Coordinate Plane and Ordered Pairs: Axes, Quadrants, and Plotting Points

An ordered pair is a way to locate a point on the rectangular coordinate system, written as $(x, y)$ .
Example: to express the location with x=2 and y=13, you can write $(2, 13)$ instead of saying "x equals 2 and y equals 13".
The first coordinate (x) indicates horizontal position; the second coordinate (y) indicates vertical position.
Points are located by starting at the origin and moving horizontally (along the x-axis) then vertically (along the y-axis).
The origin is the point where both coordinates are zero: $(0, 0)$ .
When you plot a set of potential solutions, you place each point on the grid according to its coordinates.

The coordinate system consists of two perpendicular lines: the horizontal axis (x-axis) and the vertical axis (y-axis).
The intersection of the x-axis and y-axis is the origin: $(0, 0)$ .
The axes divide the plane into four regions called quadrants: Quadrant I, Quadrant II, Quadrant III, Quadrant IV.
Axes and quadrants are used to describe the location of points:
- Positive x coordinates move to the right along the x-axis; negative x coordinates move to the left.
- Positive y coordinates move up along the y-axis; negative y coordinates move down.
Example: point $(0, 1)$ lies on the y-axis above the origin (on the vertical axis).

To locate a point $(x, y)$:
- Start at the origin $(0, 0)$ .
- Move horizontally by $|x|$ units in the direction indicated by the sign of x (right if $x>0$, left if $x<0$).
- Then move vertically by $|y|$ units in the direction indicated by the sign of y (up if $y>0$, down if $y<0$).
Important convention:
- The first number is always the x-coordinate (horizontal position).
- The second number is always the y-coordinate (vertical position).
Example placements:
- $(2, 13)$ means 2 units to the right and 13 units up from the origin.
- $(0, 1)$ lies on the y-axis one unit above the origin.
- $(0, 0)$ is the origin itself.

Quadrant I: $(+x, +y)$
Quadrant II: $(−x, +y)$
Quadrant III: $(−x, −y)$
Quadrant IV: $(+x, −y)$
An equation like $x = c$ describes a vertical line: every point on this line has the same x-coordinate $c$.
An equation like $y = c$ describes a horizontal line: every point on this line has the same y-coordinate $c$.
The line $x = 0$ is the y-axis; it passes through the origin and includes all points with x-coordinate 0, not just the origin.
The line $y = 0$ is the x-axis; it passes through the origin and includes all points with y-coordinate 0, not just the origin.

Connecting new notation to prior ideas: thinking in terms of ordered pairs links to earlier work with coordinates and grid systems we already know.
When solving or collecting a set of solutions, you graph each solution as a point on the grid to visualize the solution set.
If you ask where a point lies on the origin or axes:
- The origin is $(0, 0)$ .
- A point with x=0 but y ≠ 0 lies on the y-axis (the vertical line $x = 0$).
- A point with y=0 but x ≠ 0 lies on the x-axis (the horizontal line $y = 0$).

Graphing is a foundational tool in algebra, geometry, and data visualization; it helps see relationships between variables.
Ordered pairs provide a precise way to communicate location in two dimensions, which is essential in fields like cartography, computer graphics, and physics.
Understanding the distinction between a single point (e.g., $(2,13)$) and a line (e.g., $x=c$ or $y=c$) is crucial for solving systems of equations and for graphing functions.

Point notation: $(x, y)$ identifies a location on the plane.
The first coordinate x measures horizontal position; the second coordinate y measures vertical position.
Start at the origin, then move along x, then along y to reach the point.
The axes divide the plane into four quadrants with common sign patterns: I $(+,+)$, II $(−,+)$, III $(−,−)$, IV $(+,-)$.
Vertical lines have the form $x = c$; horizontal lines have the form $y = c$.
The origin is $(0, 0)$ ; the y-axis is $x = 0$ and the x-axis is $y = 0$.