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.