The problem asks to find the composition of transformations that map quadrilateral ABCD to quadrilateral EHGF. The transformations are as follows:

1. Reflect over the x-axis: This transformation will change the coordinates of each vertex from $(x, y)$ to $(x, -y)$.
2. Translate: The problem mentions translating the points with a rule of $(x + 6, y + [ ? ])$. The question mark indicates that the translation in the $y$-coordinate is yet to be determined.

Let's analyze the reflection and translation:

• Reflection over the x-axis:

  • Each point of ABCD reflects over the x-axis, so:
    • $A'(-5, 5)$ becomes $A''(-5, -5)$
    • $B'(-2, 5)$ becomes $B''(-2, -5)$
    • $C'(-1, 2)$ becomes $C''(-1, -2)$
    • $D'(-4, 2)$ becomes $D''(-4, -2)$

• Translation by (6, ?):

  • The coordinates must be adjusted horizontally by adding 6 to the x-values.
  • The unknown vertical translation is what we need to solve for. By observing the final positions of points E, H, G, and F, we can determine the translation.

From the graph:

• Point $A''(-5, -5)$ translates to $E(1, -2)$, indicating the translation in the $y$-direction is $+3$.

Thus, the full translation rule is (x + 6, y + 3).

Final Answer:

• The composition of transformations that map quadrilateral ABCD to EHGF is:
  1. Reflect over the x-axis.
  2. Translate by (x + 6, y + 3).

Would you like further details or have any other questions?

Here are five related questions to deepen your understanding:

1. What are the general rules for reflecting a point across the y-axis instead of the x-axis?
2. How would the translation rule change if the movement was downward instead of upward?
3. What other types of transformations can be used to map shapes on a coordinate plane?
4. How would the result change if we reflected over both the x-axis and y-axis?
5. Can you find the center of rotation if ABCD was rotated to match EHGF instead of using reflection and translation?

Tip: When performing a series of transformations, the order matters. A different sequence of operations can lead to a different final position.