Chapter 9.4, Problem 35PPS

To determine

To find: The transformation that combined to map the given Figure.

Answer to Problem 35PPS

The transformation combined is rotation is $180°$ about the origin and the reflection in the x-axis.

Explanation of Solution

Given:

The given Figure is shown Figure 1.

  

Figure 1

Calculation:

Consider the given figure. The first step is to translate QRST and describe the transformation as,

  $\begin{array}{l}Q\left(-6,5\right)\to Q^{\prime }\left(6,-5\right)\\ R\left(-2,6\right)\to R^{\prime }\left(2,-6\right)\\ S\left(-1,3\right)\to S^{\prime }\left(1,-3\right)\\ T\left(-5,2\right)\to T^{\prime }\left(5,-2\right)\end{array}$

From above note the sign change of x and y coordinates. Thus, the rotation is described as $\langle x,y\rangle =\langle -x,-y\rangle$

The next step to reflect $Q^{\prime }{R}^{\prime }{S}^{\prime }{T}^{\prime }$ and describe the transformation as,

  $\begin{array}{l}{Q}^{\prime }\left(6,-5\right)\to Q^{″}\left(6,5\right)\\ R^{\prime }\left(2,-6\right)\to R^{″}\left(2,6\right)\\ S^{\prime }\left(1,-3\right)\to S^{″}\left(1,3\right)\\ T^{\prime }\left(5,-2\right)\to T^{″}\left(5,2\right)\end{array}$

Thus, the above reflection is $\langle x,y\rangle =\langle -x,-y\rangle$ .

Thus, the rotation is $180°$ about the origin and the reflection in the x-axis.