Quadrants and Signs

The two axes of the Cartesian plane do more than just label points , they slice the plane into four regions called quadrants. Each quadrant has its own sign pattern for $(x, y)$, and learning the four patterns is the fastest way to predict where a point will land before you draw a single dot.

Definitions

The four open regions of the Cartesian plane, not including the axes themselves, are the quadrants. They are numbered counter-clockwise starting from the upper right:

• Quadrant I (upper right): $x > 0$ and $y > 0$. Signs $(+, +)$.
• Quadrant II (upper left): $x < 0$ and $y > 0$. Signs $(-, +)$.
• Quadrant III (lower left): $x < 0$ and $y < 0$. Signs $(-, -)$.
• Quadrant IV (lower right): $x > 0$ and $y < 0$. Signs $(+, -)$.

The two axes themselves are not in any quadrant. A point with $x = 0$ or $y = 0$ lies on an axis, not in a quadrant.

Concept and use

The sign chart. Memorise the four sign patterns:

Once you internalise this, you can name the quadrant of a point in a single glance.

A handy mnemonic. Walking counter-clockwise from the upper right, the sign of $x$ flips when you cross the $y$-axis (going left or coming back), and the sign of $y$ flips when you cross the $x$-axis (going down or coming back).

Why we number counter-clockwise. It matches the mathematical convention for measuring angles: angles increase counter-clockwise from the positive $x$-axis. Quadrant I corresponds to $0^\circ$–$90^\circ$, Quadrant II to $90^\circ$–$180^\circ$, and so on. This convention is universal in trigonometry and higher mathematics.

Points on the axes. As noted, points with $x = 0$ are on the $y$-axis, and points with $y = 0$ are on the $x$-axis. They are borderline , not in any quadrant. The origin $(0,0)$ is on both axes simultaneously.

Mirror moves between quadrants.

• Reflecting $(x, y)$ across the $x$-axis sends it to $(x, -y)$ , Quadrant I $\leftrightarrow$ Quadrant IV, and Quadrant II $\leftrightarrow$ Quadrant III.
• Reflecting $(x, y)$ across the $y$-axis sends it to $(-x, y)$ , Quadrant I $\leftrightarrow$ Quadrant II, and Quadrant III $\leftrightarrow$ Quadrant IV.
• Reflecting across the origin sends $(x, y)$ to $(-x, -y)$ , Quadrant I $\leftrightarrow$ Quadrant III, and Quadrant II $\leftrightarrow$ Quadrant IV.

These three operations are the most common ways exam questions move points around. Knowing them turns the question into a one-step sign change.

Quadrant size. All four quadrants are infinite. They are not "small boxes"; they extend forever in their respective directions.

Worked examples

Example 1. In which quadrant does $(7, -3)$ lie?

$x > 0$, $y < 0$. That is Quadrant IV.

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

$x < 0$, $y < 0$. That is Quadrant III.

Example 3. $(-3, 0)$ , quadrant or axis?

$y = 0$, so it is on the $x$-axis, not in any quadrant.

Example 4. Reflect $(4, -7)$ across the $y$-axis. Which quadrant does the image lie in?

Reflection across the $y$-axis sends $(x, y)$ to $(-x, y)$, giving $(-4, -7)$. That is Quadrant III.

Example 5. Reflect $(2, 5)$ across the origin. Which quadrant does the image lie in?

Sends $(2, 5) \mapsto (-2, -5)$. That is Quadrant III.

Try it yourself

1. State the quadrant of each: $(3, 4), (-1, 2), (-5, -6), (8, -3), (0, 7), (-2, 0)$.
2. Reflect $(-3, 5)$ across the $x$-axis. In which quadrant does the image lie?
3. Reflect $(-3, 5)$ across the $y$-axis. In which quadrant does the image lie?
4. Reflect $(4, -2)$ across the origin. In which quadrant does the image lie?
5. Find one point in each of the four quadrants whose coordinates are non-zero integers.
6. Where does the point $(0, -3)$ lie?
7. A point has positive abscissa and negative ordinate. In which quadrant is it?
8. Which two reflections together produce a reflection across the origin?
9. Give the quadrant of $(a, b)$ if $a < 0$ and $b > 0$.
10. Is the point $(-1, 0)$ in Quadrant II or Quadrant III? Justify.

Pitfalls / Insight

• A point on the axis is not in any quadrant. Don't be tempted to say "it's in Quadrant II" just because $x < 0$; if $y = 0$, it sits on the $x$-axis.
• Quadrant numbering is counter-clockwise. Quadrant II is upper left, not lower right.
• Sign of zero. Treat $0$ as neither positive nor negative when classifying quadrants.

Insight. Each quadrant is defined by just a sign pattern. Once you can read the sign of $x$ and the sign of $y$, you have the quadrant , no plotting required. That single observation is what makes coordinate geometry quick.