Quadrants

Study Sheet

Quadrants

The four regions of the plane

Reading the Signs

Concept
The Big Idea
xy−6−5−4−3−2−112345−3−2−112I (+, +)II (−, +)III (−, −)IV (+, −)(2, 2)(−3, 1)(−2, −2)(3, −1)
One point in each quadrant; the signs of its coordinates name the quadrant.

The axes cut the plane into four quadrants, numbered anticlockwise from the top right. The signs of xx and yy tell you which one you are in.

With negative numbers on both axes, the plane has four quadrants, numbered counterclockwise starting from the upper right. The signs of the coordinates tell the quadrant: (+,+)(+, +) is I, (−,+)(-, +) is II, (−,−)(-, -) is III, (+,−)(+, -) is IV. Points on an axis have a zero coordinate and belong to no quadrant.

Tip
The pattern

I is (+,+)(+,+), II is (−,+)(-,+), III is (−,−)(-,-), IV is (+,−)(+,-).

Example
Plotting (−3,2)(-3, 2)
xy−6−5−4−3−2−11234−1123(−3, 2)
Three left, then two up.

Start at the origin. The xx-coordinate −3-3 means 33 to the left; then 22 up. Left and up is quadrant II.

Example
Points on an axis

(0,−4)(0, -4) is 44 straight down, on the yy-axis; (5,0)(5, 0) is 55 to the right, on the xx-axis. A zero coordinate puts a point on an axis, and axis points belong to no quadrant.

Side note
Where it goes wrong

Reflections: reflecting across the xx-axis changes the sign of yy (the point moves up or down), not xx. Sketch it — the axis you reflect across stays fixed.

Try it
Try it: name the quadrant

Name the quadrant or axis for (7,−1)(7, -1), (−2,−8)(-2, -8), (−5,6)(-5, 6) and (0,3)(0, 3).

Show the answer

(7,−1)(7, -1): right and down, IV. (−2,−8)(-2, -8): left and down, III. (−5,6)(-5, 6): left and up, II. (0,3)(0, 3): on the yy-axis.

In Plain Terms & More Examples

Concept
In plain terms

The signs of xx and yy tell you the quadrant: quadrants are numbered anticlockwise from the top right.

Example
Where is (−4,−1)(-4, -1)?
xy−7−6−5−4−3−2−1123−3−2−11(−4, −1)
Left 4, down 1: quadrant III.

Both are negative, so it is in quadrant III.

Example
Where is (5,−2)(5, -2)?
xy−3−2−11234567−3−2−1(5, −2)
Right 5, down 2: quadrant IV.

xx is positive and yy is negative, so it is in quadrant IV.

Side note
Where it goes wrong

Going up for a negative yy. In (5,−2)(5, -2) the −2-2 means 22 down. Positive is right and up; negative is left and down, for both coordinates.

Try it
Try it: signs and moves

Which quadrant holds the points with x<0x < 0 and y>0y > 0? Where is the point 44 left of the origin and 66 down?

Show the answer

Left and up: quadrant II\mathbf{II}. Four left and six down is (−4,−6)\mathbf{(-4, -6)}, in quadrant III.

Reflections and Distances

Concept
Flip a sign

Reflecting across the xx-axis changes the sign of yy; across the yy-axis changes the sign of xx. Distance along a horizontal or vertical line is the absolute difference of the coordinates that change.

Example
Reflect (3,−2)(3, -2)
xy−6−5−4−3−2−112345−3−2−112(3, −2)(3, 2)(−3, −2)(−3, 2)
Across the x-axis the height flips; across the y-axis the side flips; across both, both flip.

Across the xx-axis: (3,2)(3, 2). Across the yy-axis: (−3,−2)(-3, -2). Across both: (−3,2)(-3, 2).

Example
Distance across an axis
xy−6−5−4−3−2−11234567−11234(−3, 2)(5, 2)35
3 on one side of the axis plus 5 on the other.

(−3,2)(-3, 2) and (5,2)(5, 2) have the same yy, so the distance is ∣5−(−3)∣=8|5 - (-3)| = 8. On the grid: 33 units from x=−3x = -3 to the axis, then 55 more.

Side note
Where it goes wrong

Subtracting the sizes across an axis. From (−3,2)(-3, 2) to (5,2)(5, 2) is not 5−3=25 - 3 = 2: the points are on opposite sides of the yy-axis, so the distances add, 3+5=83 + 5 = 8. Subtracting the coordinates themselves, 5−(−3)5 - (-3), handles this automatically.

Try it
Try it: distance and quadrant

How far is (−4,7)(-4, 7) from (−4,−3)(-4, -3)? In which quadrant is (−5,−1)(-5, -1)?

Show the answer

Same xx, so ∣7−(−3)∣=10|7 - (-3)| = \mathbf{10}. Both coordinates negative: quadrant III\mathbf{III}.

Word Problems

Concept
Maps in four quadrants

Put the origin at a landmark, like the town square. East and north are positive, west and south negative. Distances along streets are differences of one coordinate.

Negative coordinates let a map stretch in every direction from its centre. Streets on a grid run horizontally or vertically, so a walking distance changes one coordinate at a time, and it is found with the same rule as distance on a number line.

Example
A polygon on the grid
xy−4−3−2−112345−2−112364
Width |4 − (−2)| = 6, height |3 − (−1)| = 4.

Vertices (−2,3),(4,3),(4,−1),(−2,−1)(-2, 3), (4, 3), (4, -1), (-2, -1) form a rectangle 66 wide and 44 tall: perimeter 2020, area 2424.

Example
Library to school

The library is at (−4,1)(-4, 1) and the school at (2,1)(2, 1). They have the same yy, so the walk along the street is ∣2−(−4)∣=6|2 - (-4)| = 6 blocks.

Side note
Where it goes wrong

Adding coordinates for a perimeter. The rectangle (−2,3),(4,3),(4,−1),(−2,−1)(-2, 3), (4, 3), (4, -1), (-2, -1) has sides ∣4−(−2)∣=6|4 - (-2)| = 6 and ∣3−(−1)∣=4|3 - (-1)| = 4, so its perimeter is 2(6+4)=202(6 + 4) = 20. Adding up the coordinates themselves measures nothing.

Try it
Try it: the park
xy−6−5−4−3−2−112345−4−3−2−1(−3, −3)(3, −3)
The south side of the park.

A square park has corners at (−3,−3)(-3, -3) and (3,−3)(3, -3) on its south side. Where are the north corners, and what is the park's area?

Show the answer

Side length 66, so the north corners are (−3,3)(\mathbf{-3, 3}) and (3,3)(\mathbf{3, 3}); area 36\mathbf{36}.

Fractions and Decimals on the Plane

Concept
Between the grid lines

Coordinates do not have to be whole numbers. (2.5,−1.5)(2.5, -1.5) is halfway between grid lines in both directions: 2.52.5 right, then 1.51.5 down.

Each axis is a whole number line, so every number on it has a place: halves sit halfway between grid lines, quarters a quarter of the way. The quadrant rules and the distance rule work exactly as before.

Example
Plotting (2.5,−1.5)(2.5, -1.5)
xy−2−11234−2−1(2.5, −1.5)
Halfway between grid lines, both ways.

Go 2.52.5 right, halfway between 22 and 33; then 1.51.5 down, halfway between −1-1 and −2-2. Right and down: quadrant IV.

Example
Plotting (−12,3)(-\tfrac12, 3)

Half a unit left of the origin, then 33 up: quadrant II, very close to the yy-axis.

Example
A distance

(−1.5,2)(-1.5, 2) and (3,2)(3, 2) share y=2y = 2. Distance: ∣3−(−1.5)∣=4.5|3 - (-1.5)| = 4.5 units.

Side note
Where it goes wrong

Counting from the wrong side of −1-1. −1.5-1.5 lies between −1-1 and −2-2, farther from zero than −1-1. Plotting it at −0.5-0.5, between 00 and −1-1, moves the point the wrong way.

Try it
Try it: halves

Plot (−2.5,0.5)(-2.5, 0.5). Which quadrant is it in, and how far is it from (−2.5,−3)(-2.5, -3)?

Show the answer

Left 2.52.5, up 0.50.5: quadrant II\mathbf{II}. The two points share x=−2.5x = -2.5, so the distance is ∣0.5−(−3)∣=3.5|0.5 - (-3)| = \mathbf{3.5} units.

Why It Works

Distance along a horizontal line is ∣x2−x1∣|x_2 - x_1| and along a vertical line is ∣y2−y1∣|y_2 - y_1| — absolute values, because a distance is never negative. So (−3,2)(-3, 2) and (5,2)(5, 2) are ∣5−(−3)∣=8|5 - (-3)| = 8 apart. Reflecting across the yy-axis sends (x,y)(x, y) to (−x,y)(-x, y) because the mirror keeps the height and flips the side; this is the first appearance of a transformation rule written in coordinates.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.