The 4 Quadrants of a Graph

As you've probably noticed, a Cartesian plane can be divided into 4 quadrants (can be also abbreviated to Q). Each of them are typically named from I to IV (that is, 1 to 4 in Roman numerals), as shown below.

An easy way to remember which is which is to remember that I is located at the top right of the graph, and II, III, and IV follow on clockwise (indicated by the circular arrow).

Here are some of the characteristics of I to IV.
1. I and IV (the right half of the graph): All the x-coordinates are positive.
2. II and III (the left half of the graph): All the x-coordinates are negative.
3. I and II (the top half of the graph): All the y-coordinates are positive.
4. III and IV (the bottom half of the graph): All the y-coordinates are negative.

Example 1
Which quadrant are the following coordinates in?
A (2, 1)
B (-2, 3)
C (3, -3)
D (-2, -3)
E (x, 3)
F (2, y)

To get the solution, plotting the points on a graph is helpful if you can't visualize them in your head.

From the graph, we can see that:
A is in Q I
B is in Q II
C is in Q IV
D is in Q III

In E and F, since x and y are not given, they represent all points of coordinates at x and y respectively.

Line E lies on Q I and II

Line F lies on Q I and IV

Example 2
Which quadrant(s) do (x,y) fall into such that xy < 0.

From the problem statement, you'll need to find xy that is has a result that's less than 0, i.e. negative numbers. As you know, when you multiply a positive number with a negative, your result is a negative number. So if you multiply x that has a positive value with a y that has a negative, you'll satisfy xy < 0.

E.g. if x = 5, y = -3, then xy = -15.

So we know that (x, y) xy < 0 falls into Q IV. But we're not done yet. If x is negative and y is positive, we see that this also satisfies xy < 0.

E.g. if x = -5, y = 3, then xy = -15.

So (x,y), xy < 0 also falls into Q II.

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