Transformations

Stage 10 of 23 Strand 3 of 9 6 lessons

6 illustrated lessons, each teaching the why before the how.

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Reflection

Every point crosses the mirror to the same distance.

Reflection carries each point straight across the mirror line to the same distance beyond

The gold line is the mirror, and the shape beyond it is the image.

Each vertex lands as far past the line as it started before it — count the squares.

Mirror on the y-axis: the vertex (4, 3) lands on (−4, 3). Only the x flips sign.

The mirror y = x swaps the coordinates: (x, y) lands on (y, x).

Now you

After a reflection, where does the image sit?

Reflect the point (3, 2) in the y-axis. Where does it land?

Rotation

The whole shape turns about one fixed point.

A rotation turns every point of the shape about one fixed center

A quarter turn counterclockwise about the origin. The gold dot is the center.

A 90° turn counterclockwise sends (x, y) to (−y, x) — try it on (2, 1).

A half turn sends (x, y) to (−x, −y): both signs flip, either way you turn.

Every point circles the center — except the center itself, which never moves.

Now you

Rotate (3, 0) by 90° counterclockwise about the origin. Where does it land?

Rotate (2, 1) by 180° about the origin. Where does it land?

Translation of Shapes

Every point slides by the same vector.

A translation slides every point of the shape by the same vector

Every vertex slides the same way — the gold arrow is that one shared move.

The column vector [a, b] adds to every vertex: a across, then b up.

From (1, 2), the vector [3, −1] means 3 right and 1 down: it lands on (4, 1).

Nothing turns and nothing stretches — the image is the same shape, relocated.

Now you

Translate the point (2, 3) by the vector [4, −2]. Where does it land?

Translate the point (−1, 2) by the vector [3, 3]. Where does it land?

Enlargement with a Center

Rays from one point stretch the shape.

An enlargement stretches every vertex along its ray from the center by the scale factor

Rays run from the center through each vertex, and the image sits farther out.

Scale factor k: each vertex lands k times as far from the center, along its ray.

k = 1/2 pulls every vertex halfway in. Smaller — yet still called an enlargement.

A negative k carries each ray through the center and out the far side, upside down.

Now you

What does a scale factor of 1/2 do?

Where does a negative scale factor put the image?

Combined Transformations

Two moves, and sometimes one name for the pair.

Two transformations run in turn, and sometimes one single move does the same job

First move: reflect the shape in the y-axis.

Second move: reflect that image in the x-axis. Two flips, one after the other.

One single move has the same effect: a 180° rotation about the origin.

Swapping this pair changes nothing — but for many pairs the order does matter.

Now you

Is a translation then a reflection generally the same as reflecting first?

Reflecting in the y-axis, then in the x-axis, equals which single move?

Rotational Symmetry

A turn that lands the shape on itself.

The order of rotational symmetry counts how many turns land a shape on itself in one full turn

A third of a turn lands this shape on a copy of itself — all three are the shape.

The order of rotational symmetry counts those landings in one full turn: here, 3.

A square lands on itself at every quarter turn — four landings, so order 4.

Every shape returns after one full turn, so order 1 means no extra symmetry.

Now you

What is the order of rotational symmetry of an equilateral triangle?

A shape has rotational symmetry of order 1. What does that mean?

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