Combining transformations Worked Solutions — GCSE Maths

Apply the first transformation, the reflection in the line $x = 0$, to the point.

The rule for a reflection in the line $x = 0$ is $(x,\, y) \rightarrow (-x,\, y)$, so $A(3, 1)$ moves to $(-3, 1)$.

Apply the second transformation, the translation by the vector (2, 3), to that image.

The second transformation acts on the image $(-3, 1)$, not on the original point. Its rule is $(x,\, y) \rightarrow (x + 2,\, y + 3)$.

State the coordinates of the final image.

The point ends at $(-1, 4)$. Doing the two transformations in the other order would generally land somewhere else.

Apply the first transformation, the translation by the vector (4, -1), to the point.

The rule for a translation by the vector $\begin{pmatrix} 4 \\ -1 \end{pmatrix}$ is $(x,\, y) \rightarrow (x + 4,\, y - 1)$, so $P(-2, 4)$ moves to $(2, 3)$.

Apply the second transformation, the reflection in the line $y = 0$, to that image.

The second transformation acts on the image $(2, 3)$, not on the original point. Its rule is $(x,\, y) \rightarrow (x,\, -y)$.

State the coordinates of the final image.

The point ends at $(2, -3)$. Doing the two transformations in the other order would generally land somewhere else.

Apply the first transformation, the reflection in the line $y = x$, to the point.

The rule for a reflection in the line $y = x$ is $(x,\, y) \rightarrow (y,\, x)$, so $A(5, 2)$ moves to $(2, 5)$.

Apply the second transformation, the translation by the vector (-2, 5), to that image.

The second transformation acts on the image $(2, 5)$, not on the original point. Its rule is $(x,\, y) \rightarrow (x - 2,\, y + 5)$.

State the coordinates of the final image.

The point ends at $(0, 10)$. Doing the two transformations in the other order would generally land somewhere else.

Apply the first transformation, the rotation of 180 degrees about (0, 0), to the point.

The rule for a rotation of $180^{\circ}$ about the point $(0, 0)$ is $(x,\, y) \rightarrow (-x,\, -y)$, so $B(-1, -3)$ moves to $(1, 3)$.

Apply the second transformation, the translation by the vector (3, 2), to that image.

The second transformation acts on the image $(1, 3)$, not on the original point. Its rule is $(x,\, y) \rightarrow (x + 3,\, y + 2)$.

State the coordinates of the final image.

The point ends at $(4, 5)$. Doing the two transformations in the other order would generally land somewhere else.

Apply the first transformation, the translation by the vector (1, -2), to the point.

The rule for a translation by the vector $\begin{pmatrix} 1 \\ -2 \end{pmatrix}$ is $(x,\, y) \rightarrow (x + 1,\, y - 2)$, so $A(0, 4)$ moves to $(1, 2)$.

Apply the second transformation, the rotation of 90 degrees clockwise about (0, 0), to that image.

The second transformation acts on the image $(1, 2)$, not on the original point. Its rule is $(x,\, y) \rightarrow (y,\, -x)$.

State the coordinates of the final image.

The point ends at $(2, -1)$. Doing the two transformations in the other order would generally land somewhere else.