The task involves reflecting the triangle $\Delta EFG$ to its image $\Delta E'F'G'$ over the x-axis. You need to determine the coordinates of the reflected image points $E', F', G'$, which correspond to points $E, F, G$ after reflection.

Coordinates of $\Delta EFG$:

• $E (-3, 5)$
• $F (-2, 2)$
• $G (-4, 2)$

Reflection rule:

When reflecting over the x-axis, the y-coordinate of each point changes sign, while the x-coordinate remains the same. The reflection formula is: $(x, y) \to (x, -y)$

Applying the reflection:

• For point $E (-3, 5)$, the reflection is $E' (-3, -5)$.
• For point $F (-2, 2)$, the reflection is $F' (-2, -2)$.
• For point $G (-4, 2)$, the reflection is $G' (-4, -2)$.

Completed table:

Explanation in words:

To map $\Delta EFG$ to $\Delta E'F'G'$, you reflect each vertex over the x-axis. This means for every point, the x-coordinate stays the same, while the y-coordinate changes sign. For example, point $E (-3, 5)$ is reflected to $E' (-3, -5)$, and the same rule applies to the other points.

Would you like more details or have any questions?

Additional questions:

1. What happens if you reflect $\Delta EFG$ over the y-axis instead of the x-axis?
2. How would the coordinates change if you reflected the triangle over both the x- and y-axes?
3. What is the distance between point $E$ and its reflection $E'$?
4. How can you use the reflection formula to determine if two points are symmetric with respect to the x-axis?
5. Can you describe the transformation if the triangle is reflected over a diagonal line like $y = x$?

Tip:

Reflections maintain the size and shape of a figure, but they reverse the orientation relative to the axis of reflection.