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T
\( (x, y) \to (x', y') \)

Representing Transformations
as Functions

Learn how every geometric transformation can be written as a function that maps points to points, and discover which transformations preserve distance and angle.

Geometry

What Is a Transformation?

In geometry, a transformation is any rule that moves or changes every point in the plane. Think of it as a machine: you feed in a point \((x, y)\) and get out a new point \((x', y')\).

Input
\( (x, y) \)
\( T \)
 Transformation Function / Rule Output
\( (x', y') \)

Function Notation for Transformations

Just like \( f(x) = 2x + 1 \) maps numbers to numbers, a geometric transformation maps points to points. We write:

Domain
All points \((x,y)\) in the plane: the set of possible inputs.
Range
The set of output points \((x', y')\): the image of the transformation.

Why Write Transformations as Functions?

1
Precision: A formula like \( T(x,y) = (-y, x) \) removes all ambiguity about what "rotate" means.
2
Computation: You can plug in coordinates and compute exact image points.
3
Composition: Applying two transformations in sequence is the same as composing two functions: \( (S \circ T)(x,y) \).
4
Classification: Function form reveals whether a transformation preserves distances and angles.

The Four Rigid Motions (Isometries)

A rigid motion (isometry) preserves both distance and angle measure. The image is always congruent to the pre-image.

Translation
\( T(x,y) = (x+a,\; y+b) \)
Slides every point the same direction and distance.
Reflection
\( T(x,y) = (-x,\; y) \)
(over the \(y\)-axis) Flips across a line.
Rotation
\( R_{90^\circ}(x,y) = (-y,\; x) \)
Turns about a center by a fixed angle.
Glide Reflection
A reflection followed by a translation parallel to the line of reflection.

Non-Rigid Transformations

Non-rigid transformations change the size or shape of a figure. Distances are not preserved.

Dilation
\( D_k(x,y) = (kx,\; ky) \)
Scales uniformly by factor \(k\). Preserves angles but not distances.
Horizontal Stretch
\( S(x,y) = (kx,\; y) \)
Stretches only in the \(x\)-direction. Preserves neither distances nor angles.

Rigid vs Non-Rigid: Side by Side

Translation (Rigid)
\(A\)
\(B\)
\(C\)
\(A'\)
\(B'\)
\(C'\)
 Same size and shape Horizontal Stretch (Non-Rigid) y x
\(P\)
\(Q\)
\(R\)
\(P'\)
\(Q'\)
\(R'\)
 Shape distorted

Core Rule: Distance Preservation

When this condition holds, angle measures are also preserved automatically. The image figure is congruent to the original: \( \triangle ABC \cong \triangle A'B'C' \).

Transformation Formulas at a Glance

Transformation Function Rule Preserves
Translation by \((a,b)\) \( (x+a,\; y+b) \) Distance + Angle
Reflection over \(y\)-axis \( (-x,\; y) \) Distance + Angle
Reflection over \(x\)-axis \( (x,\; -y) \) Distance + Angle
Rotation \(90^\circ\) CCW \( (-y,\; x) \) Distance + Angle
Rotation \(180^\circ\) \( (-x,\; -y) \) Distance + Angle
Dilation (factor \(k\)) \( (kx,\; ky) \) Angle only
Horizontal stretch \( (kx,\; y) \) Neither

How to Apply a Transformation

1
Identify the rule: Write the transformation as a function \( T(x,y) = (\ldots,\; \ldots) \).
2
List the vertices: Write the coordinates of every point you need to transform.
3
Substitute: Plug each point into the rule. Compute the output coordinates.
4
Plot and connect: Plot the image points, then connect them in the same order.
5
Classify: Check if distances and angles are preserved to determine rigid vs. non-rigid.

Example 1: Translation

Apply \( T(x,y) = (x+4,\; y-2) \) to triangle with vertices \( A(1,3) \), \( B(4,3) \), \( C(1,6) \).

1 2 3 4 5 6 7 8 1 2 3 4 5 6
\(A\)
\(B\)
\(C\)
\(A'\)
\(B'\)
\(C'\)
 x y

Example 2: Reflection over the \(y\)-axis

Apply \( T(x,y) = (-x,\; y) \) to points \( P(2,1) \), \( Q(5,1) \), \( R(5,4) \).

y x 0
\(P\)
\(Q\)
\(R\)
\(P'\)
\(Q'\)
\(R'\)
 line of reflection: y-axis

Example 3: Horizontal Stretch

Apply \( S(x,y) = (2x,\; y) \) to \( A(1,0) \), \( B(1,3) \), \( C(3,0) \).

x y
\(A\)
\(B\)
\(C\)
\(A'\)
\(B'\)
\(C'\)
 1 2 3 5 6

Composing Two Transformations

Applying transformation \( S \) after \( T \) is written as the composition \( (S \circ T)(x,y) = S\big(T(x,y)\big) \). You evaluate the inner function first.

Rigid + Rigid
The composition of two isometries is always an isometry. Distances and angles are preserved.
Includes Non-Rigid
If either transformation is non-rigid, the composition is generally non-rigid.

Common Mistakes

Mixing Up Order
In compositions, \( S \circ T \) means "apply \( T \) first, then \( S \)." Reversing the order gives a different result.
Confusing Dilation with Isometry
A dilation preserves angles but NOT distances. It is not a rigid motion unless \( k = \pm 1 \).
Forgetting the Sign
In \( T(x,y) = (-y,\; x) \), the negative is on \( y \), not \( x \). Swapping or dropping it gives the wrong rotation direction.

Question 1 of 5

Which function rule represents a translation 5 units left and 3 units up?

Question 2 of 5

The transformation \( T(x,y) = (3x,\; y) \) is applied to a general triangle. What is true about the image?

Question 3 of 5

What is the image of the point \( (4, -2) \) under a \(90^\circ\) counterclockwise rotation about the origin?

Question 4 of 5

Which of the following transformations does not preserve distance between points?

Question 5 of 5

Let \( T(x,y) = (x+2,\; y) \) and \( S(x,y) = (x,\; -y) \). What is \( (S \circ T)(1, 3) \)?