Transformations

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Past Paper Questions to practice - Slide​ 31

​** 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.

​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.

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.

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)​

J(x,y) --> J'(-x,-y)​

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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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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)

​Clockwise 90 degrees / Anti-Clockwise 270 degrees

(about the origin)​

​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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(about the origin)​

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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)​

​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)

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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​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.​

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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.

​P' = 2 (3,4) = (6,8)

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

R' = ​2(4,1) = (8,2)

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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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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(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).

​June 2018 P2 No. 3 (b)