Transformations: Combined and inverse transformations
Practice Questions

A surveyor in Trinidad is mapping a plot of land. She applies transformation T₁ (translation by vector \(\begin{pmatrix} 3 \\ 2 \end{pmatrix}\)) followed by transformation T₂ (translation by vector \(\begin{pmatrix} -1 \\ 4 \end{pmatrix}\)). What single translation is equivalent to T₂T₁?

1. Translation by \(\begin{pmatrix} 2 \\ 6 \end{pmatrix}\)
2. Translation by \(\begin{pmatrix} 4 \\ -2 \end{pmatrix}\)
3. Translation by \(\begin{pmatrix} -4 \\ 2 \end{pmatrix}\)
4. Translation by \(\begin{pmatrix} 2 \\ -2 \end{pmatrix}\)

A triangle ABC is first reflected in the y-axis to give triangle A'B'C', and then A'B'C' is rotated 90° clockwise about the origin to give triangle A''B''C''. Which single transformation maps triangle ABC directly onto triangle A''B''C''?

1. Reflection in the line y = x
2. Reflection in the line y = -x
3. Rotation 90° anticlockwise about the origin
4. Rotation 180° about the origin