A coordinate system, or more precisely, a Cartesian coordinate system, is a method to precisely present the position of points, whether on a plane (two-dimensional) or in three-dimensional space. In this chapter, we will focus on a coordinate system on a plane, that is, a system made up of two axes (straight lines) that are perpendicular to each other (forming a 90-degree angle between them).
To facilitate the understanding of the material, let's look at the following drawing, which gives an example of a two-dimensional coordinate system:
As mentioned, as can be seen in the drawing, it is customary to identify the horizontal axis with X and the vertical axis with Y.
The intersection point between the two axes is called the origin and is usually identified with the letter O.
Along the X horizontal axis, the numbers increase to the right of the origin and decrease to the left of the origin.
Along the Y vertical axis, the numbers increase upwards in relation to the origin and decrease downwards in relation to the origin.
Characterization of a Point in a Cartesian Coordinate System
Identifying a point in the coordinate system is done using two values, X and Y.
In the next step, we will use the drawing below and demonstrate how a point can be characterized on a two-dimensional plane.
Quadrants in a Coordinate System
The two axes divide the plane into four quadrants. As you can see in the illustration, the first quadrant is the upper right part, marked with an I, and the rest follow in a counterclockwise sequence.
Each of the quadrants is characterized by different values of X and Y, which are detailed below:
Quadrant I - positive X values and positive Y values
Quadrant II - negative X values and positive Y values
Quadrant III - negative X values and negative Y values
Quadrant IV - positive X values and negative Y values
Quadrants in a Coordinate System
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Test your knowledge
Question 1
Which point is marked on the graph?
Incorrect
Correct Answer:
\( (6,1) \)
Question 2
Which point is marked on the map?
Incorrect
Correct Answer:
\( (5,3) \)
Question 3
Choose the appropriate drawing where the dots appear:
\( (-5,1),\lparen7,0),(-4,-1) \)
Incorrect
Correct Answer:
Characterization of a Point in a Cartesian Coordinate System
Identifying a point in the coordinate system is done using two values, X and Y.
In the next step, we'll use the drawing below and demonstrate how a point can be characterized in a two-dimensional plane.
In the drawing, we see the point B. We are interested in characterizing it. To do this, first, we'll draw two vertical lines from the point B to each of the two axes, as shown in the drawing.
We see that on the horizontal X axis, the point "meets" the value 5, while on the vertical Y axis, the point "meets" the value 2.
It's common to write the values as points in parentheses, where the value of X appears as the first value on the left, while the value of Y appears as the second value, on the right.
Therefore, point B would be written as follows:
B(5,2)
Characterizing Points Located on One of the Axes
When the point is on one of the coordinates (for example, points E and G in the illustration)
Points in a Cartesian Coordinate System
Remember the following rule:
When the point is on theX axis, theY value is0, which means it has coordinates: E(4,0)
When the point is on theY axis, the value ofX is0, which means it has coordinates: G(0,3)
Do you know what the answer is?
Question 1
Choose the appropriate drawing where the dots appear:
\( (1,3),\lparen2,5),(3,3) \)
Incorrect
Correct Answer:
Question 2
Choose the appropriate drawing where the dots appear:
\( (1,0),\lparen-2,-2),(-5,4) \)
Incorrect
Correct Answer:
Question 3
Choose the appropriate drawing where the dots appear:
\( (6,6),\lparen-3,0) \)
Incorrect
Correct Answer:
Examples and Practice on Coordinate Systems
Exercise 1
Look at the following coordinate system and these are the points A,B,C,D.
Solution:
We will apply what we've learned and drop verticals from the points A,C,D to each of the axes
It should be noted that point B is on the X axis, and therefore, its Y value is 0.
Answer:
A(2,2)
B(−1,0)
C(−2,3)
D(3,−2)
Exercise 2
Plot the following points on the coordinate system:
k(0,5)
L(−1,5)
M(0,0)
N(−4,0)
P(2,4)
Solution:
For points L and P, you need to find the X values and the Y values where the intersection of these values is indeed the desired point.
The points K and N are located on the axes themselves, and therefore the following rule applies:
When the point is on the X axis, the Y value is 0
When the point is on the Y axis, the X value is 0
The point M is located at the intersection of the two axes.
Answer:
See the following drawing:
If you're interested in more information about "graphs," you can find detailed information in the following articles:
