Transformations — translation, reflection, rotation, enlargement (basic): Foundations

Everything the parent note throws at you — vectors, sign flips, $cos θ$ , scale factors — is just a different recipe for turning the pair $(x, y)$ into a new pair $(x^{'}, y^{'})$ . This page builds every symbol in that sentence from nothing, in the order you meet them.

1. The point and its two numbers: $(x, y)$

Before you can move a point, you must be able to name one.

Figure — Transformations — translation, reflection, rotation, enlargement (basic)

The picture shows two number lines crossing at the origin — the horizontal one is the $x$ -axis, the vertical one the $y$ -axis. To find $(3, 2)$ you walk 3 right, then 2 up. That "walk right, then walk up" habit is the whole of Coordinate Geometry, and every transformation formula is written in this language.

2. Sign and the four regions: quadrants

Because $x$ and $y$ can each be positive or negative, the paper splits into four regions called quadrants. Knowing signs lets you check an answer instantly.

Quadrant	$x$ sign	$y$ sign	Where
I	$+$	$+$	right, up
II	$-$	$+$	left, up
III	$-$	$-$	left, down
IV	$+$	$-$	right, down

3. The vector $(a b)$ : an instruction, not a place

Coordinates name where you are. A vector names how to move.

The amber arrow in the figure is the vector $(2 - 3)$ : 2 right, 3 down. Notice the same arrow is drawn starting from two different points — it says the same instruction from both. That is exactly why translation "moves every point by the same amount": you copy one arrow onto every vertex.

Why the topic needs it: translation is a vector. Adding a vector to a point, $(x^{'}, y^{'}) = (x + a, y + b),$ is the simplest transformation, and the pattern "old numbers $\to$ new numbers by arithmetic" is the mould every other transformation is poured into.

4. Distance: how we know shape and size are "unchanged"

The parent note keeps claiming a transformation "keeps the shape and size." To prove that, you need to measure lengths.

Where this comes from: the two points are the ends of the slanted side of a right triangle. Its horizontal leg is $(x_{2} - x_{1})$ , its vertical leg is $(y_{2} - y_{1})$ , and Pythagoras turns the two legs into the slanted length. The squaring is what erases minus signs, so direction doesn't matter — only how far apart. Full build-up sits in Distance Formula.

5. Scale factor $k$ : the one number that resizes

The sign of $k$ , made concrete. Take centre $(0, 0)$ and point $R (2, 1)$ , so the image is $(k \cdot 2, k \cdot 1)$ .

$k$	image of $R (2, 1)$	what happened
$2$	$(4, 2)$	twice as far, same side
$0.5$	$(1, 0.5)$	half as far, same side
$- 1$	$(- 2, - 1)$	same distance, opposite side of centre
$- 2$	$(- 4, - 2)$	twice as far, opposite side

A negative $k$ pushes the point through the centre to the far side — the amber points in the figure are the negative- $k$ images, mirrored across the origin from the cyan positive- $k$ ones.

Why area uses $k^{2}$ : area is (length) $\times$ (width). If length becomes $k$ times longer and width becomes $k$ times longer, area becomes $k \times k = k^{2}$ times bigger. That is the whole reason a scale factor of $2$ makes something four times the area, not twice. (The sign of $k$ makes no difference to area, since $(- k)^{2} = k^{2}$ .)

6. Perpendicular distance and the mirror line

Reflection needs one more idea before it makes sense: perpendicular distance.

In the figure the short cyan segment hits the mirror line square-on ( $9 0^{\circ}$ , shown by the small square) — that is the perpendicular distance. The longer amber path is slanted, so it measures more. When we reflect, we always use the perpendicular one.

Reflecting in the $x$ -axis flips the sign of $y$ ; in the $y$ -axis flips the sign of $x$ ; in the line $y = x$ swaps the two numbers. That swap is your first taste of an inverse — the reflection in $y = x$ turns a point into "the same numbers, roles exchanged", which is exactly what Inverse Functions do to inputs and outputs. Reflection is also the engine of Symmetry.

7. The angle $θ$ , and $cos θ$ , $sin θ$ : naming a turn

Rotation is the only transformation needing a genuinely new idea: an angle and the two functions that convert an angle into coordinate arithmetic.

Now, why do $cos$ and $sin$ show up at all? Because we need to answer: if I turn a point around the origin, where do its two numbers go? A turn mixes the sideways and up/down directions together, and $cos θ$ and $sin θ$ are precisely the two numbers that describe that mixing.

From the picture you can read off the special values the parent note uses:

$θ$	$cos θ$	$sin θ$	landing point
$0^{\circ}$	$1$	$0$	$(1, 0)$ right
$9 0^{\circ}$	$0$	$1$	$(0, 1)$ up
$18 0^{\circ}$	$- 1$	$0$	$(- 1, 0)$ left
$27 0^{\circ}$	$0$	$- 1$	$(0, - 1)$ down

Building the rotation formula, one arrow at a time

Instead of trusting the formula, let us watch it grow. The trick: any point $(x, y)$ is made from two pure building-block moves — $x$ copies of the "right" arrow $(1, 0)$ and $y$ copies of the "up" arrow $(0, 1)$ : $(x, y) = x \cdot (1, 0) + y \cdot (0, 1) .$ Rotating is fair to both parts, so if we learn where each building-block arrow lands, we can rebuild the answer.

Step A — turn the "right" arrow $(1, 0)$ . What we did: it starts at angle $0^{\circ}$ ; turning it by $θ$ puts it at angle $θ$ on the unit circle. What it looks like: the cyan arrow in the figure swings up to the landing point $(1, 0) \to (cos θ, sin θ) .$

Step B — turn the "up" arrow $(0, 1)$ . What we did: it starts at angle $9 0^{\circ}$ ; adding $θ$ lands it at angle $9 0^{\circ} + θ$ . Why: using the exact same "walk round the circle" rule. What it looks like: the amber arrow swings to $(0, 1) \to (cos (9 0^{\circ} + θ), sin (9 0^{\circ} + θ)) = (- sin θ, cos θ) .$ Read that landing point straight off the circle: a quarter-turn ahead of $(cos θ, sin θ)$ , its horizontal coordinate is $- sin θ$ and its vertical coordinate is $cos θ$ . This is where the minus sign is born.

Step C — reassemble. What we did: put the two rotated arrows back together in the same recipe, $x$ of the first plus $y$ of the second:

(x^{'}, y^{'}) = x (cos θ, sin θ) + y (- sin θ, cos θ) .

Reading the horizontal and vertical parts separately: $x^{'} = x cos θ - y sin θ, y^{'} = x sin θ + y cos θ .$ Nothing was assumed — each term is one arrow's landing spot, scaled and added. The compact grid form of this lives in Matrices; deeper trig in Trigonometry.

What about negative or large angles?

Intuition

$θ$ works for every turn, not just $0$ – $9 0^{\circ}$ The unit-circle rule "walk to angle $θ$ " never breaks, so the formula is already universal — you just keep walking (or walk backwards).

Negative $θ$ means turn clockwise. Since $cos (- θ) = cos θ$ and $sin (- θ) = - sin θ$ , a clockwise $9 0^{\circ}$ (that is $θ = - 9 0^{\circ}$ ) sends $(x, y) \to (y, - x)$ — check: $(3, 1) \to (1, - 3)$ , dropping into quadrant IV, exactly opposite the anticlockwise result.
$θ \geq 36 0^{\circ}$ means more than one full loop; a full $36 0^{\circ}$ brings you home, so $θ$ and $θ - 36 0^{\circ}$ do the same thing. E.g. $37 0^{\circ}$ acts like $1 0^{\circ}$ .
$θ = 18 0^{\circ}$ gives $cos = - 1, sin = 0$ , so $(x, y) \to (- x, - y)$ — a point flung straight through the origin, landing in the diagonally opposite quadrant.

In every case the signs of the output tell you which quadrant you ended in — your quadrant alarm from Section 2 still guards you.

Prerequisite map

The foundations stack like this — read it bottom-up, each block resting on the ones beneath it:

Equipment checklist

Cover the right side and test yourself. Each should feel obvious before you tackle the parent note.

What do the two numbers in

(x, y)

mean?

First = steps right/left, second = steps up/down, from the origin

(0, 0)

Which quadrant has

x < 0, y > 0

Quadrant II (left and up).

What does the column vector

(a b)

instruct?

Move

a

across and

b

up — an arrow (a move, not a place).

How do you translate

(x, y)

(a b)

(x + a, y + b)

Distance between

(x_{1}, y_{1})

and

(x_{2}, y_{2})

(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

What does "congruent" mean?

Same shape and same size — every length and angle matches.

Enlarge

(2, 1)

from origin with

k = - 1

(- 2, - 1)

— same distance, opposite side of centre.

Why does area scale by

k^{2}

under enlargement by

k

Both length and width multiply by

k

, and area = length

\times

width.

What is perpendicular distance?

The shortest distance to a line, measured at a right angle to it.

What are

cos θ

and

sin θ

geometrically?

The horizontal and vertical coordinates of the point at angle

θ

on the unit circle.

Where does the "up" arrow

(0, 1)

go under rotation by

θ

(- sin θ, cos θ)

— this is where the minus sign comes from.

What does

θ = - 9 0^{\circ}

do to

(3, 1)

(1, - 3)

— a clockwise quarter-turn into quadrant IV.

Recall Quick self-quiz

Give the image of $(3, 1)$ after $9 0^{\circ}$ anticlockwise rotation about the origin. ::: $(- 1, 3)$ — using $(x, y) \to (- y, x)$ ; note it jumps from quadrant I to quadrant II, which the signs confirm.