A shear transformation S maps A(2,3) onto A'(8,3). Find the matrix S. Find the images of B(5,2) and C(4,-1) under S.

You're on a roll — Step 1: Find the matrix representing the shear (a)(i). A shear transformation with the x-axis invariant has the matrix form S = 1 & k \\ 0 & 1 . Given that A(2,3) is mapped onto A'(8,3): 1 & k \\ 0 & 1 2 \\ 3 = 8 \\ 3 This gives the equations: 1(2) + k(3) = 8 2 + 3k = 8 0(2) + 1(3) = 3 3 = 3 From the first equation: 3k = 8 - 2 3k = 6 k = 2 The matrix representing the shear is: 1 & 2 \\ 0 & 1 Step 2: Find the coordinates of B' and C' (a)(ii). Using the shear matrix S = 1 & 2 \\ 0 & 1 : For B(5,2): 1 & 2 \\ 0 & 1 5 \\ 2 = 1(5) + 2(2) \\ 0(5) + 1(2) = 5 + 4 \\ 2 = 9 \\ 2 So, B' is (9,2). For C(4,-1): 1 & 2 \\ 0 & 1 4 \\ -1 = 1(4) + 2(-1) \\ 0(4) + 1(-1) = 4 - 2 \\ -1 = 2 \\ -1 So, C' is (2,-1). Step 3: Find the coordinates of triangle A''B''C'' (b)(i). Triangle A''B''C'' is the image of A'B'C' under the transformation matrix M = -2 & 0 \\ 0 & 1 . Coordinates of A' are (8,3), B' are (9,2), and C' are (2,-1). For A'(8,3): -2 & 0 \\ 0 & 1 8 \\ 3 = -2(8) + 0(3) \\ 0(8) + 1(3) = -16 \\ 3 So, A'' is (-16,3). For B'(9,2): -2 & 0 \\ 0 & 1 9 \\ 2 = -2(9) + 0(2) \\ 0(9) + 1(2) = -18 \\ 2 So, B'' is (-18,2). For C'(2,-1): -2 & 0 \\ 0 & 1 2 \\ -1 = -2(2) + 0(-1) \\ 0(2) + 1(-1) = -4 \\ -1 So, C'' is (-4,-1). Step 4: Describe the transformation M fully (b)(ii). The transformation matrix is M = -2 & 0 \\ 0 & 1 . This matrix represents a stretch with a scale factor of -2 parallel to the x-axis (with the y-axis as the invariant line). This can also be described as a reflection in the y-axis combined with a stretch of factor 2 parallel to the x-axis. Step 5: Determine a single matrix that maps triangle A''B''C'' onto triangle ABC (c). The transformation from ABC to A''B''C'' is given by the product of the matrices M and S, in that order: MS. MS = -2 & 0 \\ 0 & 1 1 & 2 \\ 0 & 1 = (-2)(1) + (0)(0) & (-2)(2) + (0)(1) \\ (0)(1