In a combined transformation written VW, apply W first and then V. Track one point through each step, and write down the position after every step.
This lesson completes the section on congruency, enlargement and combined transformations. It builds on finding the centre of enlargement.
How do I read the notation?
Read the letters from right to left. The letter nearest the point acts first.
| Notation | Meaning |
|---|---|
| VW | W first, then V |
| WV | V first, then W |
Always state the order in words in your answer, for example “a translation followed by a reflection”.
Worked example: a point through two steps
Let T be the translation by the vector (4, 1), and R be the reflection in the y-axis. Take the point P(2, 3).
RT(P): T first, then R. T moves P to (2 + 4, 3 + 1) = (6, 4). R reflects in the y-axis, so (6, 4) becomes (−6, 4).
TR(P): R first, then T. R reflects P to (−2, 3). T moves it to (−2 + 4, 3 + 1) = (2, 4).
The two images are (−6, 4) and (2, 4). The order matters.
The mistake that reads left to right
A common slip is to apply the left letter first. For RT the student reflects first and gets (2, 4), which is the answer for TR.
A reliable check is to write the point after each step. If the first step shown is not the one nearest the point in the notation, the order has been reversed.
When does the order not matter?
Test with a point. Let M be the reflection in the x-axis, and N be the enlargement with centre (0, 0) and scale factor 3. Take A(1, 2).
MN(A): N first gives (3, 6), then M gives (3, −6). NM(A): M first gives (1, −2), then N gives (3, −6).
Both give (3, −6), so this pair gives the same image in either order. That is a property of this pair and not a general rule.
Check yourself
Find MN(B) and NM(B) for B(2, −1), with M and N as above. Are they equal?
Answer
MN(B): N first gives (6, −3), then M gives (6, 3).
NM(B): M first gives (2, 1), then N gives (6, 3).
The results are equal, which agrees with the pair above. This pair commutes, but always test rather than assume.
What to study next
Test all four skills with the section practice set. You can also try your own points in the transformation coordinates explorer.
If you want a teacher to check the order of your steps on new questions, see online one-to-one Mathematics tuition.