The coordinate plane draws a horizontal x-axis and a vertical y-axis, so the position of any point can be written as a pair of numbers.
Just as a map names a column and a row, the point (3, 2) is 3 to the right of the origin and 2 up.
Once a position is written as numbers, the signs alone tell you which quadrant it is in and where its reflections in the axes or the origin lie.
On this page you move point P to check its quadrant and its three reflections, and take a first look at finding the distance between two points with a right triangle.
Coordinates: a position as two numbers
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Move point P to see its quadrant and its reflections in the x-axis, the y-axis and the origin
3
2
🗺️ What coordinates are
①The coordinate plane is two number lines crossing at right angles at the origin O. The horizontal one is the x-axis and the vertical one the y-axis
②A point is written as (x-coordinate, y-coordinate), in that order. P(3, 2) is 3 to the right of the origin and 2 up
③The x-coordinate is + to the right and − to the left; the y-coordinate is + upwards and − downwards
④The origin O is (0, 0). The dashed lines in the picture drop from P to the two axes, and following them is how you read the coordinates
Telling the quadrants apart
Telling quadrants apart
I: (+, +) II: (−, +) III: (−, −) IV: (+, −)
The signs of the x- and y-coordinates alone fix the quadrant
🔍 The four quadrants
①Quadrant I is top right: x > 0 and y > 0
②Quadrant II is top left: x < 0 and y > 0
③Quadrant III is bottom left: x < 0 and y < 0
④Quadrant IV is bottom right: x > 0 and y < 0
⑤A point on an axis belongs to no quadrant. On the x-axis y = 0, and on the y-axis x = 0. Move P onto an axis in the picture and the sentence below changes to "on the x-axis" or "on the y-axis"
Reflections in the axes and the origin
Reflection in the x-axis
(a, b) → (a, −b)
Where the point lands if you fold along the x-axis. Only the sign of the y-coordinate changes
Reflection in the y-axis
(a, b) → (−a, b)
Where the point lands if you fold along the y-axis. Only the sign of the x-coordinate changes
Reflection in the origin
(a, b) → (−a, −b)
The point turned 180° about the origin. Both signs change
💡 Keeping reflections straight
①In a reflection in the x-axis, the x-axis is the fold line. The x-coordinate stays the same and only the y-coordinate changes sign
②A reflection in the y-axis is the other way round: only the x-coordinate changes sign
③A reflection in the origin is the same as reflecting in the x-axis and then the y-axis, so both signs change
④Set the coordinates of the three reflections in the legend under the picture beside those of P, and you can see at once which sign changed
Distance between two points (preview)
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Distance between A and B: the horizontal and vertical differences and a right triangle
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-1
Distance between two points
d = √((x2 - x1)2 + (y2 - y1)2)
It is the hypotenuse of a right triangle whose two shorter sides are the horizontal and vertical differences. Pythagoras' theorem comes later
🔍 Why a right triangle?
①Draw a horizontal line from A and a vertical line from B; they meet at a right angle and make a right triangle
②The horizontal side is the difference in x-coordinates, |x₂ − x₁|, and the vertical side the difference in y-coordinates, |y₂ − y₁|
③The distance between the points is the hypotenuse, found from hypotenuse² = horizontal² + vertical² (Pythagoras' theorem)
④With the starting values A(3, 2) and B(4, −1), the sides are 1 and 3, so the distance is √10 ≈ 3.16
Check with examples
Example 1
In which quadrant is the point P(−3, 2)?
1
Look at the signs of the x- and y-coordinates.
x = −3 < 0, y = 2 > 0
2
A point with signs (−, +) is top left, which is Quadrant II.
(−, +) ⇒ Quadrant II
▸ Quadrant II
Move P to (−3, 2) in the picture and it sits in the top-left region, and the sentence below changes to "Quadrant II".
Example 2
Find the coordinates of the reflection of A(2, −5) in the x-axis.
1
A reflection in the x-axis changes only the sign of the y-coordinate.
(x, y) → (x, −y)
2
Put in the coordinates of A.
(2, −5) → (2, 5)
▸ (2, 5)
The reflection of the same point in the y-axis is (−2, −5), and in the origin (−2, 5). Side by side, the three make the difference clear.
Exam checklist
Coordinate plane essentials
point P(a, b): (x-coordinate, y-coordinate), in that order
(2, 3) and (3, 2) are different points
Grade 7 school test style
The point (a, b) is in Quadrant III. Which quadrant is the point (−a, b) in?
①Quadrant I
②Quadrant II
③Quadrant III
④Quadrant IV
⑤On the x-axis
▸ ④ Quadrant IV
1
The point is in Quadrant III, so a < 0 and b < 0.
a < 0, b < 0
2
−a > 0 and b < 0, so the signs are (+, −).
(−a, b) = (+, −) ⇒ Quadrant IV
🎯 Exam checklist
①Write coordinates in the order (x, y)
②Points on the x-axis have y = 0 and points on the y-axis have x = 0, and neither belongs to a quadrant
③Tell quadrants apart by the signs of the two coordinates
④Before finding a reflection, check whether it is in the x-axis, the y-axis or the origin
⑤A grid square is not always 1, so read the numbers on the axes first