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### 1. Translation Component from \( B(-3, -4) \) to \( B'(-2, 1) \)

The translation component, \( T \), is the difference between the coordinates of \( B' \) and \( B \). To find \( T \), use the formula:

\[
T = B' - B = (-2, 1) - (-3, -4) = (1, 5)
\]

Thus, the translation vector is:
\[
T = \begin{pmatrix} 1 \\ 5 \end{pmatrix}
\]

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### 2. Translation of Points \( M(-2, -3), N(4, -5) \)

You need to translate points using the given translation vector \( T = \begin{pmatrix} 3 \\ 1 \end{pmatrix} \).

For \( M(-2, -3) \):
\[
M' = (-2, -3) + \begin{pmatrix} 3 \\ 1 \end{pmatrix} = (1, -2)
\]

For \( N(4, -5) \):
\[
N' = (4, -5) + \begin{pmatrix} 3 \\ 1 \end{pmatrix} = (7, -4)
\]

Next, for the point \( D'(-5, -3) \) with a different translation vector \( T = \begin{pmatrix} -1 \\ 0 \end{pmatrix} \):
\[
D'' = (-5, -3) + \begin{pmatrix} -1 \\ 0 \end{pmatrix} = (-6, -3)
\]

For \( E'(0, -4) \) with the same translation vector:
\[
E'' = (0, -4) + \begin{pmatrix} -1 \\ 0 \end{pmatrix} = (-1, -4)
\]

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### 3. Translation of the Curve \( f(x) = 3 - 2x \)

The curve is first translated by \( T_2 = \begin{pmatrix} -2 \\ 3 \end{pmatrix} \). This means we adjust both the \( x \)-values and \( y \)-values accordingly. The equation after translation along \( x \) by \( -2 \) is:

\[
f(x+2) = 3 - 2(x+2) = 3 - 2x - 4 = -2x - 1
\]

Next, apply the translation \( T_1 = \begin{pmatrix} 1 \\ -1 \end{pmatrix} \), which affects the \( y \)-axis:

\[
f'(x+2) = (-2x - 1) - 1 = -2x - 2
\]

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### 4. Translation of the Curve \( g(x) = 3x^2 - x + 5 \)

The curve is first translated by \( T_1 = \begin{pmatrix} -2 \\ 0 \end{pmatrix} \). This only affects the \( x \)-values:

\[
g(x+2) = 3(x+2)^2 - (x+2) + 5 = 3(x^2 + 4x + 4) - (x+2) + 5
\]
\[
= 3x^2 + 12x + 12 - x - 2 + 5 = 3x^2 + 11x + 15
\]

Next, apply the translation \( T_2 = \begin{pmatrix} 1 \\ 2 \end{pmatrix} \), which adds 2 to the \( y \)-value:
\[
g'(x+2) = 3x^2 + 11x + 15 + 2 = 3x^2 + 11x + 17
\]

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### 5. Composition of Translations for \( f(x) = 2x + 5 \) and \( g(x) = 1 - 3x \)

You need to first find the composite function \( g(f(x)) \), then apply the translations \( T_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix} \) and \( T_2 = \begin{pmatrix} -1 \\ 3 \end{pmatrix} \).

1. **Composite Function** \( g(f(x)) \):
\[
f(x) = 2x + 5
\]
\[
g(f(x)) = 1 - 3(2x + 5) = 1 - 6x - 15 = -6x - 14
\]

2. **Apply Translation \( T_1 \)**:
\[
g(f(x+1)) = -6(x+1) - 14 = -6x - 6 - 14 = -6x - 20
\]

3. **Apply Translation \( T_2 \)**:
\[
g'(f(x+1)) = (-6x - 20) + 3 = -6x - 17
\]

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Would you like more details on any of these solutions, or help with a specific question? Here are some related questions to deepen your understanding:

1. How do translation transformations affect parametric equations in general?
2. Can you derive the inverse of the translation matrix and explain its significance?
3. How would the curves look geometrically after applying the translations? Can you sketch them?
4. If multiple transformations (rotation, reflection) were applied, how would the process change?
5. What are the applications of these transformations in physics and engineering?

### Tip:
When working with transformations, always apply them step-by-step, making sure to handle the translation components of both the \( x \)-axis and \( y \)-axis separately to avoid confusion.

1) Cari komponen translasi T dari B(-3, -4) → B'(-2,1). 2) Tentukan peta dari titik M(-2, -3), N(4, -5) dengan komponen translasi T = (3,1) dan prapeta dari titik D'(-5, -3), E'(0, -4) dengan komponen translasi T = (-1,0). 3) Tentukan bayangan dari persamaan kurva f(x) = 3 - 2x yang ditranslasi oleh T2 = (-2,3) dan dilanjutkan translasi T1 = (1,-1). 4) Tentukan bayangan dari persamaan kurva g(x) = 3x² - x + 5 yang ditranslasi oleh T1 = (-2,0) dan dilanjutkan translasi T2 = (1,2). 5) Diketahui f(x + 2) = 2x + 5 dan g(x) = 1 - 3x. Tentukan bayangan dari persamaan kurva (gof)(x) oleh translasi T1 = (1,1) dan dilanjutkan translasi T2 = (-1,3). Gambar proses translasi yang terjadi.

Solve translation problems involving coordinate transformations and curve translations. This problem includes translations of points, composite functions, and step-by-step geometric transformations, suitable for Grades 10-12.

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