https://file.aiquickdraw.com/mathbot/file/1724645544687ge8ngt2j.jpg

To solve the problem provided in the image, we need to go through two main transformations: a translation and a reflection.

### Step 1: Translation
The point \( S(2, -5) \) is translated using the translation vector \( T = \begin{pmatrix} 1 \\ 3 \end{pmatrix} \). 

To translate a point by a vector \( T = \begin{pmatrix} a \\ b \end{pmatrix} \), you add the components of the vector to the coordinates of the point.

\[
S'(x', y') = S(x + a, y + b)
\]

For the given translation:
\[
x' = 2 + 1 = 3
\]
\[
y' = -5 + 3 = -2
\]

So, the coordinates of the point after translation are \( S'(3, -2) \).

### Step 2: Reflection over the line \( x = 3 \)
Next, we reflect the translated point \( S'(3, -2) \) over the line \( x = 3 \).

When reflecting a point \( (x_1, y_1) \) over the vertical line \( x = c \), the x-coordinate changes as follows:
\[
x_2 = 2c - x_1
\]
The y-coordinate remains the same:
\[
y_2 = y_1
\]

For the line \( x = 3 \) and the point \( S'(3, -2) \):
\[
x_2 = 2(3) - 3 = 6 - 3 = 3
\]
\[
y_2 = -2
\]

Thus, the coordinates of the point after the reflection are \( (3, -2) \).

### Final Answer
The final coordinates of the image of point \( S \) after the translation and reflection are \( \mathbf{(3, -2)} \).

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Would you like more details or have any questions? 

Here are five related questions for further exploration:

1. How would the solution change if the point \( S \) was reflected over a line such as \( y = -2 \)?
2. What would the final coordinates be if the original point \( S(2, -5) \) was translated by a different vector, say \( \begin{pmatrix} -4 \\ 6 \end{pmatrix} \)?
3. What is the general formula for reflecting a point over a vertical line \( x = c \)?
4. How would the solution change if the reflection was over the y-axis instead of the line \( x = 3 \)?
5. How do translations and reflections affect the distance of a point from the origin?

**Tip:** When performing multiple transformations, always follow the order given in the problem, as it can affect the final result.

Explore how to solve a point transformation problem involving translation and reflection. Learn step-by-step how to translate a point and reflect it over a vertical line. Suitable for high school students studying coordinate geometry.

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