)$ has a negative $x$ -coordinate and negative $y$ -coordinate, therefore it will be in the lower-left quadrant.

coordinates in four quadrants example studysmarter

There's more to coordinates in four quadrants than just plotting them on a four-quadrant coordinate plane. Let's take a look at some more examples.

Examples of Coordinates in Four Quadrants

(1)

What is the distance between points $A$ and $B$ ?

coordinates in four quadrants example studysmarter

Solution:

Reading from the coordinate plane point $A$ has coordinates $(8, 7)$ , and point $B$ has coordinates $(- 5, - 4) .$

We can find the distance between the two points by using Pythagoras' theorem, $d^{2} = a^{2} + b^{2}$ .

$d = \sqrt{a^{2} + b^{2}}$

$d = \sqrt{(8 - (- 5)) + (7 - (- 4))}$

$d = \sqrt{13 + 11}$

$d = \sqrt{24} \approx 4.9$

(2)

Is point $C$ or $D$ closer to the point $(0, 0)$ ?

coordinates in four quadrants example studysmarter

Solution:

Reading off the coordinate plane, point $C$ has the coordinates $(3, - 5)$ and $D$ has coordinates $(- 6, 2)$ .

Firstly, we must find out how far $C$ is from the point $(0, 0)$ .

$d_{c} = \sqrt{3^{2} + {(- 5)}^{2}}$

$d_{c} = \sqrt{9 + 25}$

$d_{c} = \sqrt{34} \approx 5.8$

And then we find out how far $D$ is from the $(0, 0) .$

$d_{d} = \sqrt{{(- 6)}^{2} + 2^{2}}$

$d_{d} = \sqrt{36 + 4}$

$d_{d} = \sqrt{40} \approx 6.3$

Coordinates in Four Quadrants - Key takeaways

The full coordinate plane is divided into four quadrants by the horizontal and vertical axes.
The quadrant that any point is located within can be determined by the sign of each of that point's coordinates.

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Frequently Asked Questions about Coordinates in Four Quadrants

What is the meaning of coordinates in four quadrants?

When each coordinate axis is extended to include both negative and positive numbers, they split the coordinate plane into four quadrants where coordinates can be plotted.

What is an example of coordinates in 4 quadrants?

Points such as (–1, 4) and (5, –6) are examples of coordinates that lie on the 2nd and 4th quadrants of the four-quadrant coordinate plane respectively.

How do you graph coordinates in 4 quadrants?

Coordinates in four quadrants can be plotted by finding the correct value on each axis, and marking the point on the plane where those two values intersect.

How do you read coordinates in 4 quadrants?

Coordinates in four quadrants are made up of an x-coordinate and y-coordinate of the form (x, y).

The x-coordinate is how far along the horizontal axis a point is, and the y-coordinate is how far along the vertical axis a point is.

Does a coordinate plane have 4 quadrants?

When both the x- and y-axes are extended to include negative numbers, the coordinate plane has four quadrants.

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