Regions of the Plane

2 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

The two axes cut the plane into four parts. Each part is called a quadrant.

They are numbered I, II, III and IV. The numbering starts at the top right and goes counter-clockwise. To find which quadrant a point is in, you only need the signs of its two coordinates.

Rule

Sign conventions by region:

Quadrant	Signs	Where
I	$(+, +)$	upper right
II	$(-, +)$	upper left
III	$(-, -)$	lower left
IV	$(+, -)$	lower right
A point lying exactly on an axis belongs to no quadrant.

How it is used

Three standard moves, in rising difficulty:

Read off. Signs of $(x, y)$ name the quadrant directly: $B(4, -3)$ is $(+, -)$ , Quadrant IV.
Sign algebra. If you know the quadrant, you know the signs of $x$ and $y$ . From those you can work out the sign of an expression such as $-xy$ , using the ordinary rules for multiplying signs.
Conditions to inequalities. Say $M(x-4, \; 7-x)$ is in Quadrant IV. Quadrant IV means the first coordinate is positive and the second is negative. So $x - 4 > 0$ and $7 - x < 0$ . Solve both and take the values that work for each.

Worked example

$P(x, y)$ lies in Quadrant II. In which quadrant is $Q(-xy, \; x)$ ? Quadrant II means $x < 0$ and $y > 0$ . First coordinate: $-xy = -(-)(+) = (+)$ - positive. Second coordinate: $x$ - negative. So $Q$ has sign pattern $(+, -)$ : Quadrant IV.

Explore

Drag the point through all four quadrants and watch the label change. Now put it exactly on an axis. The label says it belongs to no quadrant.

Regions of the plane

Try this. Cross each axis and watch the quadrant change. At x = 0 or y = 0, the point is on an axis, not inside a quadrant.

Point A · x-2Point A · y3

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

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.

Which point satisfies

x<0

and

y>0

$(-1,2)$

$(1,2)$

$(-1,-2)$

Hint 1 · Find a starting point

Both conditions must hold for the same point.

Hint 2 · Take the next step

The x-coordinate must be negative and the y-coordinate positive.

Show the reasoning

Answer: $(-1,2)$

Only (−1, 2) is left of the vertical axis and above the horizontal axis. The word “and” means intersection, not either condition alone.

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.

Optional marks, not a grade. Saved in this browser only. Open notebook →