Transformations

TRANSFORMATIONS

​By: kasskassmaths

​General Mathematics Syllabus

Section #8, Objectives ​7 - 11

TABLE OF CONTENTS

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​Transformations - Slide 3

Translation - Slide 4

Reflection - Slide 8

Rotation - Slide 14

Enlargement - Slide 18

Past Paper Question - Slide 25

Past Paper Questions to practice - Slide​ 31

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Transformations

A transformation may be defined as an operation such that each point in the pre-image has a unique image and each point in the image is the image of exactly one point. It is often referred to as a mapping.

A transformation is said to describe the relation between any point (pre-image point) and its image point. Its a one-to-one relation / mapping of all points on the object onto corresponding points on the image. The object under a transformation is the plane figure that is undergoing a change in position of the object or pre-image.​​

​** 4 Types of Transformations are :

1) Translation

2) Reflection

3) Rotation

4)​ Enlargement

In a given question, an object can undergo ​a combination (max 2) of any of these 4 types.

TRANSLATION (Displacement)

​It can also be stated that the Pre-image moved 5 units along the x axis, from left to right and 1 unit along the y axis, upwards.

REFLECTION

A reflection is a way of transforming a shape as a plane mirror does. In a plane, the result of reflecting an object in a mirror line or an axis of reflection is called its mirror image. The object and image are symmetrical about the mirror line.

There are 7 cases of reflection, where the axis

of symmetry is ​different for each case.

When describing a reflection that took place,

include the ​equation of the axis of reflection.

​1. Flipped in the x axis

F(x,y) --> F' (x,-y)​

Both up to down and vice versa​

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​2. Flipped in the y axis

G(x,y)​--> G'(-x,y)

Both left to right and vice versa​

​3. Flipped in the line y = x​

H(x,y) --> H'(y,x)​

​4. Flipped in the line y = -x

I(x,y) --> I'(-y,-x)​

​5. Flipped in the origin

J(x,y) --> J'(-x,-y)​

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6. Flipped in the line x = k

(vertical line)

L(x,y) --> L'(2k-x,y)​

For k = -2​ :

A(-6,2) -> A'(1,2)

B(-3,1) -> B'(-1,1)​

C(-2,5) -> C'(-2,5)​

​When this is not given and need to be found, one can draw a line connecting the image to its object and find the perpendicular bisector(this will be the axis of symmetry).​

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​7.​ Flipped in the line y = c (horizontal line)

K(x,y) -> K'(x, 2c-y)

For c = 4​

​R(1,1) -> R'​(1,7)

S (3,3) -> S' (3,5)

​T (4,1) -> T' (4,7)

ROTATION

This is a transformation in which every point turns through the same angle about the same centre in the same direction. In a plane, rotation about a single point is called the centre of rotation (this doesn't change its position after the rotation). It is the invariant point. A rotation can be clockwise or anti-clockwise.

A negative angle (-θ°) means a rotation of θ° clockwise.

A positive angle (θ°) means a rotation of θ° anti-clockwise.

​Clockwise 90 degrees / Anti-Clockwise 270 degrees

(about the origin)​

F(x,y)-> F'(y,-x)

​KLMN -> K'L'M'N'

K(-4,-4) -> K'(-4,4)

L(0,-4) -> L'(-4,0)

M(0,-2) -> M'(-2,0)

N​(-4,-2) -> N'(-2,4)

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Anti-Clockwise 90 degrees

/ Clockwise 270 degrees

(about the origin)​

A(x,y)-> A'(-y,x)

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FGH-> F'G'H'

​F(-4,-2) -> F'(2,-4)

G(-2,-2) -> G'(2,-2)

​​H(-3,1) -> H'(-1,-3)

​Clockwise 180 degrees / Anti - Clockwise 180 degrees

(about the origin)​

H(x,y)-> H'(-x,-y)

​PQRS -> P'Q'R'S'

P(-2,-2) -> P'(2,2)

Q(1,-2) -> Q'(-1,-2)

R(2,-4) -> R'(-2,4)

S(-3,-4) -> S'​(3,4)

​Note, for rotations whether about the origin or not, the distance of an object and its image from the centre of rotation are equal.

This is one way how the ​centre of rotation is located.

Another way is by ​joining the object points to its respective image points and construct their perpendicular bisectors. Extending these lines will give a common point that they all pass through (the center of rotation)

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When asked to describe a rotation that occurred on a graph, ​include the angle / degrees (90,180, 270, 360), direction (clockwise / anticlocwise) and centre of rotation (origin or (x,y)).

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For clockwise and anti-clockwise​ 360 degrees rotation, P(x,y) -> P' (x,y).

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Enlargement (size transformations)

​Points to note :

​1)Enlargements don't maintain the lengths. (pre-image sides ≠ image sides)

​**2)The ratio of the length(s) on the image to the corresponding length(s) on the pre-image is a constant called the scale factor (k). This is one way used to find the scale factor.

3)​ The order of the points remain the same.

4) When describing an enlargement that took place include the center of enlargement and scale factor.​

​When the origin is the centre of enlargement, then the pre-image P(x,y), when increased or reduced by a scale factor of k, is mapped onto P' (kx , ky),that is, the pre-image or coordinate(s) is multiplied by the scale factor.

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When there's a centre of enlargement X(a,b), then the pre-image P(x,y), when increased or reduced by a scale factor of k, is mapped onto ​k × PX, that is, the length or distance of the pre-image to the centre of enlargement is multiplied by the scale factor.

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One way To find the centre of enlargement for a given enlargement, ​join the image points to the corresponding object points and produce the lines either forwards or backwards (depending on question) until they meet at a common point. This is the centre of enlargement.

​Triangle PQR has vertices P (3,4) , Q(5,3) and R(4,1). Determine the image of triangle PQR at scale factor 2 with the origin as the center of enlargement. Display on a graph.

​P' = 2 (3,4) = (6,8)

Q' = 2(5,3) = (10,6)

R' = ​2(4,1) = (8,2)

​Triangle PQR has vertices P (6,6) , Q(8,2) and R(8,6). Determine the image of triangle PQR at scale factor ½ with the coordinate X(1,1) as the center of enlargement. Display on a graph.

From Graph : ​

P' = ​½ PX = (3.5,3.5)

Q' = ​½ QX = (4.5,1.5)

R' = ½ RX = (4.5,3.5)

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When describing the geometric transformation(s) made between an object and its image recall these terms.

1.Orientation: Orientation refers to the arrangement of points, relative to one another, after a transformation has occurred. 

2.Congruency ​: A congruence transformation is a moved figure that retains the same size, shape, angles, and side lengths of the original image. The final figure is exactly equal to the original image, also called the pre-image. (Rotations, Reflections & Translations)

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May / June 2022 P2 #3(a) & (b)

​(a) i) A rotation of 180 degrees either clockwise or anti-clockwise, about the centre of rotation (7,7), maps P to Q. Since the object and its image share a common side, (the side that joins (7,6) and (7,8)) and the midpoint of that side is (7,7), the centre of rotation is (7,7).

(ii) A reflection in the line x = 1, maps P to R. R, the image of P is congruent to the object or its pre-image P and the perpendicular bisector of the line joining P and R is x = 1.​

(iii)​ P maps onto S by an enlargement of scale factor 2 and centre of enlargement

(7, 11). The object and its image is similar, having the same shape but different sizes, hence an enlargement occurred. Using one side of the image (8 units) and its corresponding side in the object (4 units), the scale factor is 8 / 4 = 2. Joining the image points to the corresponding object points and extending until a common point is met gave the (7,11).

​(b) Image T is located where P was moved 2 units to the left and 3 places upwards.

July 2021 P2 #3(b)

Past Paper Questions To Practice

​June 2019 No. 3(b)

​June 2018 P2 No. 3 (b)