The Cartesian Plane
Plot ordered pairs on the Cartesian plane, navigate all four quadrants and begin graphing linear relationships.
The Coordinate System
The Cartesian plane (named after mathematician René Descartes) is a flat grid formed by two number lines crossing at right angles at the origin (0, 0). Every point on the plane is described by an ordered pair (x, y).
Ordered Pairs (x, y)
Every point on the Cartesian plane is written as an ordered pair (x, y). The x-coordinate (horizontal) always comes first, the y-coordinate (vertical) comes second. Think: "walk before you climb."
Plotting (4, −2)
- 1Start at the origin (0, 0).
- 2Move 4 units right along the x-axis (positive x).
- 3Move 2 units down (negative y).
- 4Mark the point. It lands in Quadrant 4.
Graphing Linear Relationships
A linear relationship produces a straight line when graphed. To graph it, create a table of values by substituting x-values into the rule, plot the points, then draw a straight line through them.
Example: Graph y = x + 2
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y = x + 2 | 0 | 1 | 2 | 3 | 4 |
Points to plot: (-2, 0), (-1, 1), (0, 2), (1, 3), (2, 4)
Key Vocabulary
Origin
The point (0, 0) where the x-axis and y-axis cross. The starting point of the coordinate system.
Ordered Pair
A pair of numbers (x, y) used to locate a point. The x-coordinate comes first, then the y-coordinate.
Quadrant
One of the four regions of the Cartesian plane, created by the x and y axes. Numbered Q1–Q4 anti-clockwise from top-right.
Linear Relationship
A relationship between x and y that forms a straight line when graphed. Described by a rule such as y = 2x + 1.
Worked Examples
Plot A(3, 4), B(-2, 1) and C(0, -3). State which quadrant each is in.
A(3, 4): x positive, y positive → Quadrant 1
B(-2, 1): x negative, y positive → Quadrant 2
C(0, -3): lies on the y-axis (not in any quadrant)
Find the missing y-value for y = 2x − 1 when x = 3.
Substitute x = 3: y = 2(3) − 1 = 6 − 1 = 5
The point is (3, 5).
A point is 3 units left and 4 units up from the origin. Write its coordinates.
3 units left = x = −3
4 units up = y = +4
Coordinates: (−3, 4) → Quadrant 2
Knowledge Check
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Key Concepts Summary
- ●The Cartesian plane has a horizontal x-axis and a vertical y-axis meeting at the origin (0, 0).
- ●Points are written as ordered pairs (x, y) — x first, y second.
- ●The plane is divided into 4 quadrants: Q1 (+,+), Q2 (-,+), Q3 (-,-), Q4 (+,-).
- ●A linear relationship is graphed by making a table of values, plotting points, and joining them in a straight line.
- ●Points on the x-axis have y = 0; points on the y-axis have x = 0.