Questions: Write a rule for the following transformation: K(-3,4), L(-3,5), M(1,5), N(1,4) to K'(0,-3), L'(0,-2), M'(4,-2), N'(4,-3) dilation of 0.25 about the origin rotation 90 degrees counterclockwise about the origin reflection across the y-axis translation: 3 units right and 7 units down

Solution Steps

The original points $K(-3, 4), L(-3, 5), M(1, 5), N(1, 4)$ are dilated by a factor of $0.25$. The dilated coordinates are calculated as follows: $$K' = (-3 \times 0.25, 4 \times 0.25) = (-0.75, 1), \quad L' = (-3 \times 0.25, 5 \times 0.25) = (-0.75, 1.25)$$ $$M' = (1 \times 0.25, 5 \times 0.25) = (0.25, 1.25), \quad N' = (1 \times 0.25, 4 \times 0.25) = (0.25, 1)$$ Thus, the dilated points are: $$K'(-0.75, 1), \quad L'(-0.75, 1.25), \quad M'(0.25, 1.25), \quad N'(0.25, 1)$$

The dilated points are then rotated $90^\circ$ counterclockwise about the origin using the rotation matrix: $$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$ The rotated coordinates are: $$K'' = (1, 0.75), \quad L'' = (1.25, 0.75), \quad M'' = (1.25, -0.25), \quad N'' = (1, -0.25)$$

Next, the rotated points are reflected across the $y$-axis. The reflected coordinates are: $$K''' = (-1, 0.75), \quad L''' = (-1.25, 0.75), \quad M''' = (-1.25, -0.25), \quad N''' = (-1, -0.25)$$

Finally, the reflected points are translated $3$ units to the right and $7$ units down. The translated coordinates are: $$K'''' = (-1 + 3, 0.75 - 7) = (2, -6.25), \quad L'''' = (-1.25 + 3, 0.75 - 7) = (1.75, -6.25)$$ $$M'''' = (-1.25 + 3, -0.25 - 7) = (1.75, -7.25), \quad N'''' = (-1 + 3, -0.25 - 7) = (2, -7.25)$$

Final Answer