The problem describes a series of translations for triangle $ABC$ with vertices $A(-2, -2)$, $B(-3, 1)$, and $C(1, 1)$. The translations occur in two steps:

1. The first translation moves the points by $(x, y) \to (x - 1, y + 3)$, producing triangle $A'B'C'$.
2. The second translation moves triangle $A'B'C'$ by $(x, y) \to (x + 4, y - 1)$, producing triangle $A''B''C''$.

Part (a): Find the coordinates for the vertices of triangle $A''B''C''$.

1. First translation (for triangle $A'B'C'$):

   • For vertex $A(-2, -2)$: $(-2, -2) \to (-2 - 1, -2 + 3) = (-3, 1)$
   • For vertex $B(-3, 1)$: $(-3, 1) \to (-3 - 1, 1 + 3) = (-4, 4)$
   • For vertex $C(1, 1)$: $(1, 1) \to (1 - 1, 1 + 3) = (0, 4)$

   So, after the first translation, the vertices of triangle $A'B'C'$ are: $A'(-3, 1), B'(-4, 4), C'(0, 4)$

2. Second translation (for triangle $A''B''C''$):

   • For vertex $A'(-3, 1)$: $(-3, 1) \to (-3 + 4, 1 - 1) = (1, 0)$
   • For vertex $B'(-4, 4)$: $(-4, 4) \to (-4 + 4, 4 - 1) = (0, 3)$
   • For vertex $C'(0, 4)$: $(0, 4) \to (0 + 4, 4 - 1) = (4, 3)$

   So, the vertices of triangle $A''B''C''$ are: $A''(1, 0), B''(0, 3), C''(4, 3)$

Part (b): Write a rule for one translation that maps triangle $ABC$ to triangle $A''B''C''$.

Instead of performing two translations separately, we can combine them into one:

1. The first translation is $(x, y) \to (x - 1, y + 3)$.
2. The second translation is $(x, y) \to (x + 4, y - 1)$.

We can combine these transformations as follows: $(x, y) \to ((x - 1) + 4, (y + 3) - 1) = (x + 3, y + 2)$

So, the combined translation that maps triangle $ABC$ directly to triangle $A''B''C''$ is: $(x, y) \to (x + 3, y + 2)$

Do you want more details or have any other questions? Here are some related questions:

1. What would happen if you only apply the second translation to triangle $ABC$?
2. Can you reverse the process to find the original triangle from $A''B''C''$?
3. How can you describe the translations in vector form?
4. What would the vertices of triangle $A'B'C'$ be if the first translation was different?
5. How do these translations affect the area and orientation of the triangle?

Tip: When combining multiple transformations, always apply one transformation at a time and simplify the overall movement into a single translation if possible.