To determine the series of transformations that map Figure $I$ onto Figure $J$, we need to carefully analyze the positions and orientations of the two figures.

Step-by-Step Process:

1. Reflection across the x-axis:

   • Figure $J$ appears to be an upside-down version of Figure $I$, so a reflection over the x-axis would map the shape from above the x-axis (where Figure $I$ is) to below it (where Figure $J$ is).

2. Translation:

   • After reflecting Figure $I$ over the x-axis, we can see that Figure $J$ is shifted downward and to the right. This suggests a translation is needed.
   • Translation can be calculated by looking at the movement of a specific point from Figure $I$ to Figure $J$.
     • For example, if we consider a point on Figure $I$, such as $(-1, 6)$, after reflecting over the x-axis, it will be at $(-1, -6)$. We then need to translate it to its corresponding point on Figure $J$, which could be something like $(-1, -3)$, so this requires a vertical translation upward by 3 units.

Thus, the series of transformations that would map Figure $I$ onto Figure $J$ would be:

1. Reflect Figure $I$ over the x-axis.
2. Translate the resulting figure downward (or upward) by the necessary amount to align with Figure $J$.

Would you like more details on any of these transformations, or how to calculate them specifically?

Related Questions:

1. What is the general rule for reflection over the x-axis in coordinate geometry?
2. How can we calculate the exact translation needed after a reflection?
3. Can there be a rotation involved in this transformation process?
4. How can transformations be combined into a single matrix operation?
5. What is the effect of reflecting over both the x-axis and y-axis in sequence?

Tip:

When analyzing transformations, always break them down into the simplest operations (reflection, rotation, translation) to avoid confusion.