Coordinate Geometry • Topic 1 of 5

The Coordinate Plane

The plane is built from two perpendicular number lines: the horizontal x-axis and the vertical y-axis, crossing at the origin (0, 0). A point is an ordered pair (x, y) where x is read first (right/left) and y second (up/down) — reversing them is the most common beginner error. The axes split the plane into four quadrants, numbered counter-clockwise starting top-right: Quadrant I has (+, +), II has (−, +), III has (−, −), and IV has (+, −). A point on an axis belongs to no quadrant — for example (0, 5) sits on the y-axis. The GRE uses this for quick sign reasoning: if xy > 0 the point is in Quadrant I or III; if xy < 0 it is in II or IV. Many "which quadrant" and Quantitative Comparison questions are answered purely from the signs of x and y, with no calculation at all.

The xy-plane split into four quadrants, with the points (3, 2) and (-2, 1) plottedThe Coordinate Planexy-5-4-3-2-112345-5-4-3-2-112345OIIIIIIIV(3, 2)(−2, 1)Read x first (across), then y (up/down).

✅ Solved examples

1. In which quadrant does the point (−4, 6) lie?
x is negative, y is positive → Quadrant II.
2. A point has xy < 0 and y > 0. Which quadrant is it in?
If y > 0 and the product xy < 0, then x < 0. Negative x, positive y → Quadrant II.
3. Point (a, b) is in Quadrant III. What are the signs of a and b?
Quadrant III is (−, −), so a < 0 and b < 0.
4. Quantitative Comparison. Point P = (−3, −7). Quantity A: the x-coordinate of P. Quantity B: the y-coordinate of P. Which is greater?
x = −3, y = −7. Since −3 > −7, Quantity A is greater. (On the number line −3 lies to the right of −7.)

✏️ Practice — try these, take hints as needed

1. In which quadrant does (5, −2) lie?
x positive, y negative.
Positive x, negative y.
Bottom-right.
Quadrant IV
2. On which axis does the point (0, −8) lie?
x-coordinate is 0.
x = 0 means on the vertical axis.
That is the y-axis.
The y-axis
3. If a point (x, y) has x < 0 and y < 0, which quadrant?
Both negative.
Bottom-left.
Count counter-clockwise from I.
Quadrant III
4. A point lies in Quadrant IV. What are the signs of its coordinates?
Quadrant IV is bottom-right.
x positive.
y negative.
x > 0, y < 0
5. If xy > 0 and x < 0, which quadrant is (x, y) in?
xy > 0 means same signs.
x < 0 forces y < 0.
Both negative.
Quadrant III

📝 Topic test — 8 questions

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