Transformations Tutorial

Introduction to Transformations:

Transformations are operations that change the position, orientation, or size of a shape on a coordinate plane. There are four main types of transformations:

• Translation
• Reflection
• Rotation
• Dilation

Each type of transformation has a specific effect on the shape and can be described using words and numbers.

Translation:

Translation moves a shape from one location to another without changing its size or orientation. It shifts the shape along the x-axis and/or y-axis. The notation (x, y) + (a, b) represents a translation of 'a' units horizontally and 'b' units vertically.

Reflection:

Reflection flips a shape over a line (line of reflection) to create a mirror image. Points on the shape are equidistant from the line before and after the reflection.

Rotation:

Rotation turns a shape around a point (center of rotation) by a certain angle. Positive angles indicate counterclockwise rotation, negative angles indicate clockwise. The notation (x, y) → (x', y') represents a rotation.

Dilation:

Dilation changes the size of a shape while keeping its shape and orientation. It scales the shape by a factor 'k'. Notation: (x, y) → (kx, ky).

Example:

Point A(3, 2) after transformations:

• Translation by (2, 4): A' → (5, 6)
• Reflection over x-axis: A' → (3, -2)
• Rotation by 90° counterclockwise: A' → (-2, 3)
• Dilation by a scale factor of 2: A' → (6, 4)

Practice Questions:

1. Describe the transformation that moves a triangle from (2, 3) to (5, 8).
   Answer: Translation by (3, 5).
2. If a shape is reflected over the y-axis, what happens to the x-coordinates of its points?
   Answer: The x-coordinates are negated (multiplied by -1).
3. Rotate the point B(4, -1) by 180° counterclockwise.
   Answer: B' → (-4, 1)
4. Apply a dilation with a scale factor of 0.5 to the point C(-6, 2).
   Answer: C' → (-3, 1)