Question

the vertices a(-2,-1), b(-3,2), c(-1,3), and d(0,0) form a parallelogram. the vertices a(-1,-2), b(2,-3), c(3,-1), and d(0,0) are the image of the parallelogram after a sequence of transformations. which sequence of transformations could produce the image from the pre - image?

1. a reflection over the y - axis and then a 90° clockwise rotation about the origin
2. a reflection over the x - axis and then a reflection over the y - axis
3. a 90° clockwise rotation about the origin and then a reflection over the y - axis
4. a 90° counterclockwise rotation about the origin and then a reflection over the x - axis

Explanation:

Step1: Apply rotation formula

The formula for a \(90^{\circ}\) clockwise rotation about the origin \((x,y)\to(y, - x)\). For point \(A(-2,-1)\), after rotation, it becomes \((-1,2)\).

Step2: Apply reflection formula

The formula for reflection over the \(y\) - axis is \((x,y)\to(-x,y)\). For the point \((-1,2)\) after reflection over the \(y\) - axis, it becomes \((1,2)\).

Let's check each option:
- **Option 1 (a reflection over the \(y\) - axis and then a \(90^{\circ}\) clockwise rotation about the origin)**:
- For \(A(-2,-1)\):
- Reflection over \(y\) - axis: \((2,-1)\)
- \(90^{\circ}\) clockwise rotation about the origin: \((-1,-2)\) (not \(A'(-1,-2)\))
- **Option 2 (a reflection over the \(x\) - axis and then a reflection over the \(y\) - axis)**:
- For \(A(-2,-1)\):
- Reflection over \(x\) - axis: \((-2,1)\)
- Reflection over \(y\) - axis: \((2,1)\) (not \(A'(-1,-2)\))
- **Option 3 (a \(90^{\circ}\) clockwise rotation about the origin and then a reflection over the \(y\) - axis)**:
- For \(A(-2,-1)\):
- \(90^{\circ}\) clockwise rotation about the origin: \((-1,2)\)
- Reflection over \(y\) - axis: \((1,2)\) (not \(A'(-1,-2)\))
- **Option 4 (a \(90^{\circ}\) counter - clockwise rotation about the origin and then a reflection over the \(x\) - axis)**:
- The formula for a \(90^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(-y,x)\). For \(A(-2,-1)\), after \(90^{\circ}\) counter - clockwise rotation about the origin: \((1,-2)\)
- The formula for reflection over the \(x\) - axis is \((x,y)\to(x,-y)\). For the point \((1,-2)\) after reflection over the \(x\) - axis: \((1,2)\) (not \(A'(-1,-2)\))

Let's use the transformation matrix approach.
The rotation matrix for a \(90^{\circ}\) clockwise rotation about the origin is \(R=\begin{bmatrix}0&1\\-1&0\end{bmatrix}\), and the reflection matrix over the \(y\) - axis is \(M=\begin{bmatrix}-1&0\\0&1\end{bmatrix}\)
The combined transformation matrix \(T = M\times R=\begin{bmatrix}-1&0\\0&1\end{bmatrix}\times\begin{bmatrix}0&1\\-1&0\end{bmatrix}=\begin{bmatrix}0&-1\\-1&0\end{bmatrix}\)
For a point \(\begin{bmatrix}x\\y\end{bmatrix}\), the transformed point is \(T\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}0&-1\\-1&0\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}-y\\-x\end{bmatrix}\)

For \(A(-2,-1)\), \(\begin{bmatrix}-(-1)\\-(-2)\end{bmatrix}=\begin{bmatrix}1\\2\end{bmatrix}\) (wrong)

The rotation matrix for a \(90^{\circ}\) counter - clockwise rotation about the origin is \(R_c=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\), and the reflection matrix over the \(x\) - axis is \(M_x=\begin{bmatrix}1&0\\0&-1\end{bmatrix}\)
The combined transformation matrix \(T_c = M_x\times R_c=\begin{bmatrix}1&0\\0&-1\end{bmatrix}\times\begin{bmatrix}0&-1\\1&0\end{bmatrix}=\begin{bmatrix}0&-1\\-1&0\end{bmatrix}\) (same as above, wrong)

Let's check by another method:
We know that \(A(-2,-1)\to A'(-1,-2)\)
If we first do a \(90^{\circ}\) clockwise rotation about the origin: \((x,y)\to(y, - x)\), for \(A(-2,-1)\) we get \((-1,2)\)
Then reflect over the \(y\) - axis \((x,y)\to(-x,y)\), we get \((1,2)\) (wrong)

If we first do a \(90^{\circ}\) counter - clockwise rotation about the origin: \((x,y)\to(-y,x)\), for \(A(-2,-1)\) we get \((1,-2)\)
Then reflect over the \(x\) - axis \((x,y)\to(x,-y)\), we get \((1,2)\) (wrong)

Let's check the option: a reflection over the \(y\) - axis and then a \(90^{\circ}\) clockwise rotation about the origin
- Reflection over \(y\) - axis: \((x,y)\to(-x,y)\), for \(A(-2,-1)\) we get \((2,-1)\)
- \(90^{\circ}\) clockwise rotation about the origin: \((x,y)\to(y,-x)\), for \((2,-1)\) we get \((-1,-2)\)

Answer:

The first option (a reflection over the \(y\) - axis and then a \(90^{\circ}\) clockwise rotation about the origin)