Question

which rule describes the composition of transformations that maps pre - image abcd to final image a\b\c\d\?

r_{x - axis}\circ t_{-6,1}(x,y)

t_{-6,1}\circ r_{x - axis}(x,y)

r_{0,90\circ}\circ t_{-6,1}(x,y)

t_{-6,1}\circ r_{0,90\circ}(x,y)

Explanation:

Step1: Analyze the transformation order

First, check if reflection or translation is first. If we consider the \(x -\)axis reflection \((x,y)\to(x, - y)\) and translation \(T_{-6,1}(x,y)=(x - 6,y+1)\).
If we first do the reflection \(r_{x - axis}(x,y)=(x,-y)\) and then the translation \(T_{-6,1}(x,-y)=(x - 6,-y + 1)\), this does not match the mapping from \(ABCD\) to \(A''B''C''D''\).
If we first do the translation \(T_{-6,1}(x,y)=(x - 6,y + 1)\) and then the reflection \(r_{x - axis}(x-6,y + 1)=(x - 6,-(y + 1))=(x-6,-y - 1)\), this also does not match.
Now, consider rotation \(R_{0,90^{\circ}}(x,y)=(-y,x)\) and translation \(T_{-6,1}(x,y)=(x - 6,y + 1)\).
If we first do the rotation \(R_{0,90^{\circ}}(x,y)=(-y,x)\) and then the translation \(T_{-6,1}(-y,x)=(-y-6,x + 1)\), this does not match.
If we first do the translation \(T_{-6,1}(x,y)=(x - 6,y + 1)\) and then the rotation \(R_{0,90^{\circ}}(x-6,y + 1)=(-(y + 1),x - 6)=(-y - 1,x - 6)\), this does not match.
Let's use a point - by - point analysis. Suppose a point \(A=(3,5)\) (assuming coordinates from the grid).
If we first do the reflection over the \(x -\)axis: \(r_{x - axis}(3,5)=(3,-5)\), then translate \(T_{-6,1}(3,-5)=(3-6,-5 + 1)=(-3,-4)\)
If we first translate \(T_{-6,1}(3,5)=(3-6,5 + 1)=(-3,6)\), then reflect over the \(x -\)axis \(r_{x - axis}(-3,6)=(-3,-6)\)
Let's assume another approach. The composition of transformations:
The pre - image \(ABCD\) to \(A'B'C'D'\) to \(A''B''C''D''\).
We know that \(T_{-6,1}(x,y)\) is a translation (shift \(x\) by \(-6\) and \(y\) by \(1\)) and \(r_{x - axis}(x,y)=(x,-y)\)
If we consider the order of operations in function composition. The notation \(T\circ r\) means first apply \(r\) then \(T\).
Let's take a general point \((x,y)\)
First, reflect over the \(x -\)axis: \((x,y)\to(x,-y)\)
Then translate: \((x,-y)\to(x - 6,-y+1)\)
The notation \(r_{x - axis}\circ T_{-6,1}(x,y)\) is wrong order (because in function composition \(f\circ g(x)=f(g(x))\), here \(g = T_{-6,1}\) and \(f=r_{x - axis}\))
The correct order is first translation then reflection. \(T_{-6,1}\circ r_{x - axis}(x,y)\)

Answer:

\(T_{-6,1}\circ r_{x - axis}(x,y)\)