Rotation of shapes on a coordinate grid with labeled examples.

To solve these rotation problems, we need to apply the standard rules for rotating points on a coordinate plane around the origin $(0,0)$.

Here are the rules we will use:
1. $90^\circ$ Counter-Clockwise (or $270^\circ$ Clockwise): $(x, y) \rightarrow (-y, x)$
2. $90^\circ$ Clockwise (or $270^\circ$ Counter-Clockwise): $(x, y) \rightarrow (y, -x)$
3. $180^\circ$ Rotation: $(x, y) \rightarrow (-x, -y)$

Let's find the new coordinates for each shape step-by-step.

1) $90^\circ$ counter-clockwise rotation

* Rule: Swap $x$ and $y$, then make the new $x$ negative. $(x, y) \rightarrow (-y, x)$
* Original Points: The triangle has vertices at $(1, 1)$, $(4, 1)$, and $(1, 3)$.
* Calculations:
* $(1, 1) \rightarrow (-1, 1)$
* $(4, 1) \rightarrow (-1, 4)$
* $(1, 3) \rightarrow (-3, 1)$
* New Vertices: $(-1, 1), (-1, 4), (-3, 1)$

2) $90^\circ$ clockwise rotation

* Rule: Swap $x$ and $y$, then make the new $y$ negative. $(x, y) \rightarrow (y, -x)$
* Original Points: The triangle has vertices at $(1, 1)$, $(4, 1)$, and $(1, 3)$.
* Calculations:
* $(1, 1) \rightarrow (1, -1)$
* $(4, 1) \rightarrow (1, -4)$
* $(1, 3) \rightarrow (3, -1)$
* New Vertices: $(1, -1), (1, -4), (3, -1)$

3) $180^\circ$ rotation

* Rule: Change the sign of both coordinates. $(x, y) \rightarrow (-x, -y)$
* Original Points: The L-shape has vertices at $(1, 1)$, $(3, 1)$, $(3, 2)$, $(2, 2)$, $(2, 3)$, and $(1, 3)$.
* Calculations:
* $(1, 1) \rightarrow (-1, -1)$
* $(3, 1) \rightarrow (-3, -1)$
* $(3, 2) \rightarrow (-3, -2)$
* $(2, 2) \rightarrow (-2, -2)$
* $(2, 3) \rightarrow (-2, -3)$
* $(1, 3) \rightarrow (-1, -3)$
* New Vertices: $(-1, -1), (-3, -1), (-3, -2), (-2, -2), (-2, -3), (-1, -3)$

4) $90^\circ$ counter-clockwise rotation

* Rule: Swap $x$ and $y$, then make the new $x$ negative. $(x, y) \rightarrow (-y, x)$
* Original Points: The L-shape has vertices at $(1, 1)$, $(3, 1)$, $(3, 2)$, $(2, 2)$, $(2, 3)$, and $(1, 3)$.
* Calculations:
* $(1, 1) \rightarrow (-1, 1)$
* $(3, 1) \rightarrow (-1, 3)$
* $(3, 2) \rightarrow (-2, 3)$
* $(2, 2) \rightarrow (-2, 2)$
* $(2, 3) \rightarrow (-3, 2)$
* $(1, 3) \rightarrow (-3, 1)$
* New Vertices: $(-1, 1), (-1, 3), (-2, 3), (-2, 2), (-3, 2), (-3, 1)$

5) $90^\circ$ clockwise rotation

* Rule: Swap $x$ and $y$, then make the new $y$ negative. $(x, y) \rightarrow (y, -x)$
* Original Points: The rectangle has vertices at $(1, 1)$, $(4, 1)$, $(4, 3)$, and $(1, 3)$.
* Calculations:
* $(1, 1) \rightarrow (1, -1)$
* $(4, 1) \rightarrow (1, -4)$
* $(4, 3) \rightarrow (3, -4)$
* $(1, 3) \rightarrow (3, -1)$
* New Vertices: $(1, -1), (1, -4), (3, -4), (3, -1)$

6) $180^\circ$ rotation

* Rule: Change the sign of both coordinates. $(x, y) \rightarrow (-x, -y)$
* Original Points: The rectangle has vertices at $(1, 1)$, $(4, 1)$, $(4, 3)$, and $(1, 3)$.
* Calculations:
* $(1, 1) \rightarrow (-1, -1)$
* $(4, 1) \rightarrow (-4, -1)$
* $(4, 3) \rightarrow (-4, -3)$
* $(1, 3) \rightarrow (-1, -3)$
* New Vertices: $(-1, -1), (-4, -1), (-4, -3), (-1, -3)$

Final Answer:
1) New vertices: $(-1, 1), (-1, 4), (-3, 1)$
2) New vertices: $(1, -1), (1, -4), (3, -1)$
3) New vertices: $(-1, -1), (-3, -1), (-3, -2), (-2, -2), (-2, -3), (-1, -3)$
4) New vertices: $(-1, 1), (-1, 3), (-2, 3), (-2, 2), (-3, 2), (-3, 1)$
5) New vertices: $(1, -1), (1, -4), (3, -4), (3, -1)$
6) New vertices: $(-1, -1), (-4, -1), (-4, -3), (-1, -3)$