The Coordinate Plane

2 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

Two number lines cross at right angles. The horizontal one is the $x$ -axis and the vertical one is the $y$ -axis. They meet at a point called the origin.

Together they turn the flat plane into a grid with addresses. Every point has exactly one address, and every address points to exactly one place.

Rule

The plane is the set of all ordered pairs:

\mathbb{R} \times \mathbb{R} = \mathbb{R}^2

A point is written $P(x, y)$ : $x$ is the abscissa - the horizontal position. $y$ is the ordinate - the vertical position. The order matters: $(1, 3)$ and $(3, 1)$ are different points.

How it is used

Reading the coordinates of a point answers three common questions.

Distance to the axes. $|x|$ is the perpendicular distance to the $y$ -axis; $|y|$ is the distance to the $x$ -axis.
Is it on an axis? Points on the $x$ -axis look like $(a, 0)$ ; points on the $y$ -axis look like $(0, a)$ ; the origin is $(0, 0)$ .
Which side? The signs of $x$ and $y$ say left/right of the $y$ -axis and above/below the $x$ -axis.

Worked example

Find the perpendicular distances from $P(-2, 3)$ to the coordinate axes. Distance to the $y$ -axis: $|-2| = 2$ units. Distance to the $x$ -axis: $|3| = 3$ units. The signs say where it sits: $x < 0$ , $y > 0$ - two left, three up.

Explore

Drag the point around the grid. Its address $(x, y)$ updates as you move. The dashed lines show its distance to each axis. Now put the point exactly on an axis. One coordinate becomes zero.

The coordinate plane

Try this. Move x while holding y fixed: horizontal position changes; distance from the x-axis stays fixed. Then try x = 0.

Point A · x-2Point A · y3

P = (-2, 3) · Distance to x-axis: 3 · Distance to y-axis: 2

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

Where is the point

(-2,3)

3 left, 2 up

2 right, 3 down

2 left, 3 up

Hint 1 · Find a starting point

Read the horizontal coordinate first.

Hint 2 · Take the next step

A negative x moves left; a positive y moves up.

Show the reasoning

Answer: 2 left, 3 up

The first coordinate is horizontal and the second is vertical, so move 2 left and then 3 up from the origin.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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