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Toolbox · Coordinate Plane

Coordinate Plane

Zoomable, pannable coordinate plane: plot several points and equations live, click the axes, the origin or a quadrant to make it blink three times, with quadrant tests, distance, midpoint and slope worked out step by step

Formula

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2},\quad M\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right),\quad k=\frac{y_2-y_1}{x_2-x_1}

Distance, midpoint and slope are the three formulas used most often; the line equation follows from point-slope form.

Click to locate

Unit lengthpx

Wheel to zoom · drag to pan · double-click to add a point · drag a point to move it

Click an axis, the origin or a quadrant: that part of the plane blinks three times. Clicking it directly on the plane works too.

Points

3 / 12

A(-3, 2)Quadrant II
B(2, 3)Quadrant I
C(4, -2)Quadrant IV

Measure two points

→

d5.099M(-0.5, 2.5)k0.2

Equations / functions

2 / 6

y=2x+1
y=x^2-4

Presets

A(-3, 2)

PositionQuadrant II

Distance ∣OP∣3.606

Across x-axis(-3, -2)Across y-axis(3, 2)Across origin(3, -2)

Steps

4 / 4

1Read coordinates
$P(-3,\ 2)$
Coordinates are written x first, y second: x runs right, y runs up.
2Locate
$x=-3,\quad y=2\ \Rightarrow\ Quadrant II$
Only the signs matter: x = -3, y = 2, so the point lies in Quadrant II.
3Distance to origin
$|OP|=\sqrt{(-3)^2+(2)^2}=\sqrt{13}=3.606$
Join the point to the origin: the legs are ∣x∣ and ∣y∣, so the hypotenuse is ∣OP∣ = 3.606.
4Reflections
$P_x(-3,\ -2),\quad P_y(3,\ 2),\quad P_O(3,\ -2)$
Reflecting across the x-axis flips y; across the y-axis flips x; across the origin flips both.

Concept

The Cartesian plane is built from two perpendicular number lines meeting at a shared origin: the horizontal x-axis and the vertical y-axis.

An ordered pair (x, y) matches exactly one point, and vice versa - that is what coordinates mean, and the order matters.

The axes cut the plane into four quadrants, numbered I to IV anticlockwise from the top right; points on an axis belong to none of them.

Going further

Why it is called Cartesian: In the 17th century Descartes watched a fly on his ceiling and realised two numbers could pin down its position, turning geometry into algebra.
A graph is a photograph of an equation: Every solution (x, y) of y = kx + b lies on one straight line; drawing it makes intercepts, trends and extrema visible at a glance.
From two dimensions to three: Add a z-axis through the origin perpendicular to the plane and you get 3D coordinates, so a point becomes (x, y, z).
Unit length and distortion: Changing the unit length is just changing rulers, so coordinates stay the same - but if the two axes use different scales, shapes get stretched.

Common mistakes

Always write x first: (3, -2) and (-2, 3) are completely different points.
Points on an axis belong to no quadrant - a favourite trap in exams.
To find where two curves meet, set the two expressions equal, solve for x, then substitute back for y.
Check the range before plotting: y = 1/x breaks near x = 0 because of the asymptote, not because the curve is joined.

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