Question

determine a series of transformations that would map polygon abcde onto polygon abcde.

Explanation:

). D' to E': x from 5 to 1 (4 units left), y from -4 to -2 (2 units up). Wait, no. Alternatively, the first transformation is a vertical stretch by 2, then a translation. Let's try:

Step 1: Dilation (scale) with scale factor 2, centered at (1,0). For point A(1,3): distance from center is 3, scale by 2: 6, so (1,6). Then translate down 12: (1,-6) = A'. E(1,5): distance from center is 5, scale by 2: 10, translate down 12: (1,-2) = E'. B(2,2): distance from center (1,0) is $\sqrt{(1)^2 + (2)^2}=\sqrt{5}$, scale by 2: $\sqrt{20}$, but B' is (3,-8). No, that's not.

Wait, maybe the answer is: First, a dilation with a scale factor of 2, then a translation down by 12 units and right by 0 (for x) or some x. But looking at the coordinates, A(1,3) → A'(1,-6): y changes from 3 to -6, which is 3(-2). So scale factor -2? Wait, 3(-2)=-6. E(1,5)(-2)=-10, but E' is (1,-2). No. Wait, A(1,3) to A'(1,-6): multiply y by -2. E(1,5) to E'(1,-2): 5(-0.4)=-2. No, not consistent.

Wait, maybe the transformation is: Reflect over the x-axis, then dilate by a scale factor of 2, then translate down by 6 units? A(1,3)→(1,-3)→(1,-6)→(1,-12)? No.

Wait, let's look at the y-coordinates of A and A': 3 and -6. 3 (-2) = -6. E: 5 (-0.4) = -2. No, not same scale. D: 4 * (-1) = -4. D' is (5,-4), so D(3,4)→(5,-4): x+2, y-8. C(6,3)→(10,-5): x+4, y-8. B(2,2)→(3,-8): x+1, y-10. E(1,5)→(1,-2): x+0, y-7. A(1,3)→(1,-6): x+0, y-9.

This is confusing. Maybe the intended answer is: First, a dilation (scale) with scale factor 2, then a translation down by 12 units. Let's check A: 32=6, 6-12=-6. Correct. E:52=10, 10-12=-2. Correct. B:22=4, 4-12=-8. Correct. D:42=8, 8-12=-4. Correct. C:32=6, 6-12=-6. But C' is (10,-5). Oh, x-coordinate: C(6,3)→(10,-5). 6 to 10 is +4, 3 to -5 is -8. 4 is 6 (2/3) + 0? No. Wait, maybe the scale factor is 2 for x and y? A(1,3)→(2,6)→(1,-6): x-1, y-12. E(1,5)→(2

Answer:

). D' to E': x from 5 to 1 (4 units left), y from -4 to -2 (2 units up). Wait, no. Alternatively, the first transformation is a vertical stretch by 2, then a translation. Let's try:

Step 1: Dilation (scale) with scale factor 2, centered at (1,0). For point A(1,3): distance from center is 3, scale by 2: 6, so (1,6). Then translate down 12: (1,-6) = A'. E(1,5): distance from center is 5, scale by 2: 10, translate down 12: (1,-2) = E'. B(2,2): distance from center (1,0) is $\sqrt{(1)^2 + (2)^2}=\sqrt{5}$, scale by 2: $\sqrt{20}$, but B' is (3,-8). No, that's not.

Wait, maybe the answer is: First, a dilation with a scale factor of 2, then a translation down by 12 units and right by 0 (for x) or some x. But looking at the coordinates, A(1,3) → A'(1,-6): y changes from 3 to -6, which is 3(-2). So scale factor -2? Wait, 3(-2)=-6. E(1,5)(-2)=-10, but E' is (1,-2). No. Wait, A(1,3) to A'(1,-6): multiply y by -2. E(1,5) to E'(1,-2): 5(-0.4)=-2. No, not consistent.

Wait, maybe the transformation is: Reflect over the x-axis, then dilate by a scale factor of 2, then translate down by 6 units? A(1,3)→(1,-3)→(1,-6)→(1,-12)? No.

Wait, let's look at the y-coordinates of A and A': 3 and -6. 3 (-2) = -6. E: 5 (-0.4) = -2. No, not same scale. D: 4 * (-1) = -4. D' is (5,-4), so D(3,4)→(5,-4): x+2, y-8. C(6,3)→(10,-5): x+4, y-8. B(2,2)→(3,-8): x+1, y-10. E(1,5)→(1,-2): x+0, y-7. A(1,3)→(1,-6): x+0, y-9.

This is confusing. Maybe the intended answer is: First, a dilation (scale) with scale factor 2, then a translation down by 12 units. Let's check A: 32=6, 6-12=-6. Correct. E:52=10, 10-12=-2. Correct. B:22=4, 4-12=-8. Correct. D:42=8, 8-12=-4. Correct. C:32=6, 6-12=-6. But C' is (10,-5). Oh, x-coordinate: C(6,3)→(10,-5). 6 to 10 is +4, 3 to -5 is -8. 4 is 6 (2/3) + 0? No. Wait, maybe the scale factor is 2 for x and y? A(1,3)→(2,6)→(1,-6): x-1, y-12. E(1,5)→(2