Core Concepts
In this article, we will detail the origin and components of a coordinate plane. We will also discuss the applications and impacts of the discovery of the coordinate plane.
What is a coordinate plane?
A coordinate plane can be defined as a two-dimensional plane that is made up of two intersecting axes. However, before we discuss each part of the coordinate plane, let’s explore how this concept came to be!
The Origins of a Coordinate Plane
René Descartes, often credited as the father of modern philosophy, was also an incredible mathematician. Legend has it that, late one night, he laid in bed while a fly buzzed around on the ceiling. Curiously, he wondered if there was a way to describe the fly’s location. This is the hypothesized origin for Descartes’s formulation of what we know today as the Cartesian coordinate system!
Officially introduced in the late 17th century, this system forged a link between geometry and algebra. This link is formally known as analytic geometry today. While the concept of using grids to represent location had already existed in spaces like cartography, the breakthrough of Descartes’s work is truly in its application. With his work, he was able to define and clarify the relationship between points on a grid and an algebraic equation. Now, with this revolutionary idea of debated origin, we can explain shapes like circles, squares, and triangles using algebraic equations!
It is also interesting to note that, at the time, there was some contest to the credit given to Descartes. This is because, one year prior, Pierre de Fermat had circulated a manuscript that illustrated the same concepts as Descartes! Regardless of who deserves the credit for devising the coordinate plane, its impact on modern math is undeniable.
Components of a Coordinate Plane
Coordinate planes are made up of four core parts. In this section, we’ll discuss each one in detail.
The Axes
First, let’s discuss the axes. Fundamentally, the axes are vertical and horizontal number lines that lie on the coordinate plane. The horizontal number line is commonly called the x-axis and the vertical number line is called the y-axis.
These two axes are perpendicular to each other, forming a quadrisected rectangle. Based on the needs of a given math problem, the bounds of the axes can be increased or decreased to display different portions of the number lines.
Axes are useful for orienting you as you navigate the coordinate plane. The numbers on each axis help identify the location of specific coordinates. Before we talk about coordinates in more depth, let’s pause to discuss other components of the coordinate plane.
The Quadrants
The coordinate plane is split into four quadrants due to the intersection of the x– and y-axes. Each quadrant is assigned a Roman numeral to represent it. The Roman numerals begin in the upper-right quadrant and increase as we move counter-clockwise within the coordinate plane.
Each quadrant is characterized by a combination of positive or negative x and y values. Quadrant I represents all points with a positive x value and a positive y value. Quadrant II includes all points that have a negative x value and a positive y value. The third quadrant, Quadrant III, has all points with a negative x value and a negative y value. Lastly, Quadrant IV encompasses all points with a positive x value and a negative y value.
Why do we acknowledge the individual quadrants? By recognizing the quadrants, we can gain pertinent information about the coordinates. Just by knowing the quadrant to which a point belongs, we can conceptualize its approximate location within the coordinate plane.
As we mentioned before, the quadrants are defined by the intersection of the x– and y-axes. This intersection is important for another reason, too: it’s the location of a special feature that shapes the entire coordinate plane. Let’s read on to learn more!
The Origin
Where the x– and y-axes intersect is an important factor of the coordinate plane. This intersection point is called the origin, and it is represented by the coordinates (0, 0).
The origin is a constant in the coordinate plane. However, it’s important to note it may not be visualized at times, such as if the graph is extremely zoomed in. An example of when this situation may happen is when the line of interest does not cross the origin. In these cases, the origin might lie beyond the zoomed-in region because the origin isn’t relevant to this line of interest.
In this section, we mentioned that the origin’s coordinates are (0, 0). But what does this mean? The fourth component of a coordinate plane will explain this!
The Coordinates
Utilizing the axes, we are able to label individual points using coordinates. Symbolized by the notation ( x , y ), coordinates are how we identify a point’s position on the coordinate plane. The x value indicates where along the x-axis the point is located. Similarly, the y value represents the point’s location in relation to the y-axis.
To better illustrate how this notation appears in action, let’s look at these visual representations. Below is the visual representation of (6, 5) in the left image, and (5, 6) in the right image.
As seen in these graphs, there is a difference between the points’ locations, even though they use the exact same numbers! This is because, in a set of coordinates, the order of the x and y values is very important. The order of coordinates guides how we go about identifying where a point is located, which we will discuss next.
How to Use a Coordinate Plane
The way in which we approach identifying a coordinate is crucial to making sure we arrive at the correct point. In order to identify a point of the coordinate plane, we must begin at the origin. As an example, let’s try and find the point (4, 8). Place your pen or finger at the origin, located at (0, 0).
Next, we’ll locate the x value. In our given set of coordinates, the x coordinate is positive 4. With your pen or finger already at the origin, trace a line rightward along the x-axis until you reach the x value of 4. We move rightward in this case because the x coordinate is positive. Remember that positive x values are located in either Quadrant I or Quadrant IV, depending on the y value.
If it’s easier to visualize this way, you can physically draw a line up from x = 4 to help you keep your place! This is depicted by the blue arrow in the image below.
Now that we’ve identified the x coordinate on our coordinate plane, we’ll repeat the process to locate the y coordinate. This involves following the y-axis upward until we reach the value of positive 8. Again, for visualization purposes, you can draw a line at y = 8, as shown by the green arrow below.
Now that we have located both the x and y components individually, we can follow the path of the graph’s line, until we reach the place where our blue and green arrows intersect. As shown below, where those blue and green arrows intersect is the point that we’re looking for!
Note that our point (4, 8) is located in Quadrant I. We could have predicted this, since our x and y coordinates are both positive in this example. Using the steps in this process, you can find any point on the coordinate plane!
Applications of Coordinate Planes
Now that we’ve explored how to identify a point on the coordinate plane, you may be wondering how this is applicable to the real world. Well, the coordinate plane, as is the case with many mathematical discoveries, reaches far beyond the boundaries of math!
The coordinate plane benefits physics, specifically kinematics, in the context of motion analysis. These planes serve as the backbone of the Global Positioning System (GPS), as well as air traffic regulation. Even architects use coordinate planes to model building designs!
Beyond these applications, coordinate planes frequently appear as a mechanism in games such as chess, Battleship, and even Final Fantasy! Even in art, a field that may seem like math’s polar opposite, adhering to the coordinate plane can help an artist keep their work visually proportional. In more advanced studies of mathematics courses, like differential equations, you can look forward to the introduction of a three-axis coordinate plane! With this tertiary axis, a world of new options opens up. For example, the third axis is extremely useful in engineering for creating 2-D and 3-D models of objects.
The applications of coordinate planes extend far and wide. They’re not limited to math alone, but they’re strong examples of how math forms the basis of many other fields.
Conclusion
At their core, coordinate planes serve to bridge the gap between geometry and algebra. They allow one to visualize the points on a line in relation to each other. Combined with more advanced mathematical concepts, the coordinate plane can be a powerful tool to analyze data and explain processes around us!
Mini-Quiz
Question 1: In which quadrant is the point (-8, 3) located?
Question 2: Identify a point that would be located in Quadrant IV.
Answers
Question 1: In which quadrant is the point (-8, 3) located?
The point (-8, 3) is located in Quadrant II.
Question 2: Identify a point that would be located in Quadrant IV.
Any coordinate that follows the pattern of (+, -), or a positive x value paired with a negative y value, falls into Quadrant IV.
