Question

the sequence of transformations, ( r_{o,90^{circ}} circ r_{x - axis} ), is applied to ( \triangle xyz ) to produce ( \triangle xyz ). if the coordinates of ( y ) are ( (3,0) ), what are the coordinates of ( y )?

Explanation:

Step1: Find the inverse transformation

The given transformation is \(R_{O,90^{\circ}}\circ r_{x - axis}\). The inverse transformation is \(r_{x - axis}\circ R_{O,- 90^{\circ}}\).

Step2: Apply the inverse transformation to \(Y''(3,0)\)

• First, apply \(R_{O,-90^{\circ}}\):

The rule for a rotation of \(-90^{\circ}\) about the origin \((x,y)\to(y,-x)\). For the point \((3,0)\), we have \(R_{O,-90^{\circ}}(3,0)=(0, - 3)\).

• Then, apply \(r_{x - axis}\):

The rule for a reflection over the \(x\) - axis is \((x,y)\to(x,-y)\). For the point \((0,-3)\), we have \(r_{x - axis}(0,-3)=(0,3)\).

Answer:

\((0,3)\)