B) A 90° counterclockwise rotation about the origin, followed by a reflection across the x-axis, followed by a translation 8 units right and 1 unit up.
Explanation:
The coordinates of the <u>points of the pre-image</u> are: (3, 1) (3, 4) (5, 7) (6, 5) (6, 2)
The coordinates of the <u>points of the image</u> are: (7,-2) (4,-2) (1,-4) (3,-5) (6,-5)
A 90° counterclockwise rotation about the origin negates the y-coordinate and switches it and the x-coordinate. Algebraically, (x,y)→(-y,x).
When this is applied to our points, we get: (3, 1)→(-1, 3) (3, 4)→(-4, 3) (5, 7)→(-7, 5) (6, 5)→(-5, 6) (6, 2)→(-2, 6)
A reflection across the x-axis negates the y-coordinate. Algebraically, (x, y)→(x, -y).
Applying this to our new points, we have: (-1, 3)→(-1, -3) (-4, 3)→(-4, -3) (-7, 5)→(-7, -5) (-5, 6)→(-5, -6) (-2, 6)→(-2, -6)
A translation 8 units right and 1 unit up adds 8 to the x-coordinate and 1 to the y-coordinate. Algebraically, (x, y)→(x+8, y+1).
Applying this to our new points, we have: (-1, -3)→(-1+8,-3+1) = (7, -2) (-4, -3)→(-4+8,-3+1) = (4, -2) (-7, -5)→(-7+8,-5+1) = (1, -4) (-5, -6)→(-5+8,-6+1) = (3, -5) (-2, -6)→(-2+8,-6+1) = (6, -5)
These match the coordinates of the image, so this is the correct series of transformations.
