Here I will explain everything you need to know to represent points on Cartesian axes and how to represent lines on coordinate axes.
First of all, you need to be very clear about the Cartesian axes or coordinate axes, which is what I will explain to you in the following point.
Índice de Contenidos
The Cartesian axes
To represent points and lines we need the Cartesian axes or coordinate axes, which are two perpendicular lines that meet the following characteristics:
- The horizontal axis is called x-axis or x-axis
- The vertical axis is called y-axis or ordinate axis
- The coordinates of any point are represented by (x,y), where x is measured on the x and y axis and y is measured on the y axis.
- The point where the two lines are cut is called coordinate origin and their coordinates are (0,0).
How to get the coordinates of a point
We will now learn how to obtain the coordinates of a point that is already represented in the coordinate axes.
For example: What are the coordinates of point A?
To obtain the x-coordinate, draw a vertical line (perpendicular to the x-axis) from the point to the x-axis. The cut-off point will be the x-coordinate:
It cuts with the x-axis in the value 4, so the x-coordinate is 4.
To obtain the y-coordinate, draw a horizontal line (perpendicular to the y-axis) from the point to the y-axis. The cut-off point will be the y-coordinate:
It cuts with the y axis in the value 3, so the y-coordinate is 3.
The coordinates of point A are (4,3).
Positive and negative values in the coordinate axes
The coordinates don’t always have to be positive. They can also be negative.
Where are the coordinates positive or negative?
On the x-axis, the values are positive to the right of the coordinate center and negative to the left of it.
On the y-axis, the values are positive when they are above the coordinate center and negative when they are below it.
Note that to obtain the coordinates of a point, the line always starts from the point to the axes.
Depending on the quadrant in which the point is located, the coordinates may be positive or negative. In the example above, the two coordinates were positive.
Let’s see what happens in the rest of the quadrants.
What would be the coordinates of points B, C and D?
For point B, we draw a vertical and a horizontal line from the point to the axes and see where they cut, just like we did for point A:
Its coordinates are (-2.1). In this case, x takes a negative value, since it is to the left of the coordinate origin.
We do the same thing with point C:
The coordinates of C are (-3,-2)
And now for point D:
The coordinates of D are (1,-3).
What are the coordinates of the following points?
How to represent a point on the coordinate axes
Now I am going to explain the opposite case to the previous one: we have the coordinates of the point and we have to represent it in the coordinate axes.
Let’s look at it with an example: Represent point A (2,3) on the coordinate axes.
In this case we have the x and y coordinates.
From the x-coordinate, i.e. from 2, we draw a vertical line and from the y-coordinate, from 3, a horizontal line. Where the two lines are cut, point A:
In this case the lines go from the axes to the point.
Another example: Represent point B (-4.2)
Represent the following points on the coordinate axes:
How to represent a straight line in the coordinate axes
Now that you know how to represent points, I will explain how to represent a line.
What do you need to draw a line on the Cartesian axes?
Imagine you want to draw a line with a pencil and ruler.
You start by pressing the pencil down on the paper and then slide it along the ruler. But the ruler can turn and take infinite directions.
What is the correct direction for your period?
You need another point to place the ruler to define the line you want to draw.
In other words, you need to join two points to get a line.
But the straight line doesn’t start and end at those two points. Those two points just mark the direction of the line. A straight line has an infinite length, so you can lengthen it as much as you want.
Now, how to draw the equation of a line on the coordinated axes?
Let’s see how to represent a line on the coordinate axes with an example. They ask you to represent this line:
As I indicated in the previous section, you need two points. Therefore, first of all, we must find those two points that belong to the line.
To do this, we choose two random x-values, which for example can be 0 and 1 and from these values we will calculate two more y-values. Each pair of values (x,y) will be a point on the line.
To calculate the values of y, each value of x, we substitute it in the equation and operate:
From where we get the first point (0,2)
From where we get the second point (1.0)
And you put this on a chart for clarity:
Now all you have to do is draw those points on the coordinate axes:
And once you have the points, you can now join them together and lengthen the line at both ends:
Could you calculate more points to represent the line?
You can calculate all the points you want, but 2 is enough because you don’t need more points to draw it.
If you want it once you have it represented, give x the value 3 and get a third point and you will see how it falls into the line.