Section 2.4: Rotations

Property

To display the values of two variables, we use two number lines. The horizontal number line is called the $x$-axis, and the vertical number line is the $y$-axis. The point where the two axes intersect is called the origin. The two axes divide the plane into four regions called quadrants, numbered 1 through 4 counter-clockwise around the origin.

Examples

• Points in Quadrant I, like $(4, 6)$, have a positive x-coordinate (a move to the right) and a positive y-coordinate (a move up).
• Points in Quadrant III, like $(-1, -5)$, have a negative x-coordinate (a move to the left) and a negative y-coordinate (a move down).
• The origin is the starting point $(0, 0)$ where the x-axis and y-axis meet.

Explanation

Think of the Cartesian coordinate system as a map for numbers. It uses two perpendicular lines, the x-axis (horizontal) and y-axis (vertical), to give a unique address to any point on a flat surface, letting us visualize equations.

Property

The center of rotation is the fixed point around which a figure rotates. All points on the figure move in circular paths around this center, and the center itself remains stationary during the rotation.

Examples

Property

Rotations can occur in two directions: clockwise (CW) follows the direction of clock hands, and counterclockwise (CCW) goes opposite to clock hands. A rotation of $\theta$ degrees clockwise is equivalent to a rotation of $(360° - \theta)$ degrees counterclockwise.

Examples