Question

18 shape a is reflected in the line with equation ( x = 2 ) to give shape b.
shape b is reflected in the line with equation ( x = 6 ) to give shape c.
describe fully the single transformation that maps shape a onto shape c.

Explanation:

Step1: Understand reflection distance

For a point $(x,y)$ reflected over $x=a$, new $x$-coordinate is $2a - x$.

Step2: Reflect shape A over $x=2$

Take a general point $(x,y)$ in A. After reflection over $x=2$, it becomes $(2\times2 - x, y)=(4 - x, y)$ (this is shape B).

Step3: Reflect shape B over $x=6$

Take the point $(4 - x, y)$ in B. After reflection over $x=6$, new $x$-coordinate is $2\times6 - (4 - x)=12 - 4 + x=8 + x$. The point becomes $(x + 8, y)$.

Step4: Identify single transformation

The transformation $(x,y)\to(x+8,y)$ is a horizontal translation.

Answer:

A translation by the vector

$$\begin{pmatrix} 8 \\ 0 \end{pmatrix}$$

(or a horizontal shift 8 units to the right)