Set of Transformations in Math- Complete Guide

What Are Transformations in Math?

A transformation is a function that moves or changes a geometric figure in a specific way. Every point in the original shape gets mapped to a new location according to a rule. The original figure is the preimage, and the result is the image.

Transformations are the backbone of geometry, computer graphics, robotics, and image processing. If you're working with shapes on a coordinate plane, you need to know these.

The Four Main Types of Transformations

There are four fundamental transformations. Each one does something different to your shape:

Translation

Translation moves every point of a figure the same distance in the same direction. The shape doesn't rotate, flip, or change size. It just slides.

How Translation Works

Add the same values to the x and y coordinates of every point. If a point is at (x, y) and you translate by (a, b), the new point is (x + a, y + b).

Example: Translate triangle with vertices (1, 2), (3, 4), (5, 2) by (2, -1)

The triangle keeps its shape, size, and orientation. Only its position changed.

Translation Vector Notation

You can write a translation as a vector: T(a, b). This tells you exactly how far to move in each direction. The x-component (a) moves left or right. The y-component (b) moves up or down.

Rotation

Rotation turns a figure around a fixed point by a given angle. The shape stays the same size and shape. Only its orientation changes.

Rotation Rules

For rotation about the origin (0, 0):

Example: Rotate point (3, 2) by 90° counterclockwise about the origin

(3, 2) → (-2, 3)

Rotation Matrix

For any angle θ, the rotation matrix is:

[cos θ -sin θ]
[sin θ cos θ]

Multiply this matrix by your coordinate vector to get the rotated point.

Reflection

Reflection flips a figure across a line. Every point on one side of the line gets mirrored to the other side at the same distance from the line.

Reflection Across Common Lines

Example: Reflect point (4, 3) across the y-axis

(4, 3) → (-4, 3)

The distance from the y-axis stays the same. Only the sign of the x-coordinate flips.

Dilation

Dilation resizes a figure by expanding or contracting it. The shape stays the same, but the size changes. A scale factor determines how much bigger or smaller the image becomes.

How Dilation Works

Multiply all coordinates by the scale factor k:

Example: Dilate point (2, 4) by a scale factor of 3

(2, 4) → (6, 12)

The center of dilation matters. If the center is not the origin, you measure distances from that center point instead.

Shear Transformation

Shear tilts a shape so that it slants sideways or upward. Parallel lines stay parallel, but angles change. It's like pushing the top of a rectangle sideways while keeping the bottom fixed.

Shear Formulas

A square sheared horizontally becomes a parallelogram. The height stays the same, but the top edge shifts.

Composition of Transformations

You can combine multiple transformations. When you do this, apply them in order. The result is a single composite transformation.

Example: Translate by (2, 0), then rotate 90° about the origin

Start with (3, 1):

  1. Translate: (3, 1) → (5, 1)
  2. Rotate 90° CCW: (5, 1) → (-1, 5)

The final result is (-1, 5).

Order Matters

Transformation A followed by transformation B is usually not the same as transformation B followed by transformation A. This is especially true for rotation and reflection.

Comparison: Transformation Types at a Glance

Transformation What It Does Preserves Shape? Preserves Size? Key Property
Translation Slides figure Yes Yes No rotation or flip
Rotation Turns figure around a point Yes Yes Angle and distance from center preserved
Reflection Flips across a line Yes Yes Orientation reversed (mirror image)
Dilation Resizes figure Yes No Scale factor determines size change
Shear Slants figure sideways No (becomes parallelogram) Yes One axis shifts proportionally

How To Apply Transformations: Step-by-Step

Here's how to transform any figure on a coordinate plane:

Step 1: Identify the Transformation Type

Figure out what operation you need. Read the problem or decide what you want to happen to your shape.

Step 2: Find the Rule

Match the transformation to its coordinate rule:

Step 3: Apply to All Vertices

Transform every vertex of your shape using the rule. Write down the new coordinates.

Step 4: Connect the Points

Draw lines connecting the transformed points in the same order as the original shape.

Step 5: Verify Properties

Check that your transformation preserved what it should:

Rigid vs. Non-Rigid Transformations

Rigid transformations preserve distance and angle measures. Translation, rotation, and reflection are rigid. The image is congruent to the original.

Non-rigid transformations change size or shape. Dilation and shear are non-rigid. The image is similar (dilation) or a distorted version (shear) of the original.

Transformation Notation

You'll see notation like T(x, y) → (x', y') or function notation f(P) = P'. The prime symbol (') marks the image points. A composition looks like R90° ∘ T(3,2), meaning translate first, then rotate.

Keep your notation consistent. Mixing formats leads to mistakes.