To solve this, we need to find the coordinates of vertex $S$ after performing the given transformations. The transformations are:

1. Reflection across $y = 0$ (the x-axis).
2. Translation 1 unit left and 2 units up.

Step-by-Step Solution:

1. Identify the Initial Coordinates of $S$: From the grid, $S$ is located at $(-3, -3)$.

2. Reflection across $y = 0$: Reflecting $S(-3, -3)$ across the x-axis will change the $y$-coordinate from $-3$ to $3$, resulting in the point: $S' = (-3, 3)$

3. Translation (1 unit left and 2 units up): Moving $S'(-3, 3)$ 1 unit to the left changes the $x$-coordinate from $-3$ to $-4$, and moving 2 units up changes the $y$-coordinate from $3$ to $5$. This results in: $S'' = (-4, 5)$

Final Answer:

The coordinates of $S$ after the sequence of transformations are: $S = (-4, 5)$

Would you like more details on any of these steps, or have any questions?

Here are some related questions to further explore this topic:

1. How does reflecting a point across the x-axis affect its coordinates?
2. How would a reflection across the y-axis change the coordinates of a point?
3. What would happen if the translation was 1 unit right and 2 units down instead?
4. How can we generalize the effect of reflections and translations on coordinates?
5. How would the transformations change if the reflection was across the line $y = x$ instead?

Tip: When reflecting a point, only the coordinate perpendicular to the axis of reflection changes its sign.