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.