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!
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 $C(-2,-3)?$
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a
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b
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c
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d
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e
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Which of the following shows the point $C(4,-2)?$
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a
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b
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c
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d
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e
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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?
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a
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$(3, -3)$ |
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b
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$(-3)$ |
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c
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$(-3, -3)$ |
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d
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$(3, 3)$ |
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e
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$(-3, 3)$ |
What are the $xy$-coordinates of the point shown in the axes above?
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a
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$(3,-1)$ |
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b
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$(-3,1)$ |
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c
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$(3,1)$ |
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d
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$(1,3)$ |
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e
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$(-1,3)$ |
The axes of the coordinate system divide the plane into four parts called quadrants. The quadrants are denoted using Roman numerals, as shown below.
In words, quadrant is called the first quadrant. The quadrants and are the second, third, and fourth quadrants, respectively.
Tip: You can remember the order of the coordinates by imagining writing a "C". You start at quadrant in the top-right, then go to quadrant in the top-left, then quadrant in the bottom-left, and finally quadrant in the bottom-right.
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.
Which quadrant does the point $(-5,3)$ belong to?
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a
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Between Quadrant $\textrm{II}$ and Quadrant $\textrm{III}$ |
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b
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Quadrant $\textrm{IV}$ |
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c
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Quadrant $\textrm{III}$ |
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d
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Quadrant $\textrm{II}$ |
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e
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Quadrant $\textrm{I}$ |
A point $A$ is shown on the axis above. To which quadrant does it belong?
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a
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Right quadrant |
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b
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First quadrant |
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c
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Second quadrant |
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d
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Third quadrant |
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e
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Fourth quadrant |
