Graphing geometric transformations on a coordinate plane.

I can't view or analyze images directly, so I can't see the specific details of your image. However, if you describe the problem or provide the text and coordinates from the image, I can help you solve it step by step.

Based on your description, it seems like you're working with dilations in geometry — transformations that enlarge or reduce a figure using a scale factor and a center of dilation.

Let me explain how to solve such problems in general, and then you can apply it to your specific cases:

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🔷 What is a Dilation?

A dilation is a transformation that changes the size of a figure (makes it larger or smaller) while keeping its shape the same. It uses:
- A scale factor (k)
- A center of dilation (a point, often the origin or another given point)

If the scale factor is:
- k > 1: enlargement
- 0 < k < 1: reduction
- k < 0: enlargement or reduction with reflection

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🔷 How to Perform a Dilation

For each vertex of the figure:
1. Find the vector from the center of dilation to the vertex.
2. Multiply this vector by the scale factor.
3. Add the result back to the center to get the new point.

Alternatively, if the center is at the origin (0,0), just multiply each coordinate by the scale factor.

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Example Problem (based on your image):

#### Problem 6: Dilate triangle XYZ using scale factor of 1.5 and (0,0) as center of dilation.
Suppose the coordinates are:
- X = (2, 4)
- Y = (6, 2)
- Z = (4, 8)

Then:
- X' = (2×1.5, 4×1.5) = (3, 6)
- Y' = (6×1.5, 2×1.5) = (9, 3)
- Z' = (4×1.5, 8×1.5) = (6, 12)

Plot these new points and connect them.

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#### Problem 7: Dilate quadrilateral WXYZ using scale factor of 0.5 with origin as center.
Assume:
- W = (-4, 2)
- X = (2, 4)
- Y = (4, -2)
- Z = (-2, -4)

Then:
- W' = (-4×0.5, 2×0.5) = (-2, 1)
- X' = (2×0.5, 4×0.5) = (1, 2)
- Y' = (4×0.5, -2×0.5) = (2, -1)
- Z' = (-2×0.5, -4×0.5) = (-1, -2)

Plot these new points.

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#### Problem 8: Dilate quadrilateral WXYZ using scale factor of 2 with (0,0) as center.
Use same method:
Multiply all coordinates by 2.

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✔ Steps to Solve Your Problems:

1. Identify the original coordinates of each vertex.
2. Apply the dilation formula:
- If center is origin: (x', y') = (k·x, k·y)
- If center is not origin: use vector method
3. Plot the new points.
4. Label the image (e.g., X', Y', etc.)

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Please provide:
- The coordinates of the original points (X, Y, Z, etc.)
- The scale factor
- The center of dilation

And I’ll walk you through solving each one!