Grade 7 & 8 Math Circles October 6, 2010 Graphs and Transformations

Transcription

1 Universit of Waterloo Facult of Mathematics entre for Education in Mathematics and omputing Grade 7 & 8 Math ircles October, 00 Graphs and Transformations efinition Transformation: a transformation is a mapping of the points of a figure that results in a change in position, shape, size, or appearance of a figure. Eamples Four Main Transformations:. Rotations rotation is a transformation in which the points of a figure are turned around a fied point. The fied point which the object is rotated about, can be a point on the shape or off the shape. Each point is rotated around the fied point b an angle either clockwise or counterclockwise. NOTE: rotations are TURNS!. Reflections reflection is a transformation where ever point is smmetricall mapped to the other side of the central line. The central line is known as the mirror line. reflection is the same size of the original shape. NOTE: reflections are FLIPS!

2 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) onnect all the points. Tricks: (, ) (, ) (,) (,-) Reflections in the -ais When the mirror line is the -ais, change the point (, ) to (, ). Reflection in the -ais When the mirror line is the -ais change the point (, ) to (, ) Eercise (a) re the following rotations, reflections, or neither? i. ii. iii (b) Make the following transformations: i. ounterclockwise ii. ounterclockwise rotation of 90 o about rotation of 80 o about the origin the origin iii. lockwise rotation of 7 o about the origin

3 (c) Make the following transformations: i. Reflection with mirror ii. Reflection with mirror line = 0 line = iii. Reflection with mirror line = Translations translation is a transformation that moves a point or figure in a straight line to another position in the same plane Ever point on an object must shift in the the eact same direction and distance. NOTE: translations are SLIES! Notation (, ) ( + a, + b), where a and b are the amount the point will be shifted b. This is read move the point a units in the -direction and b units in the -direction.. Resizing Resizing is a transformation where the shape is enlarged or reduced in size while preserving the image. Resizing is also called dilation, contraction, compression, enlargement, or epansion. The sizing factor is the amount ou will increase or decrease the size of the object b. Steps: (a) Multipl each (, ) point b the sizing factor. (b) Plot our new points on the graph (c) onnect all the points.

4 Eercise (a) re the following translations, resizing, or neither? i. ii. iii (b) Translate the following in the indicated direction: i. ( + 8, + ) ii. ( +, 8) iii. ( 0, ) (c) Resize the following b the indicated sizing factor: i. Sizing factor of ii. Sizing factor of iii. Sizing factor of ombining Transformations Rotations, reflections, translations, and resizing can be combined and applied on a single object together. NOTE: the order multiple transformations are applied in ma impact the final shape. Make sure ou appl the transformations in the outlined order.

5 Eercise (a) Make the following indicated transformations: i) Resize b a factor of. Reflect with mirror line =. Translate: (, ). ii) Reflect with mirror line = 0. Resize b a factor of. Translate: ( +, ) iii) lockwise rotation of 90 o about the origin. Translate (,-). Reflect with mirror line =. iv) Resize b a factor of Translate: (, ). Reflect with mirror line = 0. ounterclockwise rotation of 80 o the origin. about Eercise (a) For the following, list what transformations are needed from the green shape to the pink shape. i) ii)

6 Problem Set. The letter F is reflected in line. The image is then reflected in line. raw the letter F that is results in these transformations. line line. eth is sitting in her classroom where behind her she has a clock and in front of her is a mirror. To the right is the image eth sees. What is the actual time?. Perform the following transformations on the graph to the right: (a) Reflect with mirror line =0. (b) Reflect with mirror line =0. (c) Translate: (, + ). (d) lockwise rotation of 90 o about the origin List the transformations required to transform the purple triangle to the green triangle.. Perform the following transformations on the graph to the right: (a) ounterclockwise rotation of 90 o about the origin. (b) Reflect with mirror line =. (c) Translate: (, ). (d) lockwise rotation of 80 o about the origin n. List the transformations required to transform the green trapezoid to the red trapezoid

7 7 7. List the transformations needed to move alphabeticall from point to point G passing through each of the other points along the wa E F G The figure shown is folded to form a cube. Three faces meet at each corner. If the numbers of the three faces at a corner are multiplied, what is the largest possible product? laire takes a square piece of paper and folds it in half four times without unfolding, making an isosceles right triangle each time. fter unfolding the paper to form a square again, the creases on the paper would look like: a) b) c) d) e) 0. rectangle is divided into three rectangles of different areas and a square GHF. The area of rectangle EHG is cm. The area of rectangle HJF is 8cm. If all rectangles have lengths and widths which are integers and areas that are less than the area of GHF, what the area of the square GHF in cm? E G H J F. The numbers on the faces of a regular die are arranged so that opposite faces total 7 (ie. is opposite ). The four dice shown have been placed so that the two numbers on the faces touching each other alwas total 9. The face labelled P is the front of one die as shown. What is the number on the face labelled P? p

8 8. In the diagram, the numbers from to are to be arranged in the b grid so that each number, ecept and, is the sum of two of its neighbours. (Numbers in the grid are neighbours if their squares touch along a side or at the corner. For eample, the number has 8 neighbours.) Some of the numbers have alread been filed in. Which number must replace the? when the grid is complete? 0 7? 9 8. rectangular piece of paper is folded so that edge lies along edge, making a crease P. It is unfolded, and then folded again so that edge lies along edge, making a second crease Q. The two creases meet at R, forming triangles P QR and R as shown. If = cm and = 8cm, what is the area of the quadrilateral RQ in cm. P R P Q Riddle: There are different houses in five different colours in a row (white, blue, red, ellow, or green). In each house lives a person with a different nationalit (ritish, sian, German, anish, 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 plas a different sport (basketball, soccer, frisbee, tennis, or hocke). No owner can have the same pet, drink the same beverage, and pla the same sport. Given the below information who owns the fish?. The ritish person lives in the red house. 9. The sian lives in the first house.. The Swede keeps dogs as pets.. The ane drinks tea.. The green house is on the immediate left of the white house.. The green house s owner drinks coffee.. The owner who plas tennis has a bird. 7. The owner of the ellow house plas basketball. 8. The owner in the center house drinks milk. 0. The owner who plas soccer lives net to the one who has a cat.. The owner who has a horse lives net to the one who plas basketball.. The owner who plas hocke drinks juice.. The German plas frisbee.. The sian lives net to the blue house.. The own who plas soccer lives net to the one who drinks water.