Coordinates Revision Notes

The Cartesian coordinate plane is a two-dimensional surface defined by two perpendicular number lines: the horizontal x-axis and the vertical y-axis. These axes intersect at a point called the origin, which has coordinates (0, 0). The plane extends infinitely in all directions along these axes, allowing every point to be uniquely identified by a pair of numbers.

• Two perpendicular axes: x-axis (horizontal) and y-axis (vertical).
• Origin at (0, 0), where the axes intersect.
• Plane extends infinitely in all directions.

Each point on the plane is represented by an ordered pair (x, y). The first number, x, indicates the horizontal distance from the origin: positive values lie to the right, negative to the left. The second number, y, indicates the vertical distance: positive values lie above the origin, negative below.

For example, the point (3, 2) is 3 units to the right and 2 units above the origin.

• Coordinates written as (x, y).
• x-value shows horizontal position; positive right, negative left.
• y-value shows vertical position; positive up, negative down.
• Example: (3, 2) is 3 right, 2 up from origin.

To plot a point, start at the origin. Move horizontally according to the x-coordinate: right if positive, left if negative. Then move vertically according to the y-coordinate: up if positive, down if negative. Mark the point at this position.

For example, to plot (-4, -1), move 4 units left and 1 unit down from the origin.

• Start at origin (0, 0).
• Move horizontally by x units: right if positive, left if negative.
• Move vertically by y units: up if positive, down if negative.
• Mark the point at the final position.
• Example: (-4, -1) is 4 left, 1 down from origin.

The coordinate plane is divided into four regions called quadrants, numbered I to IV in an anticlockwise direction starting from the top right. Each quadrant is defined by the signs of the x and y coordinates of points within it:

Quadrant I: x > 0, y > 0 (top right) Quadrant II: x < 0, y > 0 (top left) Quadrant III: x < 0, y < 0 (bottom left) Quadrant IV: x > 0, y < 0 (bottom right)

Knowing the quadrant helps describe the location of points quickly.

• Quadrant I: x > 0, y > 0 (top right).
• Quadrant II: x < 0, y > 0 (top left).
• Quadrant III: x < 0, y < 0 (bottom left).
• Quadrant IV: x > 0, y < 0 (bottom right).
• Quadrants numbered anticlockwise starting from top right.

Understanding coordinates allows you to describe positions precisely, calculate distances between points, and analyse geometric shapes. For example, the distance between two points (x₁, y₁) and (x₂, y₂) can be found using the distance formula derived from the Pythagorean theorem:

\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

This formula requires accurate knowledge of coordinates and their meaning on the plane.

• Coordinates enable precise location description.
• Distance between points calculated using the distance formula.
• Formula: \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
• Accurate plotting and interpretation are essential for calculations.