Question

unit 6
lesson 2 homework
date
period
a sequence of transformations of qrst is shown on the
coordinate grid. qrst is rotated then dilated, where qrst
is the rotation and qrst is the dilation of qrst.
complete the descriptions:

1. qrst was

to create qrst.

1. qrst was dilated using a scale factor of

centered at to create qrst.

1. complete the table:
2. determine the coordinates if qrst is first dilated then rotated:
3. based on the results above, the sequence of transformations

is
is not

Explanation:

Step1: Determine the rotation

By observing the coordinate grid, \(QRST\) is rotated \(90^{\circ}\) clockwise about the origin \((0,0)\) to get \(Q'R'S'T'\). The rule for a \(90^{\circ}\) clockwise rotation about the origin is \((x,y)\to(y, - x)\).

Step2: Determine the dilation scale - factor and center

Let's take a point. Suppose we consider a side - length. If we assume a reference side in \(Q'R'S'T'\) and \(Q''R''S''T''\). Let's use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For example, if we consider two corresponding non - collinear points. The center of dilation is the origin \((0,0)\). If we take a point \(Q'=(0,1)\) (after rotation) and \(Q''=(0,2)\) (after dilation), the scale factor \(k = 2\) (since \(Q''\) is twice as far from the origin as \(Q'\)). The rule for dilation about the origin with scale factor \(k\) is \((x,y)\to(kx,ky)\).

Step3: Complete the table

• For rotation (\(90^{\circ}\) clockwise about the origin): \((x,y)\to(y, - x)\)
• For dilation (scale factor \(k = 2\) about the origin): \((x,y)\to(2x,2y)\)

Step4: Calculate coordinates

• Dilation (\(k = 2\) about the origin):
• \(Q(-1,0)\): \(Q'=(2\times(-1),2\times0)=(- 2,0)\)
• \(R(-1,-1)\): \(R'=(2\times(-1),2\times(-1))=(-2,-2)\)
• \(S(-2,-2)\): \(S'=(2\times(-2),2\times(-2))=(-4,-4)\)
• \(T(-3,-1)\): \(T'=(2\times(-3),2\times(-1))=(-6,-2)\)
• Rotation (\(90^{\circ}\) clockwise about the origin) of the dilated points:
• \(Q'(-2,0)\): \(Q''=(0,2)\)
• \(R'(-2,-2)\): \(R''=(-2,2)\)
• \(S'(-4,-4)\): \(S''=(-4,4)\)
• \(T'(-6,-2)\): \(T''=(-2,6)\)

Step5: Check commutativity

If we first dilate \(QRST\) with scale factor \(2\) about the origin: \(Q(-1,0)\to(-2,0)\), \(R(-1,-1)\to(-2,-2)\), \(S(-2,-2)\to(-4,-4)\), \(T(-3,-1)\to(-6,-2)\), then rotate \(90^{\circ}\) clockwise about the origin: \(Q''=(0,2)\), \(R''=(-2,2)\), \(S''=(-4,4)\), \(T''=(-2,6)\).
If we first rotate \(QRST\) (\(Q(-1,0)\to(0,1)\), \(R(-1,-1)\to(-1,1)\), \(S(-2,-2)\to(-2,2)\), \(T(-3,-1)\to(-1,3)\)) then dilate with scale factor \(2\) about the origin: \(Q''=(0,2)\), \(R''=(-2,2)\), \(S''=(-4,4)\), \(T''=(-2,6)\). The sequence of transformations (rotation and dilation about the origin) is commutative.

Answer:

1. rotated \(90^{\circ}\) clockwise about the origin
2. \(2\); \((0,0)\)

3.

Transformation	Algebraic Description
Dilation	\((x,y)\to(2x,2y)\)

4.

Preimage \(QRST\)	Dilation \(Q'R'S'T'\)	Rotation \(Q''R''S''T''\)
\(R(-1,-1)\)	\(R'(-2,-2)\)	\(R''(-2,2)\)
\(S(-2,-2)\)	\(S'(-4,-4)\)	\(S''(-4,4)\)
\(T(-3,-1)\)	\(T'(-6,-2)\)	\(T''(-2,6)\)

1. is