Question

write a coordinate rule for the composition (the blue preimage to the green image).

Explanation:

Step1: Determine the scale factor

The blue triangle has vertices, for example, \(A(-4,-3)\), \(B(-4,2)\), \(C(4,-3)\). The green triangle (pre - image to image). If we assume a vertex of the blue triangle \(B(-4,2)\) and a corresponding vertex of the green triangle (say \(X\) which is close to \((-1,1)\)). The scale factor \(k\) can be found by looking at the ratio of the coordinates. If we consider the \(x\) - coordinate of \(B(-4,2)\) and assume the corresponding point in the green figure (after dilation). Let's assume the transformation is a dilation. The ratio of the \(x\) - coordinates (taking absolute values for simplicity, since the center of dilation is the origin) for a point like \(B(-4,2)\) and its image (say if we assume the image has \(x\) - coordinate \(- 1\) and \(y\) - coordinate \(1\)). The scale factor \(k=\frac{1}{4}\) (since \(\frac{-1}{-4}=\frac{1}{4}\) and \(\frac{1}{4}=\frac{1}{4}\)).

Step2: Write the coordinate rule

For a dilation centered at the origin with scale factor \(k\), the coordinate rule is \((x,y)\to(kx,ky)\). Since \(k = \frac{1}{4}\), the rule is \((x,y)\to(\frac{1}{4}x,\frac{1}{4}y)\)

Answer:

\((x,y)\to(\frac{1}{4}x,\frac{1}{4}y)\)