Transformation - Rotation - Image worksheet - Free Printable

To solve this problem, we first need to identify the coordinates of the points on the "Pre-image" from the graph.

Looking at the grid:
* Point A is at $(1, 1)$
* Point B is at $(1, 2)$
* Point C is at $(2, 2)$
* Point D is at $(2, 3)$
* Point E is at $(4, 3)$
* Point F is at $(4, 1)$

Now we will apply the rotation rules to find the new coordinates.

Part 1: Rotate $90^\circ$ Clockwise about the origin

The rule for rotating a point $(x, y)$ $90^\circ$ clockwise is:
$$ (x, y) \rightarrow (y, -x) $$
This means you swap the numbers and make the new second number negative.

1. A': Start with $A(1, 1)$. Swap to get $(1, 1)$, then make the second one negative $\rightarrow$ $(1, -1)$
2. B': Start with $B(1, 2)$. Swap to get $(2, 1)$, then make the second one negative $\rightarrow$ $(2, -1)$
3. C': Start with $C(2, 2)$. Swap to get $(2, 2)$, then make the second one negative $\rightarrow$ $(2, -2)$
4. D': Start with $D(2, 3)$. Swap to get $(3, 2)$, then make the second one negative $\rightarrow$ $(3, -2)$
5. E': Start with $E(4, 3)$. Swap to get $(3, 4)$, then make the second one negative $\rightarrow$ $(3, -4)$
6. F': Start with $F(4, 1)$. Swap to get $(1, 4)$, then make the second one negative $\rightarrow$ $(1, -4)$

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Part 2: Rotate $180^\circ$ Counterclockwise about the origin

The rule for rotating a point $(x, y)$ $180^\circ$ (either direction) is:
$$ (x, y) \rightarrow (-x, -y) $$
This means you just change the sign of both numbers (positive becomes negative, negative becomes positive).

1. A': Start with $A(1, 1)$. Change signs $\rightarrow$ $(-1, -1)$
2. B': Start with $B(1, 2)$. Change signs $\rightarrow$ $(-1, -2)$
3. C': Start with $C(2, 2)$. Change signs $\rightarrow$ $(-2, -2)$
4. D': Start with $D(2, 3)$. Change signs $\rightarrow$ $(-2, -3)$
5. E': Start with $E(4, 3)$. Change signs $\rightarrow$ $(-4, -3)$
6. F': Start with $F(4, 1)$. Change signs $\rightarrow$ $(-4, -1)$

Final Answer:
Rotate $90^\circ$ clockwise:
1. $A' = (1, -1)$
2. $B' = (2, -1)$
3. $C' = (2, -2)$
4. $D' = (3, -2)$
5. $E' = (3, -4)$
6. $F' = (1, -4)$

Rotate $180^\circ$ counterclockwise:
1. $A' = (-1, -1)$
2. $B' = (-1, -2)$
3. $C' = (-2, -2)$
4. $D' = (-2, -3)$
5. $E' = (-4, -3)$
6. $F' = (-4, -1)$