Question

which dilation is shown by the graph?

( \bigcirc d_{\frac{1}{2}}(wxyz) )
( \bigcirc d_{\frac{1}{3}}(wxyz) )
( \bigcirc d_3(wxyz) )
( \bigcirc d_2(wxyz) )

Explanation:

Step1: Identify Coordinates

First, find coordinates of original and image points. Let's take point \( W \): original \( W(-6, 0) \), image \( W'(-2, 0) \).

Step2: Calculate Scale Factor

Scale factor \( k = \frac{\text{Image Coordinate}}{\text{Original Coordinate}} \). For \( W \): \( k = \frac{-2}{-6} = \frac{1}{3} \)? Wait, no, wait. Wait, maybe \( W' \) is image, \( W \) original? Wait, no, the smaller figure is \( W'X'Y'Z' \), larger is \( WXYZ \). So scale factor from \( WXYZ \) to \( W'X'Y'Z' \) is \( k = \frac{\text{length of } W'X'}{\text{length of } WX} \). Let's check \( W(-6,0) \), \( X(3,3) \) (wait, maybe better to check \( W(-6,0) \), \( W'(-2,0) \): distance from origin? Wait, dilation center is origin? So vector from origin to \( W \) is \( (-6,0) \), to \( W' \) is \( (-2,0) \). So scale factor \( k = \frac{-2}{-6} = \frac{1}{3} \)? Wait, no, wait the options: \( D_{\frac{1}{3}}(WXYZ) \) means scale factor \( \frac{1}{3} \), shrinking. Let's check another point: \( X(3,3) \)? Wait, original \( X \) is at (3,3)? Wait image \( X' \) is at (1,1)? Wait no, looking at graph: original \( W(-6,0) \), \( W'(-2,0) \); original \( X(3,3) \), \( X'(1,1) \); original \( Y(5,-6) \), \( Y'(\frac{5}{3}, -2) \)? Wait, no, maybe \( W(-6,0) \), \( W'(-2,0) \): \( -6 \times \frac{1}{3} = -2 \), correct. \( X(3,3) \times \frac{1}{3} = (1,1) \), which matches \( X' \). So scale factor is \( \frac{1}{3} \), so dilation is \( D_{\frac{1}{3}}(WXYZ) \). Wait, but the options: first option is \( D_{\frac{1}{2}} \), second \( D_{\frac{1}{3}} \), third \( D_3 \), fourth \( D_2 \). So calculating scale factor: take \( W(-6,0) \) to \( W'(-2,0) \): \( k = \frac{-2}{-6} = \frac{1}{3} \). So the dilation is \( D_{\frac{1}{3}}(WXYZ) \).

Answer:

\( D_{\frac{1}{3}}(WXYZ) \) (the second option: \( D_{\frac{1}{3}}(WXYZ) \))