Question

practice and problem solving
identify each transformation. then use arrow notation to describe the transformation.
8.
9.

1. a figure has vertices at (j(-2,3)), (k(0,3)), (l(0,1)), and (m(-2,1)). after a transformation, the image of the figure has vertices at (j(2,1)), (k(4,1)), (l(4, - 1)), and (m(2,-1)). draw the preimage and image. then identify the transformation.

Explanation:

Step1: Analyze question 8

The orientation of the figure has changed. It is a rotation. For a rotation, if we assume the center of rotation is the origin (not shown explicitly but a common - case assumption if not otherwise stated), and we observe the change in orientation, we can describe it in arrow - notation.
Let's assume the pre - image points are \(D(x_1,y_1), E(x_2,y_2), F(x_3,y_3), G(x_4,y_4)\) and the image points are \(D'(x_1',y_1'), E'(x_2',y_2'), F'(x_3',y_3'), G'(x_4',y_4')\). A \(90^{\circ}\) counter - clockwise rotation about the origin has the rule \((x,y)\to(-y,x)\). The arrow notation for a \(90^{\circ}\) counter - clockwise rotation of quadrilateral \(DEFG\) to \(D'E'F'G'\) is \((x,y)\to(-y,x)\).

Step2: Analyze question 9

The position of the rectangle has changed. It is a translation. If we consider a general translation rule \((x,y)\to(x + a,y + b)\). By comparing the pre - image points \(W(x_1,y_1),X(x_2,y_2),Y(x_3,y_3),Z(x_4,y_4)\) and the image points \(W'(x_1',y_1'),X'(x_2',y_2'),Y'(x_3',y_3'),Z'(x_4',y_4')\), we can find the change in \(x\) and \(y\) coordinates. Let's assume the pre - image rectangle \(WXYZ\) and the image rectangle \(W'X'Y'Z'\). If we observe the movement, we can see that the rectangle has moved down and to the left. Suppose the pre - image point \(W(x,y)\) and the image point \(W'(x + a,y + b)\). By counting the grid units (not shown in the problem but a common way to analyze translations), if we assume the rectangle has moved \(h\) units to the left and \(k\) units down, the arrow notation is \((x,y)\to(x - h,y - k)\). For example, if it moves 3 units left and 2 units down, the arrow notation is \((x,y)\to(x-3,y - 2)\).

Step3: Analyze question 10

First, find the change in \(x\) and \(y\) coordinates.
For point \(J(-2,3)\) and \(J'(2,1)\):
The change in \(x\) is \(\Delta x=2-(-2)=4\) and the change in \(y\) is \(\Delta y=1 - 3=-2\).
For point \(K(0,3)\) and \(K'(4,1)\):
The change in \(x\) is \(\Delta x = 4-0 = 4\) and the change in \(y\) is \(\Delta y=1 - 3=-2\).
For point \(L(0,1)\) and \(L'(4,-1)\):
The change in \(x\) is \(\Delta x=4 - 0=4\) and the change in \(y\) is \(\Delta y=-1 - 1=-2\).
For point \(M(-2,1)\) and \(M'(2,-1)\):
The change in \(x\) is \(\Delta x=2-(-2)=4\) and the change in \(y\) is \(\Delta y=-1 - 1=-2\).
The transformation is a translation with the arrow notation \((x,y)\to(x + 4,y-2)\).

Answer:

1. Rotation, \((x,y)\to(-y,x)\)
2. Translation, \((x,y)\to(x - h,y - k)\) (where \(h\) and \(k\) are the number of units moved left and down respectively)
3. Translation, \((x,y)\to(x + 4,y-2)\)