The problem describes a transformation of triangle $ABC$ that involves a 180° rotation about the origin followed by a translation 1 unit to the right, forming triangle $A'B'C'$.

Let's break down the transformation and solve for the true statements.

1. 180° Rotation about the Origin:

• The rule for a 180° rotation is: $(x, y) \rightarrow (-x, -y)$.

2. Translation 1 Unit to the Right:

• The translation rule: After rotation, the transformation adds 1 to the x-coordinate, meaning $(x, y) \rightarrow (x + 1, y)$.

Let's now verify each statement:

A) The transformation is given by $(x, y) \rightarrow (x - 1, -y)$:

• This is incorrect because the rule for 180° rotation is $(x, y) \rightarrow (-x, -y)$, and the translation moves the point right, so the correct transformation includes an addition to the x-coordinate, not a subtraction.

B) The transformation is given by $(x, y) \rightarrow (-x + 1, -y)$:

• This is correct. After a 180° rotation, we apply $(-x, -y)$ and then shift 1 unit right, giving $(-x + 1, -y)$.

C) The coordinates of $C'$ are (2, 2):

• Let's verify by first applying the transformations to point $C$.
  • $C(-4, -3) \rightarrow (4, 3)$ after 180° rotation.
  • Then, translate 1 unit right: $(4, 3) \rightarrow (5, 3)$.
  • So, the coordinates of $C'$ are $(5, 3)$, not $(2, 2)$. This statement is incorrect.

D) The coordinates of $B'$ are (1, -2):

• Let's check point $B$:
  • $B(-1, 3) \rightarrow (1, -3)$ after 180° rotation.
  • Then, translate 1 unit right: $(1, -3) \rightarrow (2, -3)$.
  • So, the coordinates of $B'$ are $(2, -3)$, not $(1, -2)$. This statement is incorrect.

E) The coordinates of $A'$ are (3, -3):

• Let's check point $A$:
  • $A(-3, 4) \rightarrow (3, -4)$ after 180° rotation.
  • Then, translate 1 unit right: $(3, -4) \rightarrow (4, -4)$.
  • So, the coordinates of $A'$ are $(4, -4)$, not $(3, -3)$. This statement is incorrect.

Conclusion:

The only correct statement is B). The transformation is given by $(x, y) \rightarrow (-x + 1, -y)$.

Would you like more details or clarification on any part? Here are some follow-up questions to explore further:

1. How do different types of transformations (rotations, reflections) change coordinates?
2. What is the general rule for translating points in the coordinate plane?
3. How does a 180° rotation affect points symmetrically?
4. Can you apply the transformations to other figures, such as quadrilaterals?
5. How would the transformation change if it involved reflection instead of rotation?

Tip: Always remember to apply transformations step by step, especially when dealing with multiple operations like rotations and translations!