Question

(A)using geometry vocabulary, describe a sequence of transformations that maps figure P (-1,2)(-1,4) (-4,2) (-4,4) onto figure Q(2,-2)(2,-5)(4,-2)(4,-5).(B)write the transformation mapping rules for the sequence you described in part (a).

Answer 1

Before we proceed on determining the transformation happening on this problem, it's better to see first the location of the figure by drawing it in a cartesian coordinate plane. We have

If we observe the figures and the coordinates of the plot, we can see that there is a difference of 1 on the x coordinates of P and y coordinates of Q. Therefore, the first transformation that we consider here is the movement of figure P by 1 unit to the left. We have

`\begin{gathered} P_1=(-1-1,2_{})=(-2,2) \\ P_2=(-1-1,4)=(-2,4) \\ P_3=(-4-1,2)=(-5,2) \\ P_4=(-4-1,4)=(-5,4) \end{gathered}`

This transformation changes the location of figure P into

The next transformation will be the rotation of the red dotted figure on the figure above by 90 degrees counterclockwise. With this transformation, the coordinates will transform as

`P_(ccw,90)=(-y,x)`

Hence, for the rotation, we have the new coordinates.

`\begin{gathered} P_1^(\prime)=(-2,-2) \\ P_2^(\prime)=(-4,-2) \\ P_3^(\prime)=(-2,-5) \\ P_4^(\prime)=(-4,-5) \end{gathered}`

The transformed image, which is represented as NMPO, will now be at

For the last transformation, we will be reflecting the figure NMPO over the y axis. This changes the coordinates as

`P_{\text{rotation,y}-\text{axis}}=(-x,y)`

We now have the new coordinates:

`\begin{gathered} P^(\doubleprime)_1=(2,-2)=Q_1_{}_{} \\ P_2^(\doubleprime)=(4,-2)=Q_3 \\ P_3^(\doubleprime)=(2,-5)=Q_2 \\ P_4^(\doubleprime)_{}=(4,-5)=Q_4_{} \end{gathered}`

As you can see, they have the same coordinates as figure Q.

The mapping rules for the sequence described above are as follows:

First transformation (moving one unit to the left (x-1,y))

`\begin{gathered} P_1(-1,2)\rightarrow P_1(-1-1,2)\rightarrow P_1(-2,2) \\ P_2(-1,4)\rightarrow P_1(-1-1,4)\rightarrow P_2(-2,4) \\ P_3(-4,2)\rightarrow P_1(-4-1,2)\rightarrow P_3(-5,2) \\ P_4(-4,4)\rightarrow P_1(-4-1,4)\rightarrow P_4(-5,4) \end{gathered}`

Second transformation (rotation counter clockwise (-y,x))

`\begin{gathered} P_1(-2,2)\rightarrow P^(\prime)_1(-2,-2)_{} \\ P_2(-2,4)\rightarrow P^(\prime)_2(-4,-2) \\ P_3(-5,2)\rightarrow P^(\prime)_3(-2,-5)_{} \\ P_4(-5,4)\rightarrow P^(\prime)_4(-4,-5)_{} \end{gathered}`

Third Transformation (reflection over y-axis (-x,y))

`\begin{gathered} P^(\prime)_1(-2,-2)\rightarrow P^(\doubleprime)_1(-(-2),-2)\rightarrow P^(\doubleprime)_1=(2,-2)=Q_1 \\ P^(\prime)_2(-4,-2)\rightarrow P^(\doubleprime)_1(-(-4),-2)\rightarrow P^(\doubleprime)_1=(4,-2)=Q_3 \\ P^(\prime)_3(-2,-5)\rightarrow P^(\doubleprime)_1(-(-2),-5)\rightarrow P^(\doubleprime)_1=(2,-5)=Q_2 \\ P^(\prime)_4(-4,-5)\rightarrow P^(\doubleprime)_1(-(-4),-5)\rightarrow P^(\doubleprime)_1=(4,-5)=Q_4 \end{gathered}`