Question

question determine a series of transformations that would map polygon abcd onto polygon abcd. a reflection over the y - axis followed by a dilation centered at the origin with a scale factor

Explanation:

Step1: Find coordinates of original and image points

Let's assume coordinates of \(ABCD\) (original) and \(A'B'C'D'\) (image). Suppose \(A(x_1,y_1)\), \(B(x_2,y_2)\), \(C(x_3,y_3)\), \(D(x_4,y_4)\) and \(A'(x_1',y_1')\), \(B'(x_2',y_2')\), \(C'(x_3',y_3')\), \(D'(x_4',y_4')\).
For reflection over \(y -\)axis: The rule is \((x,y)\to(-x,y)\).
For dilation centered at origin with scale factor \(k\): The rule is \((x,y)\to(kx,ky)\).
Let's check the ratio of distances. Suppose after reflection of a point \(P(x,y)\) over \(y -\)axis we get \(P'(-x,y)\). Then after dilation with scale factor \(k\), we get \(P''(-kx,ky)\).
Count the units. Suppose \(AB\) has length \(l_1\) and \(A'B'\) has length \(l_2\). The scale factor \(k=\frac{l_2}{l_1}\).
Let's assume from the grid (by counting units):
If we take a point (say \(B\)) of \(ABCD\) (upper - left polygon) and its corresponding point (\(B'\)) of \(A'B'C'D'\) (lower - right polygon).
Suppose \(B\) has coordinates \((1,3)\) (after reflection over \(y -\)axis it becomes \((- 1,3)\)) and \(B'\) has coordinates \((-3, - 9)\).
First, reflection over \(y -\)axis: \((x,y)\to(-x,y)\). Then dilation: \((-x,y)\to(k(-x),ky)\).
We can find \(k\) by \(\frac{-9}{3}=- 3\) (using \(y -\)coordinates) or \(\frac{-3}{-1}=3\) (using \(x -\)coordinates, but we take absolute value for scale factor as it's a non - negative measure of enlargement/shrinkage).

Answer:

A reflection over the \(y -\)axis followed by a dilation centered at the origin with a scale factor of \(3\).