Section 1
Plotting Points & Quadrants
A point is described by an ordered pair (x , y): move x units horizontally from the origin, then y units vertically. The signs of the two coordinates place the point in exactly one quadrant — unless one coordinate is zero, in which case the point sits on an axis.
Method
How to Plot a Point
- 1Start at the origin O (0 , 0).
- 2Read the x-coordinate: move right if positive, left if negative.
- 3Read the y-coordinate: move up if positive, down if negative.
- 4Mark the point with a dot and label it with its capital letter and coordinates.
Signs
The Four Quadrants
| Region | x | y | Example |
|---|---|---|---|
| 1st quadrant | x > 0 | y > 0 | (4 , 3) |
| 2nd quadrant | x < 0 | y > 0 | (-3 , 2) |
| 3rd quadrant | x < 0 | y < 0 | (-4 , -3) |
| 4th quadrant | x > 0 | y < 0 | (3 , -4) |
- A point of the form (a , 0) lies on the x-axis; a point of the form (0 , b) lies on the y-axis. Points on an axis belong to no quadrant.
- If x·y > 0 the point is in the 1st or 3rd quadrant; if x·y < 0 it is in the 2nd or 4th quadrant.
- The origin O (0 , 0) is the only point lying on both axes.
Practice with solutions
Worked Examples
Plot A(-2 , 3), B(4 , -1), C(0 , -3) and D(2 , 0) and name the position of each.
A(-2 , 3) — x is negative, y is positive → 2nd quadrant.
B(4 , -1) — x is positive, y is negative → 4th quadrant.
C(0 , -3) — x = 0 → lies on the y-axis (below the origin).
D(2 , 0) — y = 0 → lies on the x-axis (right of the origin).
If a > 0, in which quadrant does the point (-2a , a + 1) lie?
Test the sign of each coordinate using a > 0:
-2a < 0 (negative) , a + 1 > 0 (positive)
A negative x with a positive y means the point lies in the 2nd quadrant, for every positive value of a.
The point A(4k + 4 , -k + 3) lies on the y-axis. Where is B(-2k , 4k + 1)?
A point on the y-axis has x = 0:
4k + 4 = 0 → 4k = -4 → k = -1
Substitute k = -1 into B:
B(-2(-1) , 4(-1) + 1) = B(2 , -3)
x positive, y negative → B lies in the 4th quadrant.
Your turn
Try It Yourself
- Plot A(3 , 5), B(-2 , -4), C(0 , 5) and D(5 , 0) and state the position of each.
- The point (| -5 | , (-2)²) lies in which quadrant?
- A teacher asks about (x , y) where x·y < 0. Ahmed says 2nd quadrant, Hend says 4th quadrant — who is correct?
- If A(-3k , 2k - 4) lies on the x-axis, determine the quadrant of B(k - 6 , -3k).
Answers: 1st / on the y-axis / on the x-axis appear in the Activities section, where you can also check your plotting by clicking on a live grid.