The Cartesian coordinate system (or Cartesian plane) is a grid that we can use to understand where certain objects lie, similar to a map. This grid looks as follows:
It consists of a horizontal number line called the -axis and a vertical number line called the -axis.
The origin is the location where the and axes meet and is given the symbol
Every location on the Cartesian plane is called a point. Every point is identified using an -coordinate and a -coordinate. Together, points are expressed using an ordered pair of coordinates
For example, we can mark (or plot) the point with coordinates by starting from and then traveling units towards the right, followed units up.
We say that
the -coordinate of is ,
the -coordinate of is ,
the -coordinates (or just "the coordinates") of are
The letter is just a label for our point. It's usually a good idea to label points!
Remember that a coordinate plane is made by two number lines that meet at Since these are number lines, we can extend both to include negative numbers.
We plot an ordered pair similar to before, but if the number is negative, we reverse the direction. So, starting at the origin:
if the first number is positive, move that number of places right along the -axis. If its negative, move left.
if the second number is positive, move that number of places up along the -axis. If its negative, move down.
To demonstrate, let's find the coordinates of the point above.
To find the coordinates of we start at the origin, move units left, and then units down.
Since we move left, the -coordinate is negative, and since we move down, the -coordinate is negative.
Therefore, the coordinates of are
What is the -coordinate of the point
To find the coordinates of we start at the origin, move units left, and then units up.
The coordinates of are Therefore, the -coordinate of is
What is the $x$-coordinate of the point $R?$
|
a
|
$1$ |
|
b
|
$-3$ |
|
c
|
$-1$ |
|
d
|
$2$ |
|
e
|
$3$ |
What is the $y$-coordinate of the point $P?$
|
a
|
$-2$ |
|
b
|
$1$ |
|
c
|
$2$ |
|
d
|
$-3$ |
|
e
|
$3$ |
What are the -coordinates of the point shown below?
To find the coordinates of we inspect its position along the and axes.
The point is units to the left of the origin, in the negative direction of the -axis. So, has an -coordinate of
Additionally, the point is units below the origin, in the negative direction of the -axis. So, has a -coordinate of
The point has an -coordinate of and a -coordinate of so its coordinates are
What are the $xy$-coordinates of the point shown in the axes above?
|
a
|
$(4, 2)$ |
|
b
|
$(3, 7)$ |
|
c
|
$(7, 3)$ |
|
d
|
$(3, 5)$ |
|
e
|
$(6, 2)$ |
What are the coordinates of the point $C?$
|
a
|
$(-4,2)$ |
|
b
|
$(1,3)$ |
|
c
|
$(4,2)$ |
|
d
|
$(2,-4)$ |
|
e
|
$(4,-2)$ |
Plot the point with coordinates on the Cartesian plane.
Starting at the origin we move along the and axes to get to the point
First, we move along the -axis. The point has an -coordinate of so we move unit to the right, in the positive direction of the -axis.
Then, we move along the -axis. The point has a -coordinate of so we move units down, in the negative direction of the -axis.
Which of the following shows the point $C(-2,-3)?$
|
a
|
 |
|
b
|
 |
|
c
|
 |
|
d
|
 |
|
e
|
 |
Which of the following graphs shows the point $A$ with coordinates $(2,1)?$
|
a
|
 |
|
b
|
 |
|
c
|
 |
|
d
|
 |
|
e
|
|
The axes split the coordinate plane into four parts, called quadrants.
Going counter-clockwise, it's common to call
the top-right quadrant the first quadrant (I),
the top-left is the second quadrant (II),
the bottom-left is the third quadrant (III),
the and bottom-right is the fourth quadrant (IV).
To remember the order of the quadrants, it's helpful to think about writing the letter "C" in the graph:
To which quadrant does the point belong?
In order to identify the quadrant to which the point belongs, we start by plotting the point.
- The -coordinate of the point is so we move units to the right of the origin.
- The -coordinate of the point is so we move units down from the origin.
Based on the graph, we can tell that the point belongs to quadrant , the fourth quadrant.
A point $A$ is shown on the axis above. To which quadrant does it belong?
|
a
|
Right quadrant |
|
b
|
First quadrant |
|
c
|
Second quadrant |
|
d
|
Fourth quadrant |
|
e
|
Third quadrant |
Which of the following is true regarding the point $B?$
- It is in the fourth quadrant.
- Its $x$-coordinate is $-4.$
- Its $y$-coordinate is $3.$
|
a
|
II and III only |
|
b
|
II only |
|
c
|
I and II only |
|
d
|
I only |
|
e
|
III only |
