To map Figure $S$ onto Figure $T$, we need to identify a series of transformations (like translation, reflection, rotation, and dilation) that accomplish this. Let's analyze the figures:

Step-by-Step Analysis:

1. Orientation Check: The figures have similar orientations but are not direct mirror reflections. This suggests that a rotation is more likely needed.
2. Size Check: Both Figure $S$ and Figure $T$ appear to have the same size, indicating no dilation is involved.
3. Position Check: We observe that Figure $S$ is in Quadrant II, while Figure $T$ is in Quadrant IV. A rotation or reflection combined with a translation may achieve the mapping.

Transformation Sequence:

1. Rotation by $180^\circ$ Around the Origin: If we rotate Figure $S$ by $180^\circ$ around the origin, it will align the orientation with Figure $T$ and shift Figure $S$ to Quadrant IV.

   After this rotation:

   • The coordinates of each point in Figure $S$ will change to their negative counterparts, as $(x, y)$ becomes $(-x, -y)$.

2. Translation: Next, we need to shift the rotated figure to match Figure $T$. We can determine the specific translation vector by comparing the coordinates of a corresponding pair of points in the rotated Figure $S$ and Figure $T$.

   By examining the grid, let's choose a point:

   • Suppose after the rotation, the point $(-4, -7)$ of Figure $S$ matches the point $(4, -7)$ in Figure $T$. This requires a translation of $(+8, 0)$ along the $x$-axis.

Final Answer:

To map Figure $S$ onto Figure $T$, we apply the following transformations:

1. Rotate Figure $S$ by $180^\circ$ around the origin.
2. Translate the resulting figure $8$ units to the right.

Would you like a detailed explanation or further clarification on any of these steps?

Related Questions:

1. What is the general rule for rotating a figure $180^\circ$ around the origin?
2. How can you determine the specific translation vector between two figures?
3. What other combinations of transformations might achieve the same mapping?
4. How can you confirm that two figures are congruent after transformations?
5. What are the key differences between translations, rotations, and reflections?

Tip: When performing multiple transformations, always check the effect of each step individually to ensure that the figures align as intended.