TRANSFORMATIONS Learner note: Transformations are easy to master and you can score well in questions involving this topic. Ensure that you know the different algebraic transformation rules. LESSON OVERVIEW 1. Introduction session 5 minutes. Typical exam questions: Question 1: 0 minutes 3. Discussion of solutions 35 minutes SECTION A: TYPICAL EXAM QUESTIONS Question 1 (0 minutes) (a) On the diagram below, represent the point A( 3; ). () Page 1 of 8
(b) Now represent the following points on the diagram provided on the previous page: Point B, the rotation of point A, 90 anticlockwise about the origin. Point C, the rotation of point A, 180 about the origin. Point D, the rotation of point A, 90 clockwise about the origin. (3) (c) Write down algebraic rules to describe the above transformations. (3) (d) (e) What type of quadrilateral is figure ABCD? Explain by referring to the properties of quadrilaterals. () Figure ABCD is enlarged by a scale factor of units through the origin to form its image A B C D. On the diagram provided on the previous page, sketch the image A B C D and indicate the coordinates of the vertices. () (f) (g) Determine the ratio: Area ABCD Area A B C D ABCD is reflected about the y-axis to form its image EFGH. Write down the coordinates of E. () Determine the ratio: Perimeter ABCD Perimeter EFGH (h) Describe, using words and algebraic notation, the single transformations involved if figure ABCD is transformed by the rule: 1 1 ( x; y) x ; y 1 (3) Page of 8
SECTION B: SOLUTIONS AND HINTS Question 1 (a), (b) D (4;6) A ( 6;4) A( 3;) D(;3) B( ; 3) C(3; ) C (6; 4) B ( 4; 6) (b) B( ; 3) C(3; ) D(;3) (c) 90 anticlockwise: ( x ; y) ( y ; x) 180 anti-clockwise or clockwise: ( x ; y) ( x ; y) 90 clockwise: ( x ; y) ( y ; x) (d) ABCD is a square since: Diagonals are equal in length Diagonals bisect each other at right angles B( ; 3) C(3; ) D(;3) (3) ( x ; y) ( y ; x) ( x ; y) ( x ; y) ( x ; y) ( y ; x) (3) square properties () Page 3 of 8
(e) A ( 6;4) (f) (g) (g)() B ( 4; 6) C (6; 4) D (4;6) See diagram Area ABCD 1 1 Area A B C D 4 E(3;) Perimeter ABCD 4side AB 1 Perimeter EFGH 4side EF (since AB EF) See Section C (Enlargement and reduction rules) (h) 1 1 1 ( x; y) x; y reduction by a factor of 1 1 1 1 x; y x; y reflection about x axis 1 1 1 1 x; y x; y 1 translation of 1 unit downwards 1 1 ( x; y) x; y 1 correct coordinates indicated joining points to form enlarged square () 1 4 answer answer reduction reflection translation (3) Page 4 of 8
SECTION C: ADDITIONAL CONTENT NOTES The following is a summary of the algebraic transformation rules TRANSLATION RULES If the point ( x; y ) is translated to form the point ( x a ; y b) where a is a horizontal move and b is a vertical move, then the following rules apply: If a 0, the horizontal translation is to the right. If a 0, the horizontal translation is to the left. If b 0, the vertical translation is upward. If b 0, the vertical translation is downward REFLECTION RULES Reflection about the y-axis: ( x ; y) ( x ; y) Reflection about the x-axis: ( x ; y) ( x ; y) Reflection about the line y x: ( x ; y) ( y ; x) RULES OF ROTATION ABOUT THE ORIGIN Rotation of 90 anti-clockwise: ( x ; y) ( y ; x) Rotation of 90 clockwise: ( x ; y) ( y ; x) Rotation of 180 clockwise or anti-clockwise: ( x ; y) ( x ; y) ENLARGEMENT AND REDUCTION RULES ( x ; y) ( kx ; ky) If k 1, the image is an enlargement of the original figure. Multiply the original first and second coordinates by k units to get the coordinates of the image. The image will be k times larger than the original figure. If 0k 1, the image is a reduction of the original figure. Multiply the original first and second coordinates by k units to get the coordinates of the image. Page 5 of 8
If k 0, the image is a rotation of 180of the original figure followed by an enlargement of the original figure. Multiply the original first and second coordinates by k units to get the coordinates of the image. Area of image k Area of original or Area of image k Area of original or Area of original 1 Area of image k SECTION D: HOMEWORK Question 1 (a) On the diagram below, draw figure ABCD with the coordinates of the vertices as follows: A( 6 ; 6); B( 4 ; ); C( ; 6) and D( 4 ; 8) Page 6 of 8
(b) On the diagram, draw the image A B C D if ABCD is rotated 90 anti-clockwise. Indicate the coordinates of B and C. () (c) State the general coordinates in terms of x and y of this rotation of 90 anti-clockwise using the notation ( x; y) (d) Now draw the image A B C D if ABCD is transformed under the rule: 1 1 ( x ; y) x ; y. () (e) Write down the value of the following ratio: Area ABCD Area A B C D () (f) Draw image A B C D if A B C D is rotated 180 clockwise. Indicate the coordinates of A () (g) Draw the image EFGH if A B C D is transformed under the rule ( x ; y) ( x ; y). Indicate the coordinates of E. () (h) Write down the single algebraic rule if ABCD is reflected about the x-axis, followed by a translation of 7 units right, followed by an enlargement by a scale factor of units. (4) SECTION E: SOLUTIONS TO SESSION 16 HOMEWORK 1 3xy 6.A 5x3y 11 B 9x6y 18... A 3 10x 6y B x 4 x 4 3(4) y 6 y 6 y 3 Other methods are acceptable Multiplication of A by 3 Multiply B by - Adding to get x 4 x 4 3(4) y 6 y 3 (6) Page 7 of 8
(a) y 3 x x (3 x) x (3 x) x 3 x x 9 1x 4x 3x 11x 6 0 3x 11x 6 0 (3x )( x 3) 0 x 3 or x 3 5 y 3 or y 3 You may use the quadratic formula to solve this equation y3 x substitution multiplication standard form factors x-values y-values (8) (b) x 1 3 (1 3y) 1y 3y y 1 6y 9y (1 3y) y 9y 4y 16 0 y 4 0 y 6y (3y 4)( y 1) 0 4 y or y 1 3 x 3 or x 4 9y 17 17 0 x1 3y substitution multiplication standard form factors x-values y-values (8) The SSIP is supported by Page 8 of 8