https://file.aiquickdraw.com/mathbot/file/1730339041320f7872yuv.jpg

The question provides three transformations and asks for the transformed coordinates of points \( A(2, 5) \) and \( B(-3, 4) \) under these transformations:

1. **Transformations**:
   - \( T_1 = \begin{bmatrix} 4 \\ -1 \end{bmatrix} \)
   - \( T_2 = \begin{bmatrix} -7 \\ 5 \end{bmatrix} \)
   - \( T_3 = \begin{bmatrix} -2 \\ 6 \end{bmatrix} \)

2. **Steps to Solve**:
   - For each point, add the transformation vector to the original coordinates to get the new coordinates after each transformation.
   
Let's calculate the results:

### (a) Point \( A(2, 5) \)
1. **Transformation \( T_1 \)**:
   \[
   A' = \begin{bmatrix} 2 \\ 5 \end{bmatrix} + \begin{bmatrix} 4 \\ -1 \end{bmatrix} = \begin{bmatrix} 6 \\ 4 \end{bmatrix}
   \]
   
2. **Transformation \( T_2 \)**:
   \[
   A'' = \begin{bmatrix} 2 \\ 5 \end{bmatrix} + \begin{bmatrix} -7 \\ 5 \end{bmatrix} = \begin{bmatrix} -5 \\ 10 \end{bmatrix}
   \]

3. **Transformation \( T_3 \)**:
   \[
   A''' = \begin{bmatrix} 2 \\ 5 \end{bmatrix} + \begin{bmatrix} -2 \\ 6 \end{bmatrix} = \begin{bmatrix} 0 \\ 11 \end{bmatrix}
   \]

### (b) Point \( B(-3, 4) \)
1. **Transformation \( T_1 \)**:
   \[
   B' = \begin{bmatrix} -3 \\ 4 \end{bmatrix} + \begin{bmatrix} 4 \\ -1 \end{bmatrix} = \begin{bmatrix} 1 \\ 3 \end{bmatrix}
   \]
   
2. **Transformation \( T_2 \)**:
   \[
   B'' = \begin{bmatrix} -3 \\ 4 \end{bmatrix} + \begin{bmatrix} -7 \\ 5 \end{bmatrix} = \begin{bmatrix} -10 \\ 9 \end{bmatrix}
   \]

3. **Transformation \( T_3 \)**:
   \[
   B''' = \begin{bmatrix} -3 \\ 4 \end{bmatrix} + \begin{bmatrix} -2 \\ 6 \end{bmatrix} = \begin{bmatrix} -5 \\ 10 \end{bmatrix}
   \]

### Summary of Results
- **For \( A(2, 5) \):**
  - After \( T_1 \): \( (6, 4) \)
  - After \( T_2 \): \( (-5, 10) \)
  - After \( T_3 \): \( (0, 11) \)

- **For \( B(-3, 4) \):**
  - After \( T_1 \): \( (1, 3) \)
  - After \( T_2 \): \( (-10, 9) \)
  - After \( T_3 \): \( (-5, 10) \)

Would you like additional details on how each transformation affects the points?

Here are 5 follow-up questions:
1. How would the transformations change if the points were located in a three-dimensional space?
2. What is the effect of combining multiple transformations on a single point?
3. How can matrix multiplication be used to represent these transformations?
4. How does applying transformations affect the geometric shape of a figure in coordinate space?
5. What would be the inverse transformations to return the points to their original positions?

**Tip:** In transformations, always check whether they are translations, rotations, or scalings, as this influences the result significantly.

Diketahui T1 = [4 -1], T2 = [-7 5], dan T3 = [-2 6]. Tentukan bayangan dari:

(a) titik A(2, 5) oleh translasi T1, T2, T3
(b) titik B(-3, 4) oleh translasi T1, T2, T3

Explore how to apply translation transformations to points A(2, 5) and B(-3, 4) using translation vectors T1, T2, and T3. Understand the step-by-step process of translating points in coordinate geometry. Suitable for high school students.

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