Question

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

Explanation:

Step1: Recall the dilation formula

The dilation rule is \(P_{O,\frac{1}{3}}(x,y)\to(\frac{1}{3}x,\frac{1}{3}y)\). Let the coordinates of the pre - image of \(V'\) be \((x,y)\) and the coordinates of \(V'\) be \((x',y')\). Then \(x'=\frac{1}{3}x\) and \(y' = \frac{1}{3}y\).

Step2: Find the coordinates of \(V'\) from the graph

From the graph, the coordinates of \(V'\) are \((0,1)\). So \(x'=0\) and \(y' = 1\).

Step3: Solve for the pre - image coordinates

Since \(y'=\frac{1}{3}y\), substituting \(y' = 1\) into the equation \(1=\frac{1}{3}y\). Solving for \(y\) gives \(y = 3\) (multiply both sides of the equation \(1=\frac{1}{3}y\) by \(3\): \(y=3\times1\)). And since \(x'=\frac{1}{3}x = 0\), then \(x = 0\).

So the coordinates of the pre - image of \(V'\) (vertex \(V\) of the pre - image) are \((0,3)\).

Answer:

D. (0, 3)