Use the given vertices to graph quadrilateral \( W X Y Z \) and its image after a dilation centered at the origin with scale factor \( k=-0.5 \). \( \begin{array}{l}W(8,-2), X(6,0), Y(-6,4), Z(-2,2) \\ \text { Polygon }\end{array} \)

To graph quadrilateral \( WXYZ \) and its image after a dilation centered at the origin with a scale factor \( k = -0.5 \), follow these steps: ### 1. **Original Quadrilateral \( WXYZ \)** The given vertices of quadrilateral \( WXYZ \) are: - \( W(8, -2) \) - \( X(6, 0) \) - \( Y(-6, 4) \) - \( Z(-2, 2) \) **Plotting the Original Quadrilateral:** - **Point W(8, -2):** Move 8 units to the right (positive x-axis) and 2 units down (negative y-axis). - **Point X(6, 0):** Move 6 units to the right on the x-axis. - **Point Y(-6, 4):** Move 6 units to the left (negative x-axis) and 4 units up (positive y-axis). - **Point Z(-2, 2):** Move 2 units to the left and 2 units up. Connect the points in the order \( W \) → \( X \) → \( Y \) → \( Z \) → \( W \) to form the quadrilateral. ### 2. **Dilation Transformation** A dilation transformation changes the size and/or orientation of a figure. The formula for dilation centered at the origin is: \[ \text{If } (x, y) \text{ is a point, then its image } (x', y') \text{ after dilation is } (k \cdot x, k \cdot y) \] Given: - **Scale Factor \( k = -0.5 \):** - The negative sign indicates a reflection across the origin. - The absolute value \( |k| = 0.5 \) indicates the image will be half the size of the original. **Applying Dilation to Each Vertex:** 1. **Point \( W(8, -2) \):** \[ W' = (k \cdot 8, k \cdot (-2)) = (-0.5 \times 8, -0.5 \times -2) = (-4, 1) \] 2. **Point \( X(6, 0) \):** \[ X' = (k \cdot 6, k \cdot 0) = (-0.5 \times 6, -0.5 \times 0) = (-3, 0) \] 3. **Point \( Y(-6, 4) \):** \[ Y' = (k \cdot -6, k \cdot 4) = (-0.5 \times -6, -0.5 \times 4) = (3, -2) \] 4. **Point \( Z(-2, 2) \):** \[ Z' = (k \cdot -2, k \cdot 2) = (-0.5 \times -2, -0.5 \times 2) = (1, -1) \] ### 3. **Image Quadrilateral \( W'X'Y'Z' \)** The vertices of the dilated (image) quadrilateral \( W'X'Y'Z' \) are: - \( W'(-4, 1) \) - \( X'(-3, 0) \) - \( Y'(3, -2) \) - \( Z'(1, -1) \) **Plotting the Image Quadrilateral:** - **Point \( W'(-4, 1) \):** Move 4 units to the left and 1 unit up. - **Point \( X'(-3, 0) \):** Move 3 units to the left on the x-axis. - **Point \( Y'(3, -2) \):** Move 3 units to the right and 2 units down. - **Point \( Z'(1, -1) \):** Move 1 unit to the right and 1 unit down. Connect the points in the order \( W' \) → \( X' \) → \( Y' \) → \( Z' \) → \( W' \) to form the image quadrilateral. ### 4. **Visualization Tips** - **Reflection:** The negative scale factor (\( k = -0.5 \)) causes the image to be a reflection of the original quadrilateral across the origin. - **Scaling:** The size of the image quadrilateral is half that of the original due to the scale factor's absolute value being 0.5. - **Orientation:** Since both scaling and reflection are applied, the image quadrilateral will be smaller and mirrored relative to the original. ### 5. **Summary of Coordinates** | Original Point | Original Coordinates | Dilated Image Coordinates | |----------------|----------------------|---------------------------| | \( W \) | \( (8, -2) \) | \( W'(-4, 1) \) | | \( X \) | \( (6, 0) \) | \( X'(-3, 0) \) | | \( Y \) | \( (-6, 4) \) | \( Y'(3, -2) \) | | \( Z \) | \( (-2, 2) \) | \( Z'(1, -1) \) | By plotting both quadrilaterals on the same coordinate plane, you can clearly see the effect of the dilation transformation.