IG IDEAS MATH pcole057 \#19 Check Skills Review Save / E Listen Use the given vertices to graph quadrilateral \( R S T U \) and its image after a dilation centered at the origin with scale factor \( k=-4 \). \[ R(-7,-1), S(2,5), T(-2,-3) ; U(-3,-3) \] Polygon Undo Redo Reset

Certainly! Let's walk through the process of graphing quadrilateral \( RSTU \) and its image after a dilation centered at the origin with a scale factor of \( k = -4 \). ## **1. Understanding Dilation** - **Dilation** is a transformation that resizes a figure by a scaling factor (\( k \)) while maintaining its shape. - The **center of dilation** is the fixed point from which all points are expanded or contracted. - A **scale factor** (\( k = -4 \)) indicates: - **Magnitude (4)**: The figure will be enlarged 4 times its original size. - **Negative Sign (-)**: The figure will also be **reflected** across the center of dilation (in this case, the origin). ## **2. Original Quadrilateral \( RSTU \)** Given vertices: - \( R(-7, -1) \) - \( S(2, 5) \) - \( T(-2, -3) \) - \( U(-3, -3) \) ### **Plotting \( RSTU \):** 1. **Plot each vertex** on the Cartesian plane. 2. **Connect the vertices** in order to form quadrilateral \( RSTU \). *Since I can’t provide a visual graph here, you can plot these points on graph paper or using a graphing tool.* ## **3. Applying the Dilation** **Formula for Dilation:** \[ (x, y) \rightarrow (k \cdot x, k \cdot y) \] Given \( k = -4 \), we'll apply this to each vertex. ### **Calculations:** 1. **Point \( R(-7, -1) \):** \[ R' = (-4) \times (-7, -1) = (28, 4) \] 2. **Point \( S(2, 5) \):** \[ S' = (-4) \times (2, 5) = (-8, -20) \] 3. **Point \( T(-2, -3) \):** \[ T' = (-4) \times (-2, -3) = (8, 12) \] 4. **Point \( U(-3, -3) \):** \[ U' = (-4) \times (-3, -3) = (12, 12) \] ### **Dilated Quadrilateral \( R'S'T'U' \):** - \( R'(28, 4) \) - \( S'(-8, -20) \) - \( T'(8, 12) \) - \( U'(12, 12) \) ## **4. Plotting the Dilated Quadrilateral \( R'S'T'U' \)** 1. **Plot each dilated vertex** on the same Cartesian plane. 2. **Connect the dilated vertices** in order to form quadrilateral \( R'S'T'U' \). **Key Observations:** - The negative scale factor reflects each point through the origin. - The figure is enlarged by a factor of 4. - The orientation of the dilated quadrilateral is opposite to the original due to the negative scale factor. ## **5. Summary of Coordinates** | Original Point | Coordinates | Dilated Point \( (k = -4) \) | Coordinates | |----------------|-------------|-------------------------------|--------------| | R | (-7, -1) | \( R' = -4 \times R \) | (28, 4) | | S | (2, 5) | \( S' = -4 \times S \) | (-8, -20) | | T | (-2, -3) | \( T' = -4 \times T \) | (8, 12) | | U | (-3, -3) | \( U' = -4 \times U \) | (12, 12) | ## **6. Graphing Tips** - **Use graph paper or a digital graphing tool** for accuracy. - **Label both quadrilaterals** clearly to distinguish between the original and the dilated image. - **Check reflection:** Since \( k \) is negative, ensure that the dilated quadrilateral is a mirror image across the origin. ## **7. Final Notes** - Always apply the dilation formula to each vertex. - Remember that a negative scale factor not only scales but also reflects the figure. - The center of dilation (origin) remains unchanged during dilation. If you follow these steps, you'll accurately graph both the original quadrilateral \( RSTU \) and its dilated image \( R'S'T'U' \).