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University of Waterloo Centre for Education inFaculty of Mathematics Mathematics and Computing

Grade 7 & 8 Math Circles

October 6, 2010

Graphs and Transformations

DefinitionTransformation: a transformation is a mapping of the points of a figure that results in a changein position, shape, size, or appearance of a figure.

Examples

Four Main Transformations:

1. Rotations

• A rotation is a transformation in which the points of a figure are turned around a fixedpoint.

• The fixed point which the object is rotated about, can be a point on the shape or off theshape.

• Each point is rotated around the fixed point by an angle either clockwise or counterclock-wise.

NOTE: rotations are TURNS!

2. Reflections

• A reflection is a transformation where every point is symmetrically mapped to the otherside of the central line.

• The central line is known as the mirror line.

• A reflection is the same size of the original shape.

NOTE: reflections are FLIPS!

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Steps:

(a) Measure the distance from each point to the mirror line.

(b) Measure the same distance on the other side of the mirror line and make a dot.

(c) Connect all the points.

Tricks:

(-1, 2)

(-1,-2)

Reflections in the x-axisWhen the mirror line is the x-axis,change the point (x, y) to (x,−y).

(-1, 2) (1,2)

Reflection in the y-axisWhen the mirror line is the y-axis changethe point (x, y) to (−x, y)

Exercise 1

(a) Are the following rotations, reflections, or neither?

i. ii. iii.

x

y

2

6

4

-2

-4

-6

-2-4-6 2 4 6

x

y

1

3

2

-1

-2

-3

-1-2-3 1 2 3

4

5

-5

-4

4 5-5 -4

x

y

2

6

4

-2

-4

-6

-2-4-6 2 4 6

(b) Make the following transformations:i. Counterclockwise

rotation of 90o aboutthe origin

ii. Counterclockwiserotation of 180o aboutthe origin

iii. Clockwiserotation of 75o aboutthe origin

x

y

x

y

2

6

4

-2

-4

-6

-2-4-6 2 4 6

2

6

4

-2

-4

-6

-2-4-6 2 4 6

x

y

2

6

4

-2

-4

-6

-2-4-6 2 4 6

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(c) Make the following transformations:i. Reflection with mirror

line x = 0ii. Reflection with mirror

line x = −2

iii. Reflection with mirrorline y = 2

x

y

x

y

2

6

4

-2

-4

-6

-2-4-6 2 4 6

2

6

4

-2

-4

-6

-2-4-6 2 4 6

x

y

2

6

4

-2

-4

-6

-2-4-6 2 4 6

3. Translations

• A translation is a transformation that moves a point or figure in a straight line to anotherposition in the same plane

• Every point on an object must shift in the the exact same direction and distance.

NOTE: translations are SLIDES!

Notation(x, y)→ (x + a, y + b), where a and b are the amount the point will be shifted by.This is read ‘move the point a units in the x-direction and b units in the y-direction.’

4. Resizing

• Resizing is a transformation where the shape is enlarged or reduced in size while pre-serving the image.

• Resizing is also called dilation, contraction, compression, enlargement, or expansion.

• The sizing factor is the amount you will increase or decrease the size of the object by.

Steps:

(a) Multiply each (x, y) point by the sizing factor.

(b) Plot your new points on the graph

(c) Connect all the points.

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Exercise 2

(a) Are the following translations, resizing, or neither?

i. ii. iii.

x

y

x

y

2

6

4

-2

-4

-6

-2-4-6 2 4 6

2

6

4

-2

-4

-6

-2-4-6 2 4 6

x

y

2

6

4

-2

-4

-6

-2-4-6 2 4 6

(b) Translate the following in the indicated direction:i. (x + 8, y + 4) ii. (x + 6, y − 8) iii. (x− 10, y − 6)

x

y

x

y

2

6

4

-2

-4

-6

-2-4-6 2 4 6

2

6

4

-2

-4

-6

-2-4-6 2 4 6

x

y

2

6

4

-2

-4

-6

-2-4-6 2 4 6

(c) Resize the following by the indicated sizing factor:i. Sizing factor of 1

2ii. Sizing factor of 2 iii. Sizing factor of 1

3

x

y

2

6

4

-2

-4

-6

-2-4-6 2 4 6x

y

x

y

2

6

4

-2

-4

-6

-2-4-6 2 4 62

6

4

-2

-4

-6

-2-4-6 2 4 6

Combining TransformationsRotations, reflections, translations, and resizing can be combined and applied on a single object to-gether.

NOTE: the order multiple transformations are applied in may impact the final shape. Make sureyou apply the transformations in the outlined order.

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Exercise 3

(a) Make the following indicated transformations:

i) Resize by a factor of 2.Reflect with mirror line y = 2.Translate: (x− 2, y − 4).

ii) Reflect with mirror line x = 0.Resize by a factor of 2.Translate: (x + 5, y).

x

y

2

6

4

-2

-4

-6

-2-4-6 2 4 6

x

y

1

3

2

-1

-2

-3

-1-2-3 1 2 3

4

5

-5

-4

4 5-5 -4

iii) Clockwise rotation of 90o

about the origin.Translate (x,y-4).Reflect with mirror line x = 2.

iv) Resize by a factor of 12

Translate: (x− 3, y − 3).Reflect with mirror line x = 0.Counterclockwise rotation of 180o aboutthe origin.

x

y

2

6

4

-2

-4

-6

-2-4-6 2 4 6

x

y

1

3

2

-1

-2

-3

-1-2-3 1 2 3

4

5

-5

-4

4 5-5 -4

Exercise 4

(a) For the following, list what transformations are needed from the green shape to the pink shape.

i) ii)

x

y

2

6

4

-2

-4

-6

-2-4-6 2 4 6

x

y

1

3

2

-1

-2

-3

-1-2-3 1 2 3

4

5

-5

-4

4 5-5 -4

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Problem Set

1. The letter F is reflected in line 1. The image is then reflected in line 2. Drawthe letter F that is results in these transformations.

line 1

line 2

2. Beth is sitting in her classroom where behind her she has a clock and in frontof her is a mirror. To the right is the image Beth sees. What is the actualtime?

3. Perform the following transformations on the graph to the right:

(a) Reflect with mirror line x=0.

(b) Reflect with mirror line y=0.

(c) Translate: (x, y + 2).

(d) Clockwise rotation of 90o about the origin.

x

y

2

6

4

-2

-4

-6

-2-4-6 2 4 6

4. List the transformations required to transform the purple triangle to the greentriangle. x

y

1

3

2

-1

-2

-3

-1-2-3 1 2 3

4

5

-5

-4

4 5-5 -4

n

5. Perform the following transformations on the graph to the right:

(a) Counterclockwise rotation of 90o about the origin.

(b) Reflect with mirror line y = −1.

(c) Translate: (x− 2, y − 1).

(d) Clockwise rotation of 180o about the origin.

x

y

1

3

2

-1

-2

-3

-1-2-3 1 2 3

4

5

-5

-4

4 5-5 -4

6. List the transformations required to transform the green trapezoid to the redtrapezoid. x

y

1

3

2

-1

-2

-3

-1-2-3 1 2 3

4

5

-5

-4

4 5-5 -4

7

7. List the transformations needed to move alphabetically from point A to point G passing througheach of the other points along the way.

10 20 30 40 50 60 70 80 90 1000

10

20

30

40

50

60

70

A

F

E

DC

B

G

8. The figure shown is folded to form a cube. Three faces meet at each corner. Ifthe numbers of the three faces at a corner are multiplied, what is the largestpossible product?

6

8 11

9

5

7

9. Claire takes a square piece of paper and folds it in half four times without unfolding, makingan isosceles right triangle each time. After unfolding the paper to form a square again, thecreases on the paper would look like:

a) e)d)c)b)

10. A rectangle is divided into three rectangles of different areas and a squareGHFD. The area of rectangle AEHG is 21cm2. The area of rectangle HJCF is28cm2. If all rectangles have lengths and widths which are integers and areasthat are less than the area of GHFD, what the area of the square GHFD incm2?

AG

D

FHE

B JC

11. The numbers on the faces of a regular die are arranged so that opposite facestotal 7 (ie. 2 is opposite 5). The four dice shown have been placed so thatthe two numbers on the faces touching each other always total 9. The facelabelled P is the front of one die as shown. What is the number on the facelabelled P?

p

8

12. In the diagram, the numbers from 1 to 25 are to be arranged in the 5 by 5grid so that each number, except 1 and 2, is the sum of two of its neighbours.(Numbers in the grid are neighbours if their squares touch along a side orat the corner. For example, the number 1 has 8 neighbours.) Some of thenumbers have already been filed in. Which number must replace the ”?” whenthe grid is complete?

20

23 ?

456

7

21

1 3

9 8 2

222425

13. A rectangular piece of paper ABCD is folded so that edge CD lies along edge AD, making acrease DP . It is unfolded, and then folded again so that edge AB lies along edge AD, makinga second crease AQ. The two creases meet at R, forming triangles PQR and ADR as shown.If AB = 5 cm and AD = 8cm, what is the area of the quadrilateral DRQC in cm2.

A B

CD

A B

C

D

P

A B

CD

P

RQ

Riddle: There are 5 different houses in five different colours in a row (white, blue, red, yellow, orgreen). In each house lives a person with a different nationality (British, Asian, German, Danish,or Swedish). Each person has a certain pet (cat, fish, dog, horse, or bird), drinks a certain drink(tea, milk, water, coffee, or juice), and plays a different sport (basketball, soccer, frisbee, tennis, orhockey). No owner can have the same pet, drink the same beverage, and play the same sport. Giventhe below information who owns the fish?

1. The British person lives in the red house.

2. The Swede keeps dogs as pets.

3. The Dane drinks tea.

4. The green house is on the immediate left ofthe white house.

5. The green house’s owner drinks coffee.

6. The owner who plays tennis has a bird.

7. The owner of the yellow house playsbasketball.

8. The owner in the center house drinks milk.

9. The Asian lives in the first house.

10. The owner who plays soccer lives next to the onewho has a cat.

11. The owner who has a horse lives next to the onewho plays basketball.

12. The owner who plays hockey drinks juice.

13. The German plays frisbee.

14. The Asian lives next to the blue house.

15. The own who plays soccer lives next to the onewho drinks water.