1. **State the problem:** We have a triangle with vertices A(1,2), B(3,2), C(3,1) in the plane. We want to apply the composite linear transformation \(T_2 \circ T_3 \circ T_1\) to each vertex, where: - \(T_1(x,y) = (-x,y)\) (reflection about the y-axis), - \(T_3(x,y) = (2x + 1, y + 1)\) (affine transformation: scaling x by 2 and translating by (1,1)), - \(T_2(x,y) = (x\cos\theta - y\sin\theta, x\sin\theta + y\cos\theta)\) (rotation by angle \(\theta\)). 2. **Apply \(T_1\) to each vertex:** \[ T_1(1,2) = (-1,2), \quad T_1(3,2) = (-3,2), \quad T_1(3,1) = (-3,1) \] This reflects the triangle about the y-axis. 3. **Apply \(T_3\) to the results of \(T_1\):** \[ T_3(-1,2) = (2(-1) + 1, 2 + 1) = (-2 + 1, 3) = (-1,3), \] \[ T_3(-3,2) = (2(-3) + 1, 2 + 1) = (-6 + 1, 3) = (-5,3), \] \[ T_3(-3,1) = (2(-3) + 1, 1 + 1) = (-6 + 1, 2) = (-5,2) \] This scales the x-coordinates by 2 and translates by (1,1). 4. **Apply \(T_2\) (rotation by \(\theta\)) to the results of \(T_3\):** For each point \((x,y)\), \[ T_2(x,y) = (x\cos\theta - y\sin\theta, x\sin\theta + y\cos\theta) \] Calculate for each vertex: - For \((-1,3)\): \[ (-1)\cos\theta - 3\sin\theta, \quad (-1)\sin\theta + 3\cos\theta \] - For \((-5,3)\): \[ (-5)\cos\theta - 3\sin\theta, \quad (-5)\sin\theta + 3\cos\theta \] - For \((-5,2)\): \[ (-5)\cos\theta - 2\sin\theta, \quad (-5)\sin\theta + 2\cos\theta \] 5. **Summary of final vertices after \(T_2 \circ T_3 \circ T_1\):** \[ A' = (-\cos\theta - 3\sin\theta, -\sin\theta + 3\cos\theta), \] \[ B' = (-5\cos\theta - 3\sin\theta, -5\sin\theta + 3\cos\theta), \] \[ C' = (-5\cos\theta - 2\sin\theta, -5\sin\theta + 2\cos\theta) \] 6. **Interpretation:** - The original triangle is first reflected about the y-axis. - Then it is stretched horizontally by a factor of 2 and shifted by (1,1). - Finally, it is rotated counterclockwise by angle \(\theta\). 7. **Graphical depiction:** - The initial triangle has vertices at (1,2), (3,2), (3,1). - After \(T_1\), the triangle is mirrored across the y-axis. - After \(T_3\), the triangle is stretched and translated. - After \(T_2\), the triangle is rotated by \(\theta\). This sequence of transformations can be visualized by plotting the points stepwise.