The problem involves a transformation of quadrilateral $ABCD$ into $EHGF$. Let's break it down step by step:

1. Reflect over the $y$-axis

When you reflect a point $(x, y)$ over the $y$-axis, the new coordinates become $(-x, y)$. Applying this to the vertices of $ABCD$:

• $A(-4, 2) \to A'(+4, 2)$
• $B(-3, 4) \to B'(+3, 4)$
• $C(-1, 4) \to C'(+1, 4)$
• $D(-2, 2) \to D'(+2, 2)$

Thus, the reflection of $ABCD$ produces $A'B'C'D'$, which is now situated to the right of the $y$-axis.

2. Translate the figure

The problem specifies a translation $(x - 2, y + ?)$. To determine the value of "?"), let's compare the final positions of the reflected vertices with those of \(EHGF:

For example, consider the reflected vertex $A'(4, 2)$, which corresponds to $E(2, -1)$ after translation. Solving for the vertical translation $y + ? = -1$:

$2 + ? = -1 \implies ? = -3$

The translation is therefore: $(x - 2, y - 3)$

Final Answer:

The composition of transformations that maps $ABCD$ to $EHGF$ is:

1. Reflect over the $y$-axis.
2. Translate by $(x - 2, y - 3)$.

Would you like a detailed explanation of reflections and translations, or more examples?

5 Related Questions:

1. What is the general formula for reflecting a point over the $x$-axis and the origin?
2. How does translation affect the slope of a line segment in a figure?
3. Can you find the transformation mapping $EHGF$ back to $ABCD$?
4. What are other common types of transformations besides reflections and translations?
5. How do you determine the order of transformations if multiple compositions are applied?

Tip:

When performing a composition of transformations, always handle each step sequentially, as the order of transformations matters in most cases!