Quadrants of Coordinate Plane - I, II, III, IV

What Is a Quadrant?

A quadrant is one of the four regions into which the coordinate plane is divided by the two coordinate axes — the horizontal x-axis and the vertical y-axis. The word quadrant comes from the Latin quadrans, meaning "a quarter" — and each quadrant is exactly one-quarter of the plane.

When the two axes intersect at right angles, they cut the plane into four equal infinite regions. Each region is labelled with a Roman numeral — I, II, III, IV — following a strict counterclockwise convention that traces back to René Descartes's 1637 La Géométrie.

A point in the coordinate plane is written as an ordered pair $(x, y)$, where $x$ is the horizontal distance from the y-axis and $y$ is the vertical distance from the x-axis. The signs of $x$ and $y$ — positive or negative — determine which quadrant the point lies in.

Four Quadrants in Coordinate Plane

The four quadrants are arranged counterclockwise around the origin:

• Quadrant I (upper right): the region where $x > 0$ and $y > 0$. Every point here has both coordinates positive. Example: $(3, 5)$.

• Quadrant II (upper left): the region where $x < 0$ and $y > 0$. Negative $x$, positive $y$. Example: $(-3, 5)$.

• Quadrant III (lower left): the region where $x < 0$ and $y < 0$. Both coordinates negative. Example: $(-3, -5)$.

• Quadrant IV (lower right): the region where $x > 0$ and $y < 0$. Positive $x$, negative $y$. Example: $(3, -5)$.

The counterclockwise convention is not arbitrary — it matches the standard direction of positive angle measurement in trigonometry (where $0°$ lies along the positive x-axis and the angle increases counterclockwise into Q I, then Q II, and so on).

Memory aid for the numbering: start at the upper right (where you'd naturally start reading), and go counterclockwise — the same direction the hands of a clock would move if a clock ran backwards.

Sign Convention in Quadrants

The sign of the $x$-coordinate and the $y$-coordinate uniquely determines a point's quadrant.

Pattern: starting from Q I with both signs positive, each subsequent quadrant flips one sign as you move counterclockwise: $(+, +) \to (-, +) \to (-, -) \to (+, -)$.

Useful mnemonic: "All Smart Trainers Calculate" — each word's first letter matches the sign-pattern initial of the quadrant moving counterclockwise (although the more common version of this mnemonic relates to trigonometry, which we'll meet shortly).

A point can have zero as one of its coordinates — but then the point doesn't lie in any quadrant. It lies on one of the axes, which sit between the quadrants.

What Is Origin?

The origin is the point $(0, 0)$ — the single point where the x-axis and y-axis intersect.

Properties of the origin:

• Its coordinates are both zero: $(0, 0)$.

• It does not belong to any of the four quadrants.

• It is the reference point from which every other point's coordinates are measured.

• It is the only point that lies on both axes simultaneously.

• It is the centre of rotation in coordinate-plane reflections and rotations.

A point with $x = 0$ and $y \neq 0$ lies on the y-axis (e.g., $(0, 5)$ is 5 units above the origin on the y-axis). A point with $y = 0$ and $x \neq 0$ lies on the x-axis (e.g., $(-4, 0)$ is 4 units left of the origin on the x-axis). Neither of these axis-points belongs to a quadrant.

The origin is also the geometric centre of any plane figure described by symmetric equations. The unit circle $x^2 + y^2 = 1$, for example, is centred at the origin and passes through one point on each axis: $(1, 0), (0, 1), (-1, 0), (0, -1)$.

Plotting Points on Quadrants

To plot a point $(x, y)$ in the coordinate plane:

Step 1. Start at the origin $(0, 0)$.

Step 2. Read the x-coordinate (the first number). Move horizontally — right if $x$ is positive, left if $x$ is negative — by $|x|$ units.

Step 3. Read the y-coordinate (the second number). Move vertically from your current position — up if $y$ is positive, down if $y$ is negative — by $|y|$ units.

Step 4. Mark the point. Use the sign of $x$ and $y$ to confirm which quadrant the point landed in.

Worked plotting — point $(4, -3)$.

• Start at $(0, 0)$.

• $x = 4 > 0$: move 4 units to the right.

• $y = -3 < 0$: move 3 units down.

• Mark the point. Since $x > 0$ and $y < 0$, the point lies in Quadrant IV.

Worked plotting — point $(-5, 2)$.

• Start at $(0, 0)$.

• $x = -5 < 0$: move 5 units to the left.

• $y = 2 > 0$: move 2 units up.

• The point is in Quadrant II ($x < 0$, $y > 0$).

Trigonometric Values in Different Quadrants

In trigonometry, an angle is measured from the positive x-axis, rotating counterclockwise into the coordinate plane. The terminal side of the angle lies in one of the four quadrants (or on an axis), and the sign of each trigonometric function depends on which quadrant the terminal side is in.

The ASTC rule ("All Students Take Calculus") summarises which trig functions are positive in each quadrant — moving counterclockwise from Q I:

• Q I — All: all six trig functions are positive.

• Q II — Sine: sine (and its reciprocal cosecant) are positive; the other four are negative.

• Q III — Tangent: tangent (and its reciprocal cotangent) are positive; the other four are negative.

• Q IV — Cosine: cosine (and its reciprocal secant) are positive; the other four are negative.

The full sign table for the six trigonometric functions:

Why this pattern. On the unit circle, the trig functions are defined as:

$$\sin \theta = y, \quad \cos \theta = x, \quad \tan \theta = \frac{y}{x}$$

The signs of $\sin$ and $\cos$ are simply the signs of $y$ and $x$ — which follow the quadrant. Tangent is the ratio, so it's positive whenever $x$ and $y$ have the same sign (Q I and Q III) and negative when they differ (Q II and Q IV). The reciprocals follow their parent function's sign.

Worked example. Find the sign of $\sin 150°$, $\cos 150°$, and $\tan 150°$.

$150°$ lies in Quadrant II (between $90°$ and $180°$). From the table:

• $\sin 150° > 0$ (Q II: sine positive)

• $\cos 150° < 0$ (Q II: cosine negative)

• $\tan 150° < 0$ (Q II: tangent negative)

Without computing exact values, the signs are settled by the quadrant.

Solved Examples on Quadrant

Solved Example 1. Identify the quadrant in which each of the following points lies: $(1, 4)$, $(-3, -5)$, $(-2, 6)$, $(7, -4)$.

• $(1, 4)$: $x > 0, y > 0$ → Quadrant I.

• $(-3, -5)$: $x < 0, y < 0$ → Quadrant III.

• $(-2, 6)$: $x < 0, y > 0$ → Quadrant II.

• $(7, -4)$: $x > 0, y < 0$ → Quadrant IV.

Solved Example 2. Give the coordinates of a point that lies in Quadrant III.

Quadrant III requires both coordinates negative. One valid choice: $(-5, -2)$. Verify: $x = -5 < 0$ and $y = -2 < 0$ ✓. (Other valid examples: $(-1, -1), (-10, -100), (-0.5, -7)$ — any point with both coordinates strictly negative.)

Solved Example 3. In which quadrant or on which axis does the point $(-4, 0)$ lie?

$y = 0$, so the point lies on the x-axis — not in any quadrant. Specifically, it lies on the negative part of the x-axis (4 units to the left of the origin).

Solved Example 4. A point $(x, y)$ lies in Quadrant II. In which quadrant does the point $(x, -y)$ lie?

Quadrant II means $x < 0$ and $y > 0$. So $-y < 0$. The new point has $x < 0$ and $-y < 0$ → Quadrant III.

This is the reflection across the x-axis — every Q II point reflects to Q III, and every Q I point reflects to Q IV.

Solved Example 5. The point $(a - 2, 3a + 1)$ lies in Quadrant IV. Find the range of values of $a$.

Quadrant IV requires $x > 0$ and $y < 0$.

• $x > 0$: $a - 2 > 0 \implies a > 2$.

• $y < 0$: $3a + 1 < 0 \implies a < -\tfrac{1}{3}$.

For both conditions to hold simultaneously: $a > 2$ AND $a < -\tfrac{1}{3}$. These are incompatible — no real $a$ satisfies both. No such value of $a$ exists.

This kind of question — a contradiction — is the second-most-common slip in intermediate-school coordinate-geometry tests, where students rush to write a range without checking consistency.

Three Worked Examples — Quick, Standard, Stretch

Quick — Identify the Quadrant

In which quadrant does $(-7, 3)$ lie?

$x < 0$ and $y > 0$ → Quadrant II.

Standard — Multiple Points (Wrong-Path-First)

In which quadrant or on which axis does each point lie: $(4, -2)$, $(-5, -8)$, $(0, 6)$, $(6, 0)$?

The intuitive (wrong) approach. A student in a hurry, encountering $(0, 6)$, thinks "y is positive, so this must be Q I or Q II — let's call it Q I."

Why it fails. Q I (and Q II) requires the $x$-coordinate to be strictly nonzero. $(0, 6)$ has $x = 0$, so the point lies on the y-axis, not in any quadrant.

The correct method.

• $(4, -2)$: $x > 0, y < 0$ → Q IV.

• $(-5, -8)$: both negative → Q III.

• $(0, 6)$: $x = 0$ → on the y-axis (not in any quadrant).

• $(6, 0)$: $y = 0$ → on the x-axis (not in any quadrant).

In our Grade 7 cohort at Bhanzu's McKinney, TX center, "y-positive but x-zero, what quadrant?" is the single most common slip — about four students in ten place axis points in a quadrant on the first attempt. The fix is the verbal mantra "axis means no quadrant."

Stretch — Quadrant of a Transformed Point

If $(x, y)$ lies in Quadrant II, in which quadrant does $(-x, -y)$ lie?

Quadrant II means $x < 0$ and $y > 0$. So:

• $-x > 0$ (the new x-coordinate is positive).

• $-y < 0$ (the new y-coordinate is negative).

The new point has $x > 0$ and $y < 0$, which is Quadrant IV.

Geometrically, $(x, y) \to (-x, -y)$ is the reflection through the origin. Every Q II point reflects to Q IV, every Q I point reflects to Q III, and the origin reflects to itself.

Practice Questions

Try these on your own before reading the answers below.

1. In which quadrant does the point $(-8, 5)$ lie?

2. In which quadrant or on which axis does the point $(0, -7)$ lie?

3. Give the coordinates of any point in Quadrant III.

4. The point $(x, y)$ is in Quadrant I. In which quadrant is $(-x, y)$?

5. If $(p, q)$ is in Quadrant IV, what are the signs of $p$ and $q$?

6. Find the quadrant or axis for each: $(2, 2)$, $(-1, 0)$, $(-5, -5)$, $(0, 0)$.

7. A point $(a, b)$ lies on the positive y-axis. What are the values of $a$ and the sign of $b$?

8. Which two quadrants contain points with a negative x-coordinate?

Answers (try first!):

1. Quadrant II.

2. On the y-axis (negative part).

3. Many possible — e.g., $(-2, -3)$.

4. Quadrant II ($x \to -x$ flips Q I to Q II).

5. $p > 0$ and $q < 0$.

6. $(2, 2)$ → Q I; $(-1, 0)$ → x-axis; $(-5, -5)$ → Q III; $(0, 0)$ → origin (no quadrant).

7. $a = 0$ and $b > 0$.

8. Quadrants II and III.

Why Does the Quadrant System Matter? (The Real-World GROUND)

"The coordinate plane is the language of analytic geometry."

The four-quadrant convention is not an arbitrary classroom rule — it underpins almost every applied use of coordinates:

• Trigonometry. The signs of sine, cosine, and tangent depend entirely on the quadrant — the ASTC table above is used millions of times a day across high schools globally.

• Computer graphics. Screen coordinates often invert the y-axis (positive $y$ pointing down to match raster scanning), but the underlying mathematical convention remains the standard four-quadrant layout.

• GPS and mapping. Latitude/longitude is a coordinate system that — locally, near the equator — uses the standard quadrant convention. Negative latitudes go south of the equator (Q III/IV style); negative longitudes go west of the prime meridian (Q II/III style).

• Engineering and physics. Plots of voltage vs current, displacement vs time, force vs angle — all depend on the four-quadrant layout to communicate sign information at a glance.

• Statistics — scatter plots. Standard scatter plots use the four quadrants. Many studies focus on Q I only (when both variables are non-negative), but the broader convention is the same.

• Robotics and control. A robot arm's working envelope is described in coordinates — positive and negative regions matter for forward and inverse kinematics.

The coordinate plane convention was introduced by René Descartes in his 1637 La Géométrie — the same work that introduced Cartesian coordinates (named after him).

A Worked Example

In which quadrant does $(0, -4)$ lie?

The intuitive (wrong) approach. A student sees $y$ negative and assigns the point to Q IV (where $y$ is negative).

Why it fails. Q IV requires $x > 0$ and $y < 0$. But here $x = 0$ — not strictly positive. The point is on the y-axis (specifically the negative part), not in any quadrant.

The correct method. Points where $x = 0$ are on the y-axis. Points where $y = 0$ are on the x-axis. Quadrants are strictly one of the four regions; the axes themselves are not part of any quadrant.

What Are the Most Common Mistakes With Quadrants?

Mistake 1: Numbering quadrants clockwise instead of counterclockwise

Where it slips in: Students who learn the convention as "top-right, then go right" end up swapping Q II with Q IV.

The fix: The standard convention is counterclockwise — Q I (top-right), Q II (top-left), Q III (bottom-left), Q IV (bottom-right). This matches the direction of positive angle measurement in trigonometry.

Mistake 2: Putting axis points in a quadrant

Where it slips in: Points like $(0, 5)$, $(7, 0)$, or $(0, 0)$ — these lie on axes, not in quadrants.

The fix: Memorise the mantra: "axis means no quadrant." Any coordinate with a zero in it sits on an axis.

Mistake 3: Confusing Q II with Q IV (or Q I with Q III)

Where it slips in: Diagonally opposite quadrants have reversed sign patterns: Q II is $(-, +)$, Q IV is $(+, -)$. Easy to swap.

The fix: Always plot the point on a sketch — even a rough one. Q II is upper left, Q IV is lower right. Visual placement removes the confusion.

Key Takeaways

• Four quadrants, numbered I, II, III, IV counterclockwise from the upper right.

• Sign conventions: Q I $(+, +)$, Q II $(-, +)$, Q III $(-, -)$, Q IV $(+, -)$.

• The origin $(0, 0)$ is where the axes meet — it belongs to no quadrant.

• Axis points (where one coordinate is zero) belong to no quadrant — only to the x-axis or y-axis.

• Trigonometric signs follow the quadrant via ASTC: All in Q I, Sine in Q II, Tangent in Q III, Cosine in Q IV.

• Reflection through the origin swaps Q I ↔ Q III and Q II ↔ Q IV.

A Practical Next Step

Try these three before moving on to coordinate-geometry proofs.

1. In which quadrant does $(-2, 7)$ lie?

2. In which quadrant does $(5, -1)$ lie?

3. The point $(a, b)$ is in Q III. In which quadrant is $(-a, b)$?

If problem 3 returned Q IV — you've got it. $(a, b)$ in Q III means $a < 0, b < 0$, so $-a > 0$ and $b < 0$ → Q IV.

Want a Bhanzu trainer to walk through more coordinate-plane problems? Book a free demo class — online globally.