On Your Own. ). Another way is to multiply the. ), and the image. Applications. Unit 3 _ _

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1 Applications 1 a 90 clockwise rotation matrix: - b As can be seen by the diagram, the image of P is Q and the image of R is P The coordinate of Q can be found by symmetry y R 1 P, Thus, the 45 clockwise rotation matrix is - O M P(1, 0) x Q, - c 90 clockwise rotation matrix: - - d One way is to use the one diagram in Part b above The image of P(1, 0) is the opposite of P, namely (-, - ), and the image of R is the point Q (, - ) Another way is to multiply the 90 clockwise rotation matrix by the 45 clockwise rotation matrix So, the matrix for a 135 clockwise rotation is Transformations, Matrices, and Animation T4B

2 a A reflection across a vertical line b A(, 3) A (-, 3) B(3, -4) B (-3, -4) c (x, y) (-x, y) d Since the image of P(1, 0) is (-1, 0) and the image of Q(0, 1) is (0, 1), e the matrix representation of a reflection across the y-axis is - Flag Animation Algorithm Step 1 Set up the coordinate matrix representing the flag Step Set up the 45 counterclockwise rotation matrix Step 3 Draw flag Step 4 Compute and store the new coordinates of the flag rotated 45 Step 5 Clear the old flag and draw the rotated image Step 6 Pause Step 7 Repeat Steps 4 6 fifteen more times (16 total) Step 8 Set up the reflection across the y-axis matrix Step 9 Compute and store the new coordinates of the reflected image Step 10 Set up the 45 clockwise rotation matrix Step 11 Compute the new coordinates of the rotated image Step 1 Clear the old flag and draw the rotated image Step 13 Pause Step 14 Repeat Steps fifteen more times (16 total) NOTE The solutions to Applications Task 1 are on page T4B f The final flag image is represented by the matrix The final image is the original flag reflected across the y-axis Transformations, Matrices, and Animation T43

3 3 a The matrix representation for a reflection across the x-axis is b The matrix representation for a reflection across the line y x is c The composite transformation can be found by multiplying the matrices from Parts a and b The matrix representing the composite transformation is d The matrix is the matrix representing a 90 counterclockwise rotation with center at the origin One efficient way to recognize this is to use the relationship that under this transformation, P(1, 0) P (0, 1) (first column of matrix in Part c) and Q(0, 1) Q (-1, 0) (second column of matrix) These images represent a 90 counterclockwise rotation about the origin INSTRUCTIONAL NOTE For Part c, since the reflection across the x-axis is done first, the matrix representing it must be on the right 4 a d See student layout, matrices, and algorithms (In Part b, students should note that the choice of the origin on this coordinate grid determined the coordinates that specify their initials) 5 a (1) () Both ways result in b PQRS rotated 180 So, is the matrix of the image rectangle under the composite transformation Transformations, Matrices, and Animation T44

4 6 Students coordinate matrices for their rockets will vary An example of a triangular rocket is a Student algorithms will vary The following sample algorithm uses the coordinate matrix above along with initial scale factor of 09, initial translation of 4 units up, incremental decrease of scale factor 01, incremental increase of vertical translation by 4 units, and number of times to repeat All these choices are arbitrary, and students may vary the algorithm to include some rotation of the rocket as well Step 1 Set up the coordinate matrix representing the rocket Step Set up a 09 scale factor Step 3 Set up a vertical translation of 4 units Step 4 Draw the rocket Step 5 Compute the new coordinates of the original rocket scaled and then translated Step 6 Clear the old rocket and draw the new rocket Step 7 Pause Step 8 Decrease the scale factor by 01 Step 9 Increase the vertical translation by 4 units Step 10 Repeat Steps 5 9 six times b Responses may vary The sample algorithm in Part a repeatedly performs a size transformation of the original rocket and translates the image rocket vertically To create the appearance of launching, at each iteration the scale factor is reduced to make the original rocket appear smaller while at the same time, the translation is increased to make the image rocket move farther along the positive y-axis c For this sample algorithm, the coordinates of the rocket half way through the animation are the coordinates of the image rocket after the third repetition of Steps 5 9, Connections 7 a P ( 1 b, 1, ) and Q (- a b c d a b c d R ), so a 1, so b - c R 60 : (x, y) ( 1 x - y, x + 1 and c 3 and d 1 y) 8 R 30 1 a R 60 R R R 60 b R 30 : (x, y) ( x + 1 y, - 1 x + ) y Transformations, Matrices, and Animation T45

5 9 Frieze Pattern Step 1 Set up the coordinate matrix representing the figure Step Set up the coordinate matrix for a reflection across the x-axis Step 3 Set up a horizontal translation of 6 units Step 4 Compute the image of the figure under a reflection across the x-axis Step 5 Draw the figure and its reflected image Step 6 Compute the image of the figure and its image under the translation Step 7 Draw the translated image (Do not clear the previous drawn images) Step 8 Replace the previous figure and image with the ones computed in Step 6 Step 9 Repeat Steps 6 to 8 as needed 10 a R45 ; R They are the same matrices as those previously identified with 90 counterclockwise and 180 rotations with center at the origin b R 6 45 represents a counterclockwise rotation of 70 R3 45 represents a counterclockwise rotation of 135 The power of the matrix represents the number of successive 45 counterclockwise rotations c R 90 R 70 (R 90 ) a The transformation matrix for a reflection across the x-axis is Since I, the transformation matrix for a 0-1 reflection across the x-axis is a square root of I b Students will likely suggest matrices for any transformation that when composed with itself maps each point onto itself Reflection across the y-axis: - Reflection across y x: Reflection across y -x: rotation: c Since the image of each point under the composition of a line reflection (half-turn) with itself is the original point, any matrix that represents a line reflection would be a square root of I Thus, there are an infinite number of square roots of I Transformations, Matrices, and Animation T46