Question

given the dilation rule ( d_{o, 1/3} (x, y) \to left( \frac{1}{3}x, \frac{1}{3}y
ight) ) and the image ( stuv ), what are the coordinates of vertex ( v ) of the pre - image? ( \bigcirc (0, 0) ) ( \bigcirc left( 0, \frac{1}{3}
ight) ) ( \bigcirc (0, 1) ) ( \bigcirc (0, 3) )

Explanation:

Step1: Find coordinates of \( V' \)

From the graph, \( V' \) is at \( (0, 1) \) (since it's on the y - axis, x = 0 and y = 1).

Step2: Use dilation rule to find pre - image \( V \)

The dilation rule is \( D_{O,1/3}(x,y)\to(\frac{1}{3}x,\frac{1}{3}y) \). Let the coordinates of the pre - image \( V \) be \( (x,y) \) and the coordinates of the image \( V' \) be \( (\frac{1}{3}x,\frac{1}{3}y) \). We know that \( \frac{1}{3}x = 0 \) and \( \frac{1}{3}y=1 \).
For the x - coordinate: Solving \( \frac{1}{3}x = 0 \), we get \( x = 0 \) (multiplying both sides by 3).
For the y - coordinate: Solving \( \frac{1}{3}y = 1 \), we multiply both sides by 3, so \( y=3 \).
So the coordinates of \( V \) are \( (0,3) \).

Answer:

D. (0, 3)