Worksheet 6.2 Transformations Using Matrices

Maths Quest Maths C Year 11 for Queensland Chapter 6 Transformations using matrices WorkSHEET 6.2 1

WorkSHEET 6.2 Transformations using matrices Name: ______

1 Find the image of each of the following points (a) 6 rotated through the angles   cos 4  sin 4   R    (a) 4     4 sin 4 cos 4  1  1   2 2  (b)    6  1 1   2 2  (i) (1, 3) (ii) (0, –5) Sketch the special triangle to find the  (iii) (–3, –1) values of cos etc: 4

(i) 1 1 x    1  2 2 y  1 1  3    2 2     1  3   2 2  1  3   2 2   2  2   2   4      2   2 2 

(ii) 1 1 x    0   2 2 y  1 1  5    2 2    5   5 2   2  2  5   5 2   2   2 

(iii) x  1  1  3  2 2 y  1 1 1    2 2    3  1 2 2  2    2    3  1    4    2 2  2   2    Maths Quest Maths C Year 11 for Queensland Chapter 6 Transformations using matrices WorkSHEET 6.2 2

(b)   cos 6  sin 6  R    6      sin 6 cos 6   3 1  2  2     1 3   2 2  Sketch the special triangle to find the  values of cos etc: 6

(i) x  3  1  1  2 2 y  1 3  3    2 2     3  3   2 2  1 3 3   2  2   3  3   2 1  3 3   2 

(ii) x  3  1   0    2 2  y 1 3  5    2 2    5  2    5 3   2  (iii) x  3  1   3  2 2  y'  1 3  1    2 2     3 3  1  2  3  3   2 

2 Find the equation of the image of line (a) 4 y = 2x – 1 under each of the following   cos 2  sin 2  0 1 R   rotations:       2 sin 2 cos 2  1 0  x 0 1 x   y 1 0  y (a) 2       Maths Quest Maths C Year 11 for Queensland Chapter 6 Transformations using matrices WorkSHEET 6.2 3

 Rearrange and multiply by the inverse of the (b) 2 transformation matrix.

1 Sketch the original and image after each x 0 1 x        rotation. y 1 0  y  0 1 x      1 0 y x  y y  x Substitute these values into the original equation. x  2y  1 2y  x  1 1 1 y   x  2 2 (b)   cos 2  sin 2   0 1 R       2       sin 2 cos 2   1 0 x'  0 1 x        y' 1 0 y

Rearrange and multiply by the inverse of the transformation matrix.

1 x  0 1 x        y 1 0 y 0 1 x      1 0  y x  y y  x Substitute these values into the original equation. x  2y 1 2y  x 1 1 1 y   x  2 2 Maths Quest Maths C Year 11 for Queensland Chapter 6 Transformations using matrices WorkSHEET 6.2 4

3 Find the transformation matrix that will result 1 0  (a) M  4 in a reflection through the: y0 0 1 (a) x-axis   1 0 (b) y-axis (b) M  x0  0 1 (c) line y = x   (d) line y = xtan 0 1 (c) M yx    1 0 cos 2  sin 2  (d) M yx tan    sin 2 cos 2 

4 (a) 4 (c) (0, 5) in y = x x 1 0  3 3           y 0 1 0 0 (d) (7, –1) in y = 3x

(b) x 1 0 1 1           y  0 1  4  4

(d) x cos 2  sin 2   7   3 3    2 2    y sin 3 cos 3  1 Draw a special triangle. Maths Quest Maths C Year 11 for Queensland Chapter 6 Transformations using matrices WorkSHEET 6.2 5

x  1  3   7    2 2  y 3 1 1    2 2     7  3   2.63 2          7 3  1   6.56   2   

5 Find the image of the line y = –x – 3 under x cos 2  sin 2  x        2 reflection in the line y = 2x. y sin 2 cos 2  y Draw a triangle to find  .

 = tan–12 = –1.107.c

x cos 2.214  sin 2.214x       y sin 2.214 cos 2.214 y  0.6 0.8 x      0.8  0.6y Rearrange the equation and multiply by the inverse. x  0.6  0.8 x        y  0.8  0.6 y x  0.6x  0.8y y  0.8x  0.6y Substitute into the original equation 0.8x  0.6y  0.6  0.8y  3 1.4y  0.2x  3 1 y  x  2.14 7 Maths Quest Maths C Year 11 for Queensland Chapter 6 Transformations using matrices WorkSHEET 6.2 6

6 Find the image of the following curves 4 reflected through the: 1 0  (i) x-axis (a)(i) My = 0 =   0  1 (ii) line y = x   1 x 1 0  x (a) y = x²        y 0 1 y (b) x² + (y – 1)² = 4 1 0  x      Sketch the original and image curves. 0 1 y x  x y  y Substitute these values into the original equations y  x2 y  x2

 0 1 (a)(ii) My = x =   1 0  1 x  0 1 x        y 1 0  y  0 1 x =     1 0  y x =  y y =  x Substitute these values into the original equation:  x   y2 y2  x Maths Quest Maths C Year 11 for Queensland Chapter 6 Transformations using matrices WorkSHEET 6.2 7

(ii) (b)(i) x  x y  y Substitute into the original equation: x2  (y  1) 2  4 x2  (y  1)2  4 Vertical shift of –2

(b)(ii) x  y y  x Substitute into the original equation: (y) 2  (x 1) 2  4 y2  (x 1) 2  4 Horizontal shift of –1 with a vertical shift of –1. Maths Quest Maths C Year 11 for Queensland Chapter 6 Transformations using matrices WorkSHEET 6.2 8

7 State the transformation matrix that will produce the following dilations: 2 0 (a) 2, parallel to the x-axis (a) D2,x =   0 1 (b) –1, parallel to the y-axis 1 0  1 (b) D = (c) about the origin -1,y 0 1 4    1 0 D  4 (c) 1 0 1  4  4 

8 Find the image of the following points (i) 6 under the given dilations. x 1 0 2 2 (a)           (a) (2, 2) y 0 4 2 8 (b) (–3, 4) x 1 0  3  3 (c) (0, 6)         (b)           (d) (1, –3) y 0 4  4  16  x 1 0 0  0  (c)           (i) 4, parallel to the y-axis y 0 4 6 24 1 (ii) – , parallel to the x-axis x 1 0  1   1  (d)   2 y 0 4  3 12 (iii) 3, about the origin         (ii) 1 x  2 0 2 1 (a)           y  0 1 2  2  1 1 x  2 0  3 1 2  (b)           y  0 1  4   4  1 x  2 0 0 0 (c)           y  0 1 6 6 1 1 x  2 0  1   2  (d)           y  0 1  3  3

(iii) x 3 0 2 6 (a)           y 0 3 2 6 x 3 0  3  9 (b)           y 0 3  4  12  x 3 0 0  0  (c)           y 0 3 6 18 x 3 0  1   3  (d)           y 0 3  3  9 Maths Quest Maths C Year 11 for Queensland Chapter 6 Transformations using matrices WorkSHEET 6.2 9

9 8 Find the equations of the parabola y = x² under (a) 3 the dilations given in question 7. Sketch the x 2 0 x        original and image of each dilation. y 0 1 y Rearrange and multiply by the inverse: x 1 1 0 x        y 2 0 2 y

1 x  x 2 y  y Substitute these into the original equation: 1 y  ( x) 2 2 1  x2 4

(b) 1 x 1 0  x        y 0 1 y 1 0  x =     0 1 y x  x y  y Substitute these into the original equation:  y  x2 y  x2 Maths Quest Maths C Year 11 for Queensland Chapter 6 Transformations using matrices WorkSHEET 6.2 10

x 1 1    4 0 x (c)   =  1    y 0 4  y  1 0 x = 16 4  1  y 0 4    4 0 x =     0 4 y

x  4x y  4y Substitute these into the original equation: 4y  (4x) 2 y  4x2

10 Give the shear transformation matrices that will 2 provide changes with the following: 1 2 (a) S = (a) 2, parallel to the x-axis 2,x 0 1 1   (b) , parallel to the y-axis 1 0 3 1 (b) S , y =  1  3  3 1

11 Find the image positions of the following points (i) 4 under a shear factor of x 1 00 0 (a)          (i) 3, parallel to the y-axis y 3 10 0 1 (ii) , parallel to the x-axis x 1 02 2 (b)   4 y 3 10 6 (a) A(0, 0)        (b) B(2, 0) x 1 02 2 (c)          (c) C (2, 2) y 3 12 8 (d) D(0, 2) x 1 00 0 (d)          y 3 12 2

(ii) 1 x 1 4 0 0 (a)          y 0 10 0 1 x 1 4 2 2 (b)          y 0 10 0 Maths Quest Maths C Year 11 for Queensland Chapter 6 Transformations using matrices WorkSHEET 6.2 11

1 1 x 1 4 2 2 2  (c)          y 0 12  2  1 1 x 1 4 0  2  (d)          y 0 12 2

12 Find the equation of the parabola y = 2x² under x 1 1  x  3 1 1       a shear factor of , parallel to the x-axis y  0 1 y 3 Rearrange and multiply by the inverse. 1 x 1 3  x        y 0 1 y 1 x  x  y 3 y  y Substitute into the original equation: 1 y  2(x  y) 2 3 4 2 y  2x2  xy  y2 3 9