Question

which rule describes the composition of transformations that maps \\( \triangle b c d \\) to \\( \triangle b^{\prime \prime} c^{\prime \prime} d^{\prime \prime} \\)? ( t _ { 3, - 6 } circ r _ { y = - 4 } ( x, y ) ) ( r _ { y = - 4 } circ t _ { 3, - 6 } ( x, y ) ) ( t _ { 6, - 5 } circ r _ { y - \text { axis } } ( x, y ) ) ( r _ { y - \text { axis } } circ t _ { 6, - 5 } ( x, y ) )

Explanation:

Step1: Analyze the reflection

First, observe that the figure is reflected over the \(y -\)axis (\(r_{y - axis}\)). The reflection over the \(y -\)axis changes the \(x -\)coordinate's sign: \((x,y)\to(-x,y)\).

Step2: Analyze the translation

After reflection, we need to translate the figure. A translation \(T_{a,b}\) moves a point \((x,y)\) to \((x + a,y + b)\). Looking at the coordinates of corresponding vertices, after reflection, we need to move the figure \(6\) units to the right (\(x\) - coordinate \(+6\)) and \(5\) units down (\(y\) - coordinate \(- 5\)), so \(T_{6,-5}\).

Step3: Determine the order of composition

The composition of transformations is \(r_{y - axis}\circ T_{6,-5}(x,y)\). The order is important. We first perform the translation and then the reflection.

Answer:

\(r_{y - axis}\circ T_{6,-5}(x,y)\) (the fourth option)