Coordinate plane (first quadrant, then four quadrants) Knowledge-Based Study

Study notes

1. Foundations of the Coordinate Plane

   A coordinate plane is formed by two perpendicular number lines that intersect at a shared zero point called the origin, denoted as $(0, 0)$.

   • $x$-axis: The horizontal number line. Values increase to the right and decrease to the left.
   • $y$-axis: The vertical number line. Values increase upward and decrease downward.

   Ordered Pairs in Quadrant I

   Any location on the plane is named by an ordered pair written in the form $(x, y)$:

   • The first number, the $x$-coordinate, indicates how far to move horizontally from the origin.
   • The second number, the $y$-coordinate, indicates how far to move vertically from the origin.

   In Quadrant I, located in the upper-right section of the plane, both coordinates are strictly positive: $x > 0$ and $y > 0$. For example, to locate the point $(4, 3)$, start at $(0, 0)$, move 4 units to the right, and then move 3 units up.

   The Four Quadrants

   When both axes extend infinitely in both positive and negative directions, they divide the coordinate plane into four regions called quadrants. Quadrants are numbered counterclockwise using Roman numerals, starting from the upper-right corner.

   • Quadrant I: Both coordinates are positive $(+, +)$. Example: $(2, 5)$.
   • Quadrant II: The $x$-coordinate is negative, and the $y$-coordinate is positive $(-, +)$. Example: $(-3, 4)$.
   • Quadrant III: Both coordinates are negative $(-, -)$. Example: $(-4, -2)$.
   • Quadrant IV: The $x$-coordinate is positive, and the $y$-coordinate is negative $(+, -)$. Example: $(3, -6)$.

   Points on the Axes

   Points that lie directly on an axis do not belong to any quadrant:

   • If $y = 0$, the point lies on the $x$-axis, such as $(5, 0)$ or $(-2, 0)$.
   • If $x = 0$, the point lies on the $y$-axis, such as $(0, 4)$ or $(0, -7)$.
   • The origin $(0, 0)$ lies on both axes simultaneously.

   How to Plot Any Ordered Pair $(x, y)$

   Follow this systematic procedure to plot any point on the coordinate plane:

   1. Start at the origin $(0, 0)$.
   2. Move along the horizontal axis: Move right if the $x$-coordinate is positive; move left if it is negative. If $x = 0$, do not move horizontally.
   3. Move along the vertical direction: From your new horizontal position, move up if the $y$-coordinate is positive; move down if it is negative. If $y = 0$, stay on the $x$-axis.
   4. Mark and label the point with its coordinates or letter designation.

   Quick Check Examples

   • Point $A(-5, 2)$: Move 5 units left, then 2 units up. Located in Quadrant II.
   • Point $B(-3, -4)$: Move 3 units left, then 4 units down. Located in Quadrant III.
   • Point $C(6, -1)$: Move 6 units right, then 1 unit down. Located in Quadrant IV.
   • Point $D(0, -5)$: Move 0 units horizontally, then 5 units down. Located on the $y$-axis.