Transformations as Functions — Rigid Motions

Lesson 1 of 2: Translation, Rotation, and Reflection

In this lesson:

• What it means to describe a transformation as a function
• Coordinate rules for translations, reflections, and rotations
• Why these three transformations preserve distances and angles

Learning Objectives for This Lesson

1. Represent transformations using coordinate rules in the plane
2. Describe transformations as functions mapping input points to output points
3. Identify translations, rotations, and reflections by their coordinate rules
4. Predict transformation effects by reasoning about distance and angle preservation

What Do Slides, Flips, and Turns Share?

You've been applying transformations since middle school:

• Slide (translation): moving every point in the same direction
• Flip (reflection): mirror image across a fixed line
• Turn (rotation): every point rotates around a fixed center

What do all three have in common?

Three Ways to Represent a Transformation

• Physical: trace a figure on transparency paper; slide, flip, or turn it
• Digital: use geometry software (GeoGebra, Desmos) to drag and transform figures
• Algebraic: write a coordinate rule such as

All three representations describe the same transformation — each offers different insight.

Transformations Map Every Point to an Image

• A transformation is a function: every point maps to exactly one image
• Written as or
• The original is the pre-image; the result is the image
• The rule acts on the entire plane, not just the visible figure

From 1D to 2D — Functions on Points

• 1D: shifts every number 3 units right
• 2D: shifts every point 3 right, 2 up
• Both are functions — input in, output out

Applying a Rule to a Triangle

• Apply to each vertex
• , ,
• Every point shifts by the same vector — the entire triangle moves

Quick Check: Applying a Coordinate Rule

What is the image of under ?

What is the image of ?

Apply the rule to each coordinate before advancing.

Answer: Applying the Rule to Each Point

:

• : subtract 2 from , add 5 to
• : subtract 2 from , add 5 to
• The vector shifts every point — the whole plane moves uniformly

Translation — Precise Definition and Rule

• Vector gives the horizontal and vertical shift
• Every point moves the same direction and distance
• Shape, size, and orientation are all preserved

Worked Example — Translating a Triangle

Translate with , , by vector :

Verify: ; — distances and angles preserved.

Quick Check — Writing a Translation Rule

What coordinate rule translates every point 4 left and 2 up?

Write it in the form .

Write the rule before advancing.

Answer: Translation by (−4, +2)

• Moving left by 4: subtract 4 from
• Moving up by 2: add 2 to
• The and coordinates shift independently — each by its own amount

Reflection Maps Points Across a Mirror Line

• Over -axis: — negate
• Over -axis: — negate
• Over : — swap coordinates
• Each point and its image are equidistant from the mirror line

Worked Example — Reflecting Over the y-axis

Reflect : , , over the -axis — apply :

Verify: ; — distances and angles preserved.

Rotation — Every Point Turns the Same Angle

• Rotation: every point turns the same angle around a fixed center
• 90° CCW about origin:
• Any angle is valid — center and angle fully specify a rotation

Worked Example — Rotating 90° CCW

Rotate : , , by 90° CCW — apply :

Verify: — distances preserved; shape and size unchanged.

Quick Check — Reflecting Over the x-axis

has vertices at , , .

What are the image vertices after reflection over the -axis?

Write the image coordinates before advancing.

Answer: Points on the Axis Stay Fixed

— negate only the -coordinate:

• : , so — unchanged
• : also on the -axis — unchanged
• : becomes

Points on the line of reflection are their own images.

Translation, Reflection, Rotation — All Rigid Motions

All three preserve distances between points and measures of angles:

• Translation: distances and angles unchanged; orientation preserved
• Reflection: distances and angles unchanged; orientation reversed
• Rotation: distances and angles unchanged; orientation preserved

A transformation that preserves distances and angles is called a rigid motion (or isometry).

Quick Check — Reflection and Congruence

is reflected over the -axis to produce .

Is congruent to ? How do you know?

Name the property that justifies your answer.

Answer: Rigid Motions Preserve Congruence

Yes — is congruent to .

• Reflection is a rigid motion — it preserves all distances and all angles
• Same side lengths + same angles = congruent triangles (by definition)
• This holds for all rigid motions: translate, reflect, rotate

Key Takeaways — Transformations and Rigid Motions

• A transformation is a function: every point maps to exactly one image
• Translation: — shift every point by vector
• Reflection: negate or swap coordinates depending on the mirror line
• Rotation 90° CCW: — any angle is valid
• All three are rigid motions — they preserve distances and angles

Misconception Warnings — Watch Out

Transformations act on the entire plane — not just the visible figure

Reflection is a 2D mapping — the line of reflection is a mirror, not a fold; no point leaves the plane

Rotation works for any angle — 90° and 180° are special cases, not the only options

What Comes Next — Deck 2

Deck 2 extends the classification framework:

• Dilation — scales from a center; preserves angles but not distances
• Horizontal stretch — scales one axis only; preserves neither
• Classification table — which transformations preserve what, and why

Today's rigid motions are the foundation — Deck 2 completes the picture.