Step-by-step explanation:
1b) First, reflect ABCDE over the x-axis.
(x, y) → (x, -y)
Then, translate 5 units to the right and 3 units down.
(x, -y) → (x+5, -y-3)
1c) Instead of reflecting over the x-axis, we can reflect ABCDE over the y-axis.
(x, y) → (-x, y)
Then, rotate about the origin 180 degrees.
(-x, y) → (x, -y)
Finally, translate 5 units to the right and 3 units down.
(x, -y) → (x+5, -y-3)
2b) First, rotate MNOP 90 degrees clockwise about the origin.
(x, y) → (y, -x)
Then, scale by 1/2.
(y, -x) → (y/2, -x/2)
Finally, translate 3.5 units to the left and 2.5 units down.
(y/2 - 3.5, -x/2 - 2.5)
2c) We can form a second method by changing the order of transformations.
First translate MNOP 5 units to the right and 7 units down.
(x, y) → (x + 5, y - 7)
Then scale by 1/2.
(x + 5, y - 7) → (x/2 + 2.5, y/2 - 3.5).
Finally, rotate 90 degrees about the origin.
(y/2 - 3.5, -x/2 - 2.5)
3b) First, rotate EFGH 45 degrees counterclockwise about the origin.
(x, y) → (½√2 (x - y), ½√2 (x + y))
Then scale by ⅓√2.
(½√2 (x - y), ½√2 (x + y)) → (⅓ (x - y), ⅓ (x + y))
Finally, translate 13/3 units to the right and 1 units down.
(⅓ (x - y), ⅓ (x + y)) → (⅓ (x - y) + 13/3, ⅓ (x + y) - 1)
3c) Again, we can form a second method by changing the order of the transformations.
Let's keep the first step the same, rotating EFGH 45 degrees counterclockwise about the origin:
(x, y) → (½√2 (x - y), ½√2 (x + y))
Then translate 13/√2 units to the right and 3/√2 units down.
(½√2 (x - y), ½√2 (x + y)) → (½√2 (x - y) + 13/√2, ½√2 (x + y) + 3/√2)
Finally, scale by ⅓√2.
(½√2 (x - y) + 13/√2, ½√2 (x + y) + 3/√2) → (⅓ (x - y) + 13/3, ⅓ (x + y) + 1)
4b) First, rotate XYZ 45 degrees clockwise about the origin.
(x, y) → (½√2 (x + y), ½√2 (y - x))
Then translate 5-√2 units to the right and 4√2 units up.
(½√2 (x + y), ½√2 (y - x)) → (½√2 (x + y) + 5 - √2, ½√2 (y - x) + 4√2)
4c) Instead, let's translate XYZ 5 units to the left and 3 units up.
(x, y) → (x - 5, y + 3)
Then rotate 45 degrees clockwise about the origin:
(x - 5, y + 3) → (½√2 (x + y - 2), ½√2 (y - x + 8))
Finally, translate 5 units to the right.
(½√2 (x + y - 2), ½√2 (y - x + 8)) → (½√2 (x + y - 2) + 5, ½√2 (y - x + 8))
