Collection of six geometry exercises focusing on translating shapes across a coordinate plane.

To solve these translation problems, we need to move every point of the shape according to the instructions.

How Translation Works:
* Left/Right: Changes the x-coordinate (the first number).
* Left means subtract.
* Right means add.
* Up/Down: Changes the y-coordinate (the second number).
* Up means add.
* Down means subtract.

Let's calculate the new coordinates for each problem step-by-step.

1) Translation: 3 left and 3 down

Rule: $(x - 3, y - 3)$

Original Points:
* $(1, 4)$
* $(0, 1)$
* $(5, 1)$

New Points:
* $x: 1 - 3 = -2$, $y: 4 - 3 = 1 \rightarrow (-2, 1)$
* $x: 0 - 3 = -3$, $y: 1 - 3 = -2 \rightarrow (-3, -2)$
* $x: 5 - 3 = 2$, $y: 1 - 3 = -2 \rightarrow (2, -2)$

2) Translation: 2 left and 4 up

Rule: $(x - 2, y + 4)$

Original Points:
* $(3, -1)$
* $(3, -3)$
* $(0, -3)$
* $(0, -5)$
* $(5, -5)$
* $(5, -1)$

New Points:
* $(3-2, -1+4) \rightarrow (1, 3)$
* $(3-2, -3+4) \rightarrow (1, 1)$
* $(0-2, -3+4) \rightarrow (-2, 1)$
* $(0-2, -5+4) \rightarrow (-2, -1)$
* $(5-2, -5+4) \rightarrow (3, -1)$
* $(5-2, -1+4) \rightarrow (3, 3)$

3) Translation: 2 right and 4 down

Rule: $(x + 2, y - 4)$

Original Points:
* $(-4, 4)$
* $(-1, 4)$
* $(-2, 3)$

New Points:
* $(-4+2, 4-4) \rightarrow (-2, 0)$
* $(-1+2, 4-4) \rightarrow (1, 0)$
* $(-2+2, 3-4) \rightarrow (0, -1)$

4) Translation: 4 left

Rule: $(x - 4, y)$
*(Note: Since it only says "left", the y-values stay exactly the same)*

Original Points:
* $(1, 4)$
* $(5, 4)$
* $(4, 1)$
* $(2, 1)$

New Points:
* $(1-4, 4) \rightarrow (-3, 4)$
* $(5-4, 4) \rightarrow (1, 4)$
* $(4-4, 1) \rightarrow (0, 1)$
* $(2-4, 1) \rightarrow (-2, 1)$

5) Translation: 5 left and 4 up

Rule: $(x - 5, y + 4)$

Original Points:
* $(3, -2)$
* $(5, -2)$
* $(5, -3)$
* $(4, -3)$
* $(4, -4)$
* $(2, -4)$
* $(2, -5)$
* $(3, -5)$

New Points:
* $(3-5, -2+4) \rightarrow (-2, 2)$
* $(5-5, -2+4) \rightarrow (0, 2)$
* $(5-5, -3+4) \rightarrow (0, 1)$
* $(4-5, -3+4) \rightarrow (-1, 1)$
* $(4-5, -4+4) \rightarrow (-1, 0)$
* $(2-5, -4+4) \rightarrow (-3, 0)$
* $(2-5, -5+4) \rightarrow (-3, -1)$
* $(3-5, -5+4) \rightarrow (-2, -1)$

6) Translation: 4 left and 3 up

Rule: $(x - 4, y + 3)$

Original Points:
* $(2, 0)$
* $(4, -3)$
* $(0, -3)$

New Points:
* $(2-4, 0+3) \rightarrow (-2, 3)$
* $(4-4, -3+3) \rightarrow (0, 0)$
* $(0-4, -3+3) \rightarrow (-4, 0)$

Final Answer:
1) New vertices: $(-2, 1), (-3, -2), (2, -2)$
2) New vertices: $(1, 3), (1, 1), (-2, 1), (-2, -1), (3, -1), (3, 3)$
3) New vertices: $(-2, 0), (1, 0), (0, -1)$
4) New vertices: $(-3, 4), (1, 4), (0, 1), (-2, 1)$
5) New vertices: $(-2, 2), (0, 2), (0, 1), (-1, 1), (-1, 0), (-3, 0), (-3, -1), (-2, -1)$
6) New vertices: $(-2, 3), (0, 0), (-4, 0)$