To describe the transformation of the polygon with vertices A(3, 0), B(4, 1), C(2, 2), D(1, 4), and E(1, 1) to the new polygon with vertices F(1, -3), G(0, -2), H(-1, -4), I(-3, -5), and J(0, -5), we need to determine the nature of the transformation. Transformations can involve translation, rotation, reflection, dilation, or a combination of these. Let's break this down step by step.

Step 1: Translation

One way to determine if a translation occurred is to check the difference in coordinates between the original and transformed points.

For point A(3, 0) transforming to F(1, -3):

• The change in $x$ is $1 - 3 = -2$
• The change in $y$ is $-3 - 0 = -3$

For all other points:

• B(4, 1) to G(0, -2):
  • $x$-change is $0 - 4 = -4$
  • $y$-change is $-2 - 1 = -3$
• C(2, 2) to H(-1, -4):
  • $x$-change is $-1 - 2 = -3$
  • $y$-change is $-4 - 2 = -6$
• D(1, 4) to I(-3, -5):
  • $x$-change is $-3 - 1 = -4$
  • $y$-change is $-5 - 4 = -9$
• E(1, 1) to J(0, -5):
  • $x$-change is $0 - 1 = -1$
  • $y$-change is $-5 - 1 = -6$

The changes in the coordinates are not consistent, meaning this is not a simple translation.

Step 2: Rotation or Reflection

Since the points shift differently, let's check if a rotation or reflection is involved. A pattern emerges suggesting a transformation involving both a reflection and a shift. Specifically:

1. The polygon appears flipped and possibly rotated.
2. The shape’s overall orientation changes, and the coordinate signs suggest a reflection across the x-axis and possibly a diagonal line.

Step 3: Scaling

Since the distances between points appear different, a dilation or non-uniform scaling could also be involved, but to check this thoroughly, we'd need to calculate the distances between corresponding points before and after transformation.

Summary of Transformation:

The transformation likely involves:

• Reflection across the x-axis: This changes the signs of the y-coordinates.
• Rotation and possibly dilation, given the varying distances and orientation shift.

Questions for Further Exploration:

1. Do you want to analyze if there’s any specific rotational angle between the polygons?
2. Would you like to compute the distances between corresponding points to check for scaling?
3. Should we examine any possible shear transformation involved?
4. Would you like a visual representation of both polygons to better understand the transformation?
5. Do you want to compare specific vectors formed by the corresponding points?

Tip:

For analyzing transformations, always look for consistent patterns in how coordinates shift, especially with rotation and reflection.